The core of this monograph is the development of tools to derive wellposedness results in very general geometric settin
338 87 3MB
English Pages 528 Year 2016
Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor The HodgeLaplacian
De Gruyter Studies in Mathematics
 Edited by Carsten Carstensen, Berlin, Germany Farkas Gavril, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China KarlHermann Neeb, Erlangen, Germany
Volume 64
Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Michael Taylor
The HodgeLaplacian  Boundary Value Problems on Riemannian Manifolds
Mathematics Subject Classiﬁcation 2010 31B10, 31B25, 31C12, 35A01, 35B20, 35J08, 35J25, 35J55, 35J57, 35Q61, 35R01, 42B20, 42B25, 42B37, 45A05, 45B05, 45E05, 45F15, 45P05, 47B38, 47G10, 49Q15, 58A10, 58A12, 58A14, 58A15, 58A30, 58C35, 58J05, 58J32, 78A30 Authors Dorina Mitrea Department of Mathematics University of Missouri at Columbia Columbia, MO 65211, USA email: [email protected]
Irina Mitrea Department of Mathematics Temple University Philadelphia, PA 19122, USA email: [email protected]
Marius Mitrea Department of Mathematics University of Missouri at Columbia Columbia, MO 65211, USA email: [email protected]
Michael Taylor Department of Mathematics University of North Carolina Chapel Hill, NC 275993250, USA email: [email protected]
ISBN 9783110482669 eISBN (PDF) 9783110484380 eISBN (EPUB) 9783110483390 SetISBN 9783110484397 ISSN 01790986
Library of Congress CataloginginPublication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTPBerlin, ProtagoTEXProduction GmbH, Berlin Printing and binding: CPI books GmbH, Leck ♾ Printed on acidfree paper Printed in Germany www.degruyter.com
Preface This monograph is devoted to a natural class of boundary problems for the HodgeLaplacian, acting on differential forms. This class includes the absolute and relative boundary conditions, used in the Hodgestyle representation of absolute and relative cohomology classes of the underlying domain by harmonic forms. Continuing the program in [86] aimed at understanding the solvability properties of such boundary problems under minimal geometric and analytic regularity assumptions, here we push further the analysis developed in [50] of a layer potential attack on elliptic boundary problems on a class of domains introduced by Semmes [112] and Kenig and Toro [66], which we call regular SemmesKenigToro (SKT) domains. We initiate the study of boundary value problems for differential forms in this class of domains. In addition to the absolute and relative boundary conditions mentioned earlier, we also treat the HodgeLaplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular SKT domains, with data in L p spaces, for arbitrary p ∈ (1, ∞). In a broad perspective, our results may be regarded as a natural completion, of an optimal nature, of the work initiated by E. Fabes, M. Jodeit, and N. Rivière in [32], whose scope is extended here through the consideration of differential forms in place of scalar functions, the (variablecoefficient) HodgeLaplacian in lieu of the (constant coefficient) Laplace operator, and regular SKT subdomains of Riemannian manifolds, with arbitrary topology, replacing C 1 domains with connected compact boundaries in the ﬂat Euclidean setting. In stark contrast to the scalar case from [32], the structural richness of the higher degree case considered here allows for a much larger variety of natural boundary value problems for the HodgeLaplacian, which we formulate and study systematically via potential theoretic methods. Dorina Mitrea, Columbia, MO, USA Irina Mitrea, Philadelphia, PA, USA Marius Mitrea, Columbia, MO, USA Michael Taylor, Chapel Hill, NC, USA
Contents Preface  v 1 1.1 1.2 1.3 1.4 1.5 2 2.1 2.2 2.3 2.4 3 3.1 3.2 3.3
4 4.1 4.2 4.3 5 5.1 5.2 5.3
Introduction and Statement of Main Results  1 First Main Result: Absolute and Relative Boundary Conditions  3 Other Problems Involving Tangential and Normal Components of Harmonic Forms  11 Boundary Value Problems for HodgeDirac Operators  21 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems  24 Structure of the Monograph  43 Geometric Concepts and Tools  49 Differential Geometric Preliminaries  49 Elements of Geometric Measure Theory  67 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains  91 Tangential and Normal Differential Forms on Ahlfors Regular Sets  96 Harmonic Layer Potentials Associated with the Hodgede Rham Formalism on UR Domains  109 A Fundamental Solution for the HodgeLaplacian  109 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism  117 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism  128 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains  139 The Deﬁnition and Mapping Properties of the Double Layer  140 The Double Layer on UR Subdomains of Smooth Manifolds  169 Compactness of the Double Layer on Regular SKT Domains  173 Dirichlet and Neumann Boundary Value Problems for the HodgeLaplacian on Regular SKT Domains  185 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains  186 Invertibility Results for Layer Potentials Associated with the LeviCivita Connection  196 Solving the Dirichlet, Neumann, Transmission, Poincaré, and Robin Boundary Value Problems  204
viii  Contents
6 6.1 6.2 6.3 7 7.1 7.2 8 8.1 8.2 8.3 8.4 8.5 8.6 9
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12
Fatou Theorems and Integral Representations for the HodgeLaplacian on Regular SKT Domains  231 Convergence of Families of Singular Integral Operators  231 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains  250 Spaces of Harmonic Fields and Green Type Formulas  261 Solvability of Boundary Problems for the HodgeLaplacian in the Hodgede Rham Formalism  275 Preparatory Results  275 Solvability Results  288 Additional Results and Applications  315 de Rham Cohomology on Regular SKT Surfaces  315 Maxwell’s Equations in Regular SKT Domains  336 DirichlettoNeumann Operators for the HodgeLaplacian in Regular SKT Domains  339 Fatou Type Results with Additional Constraints or Regularity Conditions  347 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property  352 The HodgePoisson Kernel and the HodgeHarmonic Measure  367 Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis  371 Connections and Covariant Derivatives on Vector Bundles  371 The Extension of the LeviCivita Connection to Differential Forms  381 The BochnerLaplacian and Weintzenböck’s Formula  386 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting  393 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting  408 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  417 A Global Sobolev Regularity Result  444 The PV Harmonic Double Layer on a UR Domain  446 CalderónZygmund Theory on UR Domains on Manifolds  451 The Fredholmness and Invertibility of Elliptic Differential Operators  474 Compact and ClosetoCompact Singular Integral Operators  482 A Sharp Divergence Theorem  490
Contents 
9.13 9.14
Clifford Analysis Rudiments  493 Spectral Theory for Unbounded Linear Operators Subject to Cancellations  496
Bibliography  501 Index  507
ix
1 Introduction and Statement of Main Results In their 1979 inﬂuential paper [32] on layer potential techniques for boundary value problems, E. Fabes, M. Jodeit and N. Rivière have showed that for a bounded domain Ω ⊂ ℝn of class C 1 , with outward unit normal ν and boundary surface measure σ, the harmonic double layer¹ Kf(x) := lim+ ε→0
1 ω n−1
∫
⟨ν(y), y − x⟩ f(y) dσ(y), x − yn
x ∈ ∂Ω,
(1.0.1)
y∈∂Ω
x−y>ε
is a compact operator on L p (∂Ω) for each p ∈ (1, ∞). This continued a long line of work, originating with Erik Ivar Fredholm in the late 1800’s and early 1900’s whose motivation for the development of Fredholm theory was the use of such compactness results in order to treat boundary value problems via integral equation methods. In particular, this development made it possible to solve in [32] boundary value problems for the Laplacian in a bounded C 1 domain Ω ⊂ ℝn equipped with Dirichlet or Neumann boundary conditions. In the formulation of these problems, the boundary data is selected from Lebesgue spaces, the boundary traces are taken in a pointwise nontangential sense, and the size of the solution was measured through the membership of the nontangential maximal function to L p (∂Ω). Concerning the compactness of K on L p (∂Ω) , the demand that Ω ⊂ ℝn is of class 1 C may be signiﬁcantly relaxed and, in [50], the authors have identiﬁed the natural class of domains in which principal value singular integral operators with an algebraic structure similar to that of the harmonic double layer K in (1.0.1) induce compact mappings on each L p (∂Ω), p ∈ (1, ∞). The membership of Ω in the class of domains in question, dubbed regular SKT² domains in [50], is characterized by the following geometric measure theoretic conditions: Ω is an open set with compact Ahlfors regular boundary, Ω satisﬁes a twosided local John condition, and
(1.0.2)
the outward unit conormal ν of Ω belongs to VMO(∂Ω). Here and elsewhere, VMO(∂Ω) stands for the Sarason class of functions of vanishing mean oscillation on ∂Ω, relative to the “surface measure” σ := H n−1 ⌊∂Ω, where H n−1 is the (n − 1)dimensional Hausdorff measure in the ambient space (viewed as a metric space; cf. [83]). See § 2.2 for deﬁnitions of Ahlfors regularity and the John condition. Domains satisfying the conditions listed in (1.0.2) include bounded domains of class C 1 or, more generally, bounded domains locally given as the uppergraphs of continuous functions with gradients in VMO. This being said, regular SKT domains need 1 where ⟨⋅, ⋅⟩ is the standard inner product, and ω n−1 denotes the area of the unit sphere in ℝn 2 acronym for SemmesKenigToro
2  1 Introduction and Statement of Main Results
not be locally graph domains since, in the plane, this class contains the category of chordarc domains with vanishing constant. An alternative way of asserting that Ω is a regular SKT domain is via demanding that Ω is a twosided NTA domain (in the sense of D. Jerison and C. Kenig [59]), with a compact Ahlfors regular boundary, and whose outward unit conormal belongs to VMO (∂Ω). The quantitative aspects of the ﬁrst two conditions in (1.0.2) make up what we will occasionally refer to as the “large geometry constants” of Ω, while the membership in the last line in (1.0.2) may be construed as an “inﬁnitesimal ﬂatness condition” satisﬁed by the set Ω. Building on the work in [86] and [96]–[100], aimed at developing boundary integral methods for Lipschitz subdomains of Riemannian manifolds, in [50] the authors also showed how these potential/geometric measure theoretic considerations can be adapted in the case of the (scalar) LaplaceBeltrami operator on a regular SKT subdomain of a Riemannian manifold. Here the goal is to continue this line of work by considering natural boundary value problems for the HodgeLaplacian³ ∆HL := −(d + δ)2 = −(dδ + δd)
(1.0.3)
acting on differential forms of arbitrary degree in the class of regular SKT subdomains of a manifold M, assumed throughout to be compact, boundaryless, orientable, ndimensional, Riemannian manifold, with a C 2 smooth metric tensor. We stress that we do not impose any further topological condition on M (in particular, M is allowed to be disconnected). In a nutshell, the program undertaken here may be viewed as a generalization of the work of E. Fabes, M. Jodeit and N. Rivière in [32], concerning L p solvability results for all p ∈ (1, ∞), in the following four basic respects: treating higherdegree differential forms (in place of scalar functions), considering the larger class of regular SKT domains (in place of C 1 domains), avoiding imposing topological restrictions on the underlying domain (unlike [32]), and having a Riemannian manifold as ambient space (instead of the ﬂat Euclidean space). Of course, considering the latter setting forces us to work with the variable coefficient HodgeLaplacian in lieu of the constant coefficient Laplacian. This being said, as in [32], in the formulation of all our boundary problems the size of the solution is expressed through the membership of nontangential maximal functions (cf. (2.2.36)) in L p , and all boundary traces are taken in a pointwise nontangential sense (cf. (2.2.32)). Also, in order to avoid the difficulties caused by the fact that ∂Ω may utterly lack any type of smooth manifold structure, all differential forms involved are taken to be sections of exterior powers of the tangent bundle on M. One of the primary achievements of the work in this monograph is that we succeed in cataloging and treating all natural boundary value problems for the HodgeLaplacian ∆HL in the de RhamHodge formalism (associated with differential form 3 with d, δ denoting, respectively, the exterior derivative operator and its formal adjoint, the codifferential
1.1 First Main Result: Absolute and Relative Boundary Conditions  3
calculus) in the aforementioned geometrical setting. More speciﬁcally, we prove solvability results for boundary value problems seeking nullsolutions u of ∆HL in a regular SKT domain Ω ⊂ M with outward conormal ν, equipped with boundary conditions obtained by prescribing, in appropriate function spaces on ∂Ω, any two⁴ of the following expressions⁵: n.t. ν ∨ u∂Ω ,
n.t. ν ∧ u∂Ω ,
n.t. ν ∨ (du)∂Ω ,
n.t. ν ∧ (du)∂Ω ,
n.t. ν ∨ (δu)∂Ω ,
n.t. ν ∧ (δu)∂Ω .
(1.0.4)
In each case, we design function spaces for the boundary data which optimally capture the key geometric, analytic, and algebraic nature of the speciﬁc boundary conditions, ultimately yielding a rich and intricate tapestry of problems.
1.1 First Main Result: Absolute and Relative Boundary Conditions We prepare to state our ﬁrst main result of this ﬂavor. This result deals with nullsolutions of the HodgeLaplacian ∆HL from (1.0.3) in a set Ω ⊂ M of ﬁnite perimeter with (geometric measure theoretic) outward unit conormal ν, for which either n.t. { ν ∨ u∂Ω { n.t. { ν ∨ (du)∂Ω
or
n.t. { ν ∧ u∂Ω { n.t. { ν ∧ (δu)∂Ω
(1.1.1)
are prescribed in appropriate function spaces deﬁned on the boundary of the underlying domain. In all cases, the resulting boundary value problem turns out to be Fredholm solvable in the sense that the problem in question has a ﬁnite dimensional space of nullsolutions, and is solvable if and only the boundary data satisfy ﬁnitely many linearly independent compatibility conditions (in such a scenario, the index of the problem is deﬁned as the difference of these two integers). The statement of Theorem 1.1 below is structured so one can monitor how changes in the nature of the function spaces from which the boundary data are selected affect the overall properties of the solutions. Theorem 1.1 and its companions make reference to a variety of function spaces. We will give brief deﬁnitions of most of them right after the statement of Theorem 1.1, referring to § 2.4 for technical details. Here we want to single out the spaces H∨l (Ω) and H∧l (Ω) of harmonic forms, deﬁned as n.t.
H∨l (Ω) := {u ∈ H l,p (Ω) : ν ∨ u∂Ω = 0} ,
(1.1.2)
4 excluding natural redundancies 5 where ∧, ∨ stand, respectively, for the exterior and interior product of differential forms, and n.t. ∂Ω indicates taking the pointwise nontangential trace on ∂Ω
4  1 Introduction and Statement of Main Results
and
n.t.
H∧l (Ω) := {u ∈ H l,p (Ω) : ν ∧ u∂Ω = 0} ,
(1.1.3)
where 1 H l,p (Ω) := {u ∈ Cloc (Ω, Λ l T M) : N u ∈ L p (∂Ω), du = 0, δu = 0 in Ω} .
(1.1.4)
Elements of H∨l (Ω) are harmonic lforms satisfying absolute boundary conditions, or “Dirichlet” harmonic forms, and elements of H∧l (Ω) are harmonic lforms satisfying relative boundary conditions, or “Neumann” harmonic forms. These play a central role in reﬂecting the topology of Ω. In (1.1.4), N u is the nontangential maximal function of u. An important Fatou theorem established in § 6 implies that elements of H l,p (Ω) always have nontangential limits, when p ∈ (1, ∞). We will show that H∨l (Ω) and H∧l (Ω) are independent of such p. Here is our ﬁrst main theorem: Theorem 1.1. Assume Ω ⊂ M is a regular SKT domain with outward unit conormal ν and surface measure σ. For a given degree l ∈ {0, 1, . . . , n} and exponent p ∈ (1, ∞) consider the boundary value problem⁶ u ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL u = 0 in Ω, { { { (BVP1)l { N u, N(du) ∈ L p (∂Ω), { { p n.t. { l−1 { { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ T M), { { { p n.t. l { ν ∨ (du)∂Ω = g ∈ Ltan (∂Ω, Λ T M).
(1.1.5)
Then the following assertions are true: (1) A solution of (BVP1)l exists if and only if g satisﬁes the compatibility condition ⊥ n.t. ⊥ n.t. g ∈ {H∨l (Ω)∂Ω } := {ω∂Ω : ω ∈ H∨l (Ω)} ,
(1.1.6)
where the symbol {. . .}⊥ refers to the annihilator of {. . .} (here viewed as a sub ∗ space of L p (∂Ω, Λ l T M), 1/p + 1/p = 1) in L p (∂Ω, Λ l T M) = (L p (∂Ω, Λ l T M)) , and where H∨l (Ω) is the space of Dirichlet harmonic ﬁelds of degree l in Ω, with a squareintegrable nontangential maximal function⁷. The latter space has ﬁnite dimension, henceforth denoted by b l (Ω).
6 the Fatou type result established in Theorem 6.6 ensures that the nontangential pointwise traces un.t. and (du)n.t. exist at σa.e. point on ∂Ω; in concert with the third condition in the formulation of ∂Ω ∂Ω p (1.1.5), this also shows that the boundary data f and g necessarily need to belong to Ltan (∂Ω, Λ l−1 TM) p l and Ltan (∂Ω, Λ TM), respectively 7 cf. the discussion pertaining to (6.3.5), (6.3.7)
1.1 First Main Result: Absolute and Relative Boundary Conditions  5
(2) The space of solutions for the homogeneous version of (BVP1)l is precisely H∨l (Ω). In particular, (BVP1)l is Fredholm solvable, with index b l (Ω, M),
(1.1.7)
where b l (Ω, M) is deﬁned as the sum of the lth Betti numbers of all connected components of M contained in Ω. As a consequence, (BVP1)l has index zero if Ω does not contain any connected component of M (which is automatically the case if M is connected to begin with). Furthermore, if {ω j }1≤j≤b l (Ω) is a ﬁxed basis of H∨l (Ω), (1.1.8) one may include a normalization condition⁸ in the formulation of the boundary value problem (BVP1)l which ensures genuine wellposedness. Speciﬁcally, for any given numbers β j ∈ ℝ, with 1 ≤ j ≤ b l (Ω), there exists a unique solution of u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { { p { { N u, N(du) ∈ L (∂Ω), p n.t. { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M), { { { { { p n.t. n.t. ⊥ { ν ∨ (du)∂Ω = g ∈ Ltan (∂Ω, Λ l T M) ∩ {H∨l (Ω)∂Ω } , { { { { { { ∫Ω ⟨u, ω j ⟩ dVol = β j for j ∈ {1, . . . , b l (Ω)}.
(1.1.9)
Moreover, if the compatibility condition (1.1.6) is satisﬁed, then the boundary value problem (BVP1)l exhibits the following features: (3) The boundary data f, g determine du uniquely and ‖N(du)‖L p (∂Ω) ≤ C (‖f‖L p (∂Ω,Λ l−1 TM) + ‖g‖L p (∂Ω,Λ l TM) ) ,
(1.1.10)
for some ﬁnite positive constant C, independent of f and g. In addition, any solution u of (BVP1)l satisﬁes u ∈ L np/(n−1) (Ω, Λ l T M),
du ∈ L np/(n−1) (Ω, Λ l+1 T M).
(1.1.11)
Moreover, in the case p = 2, one also has⁹ u ∈ H 1/2,2 (Ω, Λ l T M),
du ∈ H 1/2,2 (Ω, Λ l+1 T M).
(1.1.12)
8 the absolute convergence of the integrals ∫Ω ⟨u, ω j ⟩ dVol is guaranteed by (1.1.11) and (6.3.23) 9 here and elsewhere, H 1/2,2 (Ω) is the L2 based Sobolev space of smoothness 1/2 in the open set Ω, deﬁned via restriction, as {uΩ : u ∈ H 1/2,2 (M)}
6  1 Introduction and Statement of Main Results (4) The boundary data f, g determine δu uniquely. Moreover, any solution u of (BVP1)l satisﬁes p,δ N(δu) ∈ L p (∂Ω) ⇐⇒ f ∈ Ltan (∂Ω, Λ l T M) (1.1.13) and, when the latter condition holds, ‖N(δu)‖L p (∂Ω) ≤ C(‖f‖L p,δ (∂Ω,Λ l−1 TM) + ‖g‖L p (∂Ω,Λ l TM) ),
(1.1.14)
tan
p,δ
for some constant C ∈ (0, ∞), independent of f and g. When f ∈ Ltan (∂Ω, Λ l T M) any solution u of (BVP1)l also satisﬁes n.t. δu ∈ L np/(n−1) (Ω, Λ l−1 T M) and (δu)∂Ω exists σa.e. on ∂Ω (1.1.15) and, in the case when p = 2, δu ∈ H 1/2,2 (Ω, Λ l−1 T M).
(1.1.16)
(5) Any solution u of (BVP1)l satisﬁes N(dδu) ∈ L p (∂Ω) ⇐⇒ N(δdu) ∈ L p (∂Ω) p,δ
⇐⇒ g ∈ Ltan (∂Ω, Λ l T M),
(1.1.17)
and, when the latter condition holds, ‖N(δdu)‖L p (∂Ω) + ‖N(dδu)‖L p (∂Ω) ≤ C‖g‖L p,δ (∂Ω,Λ l TM) ,
(1.1.18)
tan
p,δ
for some ﬁnite positive constant C, independent of g. When g ∈ Ltan (∂Ω, Λ l T M) any solution u of (BVP1)l also satisﬁes n.t. δdu ∈ L np/(n−1) (Ω, Λ l T M) and (δdu)∂Ω exists σa.e. on ∂Ω (1.1.19) and, corresponding to the case when p = 2, δdu, dδu ∈ H 1/2,2 (Ω, Λ l T M).
(1.1.20)
(6) Any solution u of (BVP1)l satisﬁes du = 0 in Ω ⇐⇒ g = 0.
(1.1.21)
Corresponding to the case when g = 0, the wellposedness of (1.1.9) then yields the unique solvability of u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { { { du = 0 in Ω, (BVP2)l { { { N u, N(δu) ∈ L p (∂Ω), { { { { p,δ n.t. { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M), { { { { { ∫Ω ⟨u, ω j ⟩ dVol = β j , j ∈ {1, . . . , b l (Ω)},
(1.1.22)
for any given family {β j }1≤j≤b l (Ω) of real numbers, with {ω j }1≤j≤b l (Ω) as in (1.1.8).
1.1 First Main Result: Absolute and Relative Boundary Conditions  7
(7) Any solution u of (BVP1)l satisﬁes p,0
δdu = 0 in Ω ⇐⇒ dδu = 0 in Ω ⇐⇒ g ∈ Ltan (∂Ω, Λ l T M).
(1.1.23)
p,0
In particular, for g belonging to Ltan (∂Ω, Λ l T M) the problem (BVP1)l becomes u ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL u = 0 in Ω, { { { { { { δdu = 0 in Ω, (BVP3)l { { N u, N(du), N(δu) ∈ L p (∂Ω), { { { { { ν ∨ un.t. = f ∈ L p,δ (∂Ω, Λ l−1 T M), { { tan ∂Ω { { { { p,0 n.t. = g ∈ Ltan (∂Ω, Λ l T M). ν ∨ (du) ∂Ω {
(1.1.24)
Once more, a normalization condition relative to (1.1.8) may be included in the formulation of the above boundary value problem in order to have genuine uniqueness. (8) Any solution u of (BVP1)l satisﬁes δu = 0 in Ω if and only if p,0 p,0 n.t. ⊥ f ∈ Ltan (∂Ω, Λ l−1 T M) ∩ {H∨l−1 (Ω)∂Ω } and g ∈ Ltan (∂Ω, Λ l T M).
(1.1.25)
Consequently, the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { { { { δu = 0 in Ω, (BVP4)l { { N u, N(du) ∈ L p (∂Ω), { { { { p n.t. { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M), { { { { { p n.t. l { ν ∨ (du)∂Ω = g ∈ Ltan (∂Ω, Λ T M),
(1.1.26)
has a solution if and only if f, g are as in (1.1.25), and the space of nullsolutions for (1.1.26) is precisely H∨l (Ω). In fact, one may ensure genuine wellposedness by including normalization conditions as before, that is, by prescribing ∫Ω ⟨u, ω j ⟩ dVol for 1 ≤ j ≤ b l (Ω), in the formulation of (1.1.26). Moreover, any solution of (BVP4)l satisﬁes du = 0 in Ω ⇐⇒ g = 0.
(1.1.27)
8  1 Introduction and Statement of Main Results As a corollary, for any given numbers β j ∈ ℝ, j ∈ {1, . . . , b l (Ω)}, the following boundary value problem is wellposed: u ∈ C 1 (Ω, Λ l T M), { { { { { { δu = 0 in Ω, { { { { { { { du = 0 in Ω, (BVP5)l { { N u ∈ L p (∂Ω), { { { { p,0 n.t. ⊥ n.t. { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M) ∩ {H∨l−1 (Ω)∂Ω } , { { { { { { ∫Ω ⟨u, ω j ⟩ dVol = β j for j ∈ {1, . . . , b l (Ω)}.
(1.1.28)
(9) Analogous statements to (1)–(8) above hold for the Hodge dual of (BVP1)l , that is, for u ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL u = 0 in Ω, { { { (BVP6)l { N u, N(δu) ∈ L p (∂Ω), (1.1.29) { { p n.t. { l+1 { { { ν ∧ u∂Ω = f ∈ Lnor (∂Ω, Λ T M), { { { p n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M). In this scenario, the role of H∨l (Ω) is played by the space of Neumann harmonic ﬁelds H∧l (Ω) = ∗H∨n−l (Ω). Finally, all results above remain valid when the hypotheses on the underlying domain are relaxed from demanding that this is a regular SKT domain to just asking that Ω is an εSKT domain, provided ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of Ω). We turn to promised deﬁnitions of various function spaces that arose in the statement of Theorem 1.1, referring to § 2.4 for more details. To start, a measurable section f over ∂Ω of Λ l T M can be written f = ftan + fnor , where ftan := ν ∨ (ν ∧ f),
fnor := ν ∧ (ν ∨ f),
(1.1.30)
with ν denoting the outward unit conormal to ∂Ω. For p ∈ [1, ∞] we set p
Ltan (∂Ω, Λ l T M) := { f ∈ L p (∂Ω, Λ l T M) : f = ftan }, p
Lnor (∂Ω, Λ l T M) := { f ∈ L p (∂Ω, Λ l T M) : f = fnor }.
(1.1.31)
Next, for p ∈ [1, ∞] we set p,δ
p
p
Ltan (∂Ω, Λ l T M) := { f ∈ Ltan (∂Ω, Λ l T M) : δ ∂ f ∈ Ltan (∂Ω, Λ l−1 T M)},
(1.1.32)
1.1 First Main Result: Absolute and Relative Boundary Conditions  9
p,δ
p
where δ ∂ : Ltan (Λ l T M) → Ltan (∂Ω, Λ l−1 T M), deﬁned in § 2.4 by a duality argument, is a variant of the standard coderivative δ when ∂Ω is smooth. Then p,0
p,δ
Ltan (∂Ω, Λ l T M) := { f ∈ Ltan (∂Ω, Λ l T M) : δ ∂ f = 0 }.
(1.1.33)
Related spaces, arising in Theorems 1.2 and 1.3, are p,d
p
p
Lnor (∂Ω, Λ l T M) := { f ∈ Lnor (∂Ω, Λ l T M) : d ∂ f ∈ Lnor (∂Ω, Λ l+1 T M)}, p,d
(1.1.34)
p
where d ∂ : Lnor (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l+1 T M) is also deﬁned in § 2.4 by a duality argument, and is a variant of the exterior derivative d when ∂Ω is smooth. Then, parallel to (1.1.33), p,0 p,d Lnor (∂Ω, Λ l T M) := { f ∈ Lnor (∂Ω, Λ l T M) : d ∂ f = 0 }. (1.1.35) We return to a discussion of our main theorems. Specifying two boundary conditions¹⁰ is a characteristic feature of virtually all boundary value problems making use of the de RhamHodge formalism. Traditionally, prescribing n.t. ν ∨ u∂Ω
and
n.t. ν ∨ (du)∂Ω
(1.1.36)
on ∂Ω amounts to equipping the HodgeLaplacian with absolute boundary conditions, whereas prescribing n.t. n.t. ν ∧ u∂Ω and ν ∧ (δu)∂Ω (1.1.37) on ∂Ω is referred to as endowing the HodgeLaplacian with relative boundary conditions. These boundary conditions are signiﬁcant for their relation to absolute and relative cohomology of Ω. The degree l ∈ {0, 1, . . . , n} of the differential form u also affects the nature of the boundary value problems presented in Theorem 1.1. In relation to the scalar case, it is reassuring to note that (1.1.36) contains (in slight disguise) both the Dirichlet and Neumann boundary conditions for the LaplaceBeltrami operator ∆LB u =
1 n ∂ ∂u ∑ [g jk √g ], ∂x k √g j,k=1 ∂x j
(1.1.38)
canonically associated with the metric tensor g = ∑nj,k=1 g jk dx j ⊗ dx k on the manifold M. Concretely, for l = n the ﬁrst boundary condition in (BVP1)l is equivalent to the Dirichlet condition on ∂Ω p n.t. n.t. u∂Ω = ν ∧ (ν ∨ u∂Ω ) = ν ∧ f ∈ Lnor (∂Ω, Λ n T M) ≡ L p (∂Ω) dVol,
(1.1.39)
whereas the second boundary condition in (BVP1)l is trivially satisﬁed given that p du = 0 and Ltan (∂Ω, Λ n T M) = {0}. Thus, with natural identiﬁcations, the formulation 10 each of which amounting to a “half” of what would normally constitute a “full” boundary condition, by targeting either the tangential or the normal component of such an expression
10  1 Introduction and Statement of Main Results of (BVP1)n streamlines to the L p Dirichlet boundary value problem for the LaplaceBeltrami operator acting on scalar functions in Ω: u ∈ C 0 (Ω), { { { { { { ∆LB u = 0 in Ω, { { { N u ∈ L p (∂Ω), { { { n.t. p { u∂Ω = f ∈ L (∂Ω).
(1.1.40)
In relation to this, the regularity result from item (4) of Theorem 1.1 gives that any solution u of (1.1.40) satisﬁes N(∇u) ∈ L p (∂Ω) if and only if the boundary datum f p belongs to the Sobolev space L1 (∂Ω). Formulated as such, the version of (1.1.40) is typically referred to as the Regularity problem for the LaplaceBeltrami operator. Also, corresponding to l = 0, the ﬁrst boundary condition in (BVP1)l is obviously n.t. veriﬁed since, on the one hand we have ν ∨ u∂Ω = 0, while on the other hand we have
Ltan (∂Ω, Λ−1 T M) = {0} by degree considerations. At the same time, when l = 0 the second boundary condition in (BVP1)l becomes equivalent to the Neumann condition for scalarvalued functions since, if ∂/∂ν denotes the normal derivative on ∂Ω, we have p,δ
∂u n.t. n.t. = ⟨ν, (du)∂Ω ⟩ = ν ∨ (du)∂Ω = g ∈ L p (∂Ω). ∂ν
(1.1.41)
This shows that the formulation of (BVP1)0 streamlines to the L p Neumann boundary value problem for the LaplaceBeltrami operator in Ω: u ∈ C 1 (Ω), { { { { { { { ∆LB u = 0 in Ω, { { N u, N(∇u) ∈ L p (∂Ω), { { { { { ∂u = f ∈ L p (∂Ω). { ∂ν
(1.1.42)
For Lipschitz domains in the ﬂat, Euclidean setting, these particular limiting cases have been treated in the work of B. Dahlberg [19], E. Fabes, M. Jodeit and N. Rivière [32], D. S. Jerison and C. E. Kenig [57], G. Verchota [125], B. Dahlberg and C. E. Kenig [21]. Actually, these papers deal only with domains with connected boundary, a restriction that has been eliminated by D. Mitrea in [78]. Subsequently, such results have been extended to arbitrary Lipschitz domains in Riemannian manifolds by M. Mitrea and M. Taylor in a series of papers [96, 98–100], dealing with increasingly less restrictive smoothness conditions on the metric tensor. The formulation of the problem (BVP2)l , albeit in a much more restrictive geometric and analytic setting, goes back to W. V. D. Hodge whose pioneering work in [47, 48] has attracted a vast amount of attention over the years. As a result, a fairly sophisticated theory has emerged in the context of smooth domains in Riemannian manifolds through the work of G. F. D. Duff, P. R. Garabedian, W. T. B. Kelvin, J. J. Kohn,
1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms

11
and D. C. Spencer in [25, 27–29, 38, 39, 63, 70, 114]. When all structures involved are smooth, problems related to (BVP1)l –(BVP6)l have been treated by P. E. Conner [17], G. F. D. Duff [27], P. R. Garabedian and D. C. Spencer [39], C. B. Morrey [103, 104], G. Schwarz [109], A. W. Tucker [124]. See also Chapter 5 of [118] for an exposition. For arbitrary Lipschitz domains in the ﬂat Euclidean setting and arbitrary degrees l ∈ {0, 1, . . . , n}, similar problems have been addressed in the work of D. Mitrea and M. Mitrea [77, 80]. Subsequently, these results have been extended to the context of Lipschitz subdomains of Riemannian manifolds by D. Mitrea, M. Mitrea, and M. Taylor in [86]. The approach in [86] is based on certain Rellich identities for differential forms, building on earlier work in [56, 84, 92]. A salient feature of this method is the existence of a smooth transversal vector ﬁeld to the Lipschitz domain in question. This feature has been shown in [51] to actually characterize the class of Lipschitz domains, so any Rellichidentity based approach is inherently limited to the class of Lipschitz domains. The range 2 − ϵ(Ω) < p < 2 + ϵ(Ω), where ϵ(Ω) ∈ (0, 1) is a small number depending on the Lipschitz character of the Lipschitz domain Ω,
(1.1.43)
dealt with in [86] is optimal in the class of (all smooth manifolds and) all Lipschitz domains if one insists on either allowing arbitrary degrees l ∈ {0, 1, . . . , n}, or staying away from the endpoints, i.e., for 1 ≤ l ≤ n − 1. However, it is not clear from the work in [86] that the range (1.1.43) naturally enlarges as the regularity of the domain improves; in particular, in analogy with work carried out in [32], one would like to have the full range 1 < p < ∞ if Ω is actually a domain of class C 1 . While the methods in [86] do not give this, our present techniques, based on compactness and Fredholm theory, apply to an even more general category of domains¹¹ than the class of C 1 domains. See also [52] for results on absolute and relative boundary conditions on the Lipschitz domains with VMO normals, achieved by development of a symbol calculus.
1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms In stark contrast to the case of the scalar LaplaceBeltrami operator, the structural richness of the higher degree case allows for a considerably larger variety of natural boundary conditions for the HodgeLaplacian. The second natural boundary problem for the HodgeLaplacian in a domain Ω ⊂ M seeks nullsolutions u of the operator ∆HL in Ω for which n.t. { ν ∨ (du)∂Ω (1.2.1) { n.t. ν ∧ (δu) ∂Ω { 11 which is essentially optimal from the harmonic analysis point of view
12  1 Introduction and Statement of Main Results
are prescribed in suitable function spaces deﬁned on the boundary of the underlying domain. This makes the object of the next theorem. Theorem 1.2. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and, for a given degree l ∈ {0, 1, . . . , n} and exponent p ∈ (1, ∞), consider the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL u = 0 in Ω, { { { (BVP7)l { N u, N(du), N(δu) ∈ L p (∂Ω), { { p n.t. { l { { { ν ∨ (du)∂Ω = f ∈ Ltan (∂Ω, Λ T M), { { { p n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M).
(1.2.2)
Then the following statements are true: (1) A solution of (BVP7)l exists if and only if the compatibility condition n.t. ⊥ f − g ∈ {H l,p (Ω)∂Ω }
(1.2.3)
is satisﬁed (where 1/p + 1/p = 1 and the superscript “⊥” refers to the annihilator in L p (∂Ω, Λ l T M)). (2) The space of solutions for the homogeneous version of (BVP7)l is precisely H l,p (Ω). In particular, du and δu are uniquely determined by the boundary data and, further, ‖N(du)‖L p (∂Ω) + ‖N(δu)‖L p (∂Ω) ≤ C (‖f‖L p (∂Ω,Λ l TM) + ‖g‖L p (∂Ω,Λ l TM) )
(1.2.4)
for some ﬁnite positive constant C, independent of f and g. Also, in the case p = 2 any solution u of (BVP7)l satisﬁes u ∈ H 1/2,2 (Ω, Λ l T M), du ∈ H 1/2,2 (Ω, Λ l+1 T M), δu ∈ H
1/2,2
(Ω, Λ
l−1
(1.2.5)
T M).
In addition, if the compatibility condition (1.2.3) is satisﬁed then the following additional properties hold: (3) For any solution u of (BVP7)l one has p,δ
f ∈ Ltan (∂Ω, Λ l T M) ⇐⇒ N(δdu) ∈ L p (∂Ω) p,d
⇐⇒ N(dδu) ∈ L p (∂Ω) ⇐⇒ g ∈ Lnor (∂Ω, Λ l T M).
(1.2.6)
(4) For any solution u of (BVP7)l one has p,0
f ∈ Ltan (∂Ω, Λ l T M) ⇐⇒ δdu = 0 in Ω ⇐⇒ dδu = 0 in Ω p,0
⇐⇒ g ∈ Lnor (∂Ω, Λ l T M).
(1.2.7)
1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms

13
In the case when the latter condition holds, (BVP7)l becomes u ∈ C 1 (Ω, Λ l T M), { { { { { { δdu = dδu = 0 in Ω, { { { { (BVP8)l { N u, N(du), N(δu) ∈ L p (∂Ω), { { p,0 n.t. { { ν ∨ (du)∂Ω = f ∈ Ltan (∂Ω, Λ l T M), { { { { { p,0 n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M).
(1.2.8)
(5) Any solution u of (BVP7)l satisﬁes du = 0 in Ω ⇐⇒ f = 0.
(1.2.9)
In the scenario when the latter condition holds, (BVP7)l becomes u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { (BVP9)l { du = 0 in Ω, { { { { { N u, N(δu) ∈ L p (∂Ω), { { { p,0 n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M).
(1.2.10)
(6) Any solution u of (BVP7)l satisﬁes δu = 0 in Ω ⇐⇒ g = 0.
(1.2.11)
In this case, (BVP7)l becomes the Hodgedual of (BVP9)n−l , i.e., u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { (BVP10)l { δu = 0 in Ω, { { { { { N u, N(du) ∈ L p (∂Ω), { { { p,0 n.t. l { ν ∨ (du)∂Ω = f ∈ Ltan (∂Ω, Λ T M).
(1.2.12)
Finally, all these results continue to hold when the hypotheses on the underlying domain are relaxed from demanding that Ω is a regular SKT domain to just asking that this is an εSKT domain, for some ε > 0 which is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of Ω). The algebraic structure of the HodgeLaplacian ∆HL = −dδ − δd suggests that a natural “conormal derivative” on ∂Ω is¹² n.t. n.t. u → ν ∨ (du)∂Ω − ν ∧ (δu)∂Ω . 12 see item (3) in Lemma 2.8
(1.2.13)
14  1 Introduction and Statement of Main Results n.t. n.t. Given that the two constitutive pieces, ν ∨ (du)∂Ω and ν ∧ (δu)∂Ω , are tangential and normal forms, respectively, they may be recovered individually when the full conormal derivative is prescribed on the boundary. As such, Theorem 1.2 may be regarded as the natural conormal (or HodgeNeumann) derivative problem for the HodgeLaplacian ∆HL . In fact, if l = 0 then (BVP7)l reduces precisely to the Neumann problem for the LaplaceBeltrami operator. In this regard we wish to point out that, in the smooth context, problems related to (BVP7)l have been studied by G. F. D. Duff [26], P. E. Conner [17], C. B. Morrey [103], D. C. Spencer [114], and G. Schwarz [109]. Other higherdegree problems for the HodgeLaplacian exhibiting natural boundary conditions of interest are presented in the following four theorems. In the ﬁrst such problem, one prescribes { ν ∨ (δu)n.t. { ∂Ω (1.2.14) { { n.t. ν ∧ (du)∂Ω { in certain natural function spaces deﬁned on the boundary of the underlying domain. Theorem 1.3. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and, for a given degree l ∈ {0, 1, . . . , n} and exponent p ∈ (1, ∞), consider the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { { dδu = δdu = 0 in Ω, { { { (BVP11)l { N u, N(du), N(δu) ∈ L p (∂Ω), { { p,0 n.t. { l−2 { { { ν ∨ (δu)∂Ω = f ∈ Ltan (∂Ω, Λ T M), { { { p,0 n.t. l+2 { ν ∧ (du)∂Ω = g ∈ Lnor (∂Ω, Λ T M).
(1.2.15)
Then there exists a ﬁnite codimensional subspace of p,0
p,0
Ltan (∂Ω, Λ l−2 T M) ⊕ Lnor (∂Ω, Λ l+2 T M),
(1.2.16)
whose codimension is bounded by a ﬁnite integer independent of p, with the property that the problem (BVP11)l is solvable if and only if the boundary datum (f, g) belongs to the said subspace. In fact, a solution to (BVP11)l exists if and only if p,δ
f ∈ δ ∂ [Ltan (∂Ω, Λ l−1 T M)]
and
p,d
g ∈ d ∂ [Lnor (∂Ω, Λ l+1 T M)].
(1.2.17)
Moreover, the space of solutions of the homogeneous version of (BVP11)l may be described as the collection of differential forms u satisfying u ∈ C 1 (Ω, Λ l T M), { { { { { { dδu = δdu = 0 in Ω, { { { N u, N(du), N(δu) ∈ L p (∂Ω), { { { n.t. p,0 p,0 l−1 l+1 { u∂Ω ∈ ν ∧ Ltan (∂Ω, Λ T M) ⊕ ν ∨ Lnor (∂Ω, Λ T M).
(1.2.18)
At least when Ω ≠ M, ∂Ω ≠ ⌀, and l ∈ {1, . . . , n − 1}, this space is inﬁnite dimensional.
1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms

15
Finally, all the above results remain valid in the case when Ω is an εSKT domain, assuming that the parameter ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of the domain Ω). It turns out that the ﬁrst three conditions in the formulation of (1.2.15) imply the pointn.t. n.t. wise existence of (δu)∂Ω and (du)∂Ω on ∂Ω and also force p,0 n.t. ν ∨ (δu)∂Ω ∈ Ltan (∂Ω, Λ l−2 T M),
(1.2.19)
p,0 n.t. ν ∧ (du)∂Ω ∈ Lnor (∂Ω, Λ l+2 T M). p,0
In turn, these explain the necessity of the memberships of f and g in Ltan (∂Ω, Λ l−2 T M) p,0 and Lnor (∂Ω, Λ l+2 T M), respectively, in the formulation of the boundary value problem (BVP11)l . The second problem mentioned earlier deals with either n.t. { ν ∨ u∂Ω { n.t. { ν ∧ (du)∂Ω
or
n.t. { ν ∧ u∂Ω { n.t. { ν ∨ (δu)∂Ω
(1.2.20)
prescribed in natural spaces on ∂Ω, in the case when the harmonic form u is actually annihilated by both dδ and δd in Ω. Theorem 1.4. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and, for a given degree l ∈ {0, 1, . . . , n} and exponent p ∈ (1, ∞), consider the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { { dδu = δdu = 0 in Ω, { { { (BVP12)l { N u, N(du) ∈ L p (∂Ω), { { p n.t. { l−1 { { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ T M), { { { p,0 n.t. l+2 { ν ∧ (du)∂Ω = g ∈ Lnor (∂Ω, Λ T M).
(1.2.21)
In relation to this boundary value problem, the following conclusions hold: p,0 (1) There exists a ﬁnite codimensional subspace of Lnor (∂Ω, Λ l+2 T M), with codimension bounded by a ﬁnite integer independent of p, such that the problem (BVP12)l is solvable if and only if the boundary datum g belongs to it. In fact, the boundary value problem (BVP12)l admits a solution if and only if p,d
g ∈ d ∂ [Lnor (∂Ω, Λ l+1 T M)].
(1.2.22)
16  1 Introduction and Statement of Main Results (2) Any solution u of the homogeneous version of (BVP12)l satisﬁes δu = 0 in Ω. Moreover, the space of solutions for the homogeneous version of (1.2.21) has dimension ≤ b l (Ω) + b n−l−1 (Ω) (hence, is ﬁnite dimensional). As a consequence, the boundary value problem (1.2.21) is Fredholm solvable, with index bounded independently of p. In fact, the space of nullsolutions of (BVP12)l consists precisely of differential forms u satisfying u ∈ C 1 (Ω, Λ l T M), { { { { { { δu = 0 and δdu = 0 in Ω, (1.2.23) { { { N u, N(du) ∈ L p (∂Ω), { { { n.t. p,0 l+1 { u∂Ω ∈ ν ∨ Lnor (∂Ω, Λ T M). (3) Granted the compatibility condition (1.2.22), any solution u of (BVP12)l satisﬁes p,δ
N(δu) ∈ L p (∂Ω) ⇐⇒ f ∈ Ltan (∂Ω, Λ l−1 T M),
(1.2.24)
as well as p,0 n.t. ⊥ δu = 0 in Ω ⇐⇒ f ∈ Ltan (∂Ω, Λ l−1 T M) ∩ {H∨l−1 (Ω)∂Ω } .
(1.2.25)
In particular, the boundary value problems u ∈ C 1 (Ω, Λ l T M), { { { { { { dδu = δdu = 0 in Ω, { { { { (BVP13 )l { N u, N(du), N(δu) ∈ L p (∂Ω), { { p,δ n.t. { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M), { { { { { p,d n.t. l+1 { ν ∧ (du)∂Ω = g ∈ d ∂ [Lnor (∂Ω, Λ T M)],
(1.2.26)
and u ∈ C 1 (Ω, Λ l T M), { { { { { { δu = 0 and δdu = 0 in Ω, { { { { (1.2.27) (BVP13 )l { N u, N(du) ∈ L p (∂Ω), { { p,0 ⊥ n.t. n.t. { l−1 l−1 { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ T M) ∩ {H∨ (Ω)∂Ω } , { { { { { p,d n.t. l+1 { ν ∧ (du)∂Ω = g ∈ d ∂ [Lnor (∂Ω, Λ T M)], are always solvable. (4) Similar results as in items (1)–(3) above are valid for the Hodge dual of (BVP12)l , i.e., for u ∈ C 1 (Ω, Λ l T M), { { { { { { dδu = δdu = 0 in Ω, { { { { (1.2.28) (BVP14)l { N u, N(δu) ∈ L p (∂Ω), { { p n.t. { { ν ∧ u∂Ω = f ∈ Lnor (∂Ω, Λ l+1 T M), { { { { { p n.t. l−2 { ν ∨ (δu)∂Ω = g ∈ Ltan (∂Ω, Λ T M).
1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms

17
Finally, all the above results continue to hold in the case when Ω is an εSKT domain, granted ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of Ω). As we shall see later, the ﬁrst three conditions in the formulation of (1.2.21) imply n.t. n.t. the pointwise existence of u∂Ω and (du)∂Ω on ∂Ω, as well as the membership of p,0 n.t. ν ∧ (du)∂Ω in Lnor (∂Ω, Λ l+2 T M). This explains the necessity of selecting g from the latter space in the formulation of the boundary value problem (BVP12)l . This brings us to the third boundary value problem alluded to before, in which one seeks to specify either n.t. { ν ∧ u∂Ω { n.t. { ν ∨ (du)∂Ω
or
n.t. { ν ∨ u∂Ω { n.t. { ν ∧ (δu)∂Ω
(1.2.29)
in appropriate spaces on ∂Ω, for a nullsolution u of the HodgeLaplacian ∆HL in Ω. Theorem 1.5. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and surface measure σ. For a given degree l ∈ {0, 1, . . . , n} and exponent p ∈ (1, ∞), consider the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { (BVP15)l { N u, N(du) ∈ L p (∂Ω), { { p,d n.t. { { ν ∧ u∂Ω = f ∈ Lnor (∂Ω, Λ l+1 T M), { { { { { p n.t. l { ν ∨ (du)∂Ω = g ∈ Ltan (∂Ω, Λ T M).
(1.2.30)
Then the following statements are valid: (1) A solution to (BVP15)l exists if and only if the compatibility condition n.t. n.t. ∫ ⟨f, (dθ)∂Ω ⟩ dσ = ∫ ⟨g, θ∂Ω ⟩ dσ ∂Ω
(1.2.31)
∂Ω
is veriﬁed for each differential form θ satisfying¹³ (with 1/p + 1/p = 1) θ ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL θ = 0 in Ω, { { δθ = 0 in Ω, { { { { { { N θ, N(dθ) ∈ L p (∂Ω), { { { n.t. { ν ∨ θ∂Ω = 0 on ∂Ω.
(1.2.32)
n.t. n.t. 13 the conditions in (1.2.32) ensure that the nontangential boundary traces (dθ)∂Ω and θ∂Ω exists on ∂Ω
18  1 Introduction and Statement of Main Results Moreover, when (1.2.31) is satisﬁed, a solution u of (BVP15)l may be constructed with the additional property that N(δu) ∈ L p (∂Ω). (2) The space of nullsolutions of (BVP15)l is precisely the collection of differential forms u satisfying u ∈ C 1 (Ω, Λ l T M), { { { { { { dδu = 0 and du = 0 in Ω, { { N u ∈ L p (∂Ω), { { { { n.t. p l { u∂Ω ∈ Lnor (∂Ω, Λ T M). (3) Any solution u of (BVP15)l satisﬁes du = 0 in Ω if and only if p,0 n.t. ⊥ f ∈ Lnor (∂Ω, Λ l+1 T M) ∩ {H l+1,p (Ω)∂Ω } and g = 0.
(1.2.33)
(1.2.34)
Conversely, if (1.2.34) holds then a solution u of (BVP15)l may be found with the additional property that du = 0 in Ω. (4) Assuming that the compatibility condition (1.2.31) holds, it follows that any solution u of the boundary problem (BVP15)l satisﬁes N(dδu) ∈ L p (∂Ω) ⇐⇒ N(δdu) ∈ L p (∂Ω) p,δ
⇐⇒ g ∈ Ltan (∂Ω, Λ l T M).
(1.2.35)
(5) Similar results as in items (1)–(4) above hold for the Hodge dual of (BVP15)l , i.e., for u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { (BVP16)l { N u, N(δu) ∈ L p (∂Ω), (1.2.36) { { p,δ n.t. { l−1 { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ T M), { { { { { p n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M). Once more, these results remain valid in the case when Ω is an εSKT domain, assuming that ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of Ω). A couple of comments are in order here. First, later arguments are going to show that the ﬁrst three conditions in the formulation of (1.2.30) imply the pointwise existence of p,d n.t. n.t. n.t. u∂Ω and (du)∂Ω on ∂Ω, as well as the membership of ν ∧ u∂Ω to Lnor (∂Ω, Λ l+1 T M). This explains the necessity of selecting f from the latter space in the formulation of the boundary value problem (BVP15)l . Second, it turns out that the compatibility condition (1.2.31) may be formulated in terms of the NeumanntoDirichlet operator ΛND for the HodgeLaplacian, which we deﬁne as the mapping p ,0
p
ΛND : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M) p ,0 n.t. ΛND h := θ∂Ω for each h ∈ Ltan (∂Ω, Λ l T M),
(1.2.37)
1.2 Other Problems Involving Tangential and Normal Components of Harmonic Forms 
19
where θ is the unique solution of the boundary value problem (whose wellposedness is a consequence of the discussion in item (8) of Theorem 1.1): θ ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL θ = 0 in Ω, { { { { { { δθ = 0 in Ω, { { { { N θ, N(dθ) ∈ L p (∂Ω), { { { n.t. { { ν ∨ θ∂Ω = 0 on ∂Ω, { { { { { p ,0 n.t. { { ν ∨ (dθ)∂Ω = h ∈ Ltan (∂Ω, Λ l T M), { { { { { ∫Ω ⟨θ, ω j ⟩ dVol = 0 for 1 ≤ j ≤ b l (Ω).
(1.2.38)
Speciﬁcally, a careful inspection reveals that (1.2.31) is equivalent to demanding that p ,0 n.t. ⊥ g ∈ {H∨l (Ω)∂Ω } and, for each h ∈ Ltan (∂Ω, Λ l T M),
∫ ⟨ν ∨ f, h⟩ dσ = ∫ ⟨g, ΛND h⟩ dσ. ∂Ω
(1.2.39)
∂Ω
Consequently, the boundary value problem (BVP15)l is solvable if and only if (1.2.39) holds. Finally, we state the fourth boundary value problem referred to earlier, in which one seeks a nullsolution u of the HodgeLaplacian ∆HL in Ω with either n.t. { ν ∨ (δu)∂Ω { n.t. { ν ∨ (du)∂Ω
or
n.t. { ν ∧ (du)∂Ω { n.t. { ν ∧ (δu)∂Ω
(1.2.40)
prescribed in appropriate spaces on ∂Ω. Theorem 1.6. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν. Given a degree l ∈ {0, 1, . . . , n} and an exponent p ∈ (1, ∞), consider the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { (1.2.41) (BVP17)l { N u, N(du), N(δu) ∈ L p (∂Ω), { { p,0 n.t. { l−2 { ν ∨ (δu)∂Ω = f ∈ Ltan (∂Ω, Λ T M), { { { { { p n.t. l { ν ∨ (du)∂Ω = g ∈ Ltan (∂Ω, Λ T M). In relation to this boundary value problem, the following properties hold: (1) There exists a ﬁnite codimensional subspace of p,0
p
Ltan (∂Ω, Λ l−2 T M) ⊕ Ltan (∂Ω, Λ l T M),
(1.2.42)
20  1 Introduction and Statement of Main Results
whose codimension may be bounded independently of p, with the property that the problem (BVP17)l is solvable if and only if the boundary datum (f, g) belongs to it. In fact, a solution to (BVP17)l exists if and only if p,δ n.t. ⊥ f ∈ δ ∂ [Ltan (∂Ω, Λ l−1 T M)] and g ∈ {H∨l (Ω)∂Ω } . (1.2.43) (2) The space of nullsolutions of (BVP17)l may be described as the collection of differential forms u satisfying u ∈ C 1 (Ω, Λ l T M), { { { { { { du = 0 and dδu = 0 in Ω, { { N u, N(δu) ∈ L p (∂Ω), { { { { p,0 n.t. l−1 { ν ∨ u∂Ω ∈ Ltan (∂Ω, Λ T M).
(1.2.44)
At least when Ω ≠ M, ∂Ω ≠ ⌀, and l ∈ {1, . . . , n − 1}, this space is inﬁnite dimensional. Moreover, whenever f, g are as in (1.2.43), any solution u of the boundary value problem (BVP17)l exhibits the following additional features: (3) The boundary data f, g determine du uniquely. (4) Any solution u of (BVP17)l has N(dδu) ∈ L p (∂Ω) (hence also N(δdu) ∈ L p (∂Ω)) if p,δ and only if g ∈ Ltan (∂Ω, Λ l T M), in which case ‖N(δdu)‖L p (∂Ω) + ‖N(dδu)‖L p (∂Ω) ≤ C‖g‖L p,δ (∂Ω,Λ l TM) ,
(1.2.45)
tan
for some ﬁnite positive constant C, independent of g. (5) Any solution u of (BVP17)l satisﬁes du = 0 in Ω if and only if g = 0. (6) Any solution u of (BVP17)l satisﬁes δdu = 0 in Ω (hence also dδu = 0 in Ω) if and p,0 only if g ∈ Ltan (∂Ω, Λ l T M). (7) Analogous properties to (1)–(6) above hold for the Hodge dual of (BVP17)l , i.e., u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { (BVP18)l { N u, N(δu), N(du) ∈ L p (∂Ω), { { p,0 n.t. { { ν ∧ (du)∂Ω = f ∈ Lnor (∂Ω, Λ l+2 T M), { { { { { p n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M).
(1.2.46)
Finally, all results above continue to hold when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small, relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of Ω). Once again, the ﬁrst three conditions in the formulation of (1.2.41) ensure the pointn.t. n.t. n.t. wise existence of (δu)∂Ω and (du)∂Ω on ∂Ω, as well as the membership of ν ∨ (δu)∂Ω p,0 in Ltan (∂Ω, Λ l−2 T M). In particular, this explains the necessity of selecting f from the latter space in the formulation of the boundary value problem (BVP17)l . Of course, similar considerations apply to (BVP18)l .
1.3 Boundary Value Problems for HodgeDirac Operators

21
1.3 Boundary Value Problems for HodgeDirac Operators Going further, we shift focus from boundary problems associated with the second order, selfadjoint elliptic operator ∆HL = −(dδ + δd) to boundary problems associated with ﬁrstorder elliptic operators, of Dirac type. The prototypical example of a Dirac operator on the Riemannian manifold M is d + δ, where d is the exterior derivative operator and δ its formal adjoint. Adopting a more general point of view, we shall consider the family of HodgeDirac operators d + αδ,
where α ∈ ℝ \ {0},
(1.3.1)
acting in the Grassmann algebra bundle n
G := ⨁ Λ l T M.
(1.3.2)
l=0
Note that d + αδ is selfadjoint if and only if α = 1. Nonetheless, the identity (d + αδ)2 = −α∆HL
(1.3.3)
is valid for any α. We are concerned with “halfDirichlet” problems for the Diractype operator d + αδ in a subdomain Ω of M. Our main result in this regard, reﬁning work from [7] in the smooth setting, and [93] in the context of Lipschitz domains (see also [52]), reads as follows¹⁴. Theorem 1.7. Assume Ω ⊂ M is a regular SKT domain with outward unit conormal ν. Fix a parameter α ∈ ℝ \ {0} along with an exponent p ∈ (1, ∞), and consider the normal halfDirichlet problem for the Diractype operator d + αδ in Ω, formulated as u ∈ C 0 (Ω, G), { { { { { { (d + αδ)u = 0 in Ω, ( 12 BVPnor ) { { { N u ∈ L p (∂Ω), { { { p n.t. { ν ∧ u∂Ω = f ∈ Lnor (∂Ω, G).
(1.3.4)
Then the following properties hold: (1) The problem ( 12 BVPnor ) is solvable if and only if the boundary datum f satisﬁes the compatibility condition ⊥ n.t. ⊥ n.t. f ∈ {H∧ (Ω, G)∂Ω } := {ω∂Ω : ω ∈ H∧ (Ω, G)} .
Here
(1.3.5)
n
H∧ (Ω, G) := ⨁ H∧l (Ω)
(1.3.6)
l=0
14 for further clariﬁcations regarding the notation employed in the formulation of this result the reader is referred to the discussion preceding the proof of Theorem 1.7 in the last part of § 7.2
22  1 Introduction and Statement of Main Results
is a ﬁnite dimensional space, of dimension n
dim H∧ (Ω, G) = ∑ b l (Ω),
(1.3.7)
l=0
and {. . .}⊥ is the annihilator in L p (∂Ω, G) of {. . .}, for 1/p + 1/p = 1. (2) The space of solutions for the homogeneous version of ( 12 BVPnor ) is precisely H∧ (Ω, G). In particular, du and δu are uniquely determined by the boundary datum f (even if u itself is not). Furthermore, in the case when the compatibility condition (1.3.5) holds, the following regularity results are also valid for any solution u of ( 12 BVPnor ): (3) One has p,d
N(du) ∈ L p (∂Ω) ⇐⇒ N(δu) ∈ L p (∂Ω) ⇐⇒ f ∈ Lnor (∂Ω, G).
(1.3.8)
In particular, any u ∈ C 0 (Ω, G) with (d + αδ)u = 0 in Ω and N u ∈ L p (∂Ω), p,d p,δ n.t. n.t. satisﬁes ν ∧ u∂Ω ∈ Lnor (∂Ω, G) ⇐⇒ ν ∨ u∂Ω ∈ Ltan (∂Ω, G). Naturally accompanying estimates are valid in each case. (4) One has p,0 du = 0 in Ω ⇐⇒ δu = 0 in Ω ⇐⇒ f ∈ Lnor (∂Ω, G).
(1.3.9)
(1.3.10)
In particular, any u ∈ C 0 (Ω, G) with (d + αδ)u = 0 in Ω and N u ∈ L p (∂Ω), p,0 p,0 n.t. n.t. satisﬁes ν ∧ u∂Ω ∈ Lnor (∂Ω, G) ⇐⇒ ν ∨ u∂Ω ∈ Ltan (∂Ω, G).
(1.3.11)
(5) Similar results as in items (1)–(4) above are valid for the tangential halfDirichlet problem for the Diractype operator d + αδ in Ω: u ∈ C 0 (Ω, G), { { { { { { (d + αδ)u = 0 in Ω, ( 12 BVPtan ) { { { N u ∈ L p (∂Ω), { { { p n.t. { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, G).
(1.3.12)
Finally, all these results remain valid in the case when Ω is an εSKT domain, provided the parameter ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω (i.e., the large geometry constants of Ω). All our results in this work are valid in the category of εSKT domains, where the parameter ε > 0 is sufficiently small relative to the given integrability exponent p ∈ (1, ∞)
1.3 Boundary Value Problems for HodgeDirac Operators

23
and the other (typically “large”) geometric characteristics of the underlying domain (speciﬁcally, the constants intervening in the Ahlfors regularity condition and twosided local John condition). The class of εSKT domains is deﬁned analogously to (1.0.2), with the last condition relaxed to dist (ν, VMO(∂Ω)) < ε,
(1.3.13)
where ν is the outward unit conormal to Ω and the distance is measured in BMO(∂Ω). The consideration of the class of εSKT domains is signiﬁcant as it allows for this category of domains, exhibiting a suitably nuanced type of roughness, to be part of the current proceedings. In particular, this class contains domains which are locally uppergraphs of Lipschitz functions with gradients sufficiently close to VMO (hence, in particular, of small BMO norm), as well as Lipschitz domains with a sufficiently small Lipschitz constant. A remarkable feature of work within the class of εSKT domains is that one can target a speciﬁc integrability exponent p, arbitrarily prescribed in (1, ∞), in the formulation of the solvability results in Theorems 1.1–1.9 in such domains by appropriately “tuning down” the degree of roughness they are allowed to exhibit (i.e., asking that ε > 0 is suitably small). This point of view gives a more direct link between the present work and that undertaken in [86] where the case of arbitrary Lipschitz domains and p near 2 has been treated. In this vein, we also wish to mention that classical Dirichlet and Neumann boundary value problems in εSKT domains for scalarvalued functions and in terms of the (scalar) LaplaceBeltrami operator have been treated in [50, § 5]. As it turns out, the consideration of the more inclusive class of εSKT domains in place of regular SKT domains is necessary even when the ultimate goal is to prove solvability results for boundary problems formulated in the latter (smaller) class of domains. This is due to a technical step in the proof, employing an approximation procedure in which a given regular SKT domain Ω is exhausted with a family of subdomains Ω j ↗ Ω which, in general, are only known to be εSKT domains for some small ε > 0 (rather than genuine regular SKT domains). Regarding the role of the inﬁnitesimal ﬂatness condition (1.3.13) in the deﬁnition of the class of εSKT domains, we wish to note that there are many natural classes of domains satisfying the properties listed in the ﬁrst two lines of (1.0.2) (including bounded Lipschitz domains or, more generally, bounded domains locally given as the uppergraphs of continuous functions with gradients in BMO or, even more generally, twosided NTA domains with a compact Ahlfors regular boundary), and for any such domain Ω we always have dist (ν, VMO(∂Ω)) ∈ [0, 1].
(1.3.14)
However, the wellposedness of (BVP1)l for an arbitrary given p ∈ (1, ∞) typically fails in the absence of more stringent provisions regarding the smallness of dist (ν, VMO(∂Ω)). This is already visible from the nature of the range of p’s identiﬁed in (1.1.43), in the class of Lipschitz domains.
24  1 Introduction and Statement of Main Results
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems Moving on, we turn our attention to boundary value problems for the HodgeLaplacian of a more classical Dirichlet and Neumann ﬂavor, with data in ordinary L p spaces of differential forms. In the category of Lipschitz domains and for scalar functions, this is a topic which has received a great deal of attention (cf. [19, 21, 32, 57, 96, 98–100, 125], and the references therein). Concerning domains that are not necessarily Lipschitz, as early as 1936, M. Laurentiev¹⁵ proved in [73] that for a chordarc domain Ω in the plane the harmonic measure and the arclength measure on ∂Ω are A∞ equivalent. This turns out to be equivalent to the wellposedness of the Dirichlet problem with L p boundary data for p sufficiently large. By further combining this with certain conformal mapping techniques developed in [64], it has been established in [58] that one may also obtain the solvability for the dual range of integrability exponents of the Neumann and Regularity problems in planar chordarc domains. In the higher dimensional setting, if Ω is a bounded twosided NTA domain whose boundary is Ahlfors regular, by combining the A∞ equivalence of the surface and harmonic measures on ∂Ω with [59, Theorem 5.8, p. 105] it follows that (1.4.2) is wellposed if p is sufficiently large. The extent to which such results hold in the higher dimensional setting in regular SKT domains in the ﬂat Euclidean setting and for the scalar, constant coefficient Laplacian, has been investigated in [50]. Here we continue this line of work by proving the more general result in Theorem 1.8 below. Concerning notation, we remark that H s,p (Ω) stands for the L p based Sobolev space of fractional smoothness s ∈ ℝ in an open set Ω ⊂ M (deﬁned by restrictp ing to Ω distributions from H s,p (M)). Also, L1 (∂Ω) stands for the L p based Sobolev space of smoothness 1, suitably deﬁned on the boundary of an Ahlfors regular domain Ω ⊂ M; cf. the discussion in § 9.5. Let us also recall here Weitzenböck’s formula to the effect that − ∆HL = ∇∗ ∇ + Ric. (1.4.1) Above, ∇ is the LeviCivita connection acting on differential forms on M (cf. the discussion in § 9.2), and Ric is the socalled Weitzenböck operator (cf. § 9.3). More specifically, Ric is a curvature term of order zero (depending linearly on the Riemann curvature, via real coefficients) that preserves lforms, and is selfadjoint. In addition, Ric vanishes identically on scalar functions and coincides with the classical Ricci tensor¹⁶ when acting on 1forms on M.
15 whose name has also been spelled Lavrentiev in the literature 16 which explains the present choice of notation
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems  25
Theorem 1.8. Given a regular SKT domain Ω ⊂ M, along with an arbitrary integrability exponent p ∈ (1, ∞) and degree l ∈ {0, 1, . . . , n}, consider the Dirichlet boundary value problem for the HodgeLaplacian u ∈ C 0 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, (HLDp )l { { { N u ∈ L p (∂Ω), { { { n.t. p l { u∂Ω = f ∈ L (∂Ω, Λ T M).
(1.4.2)
In relation to this boundary value problem, the following statements are true: (i) The boundary value problem (1.4.2) is always solvable. Moreover, there exists a ﬁnite constant C = C(M, Ω, p) > 0 such that, for each f ∈ L p (∂Ω, Λ l T M), one can ﬁnd a solution u of (1.4.2) obeying ‖N u‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) ,
(1.4.3)
‖u‖L np/(n−1) (Ω,Λ l TM) ≤ C‖f‖L p (∂Ω,Λ l TM) .
(1.4.4)
as well as
In addition, in the case when p = 2, the aforementioned solution also satisﬁes the L2 square function estimate 1/2
2 (∫ (∇u)(x) dist (x, ∂Ω) dVol(x))
≤ C‖f‖L2 (∂Ω,Λ l TM) ,
(1.4.5)
Ω
where ∇ denotes the LeviCivita connection acting on differential forms on M, and has the global regularity property ‖u‖H 1/2,2 (Ω,Λ l TM) ≤ C‖f‖L2 (∂Ω,Λ l TM) .
(1.4.6)
Furthermore, whenever 2 < p < ∞, the said solution of (1.4.2) also satisﬁes the L p square function estimate ( ∫ [sup(r ∂Ω
(ii)
r>0
1−n
1 2
p
∫ (∇u)(x) dist (x, ∂Ω) dVol(x)) ] dσ(z)) 2
B r (z)∩Ω
1 p
≤ C‖f‖L p (∂Ω,Λ l TM) .
(1.4.7)
Any solution u of the problem (1.4.2) enjoys the regularity property p
f ∈ L1 (∂Ω, Λ l T M) ⇐⇒ N(∇u) ∈ L p (∂Ω).
(1.4.8)
Moreover, corresponding to p = 2, whenever f ∈ L21 (∂Ω, Λ l T M) a solution u of (1.4.2) may be found which satisﬁes the higherorder L2 square function estimate 1/2
2 ( ∫ (∇2 u)(x) dist (x, ∂Ω) dVol(x)) Ω
≤ C‖f‖L21 (∂Ω,Λ l TM) .
(1.4.9)
26  1 Introduction and Statement of Main Results
(iii) All results described in items (i)–(ii) remain valid if in place of the HodgeLaplacian ∆HL one considers the Schrödinger operator ∆HL − V in the formulation of (1.4.2), i.e., consider the Dirichlet boundary value problem u ∈ C 0 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHDp )l { { N u ∈ L p (∂Ω), { { { { n.t. p l { u∂Ω = f ∈ L (∂Ω, Λ T M),
(1.4.10)
for a realvalued nonnegative potential V ∈ L r (Ω), with r > n. (iv) Regarding uniqueness for the Dirichlet problem for the HodgeLaplacian, any two solutions of (1.4.2) differ by a form belonging to NΩl := { ω ∈ C 1 (Ω, Λ l T M) : ω = 0 in Ω ∩ M for each component M of M intersecting ∂Ω, and also satisﬁes dω = 0, δω = 0 in each component M of M contained in Ω }. (1.4.11) This is a linear space of ﬁnite dimension, dim NΩl , given by b l (Ω, M) := ∑ b l (M )
(1.4.12)
M ⊂Ω
where b l (M ) is the lth Betti number of M , and the sum is performed over all connected components M of M contained in Ω. As a consequence, the Dirichlet boundary value problem (1.4.2) is Fredholm solvable, with index b l (Ω, M)¹⁷. In particular, if Ω does not contain any connected component of M
(1.4.13)
(which is automatically the case if M is connected to begin with), then one has genuine uniqueness for (1.4.2). In fact, (1.4.2) is wellposed if and only if b l (M ) = 0
for each component M of M contained in Ω.
(1.4.14)
Under the assumption that¹⁸ at the level on lforms one has Ric ≥ 0 in each connected component M of M contained in Ω
(1.4.15)
17 in the sense that the problem (1.4.2) is solvable if and only the boundary datum f belongs to a space V of ﬁnite codimension in L p (∂Ω, Λ l TM), the space of nullsolutions W of (1.4.2) has ﬁnite dimension, and dim W − codim V = b l (Ω, M) 18 pointwise, in the sense of linear operators on the ﬁbers of Λ l TM
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems
 27
the space of nullsolutions for the Dirichlet problem (1.4.2) may be alternatively described as NΩl = { ω ∈ C 1 (Ω, Λ l T M) : ω = 0 in Ω ∩ M , for each component M of M intersecting ∂Ω, and satisﬁes ∇ω = 0 in each component M of M contained in Ω }. (1.4.16) Moreover, one has genuine uniqueness (hence, ultimately, genuine wellposedness) for the Dirichlet boundary value problem for the HodgeLaplacian formulated in (1.4.2) provided for each connected component M of M contained in Ω at the level of lforms one has Ric ≥ 0 everywhere in M
and Ric > 0 at some point in M .
(1.4.17)
In particular, the Dirichlet problem (1.4.2) is wellposed if at the level of lforms one has Ric > 0 on M. (v)
(1.4.18)
The space of nullsolutions for the Dirichlet boundary value problem for the Schrödinger operator ∆HL − V corresponding to a realvalued, nonnegative potential V belonging to L r (Ω) for some r > n, is precisely l := { ω ∈ NΩl : ω = 0 on M ∩ supp V for each NΩ,V
component M of M contained in Ω },
(1.4.19)
l ≤ dim NΩl = b l (Ω, M). As a which is a linear space of ﬁnite dimension, dim NΩ,V consequence, the Dirichlet boundary value problem for the Schrödinger operator l ∆HL − V is Fredholm solvable, with index dim NΩ,V . Also,
if Ω does not contain any connected component of M (which is automatically the case if M is connected to begin with), then one has genuine uniqueness for the Dirichlet problem for the Schrödinger operator ∆HL − V whenever the potential V ∈ L r (Ω), with r > n, is realvalued and nonnegative.
(1.4.20)
In fact, thanks to (1.4.19) and the unique continuation result from Proposition 2.9, one has genuine uniqueness (and, hence, genuine wellposedness) for the L p Dirichlet boundary value problem for the Schrödinger operator ∆HL − V whenever the potential V ∈ L r (Ω), with r > n, is realvalued, nonnegative, and strictly positive in a nonempty open subset of every component M of M contained in Ω. Moreover, under the assumption that V ∈ L r (Ω) with r > n, is a realvalued, nonnegative potential and, at the level of lforms, one has V + Ric ≥ 0 in each connected component M of M contained in Ω,
(1.4.21)
28  1 Introduction and Statement of Main Results
the space of nullsolutions for the Dirichlet boundary value problem for the Schrödinger operator ∆HL − V may be alternatively described as l = { ω ∈ C 1 (Ω, Λ l T M) : ω = 0 in Ω ∩ M for each component NΩ,V
M of M intersecting ∂Ω, and satisﬁes
∇ω = 0 in M as well as ω = 0 on M ∩ supp V for each component M of M contained in Ω }.
(1.4.22)
As a consequence¹⁹, one also has genuine uniqueness for the L p Dirichlet boundary value problem for the Schrödinger operator ∆HL − V provided V ∈ L r (Ω), with r > n, is a realvalued, nonnegative potential, with the property that, for each connected component M of M contained in Ω, at the level of lforms one has V + Ric ≥ 0 almost everywhere in M and also V + Ric > 0 in some subset of positive measure of M .
(1.4.23)
In particular, this is the case if at the level of lforms one has V + Ric ≥ 0 almost everywhere in M and also V + Ric > 0 in a subset of positive measure of each component of M.
(1.4.24)
(vi) In view of (1.4.8), for boundary data in Sobolev spaces the Dirichlet boundary value problem (1.4.2) becomes the Regularity problem²⁰ u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, (HLRp )l { { N u, N(∇u) ∈ L p (∂Ω), { { { { n.t. p l { u∂Ω = f ∈ L1 (∂Ω, Λ T M),
(1.4.25)
which is always solvable, and a solution may be found with the property that ‖N u‖L p (∂Ω) + ‖N(∇u)‖L p (∂Ω) ≤ C‖f‖L1p (∂Ω,Λ l TM) ,
(1.4.26)
‖u‖W 1,np/(n−1) (Ω,Λ l TM) ≤ C‖f‖L1p (∂Ω,Λ l TM) ,
(1.4.27)
and²¹
for some constant C ∈ (0, ∞) independent of f . Once again, any two solutions of (1.4.25) differ by a form ω belonging to the space (1.4.11). In particular, (1.4.25) is wellposed if and only if (1.4.13) holds. Also, the Regularity boundary value problem (1.4.25) is Fredholm solvable, with index b l (Ω, M).
19 of (1.4.22) and the unique continuation result from Proposition 9.4 20 it follows a posteriori from Corollary 8.23 that the condition Nu ∈ L p (∂Ω) is superﬂuous in (1.4.25)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems
 29
In fact, similar results hold for the Schrödinger operator ∆HL − V for a realvalued, nonnegative potential V ∈ L r (Ω) with r > n, i.e., for the Regularity boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, (SCHRp )l { (1.4.28) { { N u, N(∇u) ∈ L p (∂Ω), { { { n.t. p l { u∂Ω = f ∈ L1 (∂Ω, Λ T M). (vii) All results described in items (i)–(vi) remain valid in the case when Ω is an εSKT domain, assuming that ε > 0 is sufficiently small relative to the given p, the Ahlfors regularity constants, and the local John constants of Ω (as well as the potential V whenever applicable). When Ω is a Lipschitz domain, the Dirichlet problem (1.4.2) was treated in [86] for p near 2, via an approach based on Rellich estimates, which is entirely restricted to the class of Lipschitz domains. The last theorem also should be compared with the recent results from [52] obtained via the development of a symbol calculus for general elliptic operators albeit for the smaller class of domains which are simultaneously regular SKT and Lipschitz²². For example, when specialized to the case of the HodgeLaplacian (1.0.3), Theorem 4.3 from [52] yields²³ the wellposedness of the Dirichlet problem (1.4.2) for any l ∈ {0, 1, . . . , n} and p ∈ (1, ∞), provided Ω belongs to the intersection of the class of regular SKT domains with the class of Lipschitz domains. When l = 0 (i.e., in the scalar case), (1.4.2) reduces precisely to the L p Dirichlet boundary value problem for the LaplaceBeltrami operator in Ω formulated in (1.1.40). The wellposedness of the latter problem for each p ∈ (1, ∞) when Ω is a bounded C 1 domain with connected boundary in the ﬂat Euclidean setting is one of the central results in [32]. This scalar wellposedness result (i.e., the case l = 0 of (1.4.2)) has been subsequently generalized to regular SKT domains on compact Riemannian manifolds in [50]. In the same geometric setting, the case l = 0 of the Regularity problem (1.4.25) has been treated in [95].
22 these are precisely Lipschitz domains with VMO normals 1,2 23 the condition that any u ∈ H0 (Ω, Λ l TM) with ∆HL u = 0 in Ω necessarily vanishes identically in Ω, required for the applicability of Theorem 4.3 from [52], is seen with the help of the unique continuation result of N. Aronszajn, K. Krzywicki and J. Szarski (recorded here as Proposition 2.9)
30  1 Introduction and Statement of Main Results
In this vein, it is worth digressing momentarily in order to record the following diagram detailing the various relationships amongst some of the classes of domains appearing in the present work: {C 1 domains} ⊊ {domains locally given as uppergraphs of functions with gradients in VMO ∩ L∞ } = {Lipschitz domains with VMO normals} = {Lipschitz domains} ∩ {regular SKT domains} ⊊ {regular SKT domains} = ⋂ε>0 {εSKT domains} ⊊ ⋃ε>0 {εSKT domains} ⊊ {twosided NTA domains} ∩ {Ahlfors regular domains} = {twosided NTA domains with Ahlfors regular boundaries} ⊊ {uniformly rectiﬁable (aka UR) domains} ∩ {domains satisfying a twosided local John condition} ⊊ {UR domains} ⊊ {Ahlfors regular domains} ⊊ {(locally) ﬁnite perimeter domains}.
(1.4.29)
All these classes of domains have been shown in [51] to be invariant under continuously differentiable diffeomorphisms, hence they can be meaningfully considered on C 1 manifolds. The approach to establishing the wellposedness result described in Theorem 1.8 relies crucially on designing appropriate layer potentials. These are of a different nature than those used to treat (BVP1)l − (BVP16)l and, in place of (1.0.3), are welladapted to the (quasi)factorization of the HodgeLaplacian offered by Weitzenböck’s formula (1.4.1). The upshot of this method is that it allows us to also consider Neumann type problems of the sort described in Theorem 1.9 below²⁴. Before stating it, one piece of notation deserves clariﬁcation. Speciﬁcally, given an Ahlfors regular domain Ω ⊂ M with unit conormal ν, for a differential form u ∈ C 1 (Ω, Λ l T M), deﬁne its conormal derivative on ∂Ω in local coordinates as n.t. n.t. ∇ν♯ u := (ν♯ )j (∇∂ j u)∂Ω = ν k g kj (∇∂ j u)∂Ω
(1.4.30)
where ν♯ denotes the metric identiﬁcation of the covector ν ∈ T ∗ M with a vector in T M (making ν♯ the outward unit normal vector to Ω; see the discussion in § 9.1). Of course, provisions should be made to ensure that the above pointwise nontangential trace exists a.e. on ∂Ω. Fatou type results of this nature may be found in Proposition 5.12 and Theorem 6.6.
24 in the scalar case corresponding to l = 0, this reduces precisely to the L p Neumann boundary value problem for the LaplaceBeltrami operator formulated in (1.1.42)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems
 31
Theorem 1.9. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and surface measure σ = H n−1 ⌊∂Ω. Also, ﬁx a degree l ∈ {0, 1, . . . , n}, along with an exponent p ∈ (1, ∞), and pick a realvalued nonnegative potential V ∈ L r (Ω), with r > n. In this context, consider the Neumann boundary value problem for the Schrödinger operator ∆HL − V formulated as²⁵ u ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHNp )l { { N u, N(∇u) ∈ L p (∂Ω), { { { { p l { ∇ν♯ u = f ∈ L (∂Ω, Λ T M).
(1.4.31)
Concerning this boundary value problem, the following claims are valid. (i) The Neumann boundary value problem (1.4.31) is Fredholm solvable. Its index is l , the dimension of the space deﬁned in (1.4.19). Hence, if Ker (SCHNp )l dim NΩ,V denotes the space of nullsolutions of (1.4.31), one has dim Ker (SCHNp )l < ∞.
(1.4.32)
(ii) The Neumann boundary value problem (1.4.31) is solvable for a given boundary datum f ∈ L p (∂Ω, Λ l T M) if and only if ²⁶ n.t. ∫ ⟨f, υ∂Ω ⟩ dσ = 0,
∀ υ ∈ Ker (SCHNp )l ,
(1.4.33)
∂Ω
where p ∈ (1, ∞) is the Hölder conjugate exponent of p. (iii) Under the additional assumption that at the level of lforms, V + Ric ≥ 0 in Ω
(1.4.34)
it follows that Ker (SCHNp )l = { u ∈ C 1 (Ω, Λ l T M) : ∇u = 0 in Ω and u = 0 on supp (V + Ric) }.
(1.4.35)
In particular, under the assumption (1.4.34), any differential form u solving the homogeneous version of (1.4.31) is parallel in Ω (i.e., satisﬁes ∇u = 0 in Ω). (iv) The Neumann boundary value problem (1.4.31) is actually wellposed provided at the level of lforms one has V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of positive measure of each connected component of Ω.
(1.4.36)
25 Corollary 8.23 implies that condition Nu ∈ L p (∂Ω) is superﬂuous in the context of (1.4.31) 26 i.e., solvability holds if and only if the boundary datum satisﬁes ﬁnitely many compatibility conditions
32  1 Introduction and Statement of Main Results In such a scenario, there exists a ﬁnite constant C = C(M, Ω, p) > 0 such that, for each f ∈ L p (∂Ω, Λ l T M), the unique solution u of (1.4.31) satisﬁes ‖N u‖L p (∂Ω) + ‖N(∇u)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) .
(1.4.37)
(v) As a consequence corresponding to the special case when V = 0, the Neumann boundary value problem for the HodgeLaplacian²⁷ u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, (HLNp )l { { { N u, N(∇u) ∈ L p (∂Ω), { { { p l { ∇ν♯ u = f ∈ L (∂Ω, Λ T M),
(1.4.38)
is Fredholm solvable with index equal to the number b l (Ω, M) from (1.4.12). The Neumann boundary value problem (1.4.38) is solvable for a given boundary datum f ∈ L p (∂Ω, Λ l T M) if and only if n.t. ∫ ⟨f, υ∂Ω ⟩ dσ = 0
(1.4.39)
∂Ω
for every differential form υ ∈ C 1 (Ω, Λ l T M) satisfying ∆HL υ = 0 in Ω, as well as N υ, N(∇υ) ∈ L p (∂Ω) and ∇ν♯ υ = 0, where p ∈ (1, ∞) is such that 1/p + 1/p = 1. Also, if the ambient manifold M has a Weitzenböck operator Ric with the property that Ric ≥ 0 at the level of lforms everywhere in Ω (1.4.40) then any differential form belonging to C 1 (Ω, Λ l T M) solves the homogeneous version of (1.4.38) if and only if is parallel in Ω and vanishes on the portion of the support of Ric contained in Ω. Moreover, if in addition to (1.4.40) the Weitzenböck operator Ric also satisﬁes Ric > 0 at the level of lforms for some point in each connected component of Ω,
(1.4.41)
then the Neumann problem (1.4.38) is actually uniquely solvable²⁸. (vi) All the above results continue to be true in the scenario when Ω is an εSKT domain, provided ε > 0 is sufficiently small relative to the given p, the Ahlfors regularity constants and local John constants of Ω, as well as the potential V.
27 again, Corollary 8.23 shows that condition Nu ∈ L p (∂Ω) is superﬂuous in (1.4.38) 28 recall (cf. (9.3.28)) that Ric vanishes identically on scalar functions as well as nforms; hence (1.4.41) always fails when l = 0 or l = n, which is in agreement with the observation that the Neumann problem (HLNp )l always has nontrivial nullsolutions in these extreme cases (consisting, respectively, of locally constant functions, and locally constant functions times the volume form)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems  33
There is yet another natural notion of Neumann problem, which interfaces tightly with the one just discussed in Theorem 1.9 above. This is formulated at a level of regularity that is one unit lower than the problem (1.4.31), a phenomenon which is akin to the manner in which the Dirichlet problem (1.4.2) is a less regular version of the Regularity problem (1.4.25) (in the sense that the regularity of the boundary data and solutions is one unit lower in the former compared with the latter). This being said, one should be careful when formulating this less regular Neumann problem since the conormal n.t. derivative ∇ν♯ u, as deﬁned in (1.4.30), requires that (∇u)∂Ω exists at σa.e. point on ∂Ω. The issue is that we now wish to consider nullsolutions u ∈ C 0 (Ω, Λ l T M) of the Schrödinger operator ∆HL − V for which we only assume N u ∈ L p (∂Ω). In particular, this makes it impossible to consider ∇ν♯ u in a pointwise sense as in (1.4.30). To circumvent the aforementioned difficulty, we propose an alternative, weaker notion of conormal derivative, which we shall denote by ∇weak , taking differential ν♯ forms u ∈ C 0 (Ω, Λ l T M) satisfying (∆HL − V)u = 0 in Ω and N u ∈ L p (∂Ω) into the p negative exponent Sobolev space L−1 (∂Ω, Λ l T M) (cf. (9.5.44)). The speciﬁcs of this deﬁnition are as follows. Consider a regular SKT domain Ω ⊂ M with outward unit conormal ν and surface measure σ = H n−1 ⌊∂Ω. Also, ﬁx a degree l ∈ {0, 1, . . . , n}, along with an integrability exponent p ∈ (1, ∞), and pick some realvalued nonnegative potential V ∈ L r (Ω), with p r > n. In this framework, given any differential form f ∈ L1 (∂Ω, Λ l T M), the ability to solve the L p Regularity problem for the Schrödinger operator ∆HL − V in Ω (cf. item (vi) in Theorem 1.8) entails the existence of some differential form u f satisfying u f ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)u f = 0 in Ω, { { N u f , N(∇u f ) ∈ L p (∂Ω), { { { { n.t. { u f ∂Ω = f σa.e. on ∂Ω.
(1.4.42)
Moreover, any other differential form solving the above problem differs from u f l by some ω ∈ NΩ,V , the space deﬁned in (1.4.19). In concert with the Fatou type result established in Proposition 5.12, these considerations imply that the DirichlettoNeumann operator p
ΨDN : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), ΨDN f := ∇ν♯ u f ,
p
∀ f ∈ L1 (∂Ω, Λ l T M),
(1.4.43)
is unambiguously deﬁned, linear, and bounded. Bearing in mind that p ∈ (1, ∞) is arbitrary, taking the (real) transposed of the operator (1.4.43) yields, after adjusting notation, a welldeﬁned, linear and bounded operator ⊤ : L p (∂Ω, Λ l T M) → L−1 (∂Ω, Λ l T M). ΨDN p
(1.4.44)
34  1 Introduction and Statement of Main Results
We then proceed to deﬁne ∇weak : { u ∈ C 0 (Ω, Λ l T M) : (∆HL − V)u = 0 in Ω, ν♯ p and N u ∈ L p (∂Ω) } → L−1 (∂Ω, Λ l T M)
(1.4.45)
by setting n.t. ⊤ ∇weak u := ΨDN (u∂Ω ), ν♯
(1.4.46)
for every u ∈ C 0 (Ω, Λ l T M) satisfying (∆HL − V)u = 0 in Ω and N u ∈ L p (∂Ω). The basic Fatou type result from Theorem 6.6 ensures that any such differential form has a welln.t. deﬁned nontangential pointwise trace u∂Ω , belonging to L p (∂Ω, Λ l T M). Granted this, weak it follows that the operator ∇ν♯ is welldeﬁned and linear. Moreover, there exists some constant C ∈ (0, ∞) with the property that for any u as above we have weak p ∇ν♯ uL (∂Ω,Λ l TM) ≤ C‖N u‖L p (∂Ω) . −1
(1.4.47)
In fact, for any given p ∈ (1, ∞), all these considerations continue to work in the scenario when Ω is an εSKT domain, provided ε > 0 is sufficiently small relative to the given p as well as the Ahlfors regularity constants and local John constants of Ω. Here is the companion to Theorem 1.9, dealing with a brand of Neumann problem emphasizing the weak conormal derivative (1.4.46). Theorem 1.10. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν. Also, ﬁx a degree l ∈ {0, 1, . . . , n}, along with an exponent p ∈ (1, ∞), and pick a realvalued nonnegative potential V ∈ L r (Ω), with r > n. In this framework, formulate the weak Neumann boundary value problem for the Schrödinger operator ∆HL − V as u ∈ C 0 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHweakNp )l { { N u ∈ L p (∂Ω), { { { { weak p l { ∇ν♯ u = f ∈ L−1 (∂Ω, Λ T M).
(1.4.48)
In regard to this boundary value problem, the following assertions are true. (i) The weak Neumann boundary value problem (1.4.48) is Fredholm solvable. Its index l , the dimension of the space deﬁned in (1.4.19). Moreover, if one denotes is dim NΩ,V by Ker (SCHweakNp )l the space of nullsolutions of (1.4.48) then²⁹ Ker (SCHweakNp )l = Ker (SCHNp )l .
(1.4.49)
(ii) If p ∈ (1, ∞) denotes the Hölder conjugate exponent of p then the weak Neumann boundary value problem (1.4.48) is solvable for a given boundary datum p
p
f ∈ L−1 (∂Ω, Λ l T M) = (L1 (∂Ω, Λ l T M))
∗
(1.4.50)
29 recall that Ker (SCHNp )l stands for the space of nullsolutions of the Neumann problem (1.4.31)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems  35
if and only if ³⁰ p
(L1 (∂Ω,Λ l TM))∗
n.t. (f, υ∂Ω )L p (∂Ω,Λ l TM) = 0, 1
∀ υ ∈ Ker (SCHNp )l
(1.4.51)
where, generally speaking, X∗ (⋅, ⋅)X denotes the duality pairing between a linear space X and its dual X ∗ . (iii) Any solution u of the problem (1.4.48) enjoys the regularity property f ∈ L p (∂Ω, Λ l T M) ⇐⇒ N(∇u) ∈ L p (∂Ω).
(1.4.52)
Moreover, in the case when N(∇u) ∈ L p (∂Ω) the two notions of conormal derivative are in agreement, i.e., ∇weak u = ∇ν♯ u. (1.4.53) ν♯ (iv) The weak Neumann boundary value problem (1.4.48) is actually wellposed provided at the level of lforms one has V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of positive measure of each connected component of Ω.
(1.4.54)
In such a case, there exists a ﬁnite constant C = C(M, Ω, p) > 0 such that, for each p f ∈ L−1 (∂Ω, Λ l T M), the unique solution u of (1.4.48) satisﬁes p ‖N u‖L p (∂Ω) ≤ C‖f‖L−1 (∂Ω,Λ l TM) .
(1.4.55)
(v) All the above results continue to hold in the case when Ω is an εSKT domain, granted that ε > 0 is sufficiently small relative to the given p, the Ahlfors regularity constants and local John constants of Ω, as well as the potential V. The approach based on the brand of boundary layer potentials which are introduced and studied in this work continues to be effective in other instances of interest, such as the transmission boundary value problem formulated in Theorem 1.11 below. This extends work done for scalar functions in Lipschitz domains, ﬁrst in the Euclidean setting in [30], then subsequently on Riemannian manifolds in [85]. Transmission problems with L p data in Euclidean Lipschitz domains have also been treated in [74] for the Maxwell system, and in [102] for the Stokes system. Theorem 1.11. Fix an arbitrary degree l ∈ {0, 1, . . . , n} and suppose V is a potential as in (3.1.13) with the additional property that at the level of lforms, V + Ric ≥ 0 on M.
(1.4.56)
Consider a regular SKT domain Ω ⊂ M with outward unit conormal ν, and deﬁne Ω+ := Ω,
Ω− := M \ Ω.
(1.4.57)
30 i.e., solvability holds if and only if the boundary datum satisﬁes ﬁnitely many compatibility conditions
36  1 Introduction and Statement of Main Results Also, ﬁx an arbitrary integrability exponent p ∈ (1, ∞) along with a coupling parameter μ ∈ ℂ \ (−∞, 0],
μ ≠ 1.
(1.4.58)
In this context, formulate the Transmission boundary value problem for the Schrödinger operator ∆HL − V as³¹ u± ∈ C 1 (Ω± , Λ l T M), { { { { { { { (∆HL − V)u± = 0 in Ω± , { { { (SCHTp )l { N u± , N(∇u± ) ∈ L p (∂Ω), { { p { n.t. n.t. { u+ ∂Ω − u− ∂Ω = f ∈ L1 (∂Ω, Λ l T M), { { { { { + − p l { ∇ν♯ u − μ∇ν♯ u = g ∈ L (∂Ω, Λ T M).
(1.4.59)
Then the Transmission boundary value problem (1.4.59) is always solvable, and its nullspace is a ﬁnite dimensional space which may be described as NVl = {(ωΩ+ , ωΩ− ) : ω ∈ C 1 (M, Λ l T M) such that ω = 0 in M for each component M of M intersecting ∂Ω or overlapping with supp(V + Ric), and which satisﬁes (1.4.60) ∇ω = 0 in all other components M of M }. In particular, the Transmission boundary value problem (1.4.59) is Fredholm solvable with index dim NVl . As a consequence, one has genuine uniqueness (hence, ultimately, genuine wellposedness) for the Transmission boundary value problem for the Schrödinger operator formulated in (1.4.59) provided in addition to (1.4.56) at the level of lforms one also has V + Ric > 0 on some subset of M of positive measure, for each connected component M of M disjoint from ∂Ω.
(1.4.61)
In particular, the Transmission problem (1.4.59) is wellposed if at the level of lforms one has V + Ric > 0 on M.
(1.4.62)
Alternatively, the nullspace of the Transmission boundary value problem (1.4.59) may be described as NVl = {(ωΩ+ , ωΩ− ) : ω ∈ C 1 (M, Λ l T M) such that ω = 0 in M for each component M of M intersecting ∂Ω, and for each component M of M disjoint from ∂Ω satisﬁes dω = 0, δω = 0 in M and ω = 0 on the portion of (1.4.63) supp V contained in M }.
31 it follows from Corollary 8.23 that asking Nu± ∈ L p (∂Ω) is superﬂuous in (1.4.59)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems
 37
In light of the unique continuation result from Proposition 2.9, this goes to show that one has genuine uniqueness (hence, again, genuine wellposedness) for the Transmission boundary value problem for the Schrödinger operator formulated in (1.4.59) provided in addition to (1.4.56) one also has V > 0 on some nonempty open subset of M , for each connected component M of M disjoint from ∂Ω.
(1.4.64)
Consequently, the Transmission problem (1.4.59) is wellposed if V > 0 on some nonempty open subset of each connected component of M.
(1.4.65)
Moreover, completely analogous results hold for the weak Transmission boundary value problem for the Schrödinger operator ∆HL − V formulated as follows: u± ∈ C 0 (Ω± , Λ l T M), { { { { { { (∆HL − V)u± = 0 in Ω± , { { { { (SCHweakTp )l { N u± ∈ L p (∂Ω), { { n.t. n.t. { { u+ ∂Ω − u− ∂Ω = f ∈ L p (∂Ω, Λ l T M), { { { { { weak + p weak − l { ∇ν♯ u − μ∇ν♯ u = g ∈ L−1 (∂Ω, Λ T M).
(1.4.66)
In addition, any solution u± of the weak Transmission boundary value problem (1.4.66) satisﬁes the regularity property p
f ∈ L1 (∂Ω, Λ l T M) and g ∈ L p (∂Ω, Λ l T M) if and only if N(∇u+ ), N(∇u− ) ∈ L p (∂Ω).
(1.4.67)
Finally, all the above results continue to be true in the scenario when Ω is an εSKT domain, provided ε > 0 is sufficiently small relative to the given p, the coupling parameter μ, the Ahlfors regularity constants and local John constants of Ω, as well as the potential V. Next we discuss the L p PoincaréRobin boundary value problem for the Schrödinger operator ∆HL − V for integrability exponents p close to 2. To set the stage, we introduce a new piece of notation. Concretely, given an Ahlfors regular domain Ω ⊂ M with surface measure σ and a vector ﬁeld X ∈ T M∂Ω locally expressed as X = X j ∂ j , for a differential form u ∈ C 1 (Ω, Λ l T M) deﬁne its covariant derivative along X on ∂Ω in local coordinates as n.t. ∇X u := X j (∇∂ j u)∂Ω
σa.e. on ∂Ω,
(1.4.68)
assuming the latter pointwise nontangential traces exist at σalmost every point on ∂Ω.
38  1 Introduction and Statement of Main Results Theorem 1.12. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν ∈ T ∗ M and surface measure σ = H n−1 ⌊∂Ω. Fix a degree l ∈ {0, 1, . . . , n} and pick a realvalued nonnegative potential V ∈ L r (Ω), with r > n. Also, suppose X ∈ L∞ (∂Ω, T M) ∩ VMO(∂Ω, T M)
(1.4.69)
is a vector ﬁeld which is transversal to ∂Ω, in the sense that there exists some c > 0 such that (1.4.70) ⟨X, ν♯ ⟩ ≥ c at σa.e. point on ∂Ω . Then there exists an integrability exponent p o ∈ [1, 2), which depends on the above framework, with the following signiﬁcance. If po denote the Hölder conjugate of p o and some p ∈ (p o , po ) has been ﬁxed, then for any³² p
Θ : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) linear and compact operator,
(1.4.71)
the L p PoincaréRobin boundary value problem for the Schrödinger operator ∆ HL − V, formulated as³³ u ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHPRp )l { { N u, N(∇u) ∈ L p (∂Ω), { { { { n.t. p l { ∇X u + Θ(u∂Ω ) = f ∈ L (∂Ω, Λ T M),
(1.4.72)
l , the dimension of the space deﬁned in (1.4.19). is Fredholm solvable with index dim NΩ,V In particular, the said index is zero in the case in which the potential V is strictly positive in a nonempty open subset of every component of M contained in Ω. In fact, all the above results continue to be valid in the situation when Ω is an εSKT domain, provided ε > 0 is sufficiently small relative to the given p, the Ahlfors regularity constants and local John constants of Ω, as well as the potential V.
The particular case of Theorem 1.12 when Θ = 0 corresponds to the oblique derivative problem. In the context of Lipschitz domains, the latter problem has been considered in [13, 68, 96]. In the setting of Theorem 1.12, the special choice X := ν♯ (which makes (1.4.70) trivially true and also satisﬁes (1.4.69) since ν ∈ VMO(∂Ω, T ∗ M) given that Ω is a regular SKT domain) corresponds to the L p Robin boundary value problem for the
32 possibly a nonlocal operator 33 the Fatou type result from Proposition 5.12 guarantees that the boundary condition in (1.4.72) is meaningfully deﬁned, while Corollary 8.23 implies that asking Nu ∈ L p (∂Ω) is superﬂuous in (1.4.72)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems
 39
Schrödinger operator ∆HL − V formulated as³⁴ u ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHROp )l { { { N u, N(∇u) ∈ L p (∂Ω), { { { n.t. p l { ∇ν♯ u + Θ(u∂Ω ) = f ∈ L (∂Ω, Λ T M).
(1.4.73)
As such, Theorem 1.12 guarantees that this Robin problem is Fredholm solvable for p near 2. In contrast with the above analysis, it turns out that a more nuanced solvability result for the Robin boundary value problem (1.4.73) may be established, targeting arbitrary integrability exponents p ∈ (1, ∞). This makes the object of our next theorem (for related work in the Euclidean setting, the reader is referred to [41–45], and the references therein). Theorem 1.13. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and surface measure σ = H n−1 ⌊∂Ω. Fix a degree l ∈ {0, 1, . . . , n}, along with an arbitrary integrability exponent p ∈ (1, ∞), and pick some arbitrary realvalued nonnegative potential V ∈ L r (Ω), with r > n. Also, select some³⁵ p
Θ : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) linear and compact operator.
(1.4.74)
Then the Robin boundary value problem stated in (1.4.73) is Fredholm solvable, with l index dim NΩ,V , the dimension of the space introduced in (1.4.19). In particular, the said index vanishes when the potential V is strictly positive in a nonempty open subset of every component of M contained in Ω. Moreover, if in place of (1.4.74) one now considers p
Θ : L p (∂Ω, Λ l T M) → L−1 (∂Ω, Λ l T M) linear and compact operator,
(1.4.75)
then the weak Robin boundary value problem for the Schrödinger operator ∆HL − V formulated as³⁶ u ∈ C 0 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHweakROp )l { { { N u ∈ L p (∂Ω), { { { weak p n.t. l { ∇ν♯ u + Θ(u∂Ω ) = f ∈ L−1 (∂Ω, Λ T M),
(1.4.76)
34 the Fatou type result from Proposition 5.12 guarantees that the boundary condition in (1.4.73) is meaningfully deﬁned, while Corollary 8.23 shows that condition Nu ∈ L p (∂Ω) is superﬂuous in this setting 35 again, a possibly nonlocal operator 36 the Fatou type result from Theorem 6.6 ensures that the boundary condition in (1.4.76) is meaningfully deﬁned
40  1 Introduction and Statement of Main Results l is also Fredholm solvable with index dim NΩ,V . Furthermore, if Θ is both as in (1.4.74) and (1.4.75), then
Ker (SCHweakROp )l = Ker (SCHROp )l ,
(1.4.77)
and any solution u of the weak Robin problem (1.4.76) enjoys the regularity property f ∈ L p (∂Ω, Λ l T M) ⇐⇒ N(∇u) ∈ L p (∂Ω).
(1.4.78)
Finally, all the above results continue to hold in the case when Ω is an εSKT domain, provided ε > 0 is sufficiently small relative to the given p, the Ahlfors regularity constants and local John constants of Ω, as well as the potential V. In relation to the Dirichlet problem for the Schrödinger operator ∆HL − V formulated as u ∈ C 0 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, (SCHDp )l { (1.4.79) { { N u ∈ L p (∂Ω), { { { n.t. p l { u∂Ω = f ∈ L (∂Ω, Λ T M), it turns out that solvability actually holds in a larger class of domains than considered in Theorem 1.8 as long as the integrability exponent is assumed to be near 2. This remarkable phenomenon is discussed in our next theorem. Theorem 1.14. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω and whose outward unit conormal has the property that ν ∈ VMO(∂Ω, T ∗ M).
(1.4.80)
Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} and pick a realvalued nonnegative potential V ∈ L r (Ω), with r > n. Then there exist two Hölder conjugate exponents 1 ≤ p o < 2 < po ≤ ∞ such that whenever p ∈ (p o , po ) the Dirichlet problem for the Schrödinder operator ∆HL − V formulated as in (1.4.79) is always solvable. Moreover, the Neumann boundary problem for the Schrödinger operator ∆HL − V formulated in (1.4.31) is also always solvable for each p ∈ (p o , po ) if, in addition, one assumes that at the level of lforms one has V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of positive measure of each connected component of Ω.
(1.4.81)
Finally, all results above continue to hold in the case when (1.4.80) is relaxed to merely demanding that dist (ν, VMO(∂Ω, T ∗ M)) is sufficiently small, relative to the background structures, where the distance is measured in BMO.
(1.4.82)
1.4 Dirichlet, Neumann, Transmission, Poincaré, and RobinType Boundary Problems  41
In fact, as long as the integrability exponent p is near 2, a plethora of other boundary value problems are solvable in the category of UR domains considered in Theorem 1.14. For example, Theorem 5.13 and Corollary 5.14 (together with Remark 5.16) may be successfully employed to show that the L p Regularity problem (1.4.25) the weak L p Neumann problem (1.4.48), the L p Transmission problem (1.4.59) and its weak version formulated in (1.4.66), the L p PoincaréRobin problem (1.4.72), as well as the L p Robin problem (1.4.73) and its weak version formulated in (1.4.76) are all always solvable³⁷ in the above class of domains provided p is close to 2. Condition (1.4.82) which, in relation to PDE’s, has ﬁrst been considered in the Euclidean setting by V. Maz’ya, M. Mitrea, and T. Shaposhnikova in [75], helps cement the thesis that the distance dist (ν, VMO(∂Ω, T ∗ M)), measured in BMO, is a good indicator of how reasonable the underlying domain Ω is, as far as the solvability of certain basic boundary value problems in Ω is concerned. It turns out that UR domains satisfying an even weaker property than (1.4.80) constitute a class of domains for which the Dirichlet problem for the Schrödinder operator formulated as in (1.4.79) is always solvable if the integrability exponent p is sufficiently close to 2. A detailed analysis of this remarkable phenomenon is presented in our next theorem. Theorem 1.15. Let Ω be a proper UR subdomain of the manifold M, satisfying ∂(Ω) = ∂Ω. Denote by σ = H n−1 ⌊∂Ω the surface measure on ∂Ω, and by ν ∈ T ∗ M the outward unit conormal to Ω. Assume that there exists a vector ﬁeld X ∈ L∞ (∂Ω, T M) ∩ VMO(∂Ω, T M)
(1.4.83)
which is transversal to ∂Ω, in the sense that there exists some c > 0 such that ⟨X, ν♯ ⟩ ≥ c at σa.e. point on ∂Ω.
(1.4.84)
Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} and pick a realvalued nonnegative potential V ∈ L r (Ω), with r > n. Then there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞,
1/p o + 1/po = 1,
(1.4.85)
such that for each p ∈ (p o , po ) the L p Dirichlet problem for the Schrödinder operator ∆HL − V formulated as in (1.4.79) is solvable. Moreover, given any p ∈ (p o , po ) there exists a ﬁnite constant C = C(M, Ω, p) > 0 such that, for each f ∈ L p (∂Ω, Λ l T M), one can ﬁnd a solution u of (1.4.79) satisfying ‖u‖L np/(n−1) (Ω,Λ l TM) + ‖N u‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) .
37 under suitable assumptions on the potential function V
(1.4.86)
42  1 Introduction and Statement of Main Results In addition, whenever 2 < p < po , the said solution of (1.4.79) also satisﬁes the L p square function estimate 1/2 sup (r1−n ∫ ∇u2 dist (⋅, ∂Ω) dVol) ≤ C‖f‖L p (∂Ω,Λ l TM) . r>0 p L z (∂Ω) B (z)∩Ω
(1.4.87)
r
Furthermore, corresponding to the case when p = 2, the aforementioned solution also satisﬁes the L2 square function estimate 1/2 2 (∫ (∇u)(x) dist (x, ∂Ω) dVol(x)) ≤ C‖f‖L2 (∂Ω,Λ l TM) ,
(1.4.88)
Ω
where ∇ denotes the LeviCivita connection acting on differential forms on M, and has the global regularity property ‖u‖H 1/2,2 (Ω,Λ l TM) ≤ C‖f‖L2 (∂Ω,Λ l TM) .
(1.4.89)
Finally, such a solvability result actually holds assuming only that X ∈ L∞ (∂Ω, T M) is transversal to ∂Ω and has the property that dist (X, VMO(∂Ω, T M)) is sufficiently small,
(1.4.90)
relative to the relevant background structures, where the distance is measured in BMO. It is instructive to note that UR domains with outward unit conormals in VMO make a subclass of the category of domains for which the above theorem applies (since in such a case we may simply take X to be the outward normal). This being said, there are UR domains Ω with very rough outward unit conormals which, nonetheless, possess a vector ﬁeld X as in (1.4.83) which also happens to be transversal to ∂Ω. Indeed, this is the case for any Lipschitz domain, a scenario in which a smooth transversal vector ﬁeld always exists (a property which, in fact, has been shown in [51] to characterize Lipschitzianity within the class of ﬁnite perimeter domains). In particular, the solvability result described in Theorem 1.15 holds for any Lipschitz subdomain Ω of M. Although not entirely clear at a ﬁrst glance, Theorem 1.8 is intimately linked with the program aimed at systematically studying boundary value problems for the HodgeLaplacian with boundary conditions expressed in the language of the Hodgede Rham formalism associated with the differential form calculus. For example, the existence of the various nontangential boundary traces involved in the formulation of these problems is an issue of paramount importance for this work. In turn, this is settled by deriving a suitable Fatoutype theorem for the HodgeLaplacian, a process in which Theorem 1.8 plays an indispensable role. That Theorem 1.8 interfaces tightly with the results described in Theorems 1.1–1.7 is further substantiated by the following result, in which the classical Dirichlet problem for the HodgeLaplacian is reinterpreted as
1.5 Structure of the Monograph  43
the boundary problem seeking a nullsolution u of ∆HL in Ω with n.t. { ν ∨ u∂Ω { n.t. { ν ∧ u∂Ω
(1.4.91)
prescribed in tangential and normal Lebesgue spaces on ∂Ω. Theorem 1.16. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν, and select a degree l ∈ {0, 1, . . . , n}, along with an exponent p ∈ (1, ∞). Then the boundary value problem for the HodgeLaplacian u ∈ C 0 (Ω, Λ l T M), { { { { { { { ∆HL u = 0 in Ω, { { { N u ∈ L p (∂Ω), { { { p n.t. { { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M), { { { { { p n.t. l+1 { ν ∧ u∂Ω = g ∈ Lnor (∂Ω, Λ T M),
(1.4.92)
is always solvable, and the space of solutions of the homogeneous version of (1.4.92) is precisely the ﬁnite dimensional space NΩl , deﬁned in (1.4.11). In fact, one has genuine uniqueness for (1.4.92) if and only if the lth Betti number of every connected component of M contained in Ω vanishes (which is automatically the case if M is connected to begin with). Furthermore, for any solution u of the problem (1.4.92), the following regularity property holds: p
N(∇u) ∈ L p (∂Ω) ⇐⇒ ν ∧ f + ν ∨ g ∈ L1 (∂Ω, Λ l T M) p,δ
{ f ∈ Ltan (∂Ω, Λ l−1 T M), ⇒ { p,d l+1 { g ∈ Lnor (∂Ω, Λ T M).
(1.4.93)
Finally, the above results remain valid in the case when Ω is an εSKT domain, assuming that ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. In addition, in place of ∆HL one may considers the Schrödinger operator ∆HL − V for a realvalued, nonnegative, potential V ∈ L r (Ω), with r > n, which is strictly positive in a nonempty open subset of every connected component of M contained in Ω. In such a scenario, the corresponding version of (1.4.92) is genuinely wellposed, and the regularity property (1.4.93) continues to hold.
1.5 Structure of the Monograph Overall, the main issue of concern in the present work is regularity (or, rather, the lack thereof). Indeed, when all objects and structures involved are smooth, boundary value
44  1 Introduction and Statement of Main Results
problems seeking differential forms u annihilated by the HodgeLaplacian in a domain Ω and with the tangential components³⁸ of u and δu, or the normal components of u and du, or the full Dirichlet trace of u, or the conormal ∇ν♯ u, respectively, prescribed on the boundary of Ω are known to be of ShapiroLopatinski˘ı type, a scenario in which the standard elliptic theory based on pseudodifferential operator methods applies (cf., e.g., [109, 116, 118, 127]). By way of contrast, the rough setting considered here renders this approach utterly illsuited and, as an alternative, we resort to the theory of singular integral operators (with variable coefficient kernels) on uniformly rectiﬁable sets, emerging from [23, 50], which we further reﬁne and adapt to the present framework. In this regard, one of the notable achievements in this work is the development of a Fredholm theory for a category of layer potential operators generalizing (1.0.1) that are effective in the treatment of boundary value problems associated with the HodgeLaplacian in regular SKT subdomains of Riemannian manifolds, of the sort described in Theorems 1.11.16 (see the discussion in § 3 and §§ 4 and 5). As such, our work falls within the scope of the program outlined by A. P. Calderón in his 1978 ICM plenary address (cf. [11, p. 90]) in which he advocates the use of layer potentials “for much more general elliptic systems [than the scalar, ﬂat space Laplacian]”. This is also in line with the issue raised by C. E. Kenig in [65, Problem 3.2.2, p. 117] which asks whether operators of layer potential type may be inverted on appropriate Lebesgue spaces in suitable subclasses of NTA domains with compact Ahlfors regular boundaries. Moreover, our spectral radius estimates established in Theorem 3.7 are in the spirit of Problem 3.2.12 on p. 119 of Kenig’s monograph [65]. In a broader context, the basic solvability results contained in Theorems 1.1–1.16, along with the array of tools developed to establish these results, further allow one to reﬁne various aspects of the analysis of differential forms that hitherto have placed much stronger regularity demands on the underling domain. This is illustrated by the material in § 8.1 and § 8.2. In § 8.1, we develop a brand of de Rham cohomology on regular SKT surfaces (a delicate task given that such a set may lack any type of smooth manifold structure), then use it to produce Hodgelike decomposition results on regular SKT surfaces. See Theorem 8.6 and Theorem 8.9 for details. In § 8.2 we indicate how the boundary value problems for the timeharmonic Maxwell system may be formulated and solved on arbitrary regular SKT subdomains of Riemannian manifolds, for boundary data in L p based spaces with p ∈ (1, ∞) arbitrary. Of course, there are many other signiﬁcant applications not explicitly treated here. For example, the tools developed in this monograph also permit one to treat other related PDE’s (such as the HodgeDirac operator, and the HodgeSchrödinger operator ∆HL − V under certain
38 in the smooth setting, ∂Ω is a codimension one submanifold of M and it is customary to deﬁne the tangential traces on ∂Ω of differential forms originally considered in Ω via the pullback associated with the inclusion ι : ∂Ω → M
1.5 Structure of the Monograph
 45
assumptions on the potential V), as well as other types of boundary conditions (of transmission type, or Robin type) for the HodgeLaplacian. Regarding the hypotheses on the ambient manifold M, the assumption that M is compact is not essential in the sense that the vast majority of the results in this monograph hold in the case when Ω is a relatively compact subdomain of an open Riemannian manifold M. Indeed, in such a case matters may always be reduced to the compact setting by embedding Ω isometrically in a compact, boundaryless Riemannian manifold M having the same dimension as the original M. For example, starting with a compact submanifold with boundary O of M whose interior contains Ω, one can take M to be the socalled double of O, manufactured by taking two replicas of O with opposite orientations and “gluing” them together by identifying boundary points³⁹. In particular, all main results in this monograph are valid for compact subdomains of the Euclidean space ℝn . We conclude this introduction with a brief description of the contents of each chapter. Basic concepts and tools of a geometric nature that are relevant for the present work are recalled and developed in Chapter 2, starting with differential geometry in § 2.1, then moving on to geometric measure theory in § 2.2. In § 2.3 we establish a sharp integration by parts formula in Ahlfors regular domains for the exterior derivative operator, and derive a number of other useful related integral identities. The multitude of spaces of tangential and normal differential forms employed in the formulation of the boundary value problems (BVP1)l – (BVP18)l are introduced and studied in § 2.4. Our strategy in dealing with the boundary value problems (BVP1)l – (BVP18)l as well as ( 12 BVPnor ) – ( 12 BVPtan ) relies on a systematic use of a certain brand of layer potentials that are welladapted to the Hodgede Rham formalism associated with the HodgeLaplacian. Chapter 3 is largely devoted to introducing and studying such layer potentials. The starting point is the construction in § 3.1 of a suitable fundamental solution for the HodgeLaplacian. In this regard, the idea is to ﬁnd a reasonable functional analytic setting where the differential operator in question is invertible, and then consider the Schwartz kernel of its inverse. Since typically ∆HL has a nontrivial kernel, we ﬁnd it useful to work with the perturbation ∆HL − V where V is a suitably chosen scalar potential. Granted the availability of an adequate fundamental solution, we then proceed to deﬁne boundary layer potentials for the HodgeLaplacian that are compatible with the Hodgede Rham formalism in § 3.2. This section is devoted to presenting a CalderónZygmund theory for such singular integral operators in uniformly rectiﬁable domains, which is the optimal geometric setting in which results of this nature are expected to hold. The crucial task of proving compactness results and then developing a suitable Fredholm theory for the boundary layer potentials introduced in § 3.2 is subsequently carried out in § 3.3. What makes this task delicate is that precise
39 more precisely, M = O × {0, 1}/∼, where (x, 0) ∼ (x, 1) for every x ∈ ∂O
46  1 Introduction and Statement of Main Results
information is required about not only the size of the main singularity of the respective integral kernels but also their detailed algebraic structure (which, ultimately, is the source of certain key cancellations). A different brand of boundary layer potential (than those deﬁned in Chapter 3), induced by the conormal derivative associated with the (quasi)factorization of the HodgeLaplacian given by Weitzenböck’s formula in (1.4.1), is introduced and studied in Chapter 4. First, in the context of UR domains, we develop a rather rich CalderónZygmund theory for this new type of boundary layer potential in § 4.1, with a special emphasis on the properties of what we call the (LeviCivita) connection double layer potential operator (see Deﬁnition 4.1). In § 4.2 we take a second look at this double layer on UR subdomains of smooth manifolds. Then, in § 4.3, we succeed in proving that the principal value version of this double layer is a compact operator both on Lebesgue and Sobolev spaces of differential forms on boundaries of regular SKT domains in the ambient manifold. In Chapter 5 we then test the effectiveness of this brand of boundary layer potential in the treatment of the boundary value problem of Dirichlet, Regularity, and Neumann type for the HodgeLaplacian in regular SKT domains, by proving Theorems 1.8–1.16 in § 5.3, after having established key functional analytic properties (including invertibility) for these boundary layer potentials in § 5.1 and § 5.2. In turn, the wellposedness results from Chapter 5 yield key ingredients for the ongoing quest aimed at proving the solvability results announced in Theorems 1.1–1.7. Speciﬁcally, in Chapter 6 we succeed in establishing a Fatou theorem and a naturally accompanying double layer integral representation for formula for nullsolutions of the HodgeLaplacian in a regular SKT domain whose nontangential maximal function is pth power integrable (see Theorem 6.6 for speciﬁcs). Such a result underpins a good deal of the present work. In particular, it ensures that all pointwise nontangential boundary traces appearing in the formulation of Theorems 1.1–1.16 are meaningfully deﬁned, thanks to the other (earlier) conditions imposed in the statement of these boundary value problems. As a preamble to the Fatou result just mentioned, in § 6.1 we prove a rather delicate theorem pertaining to the convergence of singular integral operators considered on the boundaries of a family domains {Ω j }j∈ℕ approximating the original set Ω in a suitable fashion. In § 6.3 we consider various spaces of harmonic ﬁelds, and establish other Fatoutype theorems and naturally accompanying Greenlike integral formulas for differential forms in regular SKT domains. The proofs of all solvability results announced in Theorems 1.1–1.7 are then presented in Chapter 7, as a culmination of the work carried out thus far. Chapter 8 contains further tools, which play a signiﬁcant role in the treatment of the boundary value problems formulated in the ﬁrst part of this chapter, along with several notable consequences, of independent interest. Speciﬁcally, § 8.1 contains (among other things) a discussion pertaining to de Rham cohomology and Hodgelike decompositions on regular SKT surfaces, which is relevant in the context of Theorems 1.3 and 1.4. In § 8.2 we indicate how the classical timeharmonic Maxwell system on arbitrary regular SKT subdomains of Riemannian manifolds, and with boundary data in L p based spaces
1.5 Structure of the Monograph
 47
with p ∈ (1, ∞) arbitrary, may be subsumed within the theory developed earlier in the monograph. Next, in § 8.3 we study the DirichlettoNeumann map for the HodgeLaplacian in regular SKT domains, while in § 8.4 we revisit the topic of Fatoutype results and prove several theorems of this ﬂavor under additional regularity conditions. In § 8.5 we consider the notion of weak tangential and normal traces of differential forms in regular SKT domains satisfying Friedrichs property and study their regularity. We then close Chapter 8 by deﬁning the HodgePoisson kernel in § 8.6, where we show that in a regular SKT domain there exists a unique such object. In turn, this yields new representation formulas for the solution of the Dirichlet problem for the HodgeLaplacian, involving what we call the Hodgeharmonic measure. This monograph concludes with Chapter 9, containing useful background material from a variety of areas, along with a variety of new, more specialized results. Section 9.1 contains a brief review of covariant derivatives on general vector bundles. We further elaborate on the more specialized case of the extension of the LeviCivita connection to differential forms in Section 9.2. A detailed discussion pertaining to the BochnerLaplacian and Weintzenböck’s formula is contained in Section 9.3. The topic of L p based Sobolev spaces of order one on boundaries of Ahlfors regular domains is presented ﬁrst in the Euclidean setting in Section 9.4 then in the context of manifolds in Section 9.5. Integrating by parts on the boundaries of Ahlfors regular domains is the main topic of Section 9.6. Next, Section 9.7 is devoted to deriving a global H 1/2,2 regularity result, while in Section 9.8 we revisit the principal value harmonic double layer operator on Euclidean UR domains and clarify some of the earlier work on this topic from [50] (which plays a basic role for us here). A rather general CalderónZygmund machinery for dealing with singular integral operator in UR domains on manifolds is developed in Section 9.9. Fredholm properties of elliptic differential operators in a global context are treated in Section 9.10. The issue of establishing criteria for compactness (and closetocompactness) of such singular integral operators on L p spaces is considered in Section 9.11. In Section 9.12 we record the best available version of the Divergence Theorem on manifolds. Finally, Section 9.13 contains a brief summary of those Clifford analysis tools needed in the proof of Theorem 6.1, while in Section 9.14 we record a useful assembly of spectral theoretic results for unbounded linear operators in Hilbert spaces that satisfy certain cancellation conditions.
2 Geometric Concepts and Tools In this chapter we develop those geometric tools that are relevant to the present work. Section 2.1 deals with differential geometric preliminaries. These include material on differential forms on a Riemannian manifold M, including both exterior products u ∧ υ of differential forms u and υ, sections of Λ k T M and Λ l T M, respectively, and interior products u ∨ υ, the latter deﬁned for k ≤ l and involving the metric tensor. We also deﬁne the exterior derivative d and its formal adjoint, with respect to the Riemannian metric, δ. These are ﬁrstorder differential operators. We produce a general integration by parts identity for a broad class of ﬁrst order differential operators, which will be of further use in § 2.3. We also consider the HodgeLaplacian associated to d and δ and review some known elliptic regularity results for this operator. In § 2.2 we focus on concepts and results from geometric measure theory. We start with the class of domains with ﬁnite perimeter, and progressively specialize to the class of Ahlfors regular domains, then to the class of uniformly rectiﬁable (UR) domains, and then to the smaller classes of εSKT domains and regular SKT domains, whose characterizations also involve other classes of domains, notably NTA domains, domains satisfying a local John condition, and εReifenberg ﬂat domains. In § 2.3 we establish some sharp integration by parts formulas on Ahlfors regular domains, which will be of essential use in the analysis of layer potentials. p p Section 2.4 discusses the spaces Ltan (∂Ω, Λ l T M) and Lnor (∂Ω, Λ l T M), and their special subspaces p,δ
Ltan (∂Ω, Λ l T M),
p,d
Lnor (∂Ω, Λ l T M),
Ltan (∂Ω, Λ l T M),
p,0
(2.0.1)
p,0
(2.0.2)
and Lnor (∂Ω, Λ l T M),
previewed in Chapter 1, which played such a ubiquitous role in the statement of the main theorems given there. These are deﬁned in the setting of ﬁnite perimeter domains, though we specialize to Ahlfors regular domains to establish some of their important properties.
2.1 Differential Geometric Preliminaries Let M be a compact, boundaryless, oriented differential manifold of real dimension n ≥ 2, and denote by T M, T ∗ M its tangent and cotangent bundles, respectively. In fact, except when we make use of the Hodge star operator, we do not need M to be orientable. Assume that M is equipped with a Riemannian metric tensor n
g = ∑ g jk dx j ⊗ dx k , j,k=1
with g jk ∈ C 2 ,
(2.1.1)
50  2 Geometric Concepts and Tools
in local coordinates, so ⟨∂ j , ∂ k ⟩ = g jk = g kj ,
1 ≤ j, k ≤ n.
(2.1.2)
The existence of such a local coordinate system necessarily places a C 3 smooth differential structure on M; see [121]. For further reference, it is useful to consider the (symmetric, positive deﬁnite) n × n matrix G := (g jk )1≤j,k≤n .
(2.1.3)
As is customary, det G is also denoted by g so the corresponding volume form may be locally expressed as (2.1.4) dVol = √ g(x) dx1 ∧ ⋅ ⋅ ⋅ ∧ dx n . We also set (g jk )1≤j,k≤n := G−1 .
(2.1.5)
To proceed, given any l ∈ {0, 1, . . . , n}, for x ∈ M we denote by Λ l T x M the space of alternating lmultilinear functionals on T x M, and by Λ l T M the associated vector bundle¹. Differential forms of degree l are sections of Λ l T M. If (x1 , . . . , x n ) are local coordinates in an arbitrary coordinate patch U on M and u ∈ C 0 (M, Λ l T M) then we may write uU = ∑I=l u I dx I , where the symbol ∑ indicates that the sum is performed over ordered arrays² I = (i1 , . . . , i l ), 1 ≤ i1 < i2 < ⋅ ⋅ ⋅ < i l ≤ n, of length I = l and, for each such I, dx I := dx i1 ∧ ⋅ ⋅ ⋅ ∧ dx i l . Here, wedge stands for the usual exterior product of forms. Speciﬁcally, recall that for an ℓform u and a kform υ, the exterior product u ∧ υ is deﬁned as (ℓ + k)! u ∧ υ := Alt(u ⊗ υ) (2.1.6) ℓ!k! where the alternation of a multilinear map ω deﬁned on the mth fold Cartesian product T M × ⋅ ⋅ ⋅ × T M is deﬁned as Alt (ω)(X1 , . . . , X m ) :=
1 ∑ sign (π) ω(X π(1) , . . . , X π(m) ), m! π∈S
(2.1.7)
m
for all X1 , . . . , X m ∈ T M. Above, Sm denotes the group of all permutations of the set {1, . . . , m}, and for each π ∈ Sm we have denoted by sign (π) the sign of π. Hence, for
1 Warning: it is also common to denote this vector bundle by Λ l T ∗ M. We also agree to extend the deﬁnition of Λ l TM to all l ∈ ℤ by setting Λ l TM := 0 if l < 0, or l > n. 2 Throughout, by an array I of length l we shall understand a function f : {1, . . . , l} → {1, . . . , n}. We often identify such an array I with the list (f(1), . . . , f(l)), or (i 1 , . . . , i l ) where i1 := f(1), . . . , i l := f(l) ∈ {1, . . . , n}. In this context, we shall call the array I ordered if the function f is strictly increasing.
2.1 Differential Geometric Preliminaries

51
every X1 , . . . , Xℓ+k ∈ T M we may express (u ∧ υ)(X1 , . . . , Xℓ+k ) 1 = ∑ sign (π) u(X π(1) , . . . , X π(ℓ) )υ(X π(ℓ+1) , . . . , X π(ℓ+k) ) ℓ!k! π∈S ℓ+k
=
∑ sign (π) u(X π(1) , . . . , X π(ℓ) )υ(X π(ℓ+1) , . . . , X π(ℓ+k) )
(2.1.8)
π∈SHℓ,k
where SHℓ,k is the subset of Sℓ+k consisting of (ℓ, k)shuffles, that is, permutations π ∈ Sℓ+k such that π(1) < π(2) < ⋅ ⋅ ⋅ < π(ℓ) and π(ℓ + 1) < π(ℓ + 2) < ⋅ ⋅ ⋅ < π(ℓ + k). We continue by introducing some useful notation. Speciﬁcally, for any two (not necessarily ordered) arrays J, K of elements from {1, . . . , n}, deﬁne the generalized J Kronecker symbol ε K by setting { det ((δ jk )j∈J,k∈K ) if J = K, ε JK := { 0 otherwise, {
(2.1.9)
where δ jk := 1 if j = k, and zero if j ≠ k. It follows that 0 if either one of J, K has repetitions, { { { { { or J, K do not coincide as sets, J εK = { { { the sign of the permutation taking the { { array J onto the array K, otherwise. {
(2.1.10)
Some basic properties of these generalized Kronecker symbols are recorded below. Lemma 2.1. The following identities hold: (i) ε JK = ε KJ for all arrays J, K; (ii) ∑J ε IJ ε K = ε IK for all arrays I, K; J
KI I (iii) ε IK JK = ε KJ = ε J if the arrays I, J, K satisfy K ∩ (I ∪ J) = ⌀ as sets, where IK is the array obtained by concatenating I with K (in this order), etc.; I J ε JI for all arrays I, J, K; (iv) ε IJ K = (−1) K I J if the arrays I, J satisfy I ∩ J = ⌀ as sets, while ε IJ = 0 if I ∩ J ≠ ⌀ (v) ε IJ JI = (−1) K as sets; AJ IJ I iB IB (vi) ∑A ε iA I ε L + (−1) ∑B ε J ε L = ε iL for all arrays I, J, L and every i ∈ {1, . . . , n}.
Proof. Properties (i)–(v) are direct consequences of deﬁnitions, so we concentrate on (vi). For this, observe ﬁrst that this is trivially true when i ∉ I ∪ J as both sides are equal to zero. Next, write (with the help of (iii) and (ii) in the statement of the lemma)
AJ
iAJ
IJ
iA ∑ ε iA I ε L = ∑ ε I ε iL = ε iL A
A
when i ∈ I and i ∉ J,
(2.1.11)
52  2 Geometric Concepts and Tools
and
AJ
∑ ε iA I ε L = 0 when i ∉ I.
(2.1.12)
A
As a consequence of (2.1.11) with the roles of I and J interchanged (and with B now playing the role of A), we have (bearing (iv) in mind)
IB I(J−1) BI ∑ ε iB ∑ ε iB J ε L = (−1) J εL B
= =
B I(J−1) JI (−1) ε iL IJ (−1)I ε iL when
i ∈ J and i ∉ I,
(2.1.13)
while (2.1.12) gives
IB ∑ ε iB J εL = 0
when i ∉ J.
(2.1.14)
B
The identity in item (vi) of the lemma now readily follows from (2.1.11), (2.1.12) and (2.1.13), (2.1.14) in the case when i ∉ I ∩ J. In the case when i ∈ I ∩ J, writing J as the concatenation J − iJ + for some arrays J ± , permits us to express
AJ iA AJ ∑ ε iA I εL = ∑ εI εL A
−
iJ +
−
iAJ = ∑ (−1)A+J  ε iA I εL
− +
J
A
=
A I−1+J −  IJ − J + (−1) εL .
(2.1.15)
Thus, after adjusting notation, if i ∈ I ∩ J, with I = I − iI + we obtain from (2.1.15)
JI − I +
−
BI J−1+I  εL ∑ ε iB J ε L = (−1)
(2.1.16)
B
which further yields
IB I+I(J−1) BI (−1)I ∑ ε iB ∑ ε iB J ε L = (−1) J εL B
=
B I J+J−1+I −  JI − I + (−1) εL .
(2.1.17)
Consequently, in the case when i ∈ I ∩ J, with I = I − iI + and J = J − iJ + , from (2.1.15) and (2.1.17) we conclude that
−
AJ I iB IB I−1+J  IJ εL ∑ ε iA I ε L + (−1) ∑ ε J ε L = (−1) A
B
− +
I − iI + J − J +
−
− + −
+
−
− +
= (−1)I  ε IL I
J iJ +
−
− +
I
−
JI − I +
−
− +
−
− +
+ (−1)I J+J−1+I  ε L
+ (−1)I J+J−1+I  ε JI L −
+ (−1)I J+J−1+I  ε JI L
+J(I−1) JI − I + εL IJ ε iL ,
= (−1)I =0=
J
−
+ (−1)I J+J−1+I  ε JI L
= (−1)I−1+J  ε L
= (−1)I−1+I  ε IL I
given that I ∩ J ≠ ⌀.
J
I
− +
I
+ (−1)I J+J−1+I  ε JI L
I
(2.1.18)
2.1 Differential Geometric Preliminaries

53
In relation to the generalized Kronecker symbols, let us also note here that formula dx i (∂ j ) = δ ij ,
∀ i, j ∈ {1, . . . , n},
(2.1.19)
(expressing the fact that {dx i }1≤i≤n and {∂ j }1≤j≤n are dual bases to one another) extrapolates, in light of (2.1.9), to (dx i1 ∧ ⋅ ⋅ ⋅ ∧ dx i l )(∂ j1 , . . . , ∂ j l ) = ε IJ I := (i1 , . . . , i l )
where
J := (j1 , . . . , j l ).
and
(2.1.20)
As a consequence, given a differential form u of degree l, locally expressed as ∑I=l u I dx I , its coefficients may be recovered from the action of u on T M × ⋅ ⋅ ⋅ × T M by (2.1.21) u I = u(∂ i1 , . . . , ∂ i l ) for every ordered array I = (i1 , . . . , i l ). Hence, in the case when the ltuple I = (i1 , . . . , i l ) is not necessarily ordered,
(i ,...,i l )
u(∂ i1 , . . . , ∂ i l ) = ∑ ε J 1
uJ .
(2.1.22)
J=l
The inner product structure on the ﬁbers on T M extends naturally to T ∗ M by setting ⟨dx j , dx k ⟩x := g jk (x) = g kj (x).
(2.1.23)
The latter further induces an inner product structure on Λ l T M by selecting {ω I }I=l to be an orthonormal frame in Λ l T M provided {ω j }1≤j≤n is an orthonormal frame in T ∗ M (locally). Note that ⟨dx I , dx J ⟩x = det ((g ij (x))i∈I,j∈J ). (2.1.24) In particular, this entails ⟨dVol, dVol⟩ = 1
pointwise on M.
(2.1.25)
Next, deﬁne the Hodge star operator as the unique vector bundle morphism ∗ : Λ l T M → Λ n−l T M such that u ∧ (∗u) = u2 dVol,
(2.1.26)
where dVol is regarded as an nform on M, making use of the orientation on M. Hence, in particular, dVol = ∗1, ∗dVol = 1, and (2.1.27) u ∧ (∗υ) = ⟨u, υ⟩ dVol for all lforms u, υ. Hence, if we deﬁne the interior product between a kform υ and an lform u by setting υ ∨ u := (−1)(l+k)(n+1+k) ∗ (υ ∧ (∗u)),
(2.1.28)
54  2 Geometric Concepts and Tools
it follows from (2.1.28) and (2.1.27) that u ∨ dVol = ∗u u ∨ υ = υ ∨ u = ⟨u, υ⟩
and
for all lforms u, υ.
(2.1.29)
Other basic formulas valid within this formalism include the following (cf. [86, Lemma 4.1, p. 40] and [94, Proposition 2.1]). Lemma 2.2. Given any integers l, m, n, k ∈ {0, 1, . . . , n}, for arbitrary lform u, kform υ, 1form α, 1form β, (n − l)form θ, lform η, mform w, and (l + k)form ω, the following are true: (1) ∗ ∗ u = (−1)l(n+1) u, w ∧ (υ ∧ u) = (w ∧ υ) ∧ u, and w ∨ (υ ∨ u) = (υ ∧ w) ∨ u; (2) υ ∧ u = (−1)kl u ∧ υ and υ ∨ u = (−1)(l+k)(n+l) (∗u) ∨ (∗υ); (3) ⟨u, ∗θ⟩ = (−1)l(n+1) ⟨∗u, θ⟩ and ⟨∗u, ∗η⟩ = ⟨u, η⟩; (4) if the integer k is odd then υ ∧ (υ ∧ u) = 0 and υ ∨ (υ ∨ u) = 0; in particular, one always has α ∧ (α ∧ u) = 0 and α ∨ (α ∨ u) = 0; (5) α ∨ (u ∧ υ) = (α ∨ u) ∧ υ + (−1)l u ∧ (α ∨ υ); (6) (−1)l α ∧ (u ∨ υ) = u ∨ (α ∧ υ) − (α ∨ u) ∨ υ; (7) α ∧ (β ∨ u) + β ∨ (α ∧ u) = ⟨α, β⟩u; (8) ⟨υ ∧ u, ω⟩ = ⟨u, υ ∨ ω⟩; in addition, u ∨ (υ ∨ w) = (υ ∧ u) ∨ w; (9) ∗(υ ∧ u) = (−1)lk υ ∨ (∗u) and ∗(υ ∨ u) = (−1)k(l+1) υ ∧ (∗u). Moreover, if the 1form α is normalized such that α = 1, then also: (10) u = α ∧ (α ∨ u) + α ∨ (α ∧ u); (11) α ∧ (α ∨ u) = α ∨ u and α ∨ (α ∧ u) = α ∧ u. Proof. Properties (1)–(4) and (8), (9) are more or less direct consequences of deﬁnitions. As far as the formula in item (5) is concerned, it suffices to consider the case when α = (∂ i )♭ ∈ Λ1 T M, u = dx I ∈ Λ l T M, and υ = dx J ∈ Λ k T M. In this scenario we K have u ∧ υ = ∑K=l+k ε IJ K dx hence,
α ∨ (u ∧ υ) = ∑
IJ IJ L L ( ∑ ε iL K ε K )dx = ∑ ε iL dx ,
L=l+k−1 K=l+k
(2.1.30)
L=l+k−1
where we have made use of (2.1.38) (proved a little later, independently of the present A considerations). Going further, from (2.1.38) we obtain α ∨ u = ∑A=l−1 ε iA I dx which then implies AJ L (α ∨ u) ∧ υ = ∑ ( ∑ ε iA (2.1.31) I ε L )dx . L=l+k−1 A=l−1
B Appealing again to (2.1.38) we may write α ∨ υ = ∑B=k−1 ε iB J dx . Using this we may then write
(−1)l u ∧ (α ∨ υ) = ∑
L=l+k−1
IB L ((−1)I ∑ ε iB J ε L )dx .
(2.1.32)
B=k−1
On account of item (vi) in Lemma 2.1, property (5) now follows from (2.1.30)–(2.1.32). Next, property (6) is easily derived from (5) via Hodge duality (keeping in mind the
2.1 Differential Geometric Preliminaries

55
formulas in item (9)). Finally, property (7) can be seen by specializing (5) to the case when u = β (and then redenoting υ = u). The result presented in the lemma below is useful in the proof of Ricci’s identity, discussed later, in Proposition 9.5. Lemma 2.3. Given 1forms α, β, for each ℓ ∈ {0, 1, . . . , n} consider the operator Λℓ T M ∋ u → Rℓαβ u := α ∧ (β ∨ u) ∈ Λℓ T M.
(2.1.33)
Then, if ℓ, k ∈ {0, 1, . . . , n}, one has ℓ k Rℓ+k αβ (u ∧ υ) = (Rαβ u) ∧ υ + u ∧ (Rαβ υ),
(2.1.34)
for every u ∈ Λℓ T M and υ ∈ Λ k T M. Proof. Having ﬁxed u ∈ Λℓ T M and υ ∈ Λ k T M, using (1) in Lemma 2.2 we may write (Rℓαβ u) ∧ υ = (α ∧ (β ∨ u)) ∧ υ = α ∧ ((β ∨ u) ∧ υ),
(2.1.35)
and u ∧ (Rkαβ υ) = u ∧ (α ∧ (β ∨ υ)) = (u ∧ α) ∧ (β ∨ υ) = (−1)ℓ (α ∧ u) ∧ (β ∨ υ) = α ∧ [(−1)ℓ u ∧ (β ∨ u)].
(2.1.36)
On the other hand, since Rℓ+k αβ (u ∧ υ) = α ∧ (β ∨ (u ∧ υ)), we see from (2.1.35) and (2.1.36) that (2.1.34) follows once we show β ∨ (u ∧ υ) = (β ∨ u) ∧ υ + (−1)ℓ u ∧ (β ∨ u). This, however, is guaranteed by (5) in Lemma 2.2 (with the 1form β playing the role of α). Next, we shall prove the following basic rule concerning interior products. Proposition 2.4. In a local coordinate chart on M, for each ordered array I of length l and each i ∈ {1, . . . , n} there holds dx i ∨ dx I = ∑
n
jL
∑ ε I g ij dx L .
(2.1.37)
L=l−1 j=1
Consequently, with the ﬂat (musical) isomorphism deﬁned as in (9.1.2), one locally has
L (∂ i )♭ ∨ dx I = ∑ ε iL I dx .
(2.1.38)
L=l−1
In this endeavor, the following lemma will be useful. Lemma 2.5. Given a matrix A = (a ij )1≤i,j≤n , for any two ordered arrays I, J of numbers from {1, . . . , n} set A[IJ] := (a ij )i∈I,j∈J (2.1.39)
56  2 Geometric Concepts and Tools and, further, for each l ∈ {1, . . . , n} deﬁne³ M A,l := (det A[IJ] )I=l,J=l .
(2.1.40)
Then for each l ∈ {1, . . . , n} and any n × n matrix A it follows that (M A,l )⊤ = M A⊤,l
(2.1.41)
and, if A is invertible, the matrix M A,l is also invertible and (M A,l )−1 = M A−1 ,l = (det (A−1 )[IJ] )I=l,J=l .
(2.1.42)
Proof. Formula (2.1.41) is immediate from (2.1.39), while formula (2.1.42) is a consequence of the CauchyBinet formula which implies that⁴ M A,l M B,l = M AB,l
(2.1.43)
for any two n × n matrices A, B. For another perspective, we mention that the identity (2.1.43) is equivalent to (Λ l A)(Λ l B) = Λ l (AB). (2.1.44) See, for example, § 21 of [120]. We now turn to the proof of the Proposition 2.4. Proof of the Proposition 2.4. Fix an ordered array I of length l along with some integer i ∈ {1, . . . , n}, and recall (2.1.3)–(2.1.5) as well as (2.1.39). Write
dx i ∨ dx I = ∑ υ L dx L ,
(2.1.45)
L=l−1
for some coefficients υ := (υ L )L=l−1 . Then, for each ordered array K with K = l − 1, on the one hand we have
⟨dx i ∨ dx I , dx K ⟩ = ∑ υ L ⟨dx L , dx K ⟩ = ∑ υ L det (G−1 )[LK] L=l−1
L=l−1
= (M G−1 ,l−1 υ)K .
(2.1.46)
On the other hand,
I J ⟨dx i ∨ dx I , dx K ⟩ = ⟨dx I , dx i ∧ dx K ⟩ = ∑ ε iK J ⟨dx , dx ⟩ J=l
= ∑
J=l
ε iK J
−1
det (G )[IJ] = ∑ ε iK J (M G−1 ,l )IJ .
(2.1.47)
J=l
3 that is, M A,l is the matrix of all l × l minors of A, also referred to as the lth compound of A; cf. the discussion in [34, § 2.4, pp. 11–12] 4 another proof of (2.1.43) is given in [34, p. 11]; cf. also [54]
2.1 Differential Geometric Preliminaries

57
Collectively, (2.1.46) and (2.1.47) imply
M G−1 ,l−1 υ = ( ∑ ε iK J (M G−1 ,l )IJ ) J=l
(2.1.48) K=l−1
which, in light of (2.1.42), entails
−1
υ = (M G−1 ,l−1 ) ( ∑ ε iK J (M G−1 ,l )IJ ) J=l
K=l−1
= M G,l−1 ( ∑ ε iK J (M G−1 ,l )IJ ) J=l
.
(2.1.49)
K=l−1
Consequently, for each ordered array L with L = l − 1,
υ L = ∑ (M G,l−1 )LK ( ∑ ε iK J (M G−1 ,l )IJ ). K=l−1
(2.1.50)
J=l
To proceed, consider an ordered array J of length l containing i, say J = J1 iJ2 where J1 , J2 are ordered arrays with J1  + J2  = l − 1. Note that this forces ε iK J ≠ 0 ⇒ K = J 1 J 2
and
J1  ε iK . J = (−1)
(2.1.51)
Also, for each j denote by I j< and I j> the ordered arrays obtained by considering those elements from I which are strictly smaller and, respectively, strictly larger, than j. In particular, for each j ∈ I the ordered array I may be viewed as the concatenation I j< j I j> , hence I j := I \ {j} = I j< I j> . Expanding the determinant with respect to the ith column then yields
0}, rn r→0+
∂∗ Ω := {x ∈ ∂Ω : lim sup r→0+
(2.2.9)
where E stands for the Lebesgue measure⁷ of E ⊆ M. Then for any open set Ω ⊂ M we have (cf. [31, Theorem 1, p. 222]) Ω has ﬁnite perimeter ⇐⇒ H
n−1
(∂∗ Ω) < ∞.
(2.2.10)
In the case when Ω ⊂ M is an open set of ﬁnite perimeter, from the classical work of Federer and of De Giorgi we also have (cf. [31, Lemma 1, p. 208]) ∂∗ Ω ⊆ ∂∗ Ω
and
H
n−1
(∂∗ Ω \ ∂∗ Ω) = 0.
(2.2.11)
In particular, from (2.2.10) and (2.2.11) we conclude that for any open set Ω ⊂ M we have Ω has ﬁnite perimeter H n−1 (∂Ω) < ∞ and (2.2.12) } ⇐⇒ { H n−1 (∂Ω \ ∂∗ Ω) = 0. and satisﬁes (2.2.5) Also, for an open set Ω ⊂ M of ﬁnite perimeter, condition (2.2.5) is equivalent to H
n−1
(∂Ω \ ∂∗ Ω) = 0.
(2.2.13)
We shall work with Lebesgue spaces on ∂Ω which, unless speciﬁcally indicated otherwise, are always considered with respect to the “surface measure” σ = H n−1 ⌊∂Ω. As indicated in our next lemma, this scale of Lebesgue spaces turns out to enjoy reasonable properties in the case when the underlying domain is an open set of ﬁnite perimeter satisfying (2.2.5). Lemma 2.11. Let Ω ⊂ M be an open set of ﬁnite perimeter satisfying (2.2.13) (or, equivalently, (2.2.5)). Then the following statements are true: (i) The topological boundary, ∂Ω, is countably rectiﬁable (of dimension n − 1). In particular, ∂Ω has Lebesgue measure⁸ zero. (ii) The surface measure σ := H n−1 ⌊∂Ω is a complete⁹, ﬁnite Radon measure¹⁰ on ∂Ω. (iii) For every p ∈ (0, ∞) the natural inclusion {ϕ∂Ω : ϕ ∈ C 2 (M)} → L p (∂Ω) has dense range. 7 canonically associated with the volume element on M 8 canonically associated with the volume element on M 9 meaning that any subset of a nullset is measurable (hence also a nullset) 10 i.e., a ﬁnite Borel regular measure
(2.2.14)
70  2 Geometric Concepts and Tools (iv) Given any function f ∈ L1 (∂Ω), one has f = 0 σa.e. on ∂Ω ⇐⇒ ∫ fϕ dσ = 0 for every ϕ ∈ C 2 (M).
(2.2.15)
∂Ω
Proof. The claim in part (i) is apparent from (2.2.7), (2.2.8), and (2.2.11). We momentarily digress for the purpose of recording several useful results of a general nature (for proofs see [82], and, for the case of ﬁnite Hausdorff measure, applicable here, Chapter 12 of [119]). First, given a metric space (X, ρ) along with some number d ≥ 0, d turns out to be a Borel the associated ddimensional Hausdorff outermeasure HX,ρ regular outermeasure on X (equipped with the topology canonically induced by the metric ρ). Second, the measure induced by an outermeasure (as in Carathéodory’s theorem) is automatically complete. Third, given a topological space (X, τ), the quality of being Borel regular is hereditary, in the precise sense that μ Borel regular outermeasure on (X, τ) and A ∈ M ⇒ (μM )⌊A is a Borel regular measure on (A, τA ),
(2.2.16)
where M stands for the sigmaalgebra of μmeasurable¹¹ subsets of X. Fourth, one may easily see that completeness is also preserved when restricting a complete measure to a measurable subset of the ambient. In concert with (2.2.12), these considerations imply that σ := H n−1 ⌊∂Ω is a complete ﬁnite Radon measure on ∂Ω, as claimed in part (ii). Having established this, the same argument as in [1, Proof of Theorem 3.14, pp. 98, 99], applies and gives that for every p ∈ (0, ∞) the natural inclusion Lip (∂Ω) → L p (∂Ω) has dense range,
(2.2.17)
where Lip (∂Ω) denotes the space of Lipschitz functions deﬁned on ∂Ω. Since (cf., e.g., the discussion in [1, 83]) Lip (∂Ω) = {ϕ∂Ω : ϕ ∈ Lip (M)},
(2.2.18)
a standard molliﬁer argument then allows us to deduce (2.2.14). This justiﬁes the claim made in part (iii). To deal with the claim made in part (iv), consider an arbitrary f ∈ L1 (∂Ω) satisfying the condition recorded in the righthand side of (2.2.15). The ﬁrst observation is that, thanks to (2.2.18) and a standard molliﬁer argument, this selfimproves to ∫ fϕ dσ = 0
for every ϕ ∈ Lip (∂Ω).
(2.2.19)
∂Ω
11 recall that E ⊆ X is called μmeasurable provided μ(Y) = μ(Y ∩ E) + μ(Y \ E) for each Y ⊆ X.
2.2 Elements of Geometric Measure Theory  71
To proceed, ﬁx an arbitrary closed set C ⊂ ∂Ω and construct a sequence of functions {ϕ j }j∈ℕ ⊂ Lip (∂Ω) satisfying 0 ≤ ϕ j ≤ 1 on ∂Ω for each j ∈ ℕ, and ϕ j ↘ 1C pointwise as j → ∞.
(2.2.20)
(cf. [83, Lemma 4.14, p. 166].) Granted these, Lebesgue’s Dominated Convergence Theorem and (2.2.19) permit us to conclude that (2.2.21) ∫ f dσ = lim ∫ fϕ j dσ = 0. j→∞
C
∂Ω
Next, if we introduce A± := {x ∈ ∂Ω : ±f(x) ≥ 0},
(2.2.22)
then [1, (3.83), p. 89] gives (bearing in mind that σ is a Borel regular measure) that σ(A± ) = sup{σ(C) : C closed, C ⊆ A± }.
(2.2.23)
Fix ε > 0 arbitrary. Since the ﬁnite measure f dσ is absolutely continuous with respect to σ, it follows that there exists θ > 0 with the property that ∫f dσ < ε for each measurable E ⊆ ∂Ω with σ(E) < θ.
(2.2.24)
E
For this θ, use (2.2.23) to ﬁnd closed subsets C± of A± such that σ(A± \ C± ) < θ.
(2.2.25)
Then, thanks to (2.2.21), we may write ∫f dσ = ∫ f dσ − ∫ f dσ = ∫ f dσ − ∫ f dσ, ∂Ω
A+
A−
A+ \C+
(2.2.26)
A− \C−
and then rely on (2.2.24) and (2.2.25) to estimate ∫ f dσ ≤ ∫f dσ < ε.
A± \C±
(2.2.27)
A± \C±
The bottom line is that ∫∂Ω f dσ < 2ε hence, ultimately, f = 0 at σa.e. point on ∂Ω. This ﬁnishes the leftpointing implication in (2.2.15). Since the opposite implication is trivial, this proves the claim made in part (iv), and concludes the proof of the lemma. Moving on, recall that Σ ⊂ M is said to be an Ahlfors regular set if Σ is closed and there exist c0 , c1 ∈ (0, ∞), called Ahlfors regularity constants, such that for all x ∈ Σ, r ∈ (0, 1], c0 r n−1 ≤ H n−1 (B r (x) ∩ Σ) ≤ c1 r n−1 , (2.2.28)
72  2 Geometric Concepts and Tools where B r (x) is the (geodesic) ball of radius r centered at x. We then say that Ω is an Ahlfors regular domain if Ω is an open, nonempty, proper subset of M such that ∂Ω is an Ahlfors regular set and (2.2.13) holds.
(2.2.29)
Whenever Ω ⊂ M is an Ahlfors regular domain it follows that Ω has ﬁnite perimeter. In particular, it makes sense to talk about its outward unit normal ν and surface measure σ. In fact, as pointed out earlier there holds σ = H n−1 ⌊∂Ω. When equipped with this doubling measure and the distance induced from M, ∂Ω becomes a space of homogeneous type, in the sense of Coifman and Weiss (see [16]). This is a general type of environment where a good deal of classical harmonic analysis may be carried out. In particular, in such a setting one may consider the JohnNirenberg space BMO(∂Ω) of functions with bounded mean oscillation on ∂Ω. Also, we may deﬁne VMO(∂Ω), the Sarason class of functions of vanishing mean oscillation on ∂Ω, as the closure in BMO(∂Ω) of the set of uniformly continuous functions on ∂Ω. Later on, we shall ﬁnd the following result useful. Lemma 2.12. If Ω is a proper Ahlfors regular subdomain of M, then Ω ≠ M. Proof. Seeking a contradiction, assume Ω = M. This implies ∂Ω = Ω \ Ω = M \ Ω
(2.2.30)
which, in light of (2.2.9) and item (i) of Lemma 2.11, forces ∂∗ Ω = ⌀. Together with (2.2.13) the latter implies H n−1 (∂Ω) = 0 which, in view of (2.2.28), leads to the conclusion that ∂Ω = ⌀. In concert with (2.2.30), this ultimately allows us to conclude that Ω = M, contradicting the assumption that Ω is a proper subset of M. We now turn to the notion of nontangential boundary trace of functions deﬁned in a nonempty, proper, open set Ω ⊂ M. Fix a background parameter κ > 0 and, for each boundary point x ∈ ∂Ω, introduce the nontangential approach region Γ(x) := Γ κ (x) := {y ∈ Ω : dist (x, y) < (1 + κ) dist (y, ∂Ω)},
(2.2.31)
where all distances are taken with respect to the background geodesic metric on M. It should be noted that, under the current hypotheses, it could happen that Γ(x) = ⌀ for certain points x ∈ ∂Ω. Next, given a measurable¹² function u deﬁned a.e. in Ω (typically taking values in some vector bundle over M), we wish to consider its nontangential limit at a boundary point x ∈ ∂Ω. For this deﬁnition to be natural it is desirable to be able to approach x from within the nontangential approach region Γ(x), i.e., have x ∈ Γ(x). Assuming that this is the case, we say that u has a nontangential boundary
12 with respect to the background measure on M naturally induced by the volume element (in the sense of Riesz’s representation theorem; cf., e.g., [107, Theorem 2.14, pp. 40–41])
2.2 Elements of Geometric Measure Theory
 73
trace at x provided there exists some N(x) ⊂ Γ(x) of measure zero such that the limit¹³ n.t. (u∂Ω )(x) :=
lim
Γ(x)\N(x)∋y→x
u(y) exists.
(2.2.32)
n.t. Since it is desirable to have u∂Ω deﬁned σa.e. in ∂Ω (where, as usual, σ := H the above deﬁnition highlights the importance of having x ∈ Γ(x)
for σa.e. x ∈ ∂Ω.
n−1 ⌊∂Ω),
(2.2.33)
We shall call an open set Ω ⊆ M satisfying (2.2.33) above weakly accessible. In particular, if Ω is weakly accessible and the function u is actually continuous on Ω, the n.t. nontangential trace of u on ∂Ω exists at σa.e. point and, in fact, (u∂Ω )(x) = (u∂Ω )(x) for σa.e. x ∈ ∂Ω, i.e., the nontangential trace coincides σa.e. with the ordinary restriction to the boundary. In [50, § 2.3] it is proved that each Ahlfors regular domain is weakly accessible
(2.2.34)
in the following precise sense: for each κ ∈ (0, ∞) there exists a set A κ ⊆ ∂Ω so that H n−1 (A κ ) = 0 and x ∈ Γ κ (x) for every point x ∈ ∂Ω \ A κ .
(2.2.35)
For a ﬁxed background parameter κ > 0, let us also deﬁne the nontangential maximal operator acting on a given measurable function u deﬁned in Ω (with values in a Hermitian vector bundle over M) according to¹⁴ (N u)(x) := (Nκ u)(x) := ‖u‖L∞ (Γ κ (x)) ∈ [0, ∞],
∀ x ∈ ∂Ω,
(2.2.36)
where the essential supremum¹⁵ in the righthand side is taken with respect to the background measure on M naturally induced by the volume element. In particular, if u = u a.e. in Ω then N u = N u everywhere on ∂Ω, and u continuous in Ω ⇒ (N u)(x) = sup u(y), y∈Γ κ (x)
∀ x ∈ ∂Ω.
(2.2.37)
Moreover, it turns out that for every measurable function u deﬁned in Ω, it follows that N u is lower semicontinuous on ∂Ω.
(2.2.38)
It is also immediate from deﬁnitions that, given any measurable function u deﬁned in Ω along with any point x ∈ ∂Ω such that the nontangential limit of u at x exists, we have n.t. (u )(x) ≤ (N u)(x). (2.2.39) ∂Ω 13 if u is a continuous function in Ω to begin with, then we may take N(x) = ⌀; hence, in this case, n.t. the original deﬁnition simply becomes (u∂Ω )(x) = limΓ(x)∋y→x u(y) 14 occasionally we shall also refer to Nu as the nontangential maximal function of u 15 taken to be 0 if Γ κ (x) = ⌀
74  2 Geometric Concepts and Tools
We also mention that if Ω ⊆ M is a weakly accessible domain and u is a Lebesgue measurable function deﬁned in Ω with the property n.t. that the nontangential limit f := u∂Ω exists at σa.e. point on ∂Ω, then the function f is σmeasurable on ∂Ω.
(2.2.40)
A brief sketch of the argument is as follows. For starters, it suffices to consider realvalued u, and then one can reduce to the case where u ≥ 0 a.e. in Ω. Then, for σa.e. x ∈ ∂Ω, one can show that f(x) = lim+ N(χ δ u)(x), (2.2.41) δ→0
where χ δ is the characteristic function of the set consisting of points in Ω whose distance to ∂Ω is ≤ δ. Measurability of f then follows from (2.2.38) (bearing in mind that the measure σ is complete); see [81] for more details. The choice of the parameter κ in the deﬁnition of the nontangential maximal operator in (2.2.36) plays only a relatively minor role when measuring the size of the nontangential maximal operator on the scale of Lebesgue spaces. Speciﬁcally, the following result is a consequence of [50, Proposition 2.2, p. 2573]. Proposition 2.13. Assume Ω ⊂ M is an open set with an Ahlfors regular boundary, and deﬁne σ := H n−1 ⌊∂Ω. Also, ﬁx κ, κ ∈ (0, ∞) and p ∈ (0, ∞] arbitrary. Then there exist two ﬁnite constants C0 , C1 > 0 with the property that for each Lebesgue measurable function u deﬁned in Ω one has C0 ‖Nκ u‖L p (∂Ω) ≤ ‖Nκ u‖L p (∂Ω) ≤ C1 ‖Nκ u‖L p (∂Ω) .
(2.2.42)
In fact, the level sets of Nκ u and Nκ u have comparable σmeasures, which also implies that Nκ u < ∞ σa.e. on ∂Ω ⇐⇒ Nκ u < ∞ σa.e. on ∂Ω. (2.2.43) The ﬂexibility in choosing the parameter κ, regulating the nontangential approach regions, of the sort described in Proposition 2.13 turns out to be a very useful feature. For one thing, it implies the local boundedness result in the proposition below, which is relevant in future endeavors. Proposition 2.14. Suppose Ω ⊂ M is an open set with a nonempty Ahlfors regular boundary and set σ := H n−1 ⌊∂Ω. Then for each Lebesgue measurable function u deﬁned in Ω one has N u < ∞ σa.e. on ∂Ω ⇒ u ∈ L∞ loc (Ω).
(2.2.44)
N u ∈ L p (∂Ω) for some p ∈ (0, ∞] ⇒ u ∈ L∞ loc (Ω).
(2.2.45)
In particular, Proof. Pick an arbitrary compact set K ⊂ Ω and consider r := inf {dist (x, y) : x ∈ ∂Ω, y ∈ K} ∈ (0, ∞), R := sup {dist (x, y) : x ∈ ∂Ω, y ∈ K} ∈ (0, ∞).
(2.2.46)
2.2 Elements of Geometric Measure Theory
 75
Choosing κ > 0 such that 1 + κ > R/r then ensures that K ⊂ Γ κ (x) for every x ∈ ∂Ω. Hence, on the one hand, (Nκ u)(x) = ‖u‖L∞ (Γ κ (x)) ≥ ‖u‖L∞ (K) ,
∀ x ∈ ∂Ω.
(2.2.47)
On the other hand, from the hypotheses on u and Proposition 2.13 we deduce that Nκ u is ﬁnite σa.e. on ∂Ω. In concert with (2.2.47), this readily implies that u is essentially bounded on K, and the desired conclusion follows. Another consequence of Proposition 2.13 pertains to the independence of the nontangential approach limit on the parameter κ used to deﬁne the nontangential approach regions from which the said limit is taken. Speciﬁcally, the following result appears in [81]. Proposition 2.15. Let Ω ⊂ M be an Ahlfors regular domain, and deﬁne σ := H n−1 ⌊∂Ω. Fix κ ∈ (0, ∞) along with p ∈ (0, ∞) and assume that u : Ω → ℝ is a Lebesgue measurable function satisfying Nκ u ∈ L p (∂Ω) and for σa.e. point x ∈ ∂Ω one can ﬁnd
a Lebesgue measurable set N(x) ⊆ Γ κ (x) of measure zero such that lim(Γ κ (x)\N(x))∋y→x u(y) exists.
(2.2.48)
Then for every other κ ∈ (0, ∞) one has Nκ u ∈ L p (∂Ω) and for σa.e. point x ∈ ∂Ω one can ﬁnd
a Lebesgue measurable set N (x) ⊆ Γ κ (x) of measure zero such that lim(Γ κ (x)\N (x))∋y→x u(y) exists.
(2.2.49)
Moreover, the limits in (2.2.48) and (2.2.49) coincide σa.e. on ∂Ω. Let us also note here that, as a consequence of a general realvariable result established in [50, Proposition 3.24, p. 2647], we have the following (the reader is reminded that n stands for the dimension of M): if Ω ⊂ M is an open set with an Ahlfors regular boundary and p ∈ (0, ∞), then there exists C ∈ (0, ∞) with the property that, for any Lebesgue measurable function u deﬁned in Ω, the fact that N u belongs to L p (∂Ω) implies u ∈ L np/(n−1) (Ω) and ‖u‖L np/(n−1) (Ω) ≤ C‖N u‖L p (∂Ω) .
(2.2.50)
Moving on, we shall now give the deﬁnitions and summarize some basic properties of certain classes of locally ﬁnite perimeter domains which are relevant for the work in this monograph. Following G. David and S. Semmes [23] we make the following deﬁnition.
76  2 Geometric Concepts and Tools Deﬁnition 2.16. Call a subset Σ of M a uniformly rectifiable set provided it is Ahlfors regular and the following holds. There exist ε, C ∈ (0, ∞) such that, for each x ∈ Σ, r ∈ (0, 1], there is a Lipschitz map φ : B n−1 → M (where B n−1 is a ball of radius r r n−1 r in ℝ ) with Lipschitz constant ≤ C and satisfying H
n−1
n−1 . (Σ ∩ B r (x) ∩ φ(B n−1 r )) ≥ εr
(2.2.51)
Uniform rectiﬁability can be thought of as a quantitative version of countable rectiﬁability. Indeed, from [50, p. 2629] we know that any uniformly rectiﬁable set Σ subset of M is countably rectiﬁable (of dimension n − 1).
(2.2.52)
Following [50], we shall also make the following deﬁnition. Deﬁnition 2.17. Call a nonempty open subset Ω of M a UR (uniformly rectifiable) domain provided ∂Ω is a uniformly rectiﬁable set and (2.2.13) holds. For further use, it is useful to point out that, as is apparent from deﬁnitions, each UR domain is an Ahlfors regular domain hence, in particular, weakly accessible, and if Ω ⊂ M is a UR domain satisfying ∂Ω = ∂(Ω) then M \ Ω is also a UR domain, with the same boundary as Ω.
(2.2.53)
We next introduce the corkscrew condition which, informally speaking, amounts to the existence of “nontangential balls” at all scales and locations. Deﬁnition 2.18. Let Ω be a nonempty proper subset of M. (i) One says that Ω satisﬁes an interior corkscrew condition provided there are constants M > 1 and R > 0 (called the corkscrew constants of Ω) such that for each x ∈ ∂Ω and r ∈ (0, R) there exists y = y(x, r) ∈ Ω (called corkscrew point relative to the location x and scale r) such that dist (x, y) < r and dist(y, ∂Ω) > M −1 r. (ii) One says that Ω satisﬁes an exterior corkscrew condition if M \ Ω satisﬁes an interior corkscrew condition. (iii) We say that Ω satisﬁes a twosided corkscrew condition if Ω has both the interior and the exterior corkscrew condition. Based on deﬁnitions it is not difficult to check that any nonempty open proper subset Ω of M satisfying an interior corkscrew condition is a weakly accessible domain, in the strong sense that x ∈ Γ κ (x) for every point x ∈ ∂Ω and every κ ∈ (0, ∞).
(2.2.54)
Also, if Ω is a nonempty proper subset of M satisfying an exterior corkscrew condition, then any point on the boundary is approachable via exterior corkscrew points. c c This forces ∂Ω ⊆ (Ω c )∘ = ((Ω)∘ ) which, in turn, shows that ∂(Ω) = Ω \ (Ω)∘ = Ω ∩ ((Ω)∘ )
2.2 Elements of Geometric Measure Theory  77
contains ∂Ω. Since the opposite inclusion is always true, this argument ultimately proves that any nonempty proper subset Ω of M satisfying an exterior corkscrew condition has the property that ∂(Ω) = ∂Ω.
(2.2.55)
We also wish to note that, as apparent from (2.2.9) and (2.2.11), any nonempty open proper subset Ω of M satisfying a twosided corkscrew condition has the property that ∂Ω = ∂∗ Ω, hence also H n−1 (∂Ω \ ∂∗ Ω) = 0.
(2.2.56)
The deﬁnition below is a natural adaptation to manifolds of similar notions ﬁrst considered in the Euclidean setting in [59]. First, we introduce the Harnack chain condition which is essentially a quantiﬁed notion of path connectivity, locally near boundary points. In concert with the corkscrew condition, this is then used to deﬁne the class of nontangentially accessible domains. Deﬁnition 2.19. Let Ω be a nonempty open proper subset of M. (i) One says that Ω satisﬁes the Harnack chain condition provided there exist two constants M, R ∈ (0, ∞) with the following signiﬁcance. First, given x1 , x2 ∈ Ω, a Harnack chain from x1 to x2 in Ω is a sequence of balls B1 , . . . , B K ⊂ Ω such that x1 ∈ B1 , x2 ∈ B K and B j ∩ B j+1 ≠ ⌀ for 1 ≤ j ≤ K − 1, and such that each B j has a radius r j satisfying M −1 r j < dist(B j , ∂Ω) < Mr j . The length of the chain is K. With this piece of terminology, one then demands that if ε > 0 and x1 , x2 ∈ Ω ∩ B r/4 (z) for some z ∈ ∂Ω, r ∈ (0, R),
(2.2.57)
are such that dist (x j , ∂Ω) > ε and dist (x1 , x2 ) < 2k ε, there exists a Harnack chain B1 , . . . , B K from x1 to x2 , of length K ≤ Mk, which further has the property that the diameter of each ball B j is ≥ M −1 min (dist (x1 , ∂Ω), dist (x2 , ∂Ω)). (ii) Call Ω an interior NTA domain (or onesided NTA domain) provided Ω satisﬁes an interior corkscrew condition, as well as a Harnack chain condition. (iii) Call Ω an NTA domain provided Ω satisﬁes a twosided corkscrew condition, as well as a Harnack chain condition. (iv) Call Ω a twosided NTA domain provided both Ω and M \ Ω are NTA domains. We next recall the deﬁnition of the local John condition from [50, § 3.1]. Informally, this is a notion of nontangential path accessibility to the effect that, locally, there exists a corkscrew point from which the boundary may be accessed via rectiﬁable curves that are nontangential in the sense that the longer one travels along them away from the said corkscrew point the closer one gets to the boundary, in a quantitative, scale invariant fashion. In particular, this may be regarded as playing the role of the local starlikeness property that Lipschitz domains enjoy at boundary points.
78  2 Geometric Concepts and Tools Deﬁnition 2.20. A nonempty open set Ω ⊂ M is said to satisfy a local John condition if there exist θ ∈ (0, 1) and R > 0 (called local John constants) with the following signiﬁcance. For every x ∈ ∂Ω and r ∈ (0, R) one can ﬁnd a r ∈ B r (x) ∩ Ω such that B θr (a r ) ⊂ Ω and with the property that for each y ∈ B r (x) ∩ ∂Ω one can ﬁnd a rectiﬁable path γ y : [0, 1] → Ω, whose length is ≤ θ−1 r and such that γ y (0) = y,
γ y (1) = a r ,
dist (γ y (t), ∂Ω) > θ dist (γ y (t), y)
and ∀ t ∈ (0, 1].
(2.2.58)
Finally, Ω is said to satisfy a twosided local John condition if both Ω and M \ Ω satisfy a local John condition. Clearly, any domain satisfying a local John condition also satisﬁes a corkscrew condition. In the opposite direction, we wish to point out that any interior NTA domain satisﬁes a local John condition. Lemma 2.21. Let Ω ⊂ M be an interior NTA domain with constants M and R (as in part (i) of Deﬁnition 2.19). Suppose the points x ∈ ∂Ω and y ∈ Ω, along with the numbers r ∈ (0, R) and C ∈ (1, ∞), are such that B r/C (y) ⊂ B r (x) ∩ Ω. Then there exists some C o ∈ (1, ∞) which depends only on C and the NTA constants of Ω along with a rectiﬁable path γ x,y of length ≤ C o r, joining x with y in Ω, and such that dist (z, ∂Ω) ≥ (C o )−1 dist (z, x) for each point z ∈ γ x,y . In particular, any interior NTA domain satisﬁes a local John condition. This is proved in [50, Lemma 3.13, p. 2634] for NTA domains in the Euclidean setting but an inspection of the argument given there shows that the slightly stronger result above holds. A result related to Lemma 2.21 is proved below. Lemma 2.22. Let Ω ⊂ M be an interior NTA and ﬁx some κ ∈ (0, ∞). Then there exist parameters R ∈ (0, ∞), κ̃ ∈ (0, ∞), and C ∈ (0, ∞), with the property that whenever r ∈ (0, R), x ∈ ∂Ω,
and
y0 , y1 ∈ Γ κ (x) ∩ B r (x),
(2.2.59)
one can ﬁnd a rectiﬁable path γ ⊂ Γ ̃κ (x) joining y0 with y1 and whose length is ≤ C max {dist (x, y0 ), dist (x, y1 )}.
(2.2.60)
Proof. Let R ∈ (0, ∞) be as in part (i) of Deﬁnition 2.19, and ﬁx an arbitrary r ∈ (0, R). Also, let j0 , j1 ∈ ℕ be such that 2−j0 r ≤ dist (x, y0 ) < 2−(j0 −1) r and 2−j1 r ≤ dist (x, y1 ) < 2−(j1 −1) r.
(2.2.61)
2.2 Elements of Geometric Measure Theory  79
To ﬁx ideas, assume j0 ≤ j1 . In turn, this entails 2−j0 r ≤ max {dist (x, y0 ), dist (x, y1 )}.
(2.2.62)
For each j ∈ {j0 , . . . , j1 }, let z j be a corkscrew point relative to x at scale 2−j r. That is, for some constant C ∈ (1, ∞) which depends only on Ω, we have B2−j r/C (z j ) ⊂ B2−j r (x) ∩ Ω.
(2.2.63)
Relabel z j0 to be y0 and z j1 to be y1 . As observed at the bottom of p. 93 in [59], the Harnack chain condition implies that we may ﬁnd θ ∈ (0, 1) with the property that, for each j, there exists a polygonal path γ j joining z j−1 with z j in Ω, such that inf dist (z, ∂Ω) > θ2−j r
z∈γ j
and
length (γ j ) ≤ θ−1 2−j r.
(2.2.64)
Note that for each z ∈ γ j we have dist (z, x) ≤ length (γ j ) + dist (z j , x) ≤ θ−1 2−j r + 2−j r = (1 + θ−1 )2−j r < (1 + θ−1 )θ−1 dist (z, ∂Ω),
(2.2.65)
thanks to (2.2.64) and (2.2.63). Hence, if we set κ̃ := (1 + θ−1 )θ−1 − 1 ∈ (0, ∞),
(2.2.66)
it follows from (2.2.65) and (2.2.31) that γ j ⊂ Γ ̃κ (x) for each j.
(2.2.67)
If we now deﬁne γ to be the union of the γ j ’s, then γ is a rectiﬁable path joining y0 with y1 in Ω, with ∞
length (γ) ≤ ∑ length (γ j ) ≤ ∑ θ−1 2−j r = 2θ−1 2−j0 r j0 ≤j≤j1
≤ 2θ
−1
j=j0
max {dist (x, y0 ), dist (x, y1 )},
(2.2.68)
by (2.2.64) and (2.2.62). Also, formula (2.2.67) implies γ ⊂ Γ ̃κ (x), ﬁnishing the proof of (2.2.60). An important class of uniformly rectiﬁable sets was identiﬁed in [22]. Based on this identiﬁcation, (2.2.55), (2.2.56), (2.2.53), and Deﬁnition 2.20, we conclude the following sufficient condition for uniform rectiﬁability. Proposition 2.23. Let Ω ⊆ M be a nonempty open set satisfying a twosided corkscrew condition (which is the case if Ω satisﬁes a twosided local John condition) and whose boundary is Ahlfors regular. Then Ω is a UR domain, with the property that ∂Ω = ∂(Ω). Moreover, M \ Ω is also a UR domain, with the same boundary as Ω.
80  2 Geometric Concepts and Tools
Let us now turn our attention to deﬁning the class of regular SKT domains. We begin by considering the Euclidean ambient, then comment on how this class may be adapted to the setting of manifolds. The reader is reminded that the Hausdorff distance between two sets A, B ⊂ ℝn is deﬁned as Dist [A, B] := max {sup {dist (a, B) : a ∈ A}, sup {dist (b, A) : b ∈ B}}.
(2.2.69)
The notion of Reifenberg ﬂatness described below formalizes the notion that a given set may be locally wellapproximated by affine spaces (i.e., it may be sandwiched in between parallel hyperplanes, in a suitable quantitative scale invariant fashion). Deﬁnition 2.24. Let Σ ⊂ ℝn be a compact set and pick some ε ∈ (0, √1 ). The set Σ is 4 2 then called εReifenberg flat if there exists R > 0 such that for every x ∈ Σ and every r ∈ (0, R] one can ﬁnd a (n − 1)dimensional plane L(x, r) which contains x and such that 1 Dist[Σ ∩ B(x, r), L(x, r) ∩ B(x, r)] ≤ ε. (2.2.70) r Deﬁnition 2.25. A bounded open set Ω ⊂ ℝn is said to have the separation property if there exists R > 0 such that for every point x ∈ ∂Ω and any scale r ∈ (0, R] there exists an (n − 1)dimensional plane Π x,r containing x and a choice of unit normal vector, N x,r , to Π x,r satisfying {y + t N x,r ∈ B(x, r) : y ∈ Π x,r , t < − 4r } ⊂ Ω, {y + t N x,r ∈ B(x, r) : y ∈ Π x,r , t > 4r } ⊂ ℝn \ Ω.
(2.2.71)
Deﬁnition 2.26. Suppose Ω ⊂ ℝn is a bounded open set and pick ε ∈ (0, ε n ), where ε n ∈ (0, √1 ) is a suitably small dimensional constant. Call Ω a εReifenberg flat 4 2 domain if Ω has the separation property and ∂Ω is εReifenberg ﬂat. For example, given ε > 0, a strongly Lipschitz domain with a sufficiently small Lipschitz constant is a εReifenberg ﬂat domain. The following result is proved in [66, § 3]. Proposition 2.27. There exists a dimensional constant ε n ∈ (0, √1 ) with the property 4 2 that any open set Ω ⊂ ℝn that has the separation property, and whose boundary is a εReifenberg ﬂat set for some ε ∈ (0, ε n ), is an NTAdomain. What we presently call regular SKT (SemmesKenigToro) domains have been previously named in the literature chord arc domains with vanishing constant. The latter notion originates in the two dimensional setting, where the deﬁning condition is that the length of a boundary arc between two points does not exceed a ﬁxed multiple of the length of a chord between these points and that the quality of the approximation of the length of the arc by the length of the chord improves indeﬁnitely as the scale shrinks to zero. In higher dimensions, where this phenomenon becomes somewhat more sophisticated, this notion was introduced in S. Semmes [112] and was further developed in [66, 67]. In the higher dimensional setting, the designation “chord arc with
2.2 Elements of Geometric Measure Theory  81
vanishing constant” no longer adequately captures the essential features of such domains, and in [50] the alternative terminology “regular SKT domains” was proposed. In the Euclidean setting, the original deﬁnition of the latter category of domains from [50] is as follows: Deﬁnition 2.28. Let ε ∈ (0, ε n ), where ε n is as in Proposition 2.27. A bounded set Ω in ℝn of ﬁnite perimeter is said to be a εSKT domain if Ω is a εReifenberg ﬂat domain, ∂Ω is an Ahlfors regular set, and there exists r > 0 such that sup ( sup (− ∫ ν − ν ∆ 2 dσ)
x∈∂Ω ∆⊂∆(x,r)
1/2
) < ε,
(2.2.72)
∆
with the supremum taken over all surface balls ∆ contained in ∆(x, r) := ∂Ω ∩ B(x, r). Here, as before, ν is the measuretheoretic outward unit normal to ∂Ω and ν ∆ := ∫−∆ ν dσ. Furthermore, call a bounded open set Ω ⊂ ℝn a regular SKT domain if Ω is a εSKT domain for some ε ∈ (0, ε n ) and, in addition, − ν − ν ∆(x,r) 2 dσ) lim sup ( sup ( ∫ r→0+
x∈∂Ω
1/2
) = 0.
(2.2.73)
∆(x,r)
The following important result appears in [50]. Theorem 2.29. Let Ω ⊆ ℝn be a bounded open set that satisﬁes a twosided local John condition and whose boundary is Ahlfors regular. Then there exists a geometrical constant C o > 1 with the following signiﬁcance. Whenever there exists ε > 0, sufficiently small relative to the John and Ahlfors regularity constants of Ω, with the property that (2.2.72) holds, it follows that Ω is a ε o SKT domain, with ε o = C o ε. In particular, Ω is a ε o Reifenberg ﬂat domain and, hence, a twosided NTA domain. In addition, there exists R > 0 with the property that for every x ∈ ∂Ω and r ∈ (0, R] one has σ(∆(x, r)) − 1 ≤ C o ε, (2.2.74) n−1 υ n−1 r where σ := H n−1 ⌊∂Ω, ∆(x, r) := B(x, r) ∩ ∂Ω, and υ n−1 := π of the unit ball in ℝn−1 .
n−1 2
/Γ( n+1 2 ) is the volume
We continue by recalling a useful, natural characterization of the class of Euclidean regular SKT domains, established in [50]: Theorem 2.30. Let Ω ⊆ ℝn be a bounded open set. Then the following are equivalent: (i) Ω is a regular SKT domain; (ii) Ω is a twosided NTA domain, ∂Ω is Ahlfors regular, and ν ∈ VMO (∂Ω); (iii) Ω is a domain with an Ahlfors regular boundary, satisfying a twosided local John condition (cf. Deﬁnition 2.20), and for which ν ∈ VMO (∂Ω).
82  2 Geometric Concepts and Tools Let us also remark that if the condition ν ∈ VMO (∂Ω) in Theorem 2.30 is strengthened to ν ∈ C 0 (∂Ω), then this result becomes a characterization of C 1 domains (see [51] for details). Recall that Deﬁnition 2.28 demands that the domain is εReifenberg ﬂat, has an Ahlfors regular boundary, and its unit normal is in VMO(∂Ω). Compared with the original deﬁnition of a regular SKT domain, the last two characterizations given in Theorem 2.30 have the advantage of being more economical, in that they avoid stipulating a priori two sources of “regularity” for the boundary, namely that the unit normal has vanishing mean oscillation, plus a certain degree of Reifenberg ﬂatness. For example, in the setting of (iii) in Theorem 2.30, Reifenberg ﬂatness has been replaced by a twosided local John condition, which is not a ﬂatness/regularity condition per se. In the context of the current work, this is useful inasmuch as it is not clear that the image of a (bounded) εReifenberg ﬂat domain Ω under a C 1 diffeomorphism F is a ε Reifenberg ﬂat domain, where ε := Cε, with the constant C ∈ (0, ∞) depending on F and the geometry of Ω. Various transformational properties of classes of domains that are relevant to us have been established in [51]. Ultimately, they permit the considerations of these types of domains on manifolds (of class C 1 ). Theorem 2.31. BiLipschitz mappings preserve the class of Ahlfors regular domains, the class of domains for which the measure theoretic boundary coincides H n−1 a.e. with the topological boundary, the class of bounded domains satisfying a twosided local John condition, and the class of NTA domains (as well as the class of interior and twosided NTA domains). Moreover, if Ω ⊂ ℝn is a bounded regular SKT domain and F is a C 1 diffeomorphism ̃ := F(Ω) is also a bounded regular SKT domain. of ℝn , then Ω Similar stability results under C 1 diffeomorphisms are valid for the local version of SKT regularity deﬁned below. Deﬁnition 2.32. Let Ω be an open, nonempty, proper subspace of ℝn , and assume that x0 ∈ ∂Ω. Call Ω a regular SKT domain near x0 if the following conditions are fulﬁlled: (i) there exist r0 > 0 and λ > 0 such that B(x0 , r0 ) ∩ Ω is a set of ﬁnite perimeter (with surface measure σ and outward unit normal ν), such that H n−1 (∆ r ) ≈ r n−1 uniformly for any surface ball ∆ r ⊂ B(x0 , λr0 ) ∩ ∂Ω; (ii) there holds lim sup ( r→0+
sup ∆ r ⊆B(x0 ,λr0 )∩∂Ω
− ν(y) − ν ∆ r 2 dσ(y)) (∫
1/2
) = 0;
(2.2.75)
∆r
(iii) all points x ∈ B(x0 , λr0 ) ∩ ∂Ω satisfy a condition analogous to the twosided local John condition from Deﬁnition 2.20.
2.2 Elements of Geometric Measure Theory
 83
This is natural in the sense that a nonempty, bounded open set Ω ⊂ ℝn is a regular SKT domain if and only if Ω is a regular SKT domain near each x0 ∈ ∂Ω. As alluded to earlier, a local version of the last part of Theorem 2.31 holds as well. Speciﬁcally, we have the following result. Theorem 2.33. Let Ω ⊂ ℝn be an open set, O ⊆ ℝn an open neighborhood of Ω, and F : O → ℝn a C 1 diffeomorphism onto its image. If Ω is a regular SKT domain near x0 ∈ ∂Ω, it follows that F(Ω) is a regular SKT domain near F(x0 ). In fact, a similar deﬁnition and stability result under C 1 diffeomorphisms may be given for Euclidean domains which are εSKT near a boundary point x0 . Theorem 2.33 then allows us to deﬁne the class of regular SKT domains in an invariant fashion on manifolds, via local coordinate charts. This being said, it is useful to provide an intrinsic deﬁnition. In this regard, the starting point is the observation that the smallness of the expression in the left handside of (2.2.72) is quantitatively equivalent with the smallness of dist(ν, VMO(∂Ω)), where the distance is taken in BMO(∂Ω); see [50, Corollary 2.24, p. 2615]. Theorem 2.30 and this observation are then the motivating factors for making the following deﬁnition, of central importance to the present work. Deﬁnition 2.34. Let Ω ⊂ M be a nonempty open set. (i) Call Ω a regular SKT domain provided Ω is an Ahlfors regular domain satisfying a twosided local John condition whose outward unit conormal belongs to VMO(∂Ω). (ii) Given ε > 0, call Ω a εSKT domain provided Ω is an Ahlfors regular domain satisfying a twosided local John condition and such that dist(ν, VMO(∂Ω)) < ε, where ν is the outward unit conormal to Ω and the distance is taken in BMO(∂Ω). The deﬁnition given above differs from that proposed in [112], [66], and [50]. The equivalence Deﬁnition 2.34 with Deﬁnition 2.28 is seen from Theorem 2.29 and Theorem 2.30. Moving on, we discuss an approximation scheme which will be of relevance for our future endeavors. Proposition 2.35. Assume Ω ⊂ M is a regular SKT domain. Let σ = H n−1 ⌊∂Ω denote surface measure on ∂Ω, and ν denote the measure theoretic outward unit conormal to Ω. Then there exists a sequence of open sets Ω j ⊂ M, indexed by j ∈ ℕ, satisfying the following properties: (i) In the Hausdorff distance sense, Ω j → Ω and ∂Ω j → ∂Ω as j → ∞, and Ω j ⊆ Ω j+1 ⊆ Ω for each j ∈ ℕ,
and
Ω = ⋃ Ωj . j∈ℕ
(2.2.76)
84  2 Geometric Concepts and Tools
(ii)
Each Ω j ⊂ M is an NTA domain, of ﬁnite perimeter, and such that ∂Ω j is an Ahlfors regular set, with all constants involved independent of j. Moreover, H
n−1
(∂Ω j \ ∂∗ Ω j ) = 0,
∀ j ∈ ℕ.
(2.2.77)
In particular, each Ω j is an Ahlfors regular domain. (iii) For every j ∈ ℕ let σ j := H n−1 ⌊∂Ω j denote the surface measure on ∂Ω j . Then σ j → σ
weak∗
as Radon measures, as j → ∞.
(2.2.78)
(iv) For each j ∈ ℕ there exists a biLipschitz homeomorphism 𝛶j : ∂Ω → ∂Ω j ,
(2.2.79)
with the Lipschitz constants of 𝛶j , 𝛶−1 j bounded independently of j, such that, if Γ(x) denotes the nontangential approach region with vertex at x ∈ ∂Ω relative to Ω, then 𝛶j (x) approaches x in a nontangential fashion as j → ∞, for each x ∈ ∂Ω. That is, for each x ∈ ∂Ω one has 𝛶j (x) ∈ Γ(x) for every j ∈ ℕ (v)
and
lim 𝛶j (x) = x.
j→∞
(2.2.80)
If Γ̃ denotes the nontangential approach regions for the domain Ω, corresponding to a sufficiently large aperture, then ̃ for every j ∈ ℕ Γ j (𝛶j (x)) ⊆ Γ(x)
and
x ∈ ∂Ω,
(2.2.81)
where Γ j is used to denote the nontangential approach regions for the domain Ω j . (vi) If for each j ∈ ℕ one denotes by ν j the geometric measure theoretic unit conormal to Ω j then¹⁶ (ν j ∘ 𝛶j )(x) → ν(x) as j → ∞, (2.2.82) for every x ∈ ∂∗ Ω, hence for σa.e. x ∈ ∂Ω. (vii) For each j ∈ ℕ there exists a σmeasurable, nonnegative function J j on ∂Ω for which the following results hold. There exists some ﬁnite 𝜘 > 1 such that for every j∈ℕ 𝜘−1 ≤ J j (x) ≤ 𝜘 for σa.e. x ∈ ∂Ω, (2.2.83) and J j (x) → 1 as j → ∞,
∀ x ∈ ∂∗ Ω,
(2.2.84)
(hence, the convergence in (2.2.84) holds for σa.e. x ∈ ∂Ω) and such that for each 1 j ∈ ℕ and each f ∈ L1 (∂Ω) one has f ∘ 𝛶−1 j ∈ L (∂Ω j ) and ∫ f J j dσ = ∫ f ∘ 𝛶−1 j dσ j . ∂Ω
16 with composition interpreted as pullback
∂Ω j
(2.2.85)
2.2 Elements of Geometric Measure Theory
 85
In particular, for each σmeasurable set E ⊆ ∂Ω and each j ∈ ℕ, σ j (𝛶j (E)) = ∫ J j dσ.
(2.2.86)
E
(viii) Each 𝛶j extends to a C 1 diffeomorphism ϕ j in an open neighborhood O of ∂Ω with the property that sup ‖Dϕ j ‖L∞ (O) < ∞, (2.2.87) j∈ℕ
and lim Dϕ j (x) = I, for every x ∈ ∂∗ Ω,
j→∞
(2.2.88)
(hence, the convergence in (2.2.88) holds for σa.e. x ∈ ∂Ω). (ix) If u ∈ C 0 (Ω) satisﬁes N u ∈ L p (∂Ω) for some p ∈ (0, ∞) then for each j ∈ ℕ one also has Nj (uΩ j ) ∈ L p (∂Ω j ) where Nj is the nontangential maximal operator associated with the domain Ω j , and Nj (uΩ j )L p (∂Ω j ) ≤ C‖N u‖L p (∂Ω) (x)
for every j ∈ ℕ,
(2.2.89)
for some ﬁnite constant C = C(Ω, p) > 0, independent of u and j. For any given ε > 0, matters may be arranged so that the family {Ω j }j∈ℕ also enjoys the property that, for each j ∈ ℕ, dist (ν j , VMO(∂Ω j )) := inf {‖ν j − η‖BMO(∂Ω j ) : η ∈ VMO(∂Ω j )} < ε.
(2.2.90)
In other words, for any given ε > 0, matters may be arranged so that the approximating family {Ω j }j∈ℕ consists of εSKT domains. Finally, a similar approximation scheme works in the case when Ω is an εSKT domain, assuming that ε > 0 is sufficiently small relative the Ahlfors regularity constants and local John constants of Ω. In such a scenario matters may be arranged so that every Ω j in the approximating family is an ε SKT domain, where ε := Cε with C = C(Ω) ∈ (0, ∞) depending only on the Ahlfors regularity constants and local John constants of Ω. Proof. For Ω a δReifenberg ﬂat domain with δ > 0 sufficiently small, in the Euclidean setting, a similar approximation result has been established in [67, Appendix A.1]. The strategy in [67] is to deﬁne Ω j := ϕ j (Ω)
for each j ∈ ℕ,
(2.2.91)
where ϕ j : ℝn → ℝn ,
j ∈ ℕ,
(2.2.92)
are some suitable C ∞ diffeomorphism. In fact, these smooth mappings are biLipschitz and may be taken to be as close to being isometric as desired, in the sense that
86  2 Geometric Concepts and Tools there exists two purely dimensional constants, C n > 0 and some small α n ∈ (0, 1/C n ), with the property that for any given α ∈ (0, α n ) one can ensure that (1 − C n α)x − y ≤ ϕ j (x) − ϕ j (y) ≤ (1 + C n α)x − y, ∀ x, y ∈ ℝn ,
∀ j ∈ ℕ.
(2.2.93)
See [67, (A.1.17)]. In particular, the Lipschitz constants of ϕ j and ϕ−1 j are bounded independently of j ∈ ℕ. All relevant techniques used in the proof in [67, Appendix A.1] may be adapted to the manifold setting and for the type of domains presently considered bearing in mind that, as proved in [50, § 4.2], any regular SKT domain is a Reifenberg ﬂat domain with vanishing constant, and taking 𝛶j := ϕ j ∂Ω : ∂Ω → ∂Ω j ,
∀ j ∈ ℕ.
(2.2.94)
There are, however, several properties claimed in the present statement which are not proved, or explicitly mentioned, in [67]. These deserve closer scrutiny. The ﬁrst such property is (2.2.77). This, however, may be justiﬁed starting with (2.2.91) and then invoking the transformational properties of Lipschitz maps proved in [51]. The second property alluded to above is (2.2.81). To prove this, ﬁx j ∈ ℕ and ﬁx an arbitrary x ∈ ∂Ω. Also, pick some y ∈ Γ j (ϕ j (x)). Then y ∈ Ω j ⊂ Ω, which further implies dist (y, ∂Ω j ) ≤ dist (y, ∂Ω),
(2.2.95)
and dist (ϕ j (x), y) < (1 + κ) dist (y, ∂Ω).
(2.2.96)
Then the fact that ϕ j (x) ∈ Γ(x) entails dist (ϕ j (x), y) < (1 + κ) dist (y, ∂Ω j ) ≤ (1 + κ) dist (y, ∂Ω),
(2.2.97)
thanks to (2.2.95). Let y∗ ∈ ∂Ω be such that dist (y, ∂Ω) = dist (y, y∗ ) and observe that since ϕ j (x) ∈ Γ(x) we may rely on (2.2.97) in order to estimate dist (x, ϕ j (x)) < (1 + κ) dist (ϕ j (x), ∂Ω) ≤ (1 + κ) dist (ϕ j (x), y∗ ) ≤ (1 + κ)(dist (ϕ j (x), y) + dist (y, y∗ )) ≤ (1 + κ)((1 + κ) dist (y, ∂Ω) + dist (y, ∂Ω)) = (1 + κ)(2 + κ) dist (y, ∂Ω).
(2.2.98)
Combining this with (2.2.97) then permits us to conclude that dist (x, y) < dist (x, ϕ j (x)) + dist (ϕ j (x), y) ≤ (1 + κ)(2 + κ) dist (y, ∂Ω) + (1 + κ) dist (y, ∂Ω) = (1 + κ̃ ) dist (y, ∂Ω),
(2.2.99)
2.2 Elements of Geometric Measure Theory  87
where κ̃ := (1 + κ)(3 + κ) − 1 = κ2 + 4κ + 2 > 0.
(2.2.100)
Hence, if the nontangential approach regions Γ̃ for the domain Ω are deﬁned as in ̃ (2.2.31) with κ replaced with κ̃, it follows from (2.2.100) that y ∈ Γ(x). This establishes (2.2.81). The third property in need of a attention is the claim in item (viii). To prove it, we shall employ notation and results from [66, Appendix]. From [66, (A.1.9), (A.1.10)] we know that for each j ∈ ℕ the function ϕ j may be locally represented as ϕ j (x) = x + αr j ∑ λ ji (x)n⃗ ji
(2.2.101)
i
where {λ ji }i is a suitable family of smooth functions satisfying (cf. [66, (A.1.6)–(A.1.8)]) 0 ≤ λ ji ≤ 1,
∇λ ji  ≤
Cn , rj
∑ ∇λ ji (x) = 0, and i
∑ 1supp λ ji ≤ K n , a ﬁnite dimensional constant.
(2.2.102)
i
These properties then permit us to compute Dϕ j (x) = I + αr j ∑ ∇λ ji (x)n⃗ ji i
= I + αr j ∑ ∇λ ji (x)(n⃗ ji − ν(x)),
(2.2.103)
i
which further implies Dϕ j (x) − I ≤ αr j ∑ ∇λ ji (x) n⃗ ji − ν(x) i
≤ C n K n α sup( sup n⃗ ji − ν(x)). i
(2.2.104)
x∈supp λ ji
Since an inspection of the proof of [66, Lemma A.1.3] reveals that lim sup( sup n⃗ ji − ν(x)) = 0 for every x ∈ ∂∗ Ω,
j→∞
i
(2.2.105)
x∈supp λ ji
the conclusion in (2.2.88) follows from this, (2.2.104), and (2.2.94). Moving on, the fourth property requiring justiﬁcation is the claim in item (ix). In this regard, suppose u ∈ C 0 (Ω) has N u ∈ L p (∂Ω) for some p ∈ (0, ∞), and ﬁx an arbitrary j ∈ ℕ. Then for each x ∈ ∂Ω, Nj (uΩ j )(ϕ j (x)) =
sup y∈Γ j (ϕ j (x))
̃ u)(x). u(y) ≤ sup u(y) =: (N ̃ y∈Γ(x)
(2.2.106)
Keeping this in mind and availing ourselves of the change of variable formula (2.2.85), we may therefore estimate Nj (uΩ j )L p (∂Ω j ) = ( ∫ Nj (uΩ j )p dσ j ) ∂Ω j
1/p
88  2 Geometric Concepts and Tools 1/p
= ( ∫ Nj (uΩ j )(ϕ j (x))p J j (x) dσ(x)) ∂Ω 1/p
̃ u)(x)p dσ(x)) ≤ 𝜘1/p ( ∫ (N =𝜘
∂Ω 1/p ̃
N uL p (∂Ω) ≤ CN uL p (∂Ω) ,
(2.2.107)
where the last inequality is a consequence of [50, Proposition 2.2, p. 2573]. This shows that Nj (uΩ j ) ∈ L p (∂Ω j ) and that (2.2.89) holds for some ﬁnite constant C = C(Ω) > 0, independent of u and j. The ﬁfth property in need of a proof is recorded in item (x). Establishing this requires a few preliminary considerations. First, we shall revisit some relevant arguments from [67, Appendix A.1]. For each x ∈ ∂∗ Ω, let T x ∂Ω denote the tangent plane to ∂Ω at x, and select an orthonormal basis τ1 (x), . . . , τ n−1 (x) in T x ∂Ω. Then for each x ∈ ∂∗ Ω (2.2.108) τ1 (x) ∧ τ2 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x) = 1 and, for a choice of the sign, ν(x) = ± τ1 (x) ∧ τ2 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x).
(2.2.109)
In addition, if Dϕ j is the Jacobian matrix of ϕ j , it follows from [67, (A.1.70)] that for each j ∈ ℕ we have, for some 𝜘 ∈ (1, ∞) independent of j and x, 𝜘−1 ≤ J j (x) = Dϕ j (x)(τ1 (x)) ∧ Dϕ j (x)(τ2 (x)) ∧ ⋅ ⋅ ⋅ ∧ Dϕ j (x)(τ n−1 (x)) ≤ 𝜘,
(2.2.110)
and (cf. [67, (A.1.74)]) ν j (ϕ j (x)) = ±
Dϕ j (x)(τ1 (x)) ∧ ⋅ ⋅ ⋅ ∧ Dϕ j (x)(τ n−1 (x)) , J j (x)
(2.2.111)
with the same choice of the sign as in (2.2.109). The key estimate for us is contained in [67, (A.1.80)] which asserts that for every j ∈ ℕ we have (with α as in (2.2.93)) Dϕ j (x)(τ1 (x)) ∧ ⋅ ⋅ ⋅ ∧ Dϕ j (x)(τ n−1 (x)) − τ1 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x) ≤ C n α
(2.2.112)
at every point x ∈ ∂∗ Ω. Granted (2.2.109) and (2.2.111), we may then write for each j ∈ ℕ and x ∈ ∂∗ Ω, (2.2.113) ν j (ϕ j (x)) − ν(x) ≤ Ij (x) + IIj (x), where Dϕ j (x)(τ1 (x)) ∧ ⋅ ⋅ ⋅ ∧ Dϕ j (x)(τ n−1 (x)) − τ1 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x) Ij (x) := J j (x)
(2.2.114)
2.2 Elements of Geometric Measure Theory  89
and
τ1 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x) − τ1 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x). IIj (x) := J j (x)
(2.2.115)
Note that, thanks to (2.2.110) and (2.2.112) we have Ij (x) ≤ 𝜘 C n α,
∀ x ∈ ∂∗ Ω, ∀ j ∈ ℕ.
(2.2.116)
Also, making use of (2.2.115), (2.2.108), (2.2.110), and (2.2.112), we conclude that 1 − 1 ≤ 𝜘J j (x) − 1 IIj (x) = J j (x) = 𝜘Dϕ j (x)(τ1 (x)) ∧ ⋅ ⋅ ⋅ ∧ Dϕ j (x)(τ n−1 (x)) − τ1 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x) ≤ 𝜘Dϕ j (x)(τ1 (x)) ∧ ⋅ ⋅ ⋅ ∧ Dϕ j (x)(τ n−1 (x)) − τ1 (x) ∧ ⋅ ⋅ ⋅ ∧ τ n−1 (x) (2.2.117) ≤ 𝜘 C n α, ∀ x ∈ ∂∗ Ω, ∀ j ∈ ℕ. By combining (2.2.113), (2.2.116), and (2.2.117) we therefore arrive at the conclusion that ν j (ϕ j (x)) − ν(x) ≤ 2𝜘 C n α,
∀ x ∈ ∂∗ Ω, ∀ j ∈ ℕ.
(2.2.118)
To proceed, ﬁx an arbitrary j ∈ ℕ and, for a given function f ∈ L2 (∂Ω j ), point x ∈ ∂Ω j , and number r > 0, introduce¹⁷ 1/2
‖f‖∗ (∂Ω j ∩ B(x, r)) := sup (− ∫ f − f B 2 dσ j ) B⊆B(x,r)
,
(2.2.119)
B
where the supremum is taken over all surface balls B in ∂Ω j , i.e., sets of the form B(z, ρ) ∩ ∂Ω j with z ∈ ∂Ω j and ρ > 0, that are contained in B(x, r), and where we have set f B := σ(B)−1 ∫B f dσ j . Then from [50, Corollary 2.24, p. 2615] we know that lim [ sup ‖f‖∗ (∂Ω j ∩ B(x, r))] ≈ dist (f, VMO(∂Ω j )),
r→0+ x∈∂Ω j
(2.2.120)
uniformly for f ∈ BMO(∂Ω j ) and j ∈ ℕ. Next, having ﬁxed a point x ∈ ∂Ω j and a scale r > 0, the goal is to estimate the local BMO norm ‖ν j ‖∗ (∂Ω j ∩ B(x, r)). To this end, pick a surface ball B = B(z, ρ) ∩ ∂Ω j ⊂ B(x, r), of center z ∈ ∂Ω j and radius ρ > 0 (which is necessarily ≤ r). Then choosing ̃ := B(ϕ−1 (z), ρ̃ ) ∩ ∂Ω ensures that ρ̃ := ρ/(1 − C n α) ∈ (0, ∞) and abbreviating B j ̃ ϕ−1 j (B) ⊆ B
and
̃ uniformly in j, z, ρ. σ j (B) ≈ σ(B),
17 as usual, a barred integral indicates the average
(2.2.121)
90  2 Geometric Concepts and Tools Furthermore, introducing R := 2r/(1 − C n α) implies that ̃ ⊆ B(ϕ−1 (x), R). B j
(2.2.122)
For some C ∈ (0, ∞) which depends only on the large geometrical characteristics of Ω, we may then estimate with c an arbitrary constant, 2 (− ∫ ν j − (ν j )B dσ j )
1/2
2 = (− ∫ (ν j − c) − (ν j − c)B dσ j )
B
1/2
B
2 ≤ 2(− ∫ ν j − c dσ j )
1/2
1/2
− ν j ∘ ϕ j − c2 J j dσ) ≤ C( ∫ ϕ−1 j (B)
B
1/2
− ν j ∘ ϕ j − ν2 J j dσ) ≤ C( ∫
1/2
− ν − c2 J j dσ) + C( ∫
ϕ−1 j (B)
ϕ−1 j (B) 1/2
≤ 2C C n 𝜘3/2 α + C𝜘1/2 (− ∫ ν − c2 dσ)
,
(2.2.123)
̃ B
where we have used (iv) in the statement of the proposition, along with (2.2.118). ̃ −1 ∫̃ ν dσ then permits us to write, on account of (2.2.122) Choosing c := ν B̃ := σ(B) B and (2.2.119), 2 (− ∫ ν j − (ν j )B dσ j )
1/2
2 ≤ 2C C n 𝜘3/2 α + C𝜘1/2 (− ∫ ν − ν B̃ dσ)
1/2
̃ B
B
≤ 2C C n 𝜘
3/2
α + C𝜘
1/2
‖ν‖∗ (∂Ω ∩ B(ϕ−1 j (x), R)),
(2.2.124)
where the last quantity is deﬁned analogously to (2.2.119), relative to Ω. By taking the supremum over all surface balls B on ∂Ω j contained in B(x, r) we therefore arrive at ‖ν j ‖∗ (∂Ω j ∩ B(x, r)) ≤ 2C C n 𝜘3/2 α + C𝜘1/2 ‖ν‖∗ (∂Ω ∩ B(ϕ−1 j (x), R)).
(2.2.125)
Upon recalling that R depends linearly on r, in light of (2.2.120) this implies dist (ν j , VMO(∂Ω j )) ≤ C lim+ ‖ν j ‖∗ (∂Ω j ∩ B(x, r)) r→0
≤ 2C C n 𝜘3/2 α + C𝜘1/2 lim+ ‖ν‖∗ (∂Ω ∩ B(ϕ−1 j (x), R)) R→0
≤ 2C C n 𝜘3/2 α + C𝜘1/2 dist (ν, VMO(∂Ω)) ≤ 2C C n 𝜘3/2 α
(2.2.126)
since, by design, ν belongs to VMO(∂Ω). Hence, given any ε > 0, selecting 0 < α < min {α n , ε/(2C C n 𝜘3/2 )} yields (2.2.90).
(2.2.127)
2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 
91
To complete the proof of the proposition there remains to address the very last claim in its statement, pertaining to the case when Ω is an εSKT domain when the parameter ε > 0 is sufficiently small relative the Ahlfors regularity constants and local John constants of Ω. Assuming that this is the case, and reasoning as in (2.2.126), presently yields dist (ν j , VMO(∂Ω j )) ≤ 2C C n 𝜘3/2 α + C𝜘1/2 dist (ν, VMO(∂Ω)) ≤ 2C C n 𝜘3/2 α + C𝜘1/2 ε.
(2.2.128)
Choosing α to be of the same magnitude as ε then permits us to conclude that dist (ν j , VMO(∂Ω j )) ≤ ε ,
∀ j ∈ ℕ,
(2.2.129)
where ε := Cε, with C = C(Ω) ∈ (0, ∞) depending only on the Ahlfors regularity constants and local John constants of Ω. Hence, every Ω j in the approximating family is an ε SKT domain. We agree to write Ω j ↗ Ω as j → ∞ in order to indicate that the sequence {Ω j }j∈ℕ approximates Ω in the manner described in Proposition 2.35.
2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains Here we discuss some basic integration by parts formulas in a very general setting, which are particularly wellsuited for the goals we have in mind. Theorem 2.36. Let Ω ⊂ M be an Ahlfors regular domain, and deﬁne σ := H n−1 ⌊∂Ω. In particular, Ω is a set of ﬁnite perimeter whose measure theoretic outward unit conormal ν is deﬁned σa.e. on ∂Ω. In this setting, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} and suppose the integrability exponents 1 ≤ p, q, p , q ≤ ∞ are chosen such that q ≤ p and 1/p + 1/p = 1 = 1/q + 1/q . Then the integration by parts formula n.t. n.t. ∫⟨du, υ⟩ dVol − ∫⟨u, δυ⟩ dVol = ∫ ⟨ν ∧ u∂Ω , υ∂Ω ⟩ dσ Ω
Ω
∂Ω
n.t. n.t. = ∫ ⟨u∂Ω , ν ∨ υ∂Ω ⟩ dσ
(2.3.1)
∂Ω
holds for any differential forms u, υ satisfying p
du ∈ Lloc (Ω, Λ l+1 T M),
q
δυ ∈ Lloc (Ω, Λ l T M),
u ∈ Lloc (Ω, Λ l T M), υ ∈ Lloc (Ω, Λ l+1 T M),
q
p
(2.3.2)
92  2 Geometric Concepts and Tools
as well as
n.t. n.t. the nontangential traces u∂Ω , υ∂Ω exist σa.e. on ∂Ω, N u ⋅ N υ ∈ L1 (∂Ω), and du υ + u δυ ∈ L 1 (Ω).
(2.3.3)
The integration by parts formula for the exterior derivative operator stated above is sharp in the sense that the accompanying conditions in (2.3.3) are minimal to ensure that the terms making up (2.3.1) are meaningful to begin with. The proof of Theorem 2.36 crucially relies on a sharp version of the Divergence Theorem, proved in [81] and recalled here as Theorem 9.68 in § 9.12. Proof of Theorem 2.36. Consider the vector ﬁeld F⃗ : Ω → T M
(2.3.4)
deﬁned uniquely by the requirement that, for every x ∈ Ω and every ξ ∈ T x∗ M, ⃗
T x∗ M (ξ, F(x))T x M
= −i⟨Sym (d, ξ)u(x), υ(x)⟩x .
(2.3.5)
That F⃗ is welldeﬁned is ensured by the fact that the righthand side of (2.3.5) is linear in ξ . In fact, we locally have F⃗ = F j ∂ j with F j (x) = −i⟨Sym (d, dx j )u(x), υ(x)⟩x = ⟨dx j ∧ u(x), υ(x)⟩x .
(2.3.6)
Note that since q ≤ p, the assumptions on u, υ imply that u υ ∈ L1loc (Ω). In turn, from this and (2.3.5) it is then clear that F⃗ ∈ L1loc (Ω, T M). Next, for every scalar function ψ ∈ C 01 (Ω) we may compute (with T x M ⟨⋅, ⋅⟩T x M denoting the pointwise inner product in T x M, for each x ∈ M, and with D (Ω) (⋅, ⋅)D(Ω) denoting the distributional pairing in Ω) ⃗
D (Ω) (div F, ψ)D(Ω)
= −D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T x M ⟨(grad ψ)(x), F(x)⟩ T x M dVol(x) Ω
⃗ = − ∫ T x∗ M ((grad ψ)♭ (x), F(x)) T x M dVol(x) Ω
⃗ = − ∫ T x∗ M (dψ(x), F(x)) T x M dVol(x) Ω
= ∫⟨i Sym (d, dψ(x))u(x), υ(x)⟩x dVol(x).
(2.3.7)
Ω
Above, the third equality uses the fact that (cf. (9.1.3)) TM ⟨X, Y⟩TM
= T ∗ M (X ♭ , Y)TM
∀ X, Y ∈ T M,
(2.3.8)
employed here with X := grad ψ and Y := F,⃗ while the fourth equality utilizes (9.1.4). Also, the last equality uses (2.3.5) with ξ := dψ(x) ∈ T x∗ M. Next, by items (3) and (4) in
2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 
93
Lemma 2.8 we have d(ψu)(x) = dψ(x) ∧ u(x) + ψ(x)(du)(x) = −i Sym (d, dψ(x))u(x) + ψ(x)(du)(x),
(2.3.9)
hence i Sym (d, dψ(x))u(x) = ψ(x)(du)(x) − d(ψu)(x). Utilizing this back in (2.3.7) and making use of item (6) in Lemma 2.8 then yields ⃗
D (Ω) (div F, ψ)D(Ω)
= ∫⟨ψ(x)(du)(x), υ(x)⟩x dVol(x) Ω
− ∫⟨d(ψu)(x), υ(x)⟩x dVol(x) Ω
= ∫⟨ψ(x)(du)(x), υ(x)⟩x dVol(x) Ω
− ∫⟨ψ(x)u(x), (δυ)(x)⟩x dVol(x) Ω
= ∫{⟨(du)(x), υ(x)⟩x − ⟨u(x), (δυ)(x)⟩x }ψ(x) dVol(x). (2.3.10) Ω
This goes to show that div F⃗ = ⟨du, υ⟩ − ⟨u, δυ⟩ in D (Ω).
(2.3.11)
In particular, div F⃗ ∈ L1 (Ω) by the last condition in (2.3.3). In addition, from (2.3.5) and n.t. the ﬁrst condition in (2.3.3) we see that the nontangential trace F⃗ ∂Ω exists σa.e. on ∂Ω and ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
= ⟨−i Sym (d, ν)u(x), υ(x)⟩x = ⟨ν(x) ∧ u(x), υ(x)⟩x
for σa.e. x ∈ ∂Ω.
(2.3.12)
Finally, the fact that N(F)⃗ ≤ CN u ⋅ N υ pointwise on ∂Ω (as seen from (2.3.5)) ultimately proves that N(F)⃗ ∈ L1 (∂Ω). At this stage, the ﬁrst equality in the integration by parts formula (2.3.1) follows readily from (9.12.13), (2.3.11), and (2.3.12). The last equality in (2.3.1) is a direct consequence of item (8) in Lemma 2.2. This concludes the proof of Theorem 2.36. The energy identity established in the proposition below plays a major role in our subsequent work. To state it, recall that ∆HL := −(dδ + δd) denotes the HodgeLaplacian on the Riemannian manifold M. Proposition 2.37. Assume that Ω ⊂ M is an Ahlfors regular domain. As usual, denote by ν the outward unit conormal to Ω and let σ := H n−1 ⌊∂Ω be the surface measure on ∂Ω where, as usual, n = dim M ≥ 2. Pick an arbitrary degree l ∈ {0, 1, . . . , n} along with an integrability exponent p satisfying 2(n − 1)/n < p < ∞.
(2.3.13)
94  2 Geometric Concepts and Tools
Also, ﬁx a realvalued function V ∈ L r (Ω)
where r := np/(np − 2n + 2) ∈ (1, ∞).
(2.3.14)
In this context, suppose the differential form u satisﬁes (with p denoting the Hölder conjugate exponent of p): l (hence u ∈ L∞ loc (Ω, Λ T M) by (2.2.45)),
N u ∈ L p (∂Ω)
2,r (∆HL − V)u = 0 in Ω (hence u ∈ Hloc (Ω, Λ l T M) by (2.1.118)), n.t. n.t. n.t. there exist u∂Ω , (du)∂Ω , (δu)∂Ω at σa.e. point on ∂Ω, and N(du), N(δu) ∈ L q (∂Ω) with q := max {p , 2(n − 1)/n}.
(2.3.15)
Then the following energy identity holds¹⁸: n.t. n.t. ∫ (du2 + δu2 + Vu2 ) dVol = − ∫ ⟨ν ∨ u∂Ω , ν ∨ (ν ∧ (δu)∂Ω )⟩ dσ Ω
∂Ω
n.t. n.t. + ∫ ⟨ν ∧ u∂Ω , ν ∧ (ν ∨ (du)∂Ω )⟩ dσ.
(2.3.16)
∂Ω
Proof. We begin by observing that, thanks to (2.2.45), (2.1.118), and the hypotheses on u, we have 2,r l l u ∈ L∞ loc (Ω, Λ T M) ∩ H loc (Ω, Λ T M)
du ∈
l+1 L∞ T M), loc (Ω, Λ
δu ∈
and
l−1 L∞ T M). loc (Ω, Λ
(2.3.17)
Moreover, from (2.2.50) and the nontangential maximal function properties of u, we have u ∈ L np/(n−1) (Ω, Λ l T M), du ∈ L nq/(n−1) (Ω, Λ l+1 T M) ⊂ L2 (Ω, Λ l+1 T M), δu ∈ L
nq/(n−1)
(Ω, Λ
l−1
T M) ⊂ L (Ω, Λ 2
l−1
(2.3.18)
T M).
Consider next the vector ﬁeld F⃗ : Ω → T M deﬁned uniquely by the demand that for all x ∈ Ω and all ξ ∈ T x∗ M we have ⃗
T x∗ M (ξ, F(x))T x M
= −⟨ξ ∨ u(x), δu(x)⟩x + ⟨ξ ∧ u(x), du(x)⟩x .
(2.3.19)
The linearity in ξ of the righthand side of (2.3.19) ensures that F⃗ is welldeﬁned. It is also apparent from (2.3.19) and (2.3.17) that 1 F⃗ ∈ L∞ loc (Ω, T M) ⊂ L loc (Ω, T M).
18 all integrals involved being absolutely convergent
(2.3.20)
2.3 Sharp Integration by Parts Formulas for Differential Forms in Ahlfors Regular Domains 
95
In fact, we locally have F⃗ = F j ∂ j with F j (x) = −⟨dx j ∨ u(x), δu(x)⟩x + ⟨dx j ∧ u(x), du(x)⟩x .
(2.3.21)
Given any scalar function ψ ∈ C 01 (Ω), we may then compute ⃗
D (Ω) (div F, ψ)D(Ω)
= −D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T x M ⟨(grad ψ)(x), F(x)⟩ T x M dVol(x) Ω
⃗ = − ∫ T x∗ M ((grad ψ)♭ (x), F(x)) T x M dVol(x) Ω
⃗ = − ∫ T x∗ M (dψ(x), F(x)) T x M dVol(x) Ω
= ∫{⟨dψ ∨ u, δu ⟩ − ⟨dψ ∧ u, du ⟩} dVol,
(2.3.22)
Ω
where the last equality above is based on (2.3.19) used here with ξ := dψ(x) ∈ T x∗ M. Next, item (8) in Lemma 2.2 and item (4) in Lemma 2.8 permit us to compute (in a pointwise sense a.e. in Ω) ⟨dψ ∨ u, δu ⟩ − ⟨dψ ∧ u, du ⟩ = ⟨u, dψ ∧ δu ⟩ − ⟨d(ψu), du ⟩ + ψ⟨du, du ⟩ = ⟨u, d(ψδu)⟩ − ψ⟨u, dδu ⟩ − ⟨d(ψu), du ⟩ + ψdu2 ,
(2.3.23)
while item (6) in Lemma 2.8 (bearing in mind (2.3.17) and the fact that ψ has compact support in Ω) gives ∫⟨u, d(ψδu)⟩ dVol = ∫⟨δu, ψδu ⟩ dVol = ∫ ψδu2 dVol, Ω
Ω
(2.3.24)
Ω
and ∫⟨d(ψu), du ⟩ dVol = ∫⟨ψu, δdu ⟩ dVol = ∫ ψ⟨u, δdu ⟩ dVol. Ω
Ω
(2.3.25)
Ω
Altogether, from (2.3.22)–(2.3.25) and the fact that −dδu − δdu = ∆HL u = V u in Ω, we conclude that div F⃗ = du2 + δu2 + Vu2 in D (Ω). (2.3.26) In particular, div F⃗ ∈ L1 (Ω) by (2.3.18) and (2.3.14). Moreover, (2.3.19) and the third line n.t. in (2.3.15) imply that the nontangential trace F⃗ ∂Ω exists σa.e. on ∂Ω and ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
n.t. n.t. n.t. n.t. = − ⟨ν ∨ u∂Ω , (δu)∂Ω ⟩ + ⟨ν ∧ u∂Ω , (du)∂Ω ⟩ n.t. n.t. = − ⟨ν ∨ u∂Ω , ν ∨ (ν ∧ (δu)∂Ω )⟩ n.t. n.t. + ⟨ν ∧ u∂Ω , ν ∧ (ν ∨ (du)∂Ω )⟩,
(2.3.27)
96  2 Geometric Concepts and Tools
where the last equality is based on items (10), (8), and (4) in Lemma 2.2 (all used here with α := ν). From (2.3.19) we also see that N(F)⃗ ≤ C N u ⋅ N(du) + C N u ⋅ N(δu) pointwise on ∂Ω. Based on the conditions imposed in the ﬁrst line and the last line of (2.3.15) we may therefore conclude that N(F)⃗ ∈ L1 (∂Ω). Granted this, the energy formula (2.3.16) follows from (2.3.26) and (2.3.27), with the help the divergence formula recorded in Theorem 9.68. It is instructive to point out that a proof of (2.3.16) based on repeated integrations by parts (as in Theorem 2.36) would place more demanding conditions on u than those listed in (2.3.15).
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets Here we deﬁne various spaces of tangential and normal differential forms, which play a basic role in the present work. Deﬁnition 2.38. Let Ω ⊂ M be an open set of ﬁnite perimeter satisfying (2.2.5). Denote by ν the outward unit conormal to Ω and let σ := H n−1 ⌊∂Ω be the surface measure on ∂Ω. Let l ∈ {0, 1, . . . , n} and consider f a Λ l T Mvalued differential form on ∂Ω. Then call f tangential (to ∂Ω) if ν ∨ f = 0
at σa.e. point on ∂Ω,
(2.4.1)
f normal (to ∂Ω) if ν ∧ f = 0
at σa.e. point on ∂Ω.
(2.4.2)
and
Note that if l ∈ {0, 1, . . . , n} and f is a Λ l T Mvalued differential form on ∂Ω, then based on item (10) in Lemma 2.2 we may write f = ν ∧ (ν ∨ f) + ν ∨ (ν ∧ f).
(2.4.3)
Hence any Λ l T Mvalued differential form f on ∂Ω may be decomposed as f = ftan + fnor
(2.4.4)
where ftan := ν ∨ (ν ∧ f)
and
fnor := ν ∧ (ν ∨ f).
(2.4.5)
A direct consequence of (2.4.5) and (2.4.3) (as well as Lemma 2.2) is that ν ∨ ftan = 0,
ν ∧ fnor = 0,
ν ∧ (ν ∨ fnor ) = fnor , ν ∨ f = ν ∨ fnor ,
and
ν ∨ (ν ∧ ftan ) = ftan , ν ∧ f = ν ∧ ftan , ⟨ftan , fnor ⟩ = 0,
for each Λ l T Mvalued differential form f on ∂Ω.
(2.4.6)
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets 
97
We propose to study in greater detail the algebraic implications of the property of being tangential for the coefficients of a differential form. Speciﬁcally, we have the following result. Lemma 2.39. Suppose Ω ⊂ M is an open set of ﬁnite perimeter satisfying (2.2.5). Let ν and σ := H n−1 ⌊∂Ω be, respectively, the outward unit conormal to Ω and the surface measure on ∂Ω. Fix l ∈ {0, 1, . . . , n} and consider some Λ l T Mvalued differential form f on ∂Ω. Then for any local coordinate chart U of M one has
if f is tangential and f(x) = ∑ f I (x)dx I for x ∈ U ∩ ∂Ω I=l
n
then
jL ∑ ∑ ε I g ij (x)ν i (x)f I (x) i,j=1 I=l
=0
σa.e. x ∈ U ∩ ∂Ω
(2.4.7)
for each ordered array L of length l − 1. Proof. Suppose that in some local coordinate chart U the differential form f is written as f(x) = ∑I=l f I (x)dx I for x ∈ U ∩ ∂Ω, where each f I (x) is a scalar. If in U ∩ ∂Ω we now write ν(x) = ∑ni=1 ν i (x)dx i ∈ T x∗ M for the outward unit conormal to Ω, then for σa.e. x ∈ U ∩ ∂Ω we have n
0 = ν(x) ∨ f(x) = ( ∑ ν i (x)dx i ) ∨ ( ∑ f I (x)dx I ) I=l
i=1
n
= ∑ ∑ ν i (x)f I (x)dx i ∨ dx
I
i=1 I=l
n
jL
= ∑ ( ∑ ∑ ε I g ij (x)ν i (x)f I (x))dx L ,
(2.4.8)
L=l−1 i,j=1 I=l
where the last equality uses (2.1.37). The bottom line is that (2.4.7) holds, ﬁnishing the proof of the lemma. Assume Ω ⊂ M is an open set of ﬁnite perimeter satisfying property (2.2.5). For each degree l ∈ {0, 1, . . . , n} and each exponent p ∈ (0, ∞] introduce the following closed subspaces of L p (∂Ω, Λ l T M): p
(2.4.9)
p
(2.4.10)
Ltan (∂Ω, Λ l T M) := { f ∈ L p (∂Ω, Λ l T M) : f is tangential (to ∂Ω)}, and Lnor (∂Ω, Λ l T M) := { f ∈ L p (∂Ω, Λ l T M) : f is normal (to ∂Ω)},
where tangentiality and normality are as in Deﬁnition 2.38. In particular, for each p p exponent p ∈ (1, ∞) both Ltan (∂Ω, Λ l T M) and Lnor (∂Ω, Λ l T M) are reﬂexive Banach spaces. Also, p
Ltan (∂Ω, Λ l T M) is a closed subspace of L p (∂Ω, Λ l T M), p
Lnor (∂Ω, Λ l T M) is a closed subspace of L p (∂Ω, Λ l T M),
(2.4.11)
98  2 Geometric Concepts and Tools
and it is clear from deﬁnitions that p
p
Ltan (∂Ω, Λ n T M) = {0} = Lnor (∂Ω, Λ0 T M), p Lnor (∂Ω, Λ1 T M) p Ltan (∂Ω, Λ n−1 T M) p Ltan (∂Ω, Λ0 T M) p Lnor (∂Ω, Λ n T M)
(2.4.12)
= L (∂Ω) ν,
(2.4.13)
= L (∂Ω)ν ∨ dVol,
(2.4.14)
= L (∂Ω),
(2.4.15)
= L (∂Ω) dVol.
(2.4.16)
p p p p
In the second part of this section we introduce and study a brand of “partial” p,δ p,d Sobolev spaces, denoted by Ltan (∂Ω, Λ l T M) and Lnor (∂Ω, Λ l T M), deﬁned on the boundary of an open set Ω ⊂ M of ﬁnite perimeter satisfying (2.2.5). These function spaces will play a fundamental role in this work. Compared with the isotropic Sobolev p spaces L1 (∂Ω, Λ l T M) discussed in detail in § 9.5, they only involve certain specialized packages of derivatives, which are welladapted to the de RhamHodge formalism involving the operators d and δ. The fact that in the present geometric setting the set ∂Ω is utterly lacking any type of manifold structure makes the task of properly assigning a meaning to the aforementioned preferential combination of derivatives delicate. To cope with this very rough setting we adopt the point of view described in the deﬁnition below. Deﬁnition 2.40. Let Ω ⊂ M be an open set of ﬁnite perimeter satisfying (2.2.5), and ﬁx a degree l ∈ {0, 1, . . . , n} along with an integrability exponent p ∈ [1, ∞]. A differential p form f ∈ Ltan (∂Ω, Λ l T M) is said to have its boundary coderivative in L p if there exists a differential form in L p (∂Ω, Λ l−1 T M), denoted by δ ∂ f , with the property that ∫ ⟨dϕ, f⟩ dσ = ∫ ⟨ϕ, δ ∂ f⟩ dσ, ∂Ω
∀ ϕ ∈ C 1 (M, Λ l−1 T M).
(2.4.17)
∂Ω
A few comments are in order here. First, the reason for which it is natural to assume that f is tangential to begin with in the above deﬁnition becomes clearer from Proposition 3.3. Second, it should be observed that (2.4.17) deﬁnes δ ∂ f unambiguously since, thanks to Lemma 2.11, pairing with {ϕ∂Ω : ϕ ∈ C 1 (M, Λ l−1 T M)} separates elements in L p (∂Ω, Λ l−1 T M).
(2.4.18)
Third, condition (2.4.17) is clearly equivalent to the demand that ∫ ⟨dϕ, f⟩ dσ = ∫ ⟨ϕ, δ ∂ f⟩ dσ ∂Ω
whenever
∂Ω
ϕ ∈ C (O , Λ 1
l−1
T M) with O neighborhood of ∂Ω.
(2.4.19)
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets 
99
In fact, given a ﬁnite dimensional vector space V, condition (2.4.17) is equivalent to having ∫ ⟨dϕ, f⟩ dσ = ∫ ⟨ϕ, δ ∂ f⟩ dσ as vectors in V, ∂Ω
∂Ω l−1
ϕ ∈ C (O , V ⊗ Λ 1
whenever (2.4.20)
T M) with O ⊆ M neighborhood of ∂Ω.
In the setting of Deﬁnition 2.40 we then set p,δ
p
p
Ltan (∂Ω, Λ l T M) := {f ∈ Ltan (∂Ω, Λ l T M) : δ ∂ f ∈ Ltan (∂Ω, Λ l−1 T M)}
(2.4.21)
and equip this space with the natural graphnorm, i.e., ‖f‖L p,δ (∂Ω,Λ l TM) := ‖f‖L p (∂Ω,Λ l TM) + ‖δ ∂ f‖L p (∂Ω,Λ l−1 TM) .
(2.4.22)
tan
Again, that the insistence on having both f and δ ∂ f tangential in the deﬁnition of the p,δ space Ltan (∂Ω, Λ l T M) in (2.4.21) is natural may be seen from Proposition 3.3. Given that the assignment p,δ
ȷ : Ltan (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) ⊕ L p (∂Ω, Λ l−1 T M) ȷ(f) := (f, δ ∂ f),
p,δ
∀ f ∈ Ltan (∂Ω, Λ l T M),
(2.4.23)
is a linear isometry, its image is a closed subspace of L p (∂Ω, Λ l T M) ⊕ L p (∂Ω, Λ l−1 T M). p,δ In particular, Ltan (∂Ω, Λ l T M) is isomorphic with a closed subspace of a Banach space which is reﬂexive if p ∈ (1, ∞). From this it follows that p,δ
Ltan (∂Ω, Λ l T M) is a reﬂexive Banach space for each p ∈ (1, ∞).
(2.4.24)
Moreover, a semistandard argument, taking into account the isometry (2.4.23), then invoking the HahnBanach Theorem, and Riesz’s Representation Theorem (for duals of Lebesgue spaces) shows that, whenever p, p ∈ (1, ∞) satisfy 1/p + 1/p = 1, we have: ∗
p,δ
for every functional Λ ∈ (Ltan (∂Ω, Λ l T M)) there exist forms p
g ∈ Ltan (∂Ω, Λ l T M)
and
p
h ∈ Ltan (∂Ω, Λ l−1 T M) such that
Λ(f) = ∫⟨f, g⟩ dσ + ∫⟨δ ∂ f, h⟩ dσ, ∂Ω
(2.4.25)
p,δ
∀ f ∈ Ltan (∂Ω, Λ l T M).
∂Ω
Continuing our discussion, we remark that, by design, ν ∨ δ ∂ f = 0 and δ ∂ (δ ∂ f) = 0 on ∂Ω
p,δ
for any f ∈ Ltan (∂Ω, Λ l T M)
(2.4.26)
where, as usual, ν stands for the outward unit conormal to ∂Ω. In particular, introducing p,0 p,δ Ltan (∂Ω, Λ l T M) := {f ∈ Ltan (∂Ω, Λ l T M) : δ ∂ f = 0}, (2.4.27)
100  2 Geometric Concepts and Tools
it follows that
p,0
Ltan (∂Ω, Λ l T M) is a closed subspace both p
(2.4.28)
p,δ
of Ltan (∂Ω, Λ l T M) and of Ltan (∂Ω, Λ l T M), and the operator p,δ
p,0
δ ∂ : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l−1 T M)
(2.4.29)
is welldeﬁned, linear, and bounded. In particular, p,δ
p,0
δ ∂ [Ltan (∂Ω, Λ l T M)] ⊆ Ltan (∂Ω, Λ l−1 T M).
(2.4.30)
It is also clear from deﬁnitions that p,0
p,δ
Ltan (∂Ω, Λ0 T M) = Ltan (∂Ω, Λ0 T M) = L p (∂Ω),
(2.4.31)
p,0 Ltan (∂Ω, Λ n T M)
(2.4.32)
=
p,δ Ltan (∂Ω, Λ n T M)
= {0}.
Though the manner in which δ ∂ has been introduced in Deﬁnition 2.40 does not make its action very transparent, this should be thought as a ﬁrstorder differential operator on ∂Ω. Indeed, one may check without difficulty that the operator δ ∂ is local, in the sense that p,δ supp (δ ∂ f) ⊆ supp f, ∀ f ∈ Ltan (∂Ω, Λ l T M), (2.4.33) and satisﬁes Leibniz’s product rule δ ∂ (ψf) = ψδ ∂ f − dψ ∨ f for every
(2.4.34)
p,δ
f ∈ Ltan (∂Ω, Λ l T M) and ψ ∈ C 1 (M). p
In a parallel fashion to Deﬁnition 2.40, we shall say that a form f ∈ Lnor (∂Ω, Λ l T M) has its boundary derivative in L p if there exists an (l + 1)form in L p (∂Ω, Λ l+1 T M), denoted by d ∂ f , with the property that¹⁹ ∫ ⟨δϕ, f⟩ dσ = ∫ ⟨ϕ, d ∂ f⟩ dσ, ∂Ω
∀ ϕ ∈ C 1 (M, Λ l+1 T M).
(2.4.35)
∂Ω
With this piece of notation we then deﬁne p,d
p
p
Lnor (∂Ω, Λ l T M) := {f ∈ Lnor (∂Ω, Λ l T M) : d ∂ f ∈ Lnor (∂Ω, Λ l+1 T M)} ,
(2.4.36)
and equip this space with the natural graphnorm ‖f‖L p,d (∂Ω,Λ l TM) := ‖f‖L p (∂Ω,Λ l TM) + ‖d ∂ f‖L p (∂Ω,Λ l+1 TM) . nor
19 much as before, this condition deﬁnes d ∂ f unambiguously
(2.4.37)
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets 
101
p,d
Much as before, Lnor (∂Ω, Λ l T M) is a reﬂexive Banach space for each p ∈ (1, ∞) and, analogously to (2.4.26), we have ν ∧ d ∂ f = 0 and d ∂ (d ∂ f) = 0 on ∂Ω
(2.4.38)
p,d
for each f ∈ Lnor (∂Ω, Λ l T M). In particular, if we set p,0
p,d
Lnor (∂Ω, Λ l T M) := {f ∈ Lnor (∂Ω, Λ l T M) : d ∂ f = 0}, then
(2.4.39)
p,0
Lnor (∂Ω, Λ l T M) is a closed subspace both p
(2.4.40)
p,d
of Lnor (∂Ω, Λ l T M) and of Lnor (∂Ω, Λ l T M), the operator p,d
p,0
d ∂ : Lnor (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l+1 T M)
(2.4.41)
is welldeﬁned, linear, and bounded. In particular, p,d
p,0
d ∂ [Lnor (∂Ω, Λ l T M)] ⊆ Lnor (∂Ω, Λ l+1 T M).
(2.4.42)
Also, it is clear that p,0
p,d
(2.4.43)
p,0
p,d
(2.4.44)
Lnor (∂Ω, Λ0 T M) = Lnor (∂Ω, Λ0 T M) = {0}, Lnor (∂Ω, Λ n T M) = Lnor (∂Ω, Λ n T M) = L p (∂Ω) dVol.
Once again, d ∂ should be thought as a ﬁrstorder differential operator on ∂Ω. Indeed, d ∂ is local, in the sense that p,d
supp(d ∂ f) ⊆ supp f,
∀ f ∈ Lnor (∂Ω, Λ l T M),
(2.4.45)
and satisﬁes Leibniz’s product rule d ∂ (ψf) = ψd ∂ f + dψ ∧ f for every
(2.4.46)
p,d
f ∈ Lnor (∂Ω, Λ l T M) and ψ ∈ C 1 (M).
From Lemma 2.2 and deﬁnitions it follows that, for each degree l ∈ {0, 1, . . . , n} and exponent p ∈ [1, ∞], p
p
∗Ltan (∂Ω, Λ l T M) = Lnor (∂Ω, Λ n−l T M),
(2.4.47)
p,δ ∗Ltan (∂Ω, Λ l T M) p,0 ∗Ltan (∂Ω, Λ l T M)
=
p,d Lnor (∂Ω, Λ n−l T M),
(2.4.48)
=
p,0 Lnor (∂Ω, Λ n−l T M).
(2.4.49)
p,δ
p,d
More speciﬁcally, it turns out that if f ∈ Ltan (∂Ω, Λ l T M) then ∗f ∈ Lnor (∂Ω, Λ n−l T M) and d ∂ (∗f) = (−1)l ∗ (δ ∂ f), (2.4.50)
102  2 Geometric Concepts and Tools p,d
p,δ
whereas if g ∈ Lnor (∂Ω, Λ l T M) then ∗g ∈ Ltan (∂Ω, Λ n−l T M) and δ ∂ (∗g) = (−1)l+1 ∗ (d ∂ g). p,δ
(2.4.51)
p,d
The scales Ltan (∂Ω, Λ l T M), Lnor (∂Ω, Λ l T M) can be thought of as suitable versions of Sobolev spaces of order one on ∂Ω which are welladapted to the operators d and δ. This is more than a formal analogy as there are concrete ways of tying them up with p the scalar Sobolev spaces L1 (∂Ω), 1 < p < ∞, introduced in [50]. Speciﬁcally, from Proposition 9.41 and Proposition 9.42 it follows that for each p ∈ (1, ∞) we have p,d
(2.4.52)
p,d Lnor (∂Ω, Λ1 T M)
=
(2.4.53)
p,δ Ltan (∂Ω, Λ n−1 T M) p,δ Ltan (∂Ω, Λ n−1 T M)
=
ν∨
ν∧
p
Lnor (∂Ω, Λ1 T M) = L1 (∂Ω) ν,
=
p L1 (∂Ω), p L1 (∂Ω) ν ∨ dVol, p L1 (∂Ω) dVol.
(2.4.54) (2.4.55)
Moreover, at least if M has a trivial topology²⁰ and Ω is a UR domain, (2.4.52), (9.6.138), and Proposition 9.44, permits us to express p,0
Lnor (∂Ω, Λ1 T M) = {(∑ λ j 1Σ j )ν : λ j ∈ ℝ, Σ j connected component of ∂Ω}. (2.4.56) j
Hence, in such a scenario, we have p,0
p,0
dim Ltan (∂Ω, Λ n−1 T M) = dim Lnor (∂Ω, Λ1 T M) = number of connected components of ∂Ω,
(2.4.57)
reinforcing the idea that the scales of spaces discussed in this section carry signiﬁcant topological information concerning the underlying domain. For more on this topic, see the discussion in § 8.1. Moving on, we discuss a very useful result pertaining to the action of the boundary derivative operators δ ∂ and d ∂ . Proposition 2.41. Let Ω ⊂ M be an Ahlfors regular domain, and denote by ν its outward unit conormal. If u ∈ C 1 (Ω, Λ l T M) is such that N u, N(δu) ∈ L p (∂Ω) for some p ∈ [1, ∞] and u, δu have nontangential boundary traces at almost every point on ∂Ω, then the p,δ n.t. form ν ∨ u∂Ω belongs to Ltan (∂Ω, Λ l−1 T M) and n.t. n.t. δ ∂ (ν ∨ u∂Ω ) = −ν ∨ (δu)∂Ω .
(2.4.58)
Similarly, if u and du have nontangential boundary traces at almost any point n.t. on ∂Ω and N u, N(du) ∈ L p (∂Ω) for some p ∈ [1, ∞], then ν ∧ u∂Ω belongs to p,d
Lnor (∂Ω, Λ l+1 T M) and
n.t. n.t. d ∂ (ν ∧ u∂Ω ) = −ν ∧ (du)∂Ω .
20 speciﬁcally, if the Betti numbers b1 (M) and b2 (M) of M (cf. (3.1.9)) vanish
(2.4.59)
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets 
103
Proof. To prove identity (2.4.58), let ϕ ∈ C 1 (M, Λ l−2 T M) be arbitrary, and consider the vector ﬁeld F⃗ : Ω → T M deﬁned uniquely by asking that for all x ∈ Ω and all ξ ∈ T x∗ M there holds ⃗
T x∗ M (ξ, F(x))T x M
= ⟨(dϕ)(x), ξ ∨ u(x)⟩x + ⟨ϕ(x), ξ ∨ (δu)(x)⟩x .
(2.4.60)
Note F⃗ is welldeﬁned since the righthand side of (2.4.60) is linear in ξ . It is also clear from (2.4.60) and the assumptions on u that F⃗ ∈ L1loc (Ω, T M). In fact, we locally have F⃗ = F j ∂ j with F j (x) = ⟨(dϕ)(x), dx j ∨ u(x)⟩x + ⟨ϕ(x), dx j ∨ (δu)(x)⟩x .
(2.4.61)
For any given scalar function ψ ∈ C 01 (Ω), we may then compute ⃗
D (Ω) (div F, ψ)D(Ω)
= −D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T x M ⟨(grad ψ)(x), F(x)⟩ T x M dVol(x) Ω
⃗ = − ∫ T x∗ M ((grad ψ)♭ (x), F(x)) T x M dVol(x) Ω
⃗ = − ∫ T x∗ M (dψ(x), F(x)) T x M dVol(x) Ω
= − ∫{⟨dϕ, dψ ∨ u⟩ + ⟨ϕ, dψ ∨ δu⟩} dVol Ω
= − ∫{⟨dψ ∧ dϕ, u⟩ + ⟨dψ ∧ ϕ, δu⟩} dVol,
(2.4.62)
Ω
where penultimate equality is implied by (2.4.60) with ξ := dψ(x) ∈ T x∗ M, and the last equality used item (8) in Lemma 2.2. In addition, in view of the fact that ψ has compact support in Ω, items (4) and (6) in Lemma 2.8 permit us to write ∫⟨dψ ∧ ϕ, δu⟩ dVol = ∫⟨d(dψ ∧ ϕ), u⟩ dVol Ω
Ω
= − ∫⟨dψ ∧ dϕ, u⟩ dVol.
(2.4.63)
Ω
Combining (2.4.62) and (2.4.63) we arrive at the conclusion that div F⃗ = 0 in D (Ω).
(2.4.64)
In addition, (2.4.60) and the original assumptions on u imply that the nontangential n.t. trace F⃗ ∂Ω exists σa.e. on ∂Ω and ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
n.t. n.t. = ⟨dϕ, ν ∨ u∂Ω ⟩ + ⟨ϕ, ν ∨ (δu)∂Ω ⟩.
(2.4.65)
104  2 Geometric Concepts and Tools Since (2.4.60) also gives N(F)⃗ ≤ C N u + C N(δu) pointwise on ∂Ω, we conclude that N(F)⃗ belongs to L1 (∂Ω). In the context of Theorem 9.68, formulas (2.4.64) and (2.4.65) entail n.t. n.t. ∫ ⟨dϕ, ν ∨ u∂Ω ⟩ dσ = − ∫ ⟨ϕ, ν ∨ (δu)∂Ω ⟩ dσ, ∂Ω
∂Ω
for all ϕ ∈ C (M, Λ 1
l−2
(2.4.66)
T M).
With this in hand, (2.4.58) follows upon recalling the deﬁnition of δ ∂ . A similar proof works for (2.4.59). In turn this allows us to derive the following useful density results. Lemma 2.42. Assume Ω ⊂ M is an open set of ﬁnite perimeter satisfying (2.2.5), and denote by ν the outward unit conormal to Ω. Also, ﬁx l ∈ {0, 1, . . . , n} along with some p ∈ [1, ∞). Then p {ν ∨ ψ∂Ω : ψ ∈ C 1 (M, Λ l+1 T M)} is dense in Ltan (∂Ω, Λ l T M)
(2.4.67)
and p,δ
p
Ltan (∂Ω, Λ l T M) ⊂ Ltan (∂Ω, Λ l T M) densely.
(2.4.68)
Moreover, p
q
Ltan (∂Ω, Λ l T M) ⊂ Ltan (∂Ω, Λ l T M) densely, if 1 ≤ q ≤ p < ∞.
(2.4.69)
p
Proof. To prove (2.4.67) observe that any form f ∈ Ltan (∂Ω, Λ l T M) may be expressed as ν ∨ (ν ∧ f) and, in turn, ν ∧ f ∈ L p (∂Ω, Λ l+1 T M) may be approximated with forms belonging to the space {ψ∂Ω : ψ ∈ C 1 (M, Λ l+1 T M)} (cf. [50, Lemma 2.23, p. 2614]). This establishes (2.4.67). Given that, thanks to Proposition 2.41 we have p,δ {ν ∨ ψ∂Ω : ψ ∈ C 1 (M, Λ l+1 T M)} ⊂ Ltan (∂Ω, Λ l T M),
(2.4.70)
it is clear that (2.4.67) implies (2.4.68). Finally, (2.4.67) and (2.4.70) prove (2.4.69). In the last part of this section we are interested in phenomena of the following type. Given two disjoint Ahlfors regular domains Ω± ⊂ M with a common boundary denoted ∂Ω, and given two ldifferential forms u± deﬁned in Ω+ and Ω− , respectively, we claim that n.t. n.t. u+ ∂Ω = u− ∂Ω on ∂Ω n.t. n.t. ⇒ (du+ )∂Ω − (du− )∂Ω is normal to ∂Ω,
(2.4.71)
granted appropriate boundary control of the differential forms involved. In Proposition 2.43 below we establish slightly more general results of this ﬂavor. Later on, in Proposition 3.4, we shall see that if Ω ⊂ M is a UR domain then one can actually characp terize the space Lnor (∂Ω, Λ l+1 T M) via ldifferential forms u± in the manner suggested by (2.4.71).
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets 
105
Proposition 2.43. Suppose Ω ⊂ M is an Ahlfors regular domain satisfying ∂( Ω) = ∂Ω.
(2.4.72)
Denote by σ the surface measure on ∂Ω, and introduce Ω + := Ω and Ω− := M \ Ω. Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} along with some integrability exponent p ∈ [1, ∞]. Then any two differential forms, u+ and u− , satisfying u± ∈ C 1 (Ω± , Λ l T M), { { { { { 1 p { { N(u± ) ∈ L (∂Ω), N(du± ) ∈ L (∂Ω), { { u± n.t. , (du± )n.t. exist σa.e. on∂Ω, { ∂Ω ∂Ω { { { { n.t. n.t. { u+ ∂Ω − u− ∂Ω is normal to ∂Ω,
(2.4.73)
p n.t. n.t. (du+ )∂Ω − (du− )∂Ω ∈ Lnor (∂Ω, Λ l+1 T M).
(2.4.74)
have the property that
In particular, corresponding to u− = 0, this shows that u ∈ C 1 (Ω, Λ l T M)
} } }
} N u ∈ L1 (∂Ω), N(du) ∈ L p (∂Ω) } } }
n.t. (du) belongs to ⇒ p ∂Ω n.t. n.t. } u∂Ω , (du)∂Ω exist σa.e. on ∂Ω} Lnor (∂Ω, Λ l+1 T M). } } } } } n.t. u∂Ω is normal to ∂Ω }
(2.4.75)
Similarly, any two differential forms u± satisfying u± ∈ C 1 (Ω± , Λ l T M), { { { { { 1 p { { N(u± ) ∈ L (∂Ω), N(δu± ) ∈ L (∂Ω), { n.t. n.t. { u± ∂Ω , (δu± )∂Ω exist σa.e. on ∂Ω, { { { { { n.t. n.t. { u+ ∂Ω − u− ∂Ω is tangential to ∂Ω,
(2.4.76)
p n.t. n.t. (δu+ )∂Ω − (δu− )∂Ω ∈ Ltan (∂Ω, Λ l−1 T M).
(2.4.77)
have the property that
Proof. Property (2.4.72) ensures that ∂Ω+ = ∂Ω = ∂Ω− .
(2.4.78)
Hence, if ν denotes the outward unit conormal to Ω+ = Ω, then the outward unit conormal to Ω− is −ν. Let ϕ ∈ C 1 (M, Λ l T M) be arbitrary and consider the vector ﬁeld F⃗ : Ω → T M deﬁned uniquely by asking that for all x ∈ Ω and all ξ ∈ T x∗ M there holds ⃗
T x∗ M (ξ, F(x))T x M
= ⟨ξ ∨ ϕ(x), (du+ )(x)⟩x + ⟨ξ ∨ (δϕ)(x), u+ (x)⟩x .
(2.4.79)
106  2 Geometric Concepts and Tools Since the righthand side of (2.4.79) is linear in ξ , it follows that F⃗ is welldeﬁned. In addition, it is clear from (2.4.79) and the assumptions on u± that F⃗ ∈ L1loc (Ω, T M) . In fact, we locally have F⃗ = F j ∂ j with F j (x) = ⟨dx j ∨ ϕ(x), (du+ )(x)⟩x + ⟨dx j ∨ (δϕ)(x), u+ (x)⟩x .
(2.4.80)
For any given scalar function ψ ∈ C 01 (Ω), we may then compute ⃗
D (Ω) (div F, ψ)D(Ω)
= −D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T x M ⟨(grad ψ)(x), F(x)⟩ T x M dVol(x) Ω
⃗ = − ∫ T x∗ M ((grad ψ)♭ (x), F(x)) T x M dVol(x) Ω
⃗ = − ∫ T x∗ M (dψ(x), F(x)) T x M dVol(x) Ω
= − ∫{⟨dψ ∨ ϕ, du+ ⟩ + ⟨dψ ∨ δϕ, u+ ⟩} dVol Ω
= − ∫{⟨ϕ, dψ ∧ du+ ⟩ + ⟨δϕ, dψ ∧ u+ ⟩} dVol,
(2.4.81)
Ω
where penultimate equality is implied by (2.4.79) with ξ := dψ(x) ∈ T x∗ M, and the last equality used item (8) in Lemma 2.2. Moreover, mindful of the fact that ψ has compact support in Ω, from items (4) and (6) in Lemma 2.8 we deduce that ∫⟨ϕ, dψ ∧ du+ ⟩ dVol = − ∫⟨ϕ, d(dψ ∧ u+ )⟩ dVol Ω
Ω
= − ∫⟨δϕ, dψ ∧ u+ ⟩ dVol.
(2.4.82)
Ω
Together, (2.4.81) and (2.4.82) then ultimately prove that div F⃗ = 0 in D (Ω).
(2.4.83)
Furthermore, (2.4.79) and the assumptions on u± imply that the nontangential trace n.t. F⃗ ∂Ω exists and, at σa.e. point on ∂Ω, we have ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
n.t. n.t. = ⟨ν ∨ ϕ, (du+ )∂Ω ⟩ + ⟨ν ∨ δϕ, u+ ∂Ω ⟩.
(2.4.84)
Since (2.4.79) also gives N(F)⃗ ≤ C N(u+ ) + C N(du+ ) pointwise on ∂Ω, we conclude that N(F)⃗ ∈ L1 (∂Ω). At this stage we may invoke Theorem 9.68 and, in light of formulas (2.4.83) and (2.4.84), obtain n.t. n.t. ∫ ⟨ν ∨ ϕ, (du+ )∂Ω ⟩ dσ = − ∫ ⟨ν ∨ δϕ, u+ ∂Ω ⟩ dσ. ∂Ω
∂Ω
(2.4.85)
2.4 Tangential and Normal Differential Forms on Ahlfors Regular Sets 
107
Similar considerations in Ω− yield n.t. n.t. ∫ ⟨ν ∨ ϕ, (du− )∂Ω ⟩ dσ = − ∫ ⟨ν ∨ δϕ, u− ∂Ω ⟩ dσ. ∂Ω
(2.4.86)
∂Ω
Subtracting (2.4.86) from (2.4.85) then produces n.t. n.t. n.t. n.t. ∫ ⟨ϕ, ν ∧ ((du+ )∂Ω − (du− )∂Ω )⟩ dσ = ∫ ⟨ν ∨ ϕ, (du+ )∂Ω − (du− )∂Ω ⟩ dσ ∂Ω
∂Ω
n.t. n.t. = − ∫ ⟨ν ∨ δϕ, u+ ∂Ω − u− ∂Ω ⟩ dσ ∂Ω
n.t. n.t. = − ∫ ⟨δϕ, ν ∧ (u+ ∂Ω − u− ∂Ω )⟩ dσ ∂Ω
= 0,
(2.4.87)
n.t. n.t. thanks to the fact that, by assumption, u+ ∂Ω − u− ∂Ω is normal to ∂Ω. Given that {ϕ∂Ω : ϕ ∈ C 1 (M, Λ l T M)} separates forms in L p (∂Ω, Λ l T M),
(2.4.88)
n.t. n.t. we ultimately deduce from (2.4.87) that ν ∧ ((du+ )∂Ω − (du− )∂Ω ) = 0 at σa.e. point on ∂Ω. With this in hand, (2.4.74) readily follows. Finally, that (2.4.76) implies (2.4.77) is seen from what we have just proved and Hodge duality.
3 Harmonic Layer Potentials Associated with the Hodgede Rham Formalism on UR Domains Our strategy in dealing with the proofs of the main results is to work with boundary layer operators that are welladapted for the problems at hand. The task of developing these tools systematically is taken up in the present chapter. We study operators acting on f ∈ L p (∂Ω, Λ l T M) of the form Sl f(x) = ∫ ⟨Γ l (x, y), f(y)⟩ dσ(y),
x ∈ M \ ∂Ω,
(3.0.1)
∂Ω
along with the associated boundary limit S l f(x), as well as the principal value singũl , obtained respectively by applying to Γ l (x, y) lar integral variants, M l , N l , ̃ M l , and N either d x , δ x , δ y , or d y (and also taking either interior or exterior product with ν(x), as the situation merits). −1 Here Γ l (x, y) is the integral kernel of (∆HL − V) , where V is a scalar function constructed to make the operator invertible. This construction is made in § 3.1, where qualitative properties of Γ l (x, y), d x Γ l (x, y), etc., are derived. These will be crucial for the analysis of the operators mentioned above. Section 3.2 studies the operator properties of Sl and the associated principal value singular integral operators. The results include nontangential maximal function estimates on Sl f, dSl f , and δSl f , and formulas for nontangential limits of these operators, such as n.t. ν ∨ dSl f ∂Ω± = ∓ 12 ftan + M l f, (3.0.2) and variants. We show that if Ω is a UR domain, then p p M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M), Ml , ̃ p p ̃l : Lnor Nl , N (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l T M)
(3.0.3)
are bounded for p ∈ (1, ∞), as are (for V smooth near ∂Ω) p,δ p,δ Ml , ̃ M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M),
̃l : Lnor (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l T M). Nl , N p,d
p,d
(3.0.4)
In § 3.3 we show that the operators in (3.0.3) and (3.0.4) are actually compact, if Ω is a regular SKT domain, and discuss the invertibility of zI − M l , etc., for various complex numbers z.
3.1 A Fundamental Solution for the HodgeLaplacian Recall that H s,p (M), with s ∈ ℝ and p ∈ (1, ∞), stands for the standard scale (of fractional smoothness s, L p based) Sobolev spaces on M. Also, recall the HodgeLaplacian
110  3 Potentials for the Hodgede Rham Formalism on UR Domains
∆HL from (1.0.3). Given any realvalued, nonnegative function V ∈ L r (M), r > max {2, n/2},
(3.1.1)
it follows from Proposition 9.57 that, for each ﬁxed degree l ∈ {0, 1, . . . , n}, the operator ∆HL − V : H 1,2 (M, Λ l T M) → H −1,2 (M, Λ l T M)
(3.1.2)
is Fredholm with index zero (as well as selfadjoint). Moreover, for every differential form u ∈ H 1,2 (M, Λ l T M) we have the following global energy identity ∫ (du2 + δu2 + Vu2 ) dVol = H −1,2 (M,Λ l TM) ⟨(∆HL − V)u, u⟩H 1,2 (M,Λ l TM) ,
(3.1.3)
M
where the brackets in the righthand side stand for the duality pairing between the ∗ space H −1,2 (M, Λ l T M) = (H 1,2 (M, Λ l T M)) and H 1,2 (M, Λ l T M). In particular, (3.1.3) yields the following result: assuming the potential V is as in (3.1.1), there holds Ker (∆HL − V : H 1,2 (M, Λ l T M) → H −1,2 (M, Λ l T M)) = {u ∈ H 1,2 (M, Λ l T M) : du = 0, δu = 0, Vu = 0 on M}.
(3.1.4)
In concert with the classical unique continuation result of N. Aronszajn, K. Krzywicki and J. Szarski from [2] (recalled in Proposition 2.9) this implies that whenever V ∈ L r (M) with r > max {2, n/2} is a realvalued nonnegative potential, which is strictly positive in a nonempty open subset of every connected component of M, then ∆HL − V : H 1,2 (M, Λ l T M) → H −1,2 (M, Λ l T M) is invertible.
(3.1.5)
In fact, by Proposition 9.59, the above invertibility result selfimproves as follows: if V ∈ L r (M) with r > max{2, n/2} is a realvalued nonnegative potential, that is strictly positive in a nonempty open subset of any connected component of M, r it follows that for all p ∈ ( r−1 , r) and s ∈ [0, 1) the operator s+1,p l (M, Λ T M) → H s−1,p (M, Λ l T M) ∆HL − V : H is invertible.
(3.1.6)
By way of contrast, when V = 0 in M, it follows from (3.1.4) that Ker (∆HL : H 1,2 (M, Λ l T M) → H −1,2 (M, Λ l T M)) = H l (M),
(3.1.7)
H l (M) := {u ∈ H 1,2 (M, Λ l T M) : du = 0, δu = 0 on M}.
(3.1.8)
where
3.1 A Fundamental Solution for the HodgeLaplacian 
111
It is wellknown that H l (M), typically referred to as the space of harmonic ﬁelds on M (cf. the discussion in § 6.3), is a ﬁnite dimensional subspace of the space of nullsolutions of the HodgeLaplacian on M, and that the Betti number b l (M) := dim H l (M)
(3.1.9)
is related to the topology of M. Speciﬁcally, the classical Hodgede Rham theorem (asserting that the cohomology of a compact Riemannian manifold can be represented l by harmonic ﬁelds) gives that b l (M) is the dimension of Hsing (M; ℝ), the lth singular homology group of M over the reals. In particular (and this is going to be useful later), Poincaré duality gives that b n−l (M) = b l (M)
for all l ∈ {0, 1, . . . , n}.
(3.1.10)
The moral of this discussion is that there are natural topological obstructions to the invertibility of the HodgeLaplacian expressed by the fact that ∆HL : H 1,2 (M, Λ l T M) → H −1,2 (M, Λ l T M) is an invertible operator ⇐⇒ b l (M) = 0.
(3.1.11)
Given the goals we have in mind, it is precisely for the reasons described above that it is necessary to alter the HodgeLaplacian by considering the zeroorder perturbation¹ L := ∆HL − V on M (3.1.12) where, in place of (3.1.1), we shall now assume that the potential V is a realvalued, nonnegative function V ∈ L r (M), r > n, which is strictly positive in a nonempty open subset of any connected component of M.
(3.1.13)
For V as above, it is then apparent that the operator L may be written in local coordinates as L = L∗
n
and
n
L = ∑ ∂ j A jk ∂ k + ∑ B j ∂ j + C locally, j,k=1
j=1
with variable matrixvalued coefficients satisfying A jk ∈ C 2 ,
(3.1.14)
B j ∈ C 1 , and C ∈ L r with r > n.
Moreover, from (3.1.5) we know that L : H 1,2 (M, Λ l T M) → H −1,2 (M, Λ l T M) has an inverse.
1 that may be natural to call a HodgeSchrödinger operator
(3.1.15)
112  3 Potentials for the Hodgede Rham Formalism on UR Domains This inverse, denoted by (∆HL − V)−1 , has been studied in [86, § 6]. Among other things, it has been shown there that if Γ l (x, y) is the Schwartz kernel of (∆HL − V)−1 , viewed as mapping from H −1,2 (M, Λ l T M) onto H 1,2 (M, Λ l T M), which is the inverse of the operator ∆HL − V in (3.1.5),
(3.1.16)
then Γ l (x, y) is a symmetric double form of bidegree (l, l) exhibiting the following Hölder and Sobolev regularity: α Γ l (⋅, y) ∈ Cloc (M \ {y}, Λ l T M ⊗ Λ l T y M),
Γ l (⋅, ⋅) ∈
1+θ Cloc (M
∀ y ∈ M, ∀ α < 2 − n/r,
× M \ diag, Λ T M ⊗ Λ T M), l
l
for some θ > 0,
(3.1.17) (3.1.18)
and s,p
Γ l (⋅, y) ∈ Hloc (M \ {y}, Λ l T M ⊗ Λ l T y M),
∀ y ∈ M, ∀ s < 2, ∀ p < r.
(3.1.19)
See [50, Proposition 2.3, p. 15] in this regard. Moreover, from the discussion preceding Proposition 2.3 in [86], Γ l (⋅, y) ∈ H 1,p (M, Λ l T M ⊗ Λ l T y M)
∀ p < n/(n − 1), uniformly in y ∈ M, (3.1.20)
and Γ l (x, ⋅) ∈ H 1,p (M, Λ l T x M ⊗ Λ l T M)
∀ p < n/(n − 1), uniformly in x ∈ M,
(3.1.21)
while from (3.1.16)–(3.1.19) and (2.1.118) we conclude that there exists θ > 0 with the property that θ ∇x ∇y Γ l (⋅, y) ∈ Cloc (M \ {y}, Λ l T M ⊗ Λ l T y M),
∀ y ∈ M.
(3.1.22)
In this vein, let us also note that since ∆HL and ∗ commute (cf. item (5) of Lemma 2.8) we have ∗x ∗y Γ l (x, y) = Γ n−l (x, y), (3.1.23) where the subscripts indicate the variables in which the corresponding (Hodge star) operators are acting. The nature of the main singularity in Γ l (⋅, ⋅) along the diagonal in M × M has also been studied in [86]. To describe this, use local coordinates to write for (x, y) near an arbitrary, ﬁxed, diagonal point
IJ Γ l (x, y) = ∑ ∑ Γ l (x, y) dx I ⊗ dy J ,
(3.1.24)
I=l J=l
where Γ lJI (y, x) = Γ lIJ (x, y)
for all x ≠ y and all I, J with I = J = l.
(3.1.25)
3.1 A Fundamental Solution for the HodgeLaplacian 
113
Then, if ω n−1 is the area of the unit sphere in ℝn , for each I, J with I = J = l we have IJ
Γ l (x, y) = e0 (y, x − y)g IJ (y) + e IJ (x, y) with g IJ (y) := det((g ij (y))i∈I,j∈J )
and
n
e0 (y, z) :=
−1 ( ∑ g jk (y)z j z k ) (n − 2)ω n−1 j,k=1
e0 (y, z) :=
n 1 ln ( ∑ g jk (y)z j z k ) 4π j,k=1
−(n−2)/2
if n ≥ 3,
(3.1.26)
if n = 2,
while the less singular term satisﬁes for each ε > 0 e IJ (x, y) ≤ C ε x − y−(n−3+ε) ,
(3.1.27)
∇x e IJ (x, y) + ∇y e IJ (x, y) ≤ C ε x − y−(n−2+ε) ,
(3.1.28)
∇x ∇y e IJ (x, y) ≤ C ε x − y−(n−1+ε) ,
(3.1.29)
for some C ε ∈ (0, ∞). This follows from [86, (6.1), p. 54] and [86, Proposition 2.8, p. 21]. In this regard, we wish to momentarily digress in order to remark that more can be said under stronger regularity assumptions. Speciﬁcally, if there exists γ ∈ (0, 1) such that the ambient manifold M is of class C 2+γ , the underlying Riemannian metric has C 2+γ coefficients and, in place of (3.1.13), we now assume that V is a realvalued, nonnegative function of class C γ on M, which does not vanish identically in any connected component of M, then in addition to (3.1.27)–(3.1.29) we also have that, for each ε > 0, ∇x ∇x e IJ (x, y) ≤ C ε x − y−(n−1+ε) , (3.1.30) ∇y ∇y e IJ (x, y) ≤ C ε x − y−(n−1+ε) ,
(3.1.31)
where C ε ∈ (0, ∞). Indeed, (3.1.30) is a consequence of [86, (2.53), (2.54), p. 19], while (3.1.31) follows from (3.1.30) by relying on the observation made in [86, (2.50), p. 18]. Returning to the mainstream discussion, carried out under the assumption that V is as in (3.1.13), the following weaker version of (3.1.30) has been proved in [86, Proposition 2.6, p. 26], namely that for each ε > 0 and each q < r, 1/q
− ∇x ∇x e1 (w, y)q dw) ( ∫
≤ C q, ε x − y−(n−1+ε) .
(3.1.32)
B(x, x−y/2)
Let us also remark here that the main singularity e0 (y, x − y) in (3.1.26) satisﬁes ∂ x j ∂ y s [e0 (y, x − y)] = −∂ y j ∂ y s [e0 (y, x − y)] + O(x − y−(n−1) ),
(3.1.33)
for each j, s ∈ {1, . . . , n}, as well as n
∑ g rs (y)∂ y r ∂ y s [e0 (y, x − y)] = O(x − y−(n−1) ), r,s=1
(3.1.34)
114  3 Potentials for the Hodgede Rham Formalism on UR Domains where the last property relies on the fact that the top singularity e0 (y, ⋅) satisﬁes the “frozen” variable (namely y) partial differential equation n
∑ g rs (y)∂ z r ∂ z s [e0 (y, z)] = 0.
(3.1.35)
r,s=1
Finally, we remark that (3.1.24)–(3.1.27) imply that for each x, y ∈ M with x ≠ y we have { C dist (x, y)−(n−2) if n ≥ 3, Γ l (x, y) ≤ { (3.1.36) C α dist (x, y)−α for every α > 0 if n = 2. { In particular, this implies that Γ l (x, ⋅) ∈ L p (M, Λ l T x M ⊗ Λ l T M) ∀ p < n/(n − 2), uniformly in x ∈ M,
(3.1.37)
and Γ l (⋅, y) ∈ L p (M, Λ l T M ⊗ Λ l T y M) ∀ p < n/(n − 2), uniformly in y ∈ M.
(3.1.38)
Going further, for each ﬁxed integer l ∈ {0, 1, . . . , n}, consider the double form R l (x, y) of bidegree (l, l + 1) given by R l (x, y) := δ x (Γ l+1 (x, y)) − d y (Γ l (x, y)).
(3.1.39)
Then (3.1.18) gives θ R l (⋅, ⋅) ∈ Cloc (M × M \ diag, Λ l T M ⊗ Λ l+1 T M)
for some θ > 0.
(3.1.40)
For later purposes, it is also useful to observe here that (3.1.39) entails d y R l−1 (x, y) = d y δ x (Γ l (x, y)) = δ x d y (Γ l (x, y)) = −δ x R l (x, y).
(3.1.41)
We next claim that, in the sense of distributions, (∆HL − V)x R l (x, y) = −(dV)(x) ∨ Γ l+1 (x, y).
(3.1.42)
To prove this formula, select two arbitrary differential forms, u ∈ C 2 (M, Λ l T M) along with υ ∈ C 2 (M, Λ l+1 T M). With ((⋅, ⋅)) denoting the distributional paring on M × M, and (⋅, ⋅) denoting here the distributional paring on M, we then compute (((∆HL − V)x R l (x, y), u(x) ⊗ υ(y))) = ((R l (x, y), ((∆HL − V)u)(x) ⊗ υ(y))) = ((δ x (Γ l+1 (x, y)), ((∆HL − V)u)(x) ⊗ υ(y))) − ((d y (Γ l (x, y)), ((∆HL − V)u)(x) ⊗ υ(y))) = ((Γ l+1 (x, y), (d(∆HL − V)u)(x) ⊗ υ(y))) − ((Γ l (x, y), ((∆HL − V)u)(x) ⊗ (δυ)(y)))
3.1 A Fundamental Solution for the HodgeLaplacian 
115
= ((∆HL − V)−1 υ, d(∆HL − V)u) − ((∆HL − V)−1 (δυ), (∆HL − V)u) = ((∆HL − V)−1 υ, (∆HL − V)(du)) − ((∆HL − V)−1 υ, dV ∧ u) − (δυ, u) = (υ, du) − ((Γ l+1 (x, y), (dV ∧ u)(x) ⊗ υ(y))) − (δυ, u) = − (((dV)(x) ∨ Γ l+1 (x, y), u(x) ⊗ υ(y))).
(3.1.43)
From this, (3.1.42) follows. Going further, we propose to show that V constant ⇒ R l (⋅, ⋅) = 0.
(3.1.44)
Indeed, with u, υ as before, a computation similar in spirit to (3.1.43) gives that ((R l (x, y), u(x) ⊗ υ(y))) = (δ(∆HL − V)−1 υ − (∆HL − V)−1 (δυ), u)
(3.1.45)
and since δ commutes with (∆HL − V)−1 in the case when V is a constant, (3.1.44) follows. Continuing the conversation on this topic, we remark that based on the commutator identity (∆HL − V)d = d(∆HL − V) + (dV) ∧ ⋅ (3.1.46) and formula (3.1.42) we deduce that (∆HL − V)x (d x R l (x, y)) = − d x ((dV)(x) ∨ Γ l+1 (x, y)) + (dV)(x) ∧ δ x (Γ l+1 (x, y)) − (dV)(x) ∧ d y (Γ l (x, y)).
(3.1.47)
Likewise, from the dual version of (3.1.46), i.e., (∆HL − V)δ = δ(∆HL − V) − (dV) ∨ ⋅
(3.1.48)
and (3.1.42) we conclude that (∆HL − V)x (δ x R l (x, y)) = − δ x ((dV)(x) ∨ Γ l+1 (x, y)) − (dV)(x) ∨ δ x (Γ l+1 (x, y)) + (dV)(x) ∨ d y (Γ l (x, y)) = (dV)(x) ∨ d y (Γ l (x, y)).
(3.1.49)
Collectively, from (3.1.39)–(3.1.42), (3.1.47), (3.1.49), and elliptic regularity (cf. Proposition 2.10) we then conclude (dropping the dependence on the variables and the vector bundle) that 1 R l , d x R l , δ x R l , d y R l , δ y R l ∈ Cloc ((M \ supp (dV)) × M).
(3.1.50)
116  3 Potentials for the Hodgede Rham Formalism on UR Domains
It is useful to also carry out a similar treatment for the double form of bidegree (l + 1, l) given by Q l (x, y) := d x (Γ l (x, y)) − δ y (Γ l+1 (x, y)). (3.1.51) Concretely, from (3.1.51) we see that d x Q l (x, y) = −d x δ y (Γ l+1 (x, y)) = −δ y d x (Γ l+1 (x, y)) = −δ y Q l+1 (x, y),
(3.1.52)
while reasoning as in (3.1.43) gives that, in the sense of distributions, (∆HL − V)x Q l (x, y) = (dV)(x) ∨ Γ l (x, y).
(3.1.53)
Availing ourselves of (3.1.51)–(3.1.53) and arguing as in the case of (3.1.50) then leads to the conclusion that 1 Q l , d x Q l , δ x Q l , d y Q l , δ y Q l ∈ Cloc ((M \ supp (dV)) × M).
(3.1.54)
In fact, there is direct link between R l (x, y) and Q l (x, y). Speciﬁcally, from the symmetry of the fundamental solution Γ l (x, y) (cf. (3.1.24), (3.1.25)) and deﬁnitions it is clear that Q l (x, y) = −R l (y, x). (3.1.55) Ultimately, from (3.1.50), (3.1.54), and (3.1.55) we conclude that Rl , dx Rl , δx Rl , dy Rl , δy Rl , Ql , dx Ql , δx Ql , dy Ql , δy Ql 1 belong to Cloc ((M × M) \ (supp (dV) × supp (dV))).
(3.1.56)
Granted the regularity results from (3.1.17)–(3.1.19), and given that for every y ∈ M we have { supp [(∆HL − V)(dΓ l (⋅, y))] ⊆ (M \ supp (dV)) \ {y}, (3.1.57) { supp [(∆HL − V)(δΓ l (⋅, y))] ⊆ (M \ supp (dV)) \ {y}, { from elliptic regularity (cf. Proposition 2.10), (3.1.39), (3.1.51), and (3.1.56), we also conclude that 1 d x Γ l (x, y) ∈ Cloc (U, Λ l+1 T M ⊗ Λ l T M), 1 δ x Γ l (x, y) ∈ Cloc (U, Λ l−1 T M ⊗ Λ l T M), 1 d y Γ l (x, y) ∈ Cloc (U, Λ l T M ⊗ Λ l+1 T M),
(3.1.58)
1 δ y Γ l (x, y) ∈ Cloc (U, Λ l T M ⊗ Λ l−1 T M),
where U := {(x, y) : x, y ∈ M \ supp (dV), x ≠ y}.
(3.1.59)
We close the discussion in this section with some important notational conventions.
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism

117
Remark 3.1. Let Ω ⊂ M be an open set of ﬁnite perimeter satisfying (2.2.5) and denote by σ the surface measure on ∂Ω. Also, assume a potential V as in (3.1.13) has been given, with the additional property that supp (dV) ∩ ∂Ω = ⌀. In such a context, we shall occasionally use the symbol R l to also denote the boundary integral operator with kernel R l (x, y). That is, given any f ∈ L1 (∂Ω, Λ l+1 T M), we set R l f(x) := ∫ ⟨R l (x, y), f(y)⟩ dσ(y) (3.1.60) ∂Ω
and we shall make no notational distinction between the case when x ∈ M and x ∈ ∂Ω (which is inconsequential, thanks to the regularity of R l (x, y); cf. (3.1.56)). Moreover, by dR l , δR l , dδR l , δdR l we shall denote the integrals operator with kernel d x R l (x, y), δ x R l (x, y), d x δ x R l (x, y), and δ x d x R l (x, y) respectively (with the same caveat as above). Finally, similar conventions apply to Q l .
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism Compared with the background hypotheses in the previous section, here we strengthen the assumptions on the set Ω by taking this to be a UR domain in M, satisfying ∂( Ω) = ∂Ω.
(3.2.1)
As before, denote by ν : ∂Ω → T ∗ M the outward unit conormal to Ω, and by σ the surface measure on ∂Ω. In particular, if we introduce Ω+ := Ω
and
Ω− := M \ Ω,
(3.2.2)
then Ω± are UR domains, having ∂Ω as common boundary, and with ±ν as outward unit conormals. In this context, the single layer potential operator on ∂Ω is the integral operator Sl with kernel Γ l (x, y), acting on forms f ∈ L1 (∂Ω, Λ l T M) according to Sl f(x) := ∫ ⟨Γ l (x, y), f(y)⟩ dσ(y),
x ∈ M \ ∂Ω.
(3.2.3)
∂Ω
Then for each f ∈ L1 (∂Ω, Λ l T M) we have 1 Sl f ∈ Cloc (M \ ∂Ω, Λ l T M)
(3.2.4)
and, in the sense of distributions, (∆HL − V)Sl f = 0 in M \ ∂Ω.
(3.2.5)
118  3 Potentials for the Hodgede Rham Formalism on UR Domains Also, for each f ∈ L1 (∂Ω, Λ l T M) deﬁne the boundary version of (3.2.3) by setting S l f(x) := ∫ ⟨Γ l (x, y), f(y)⟩ dσ(y),
x ∈ ∂Ω.
(3.2.6)
∂Ω
From item (2) in Theorem 9.52 we know that for each f ∈ L1 (∂Ω, Λ l T M), n.t. n.t. = S l f = Sl f ∂Ω+ ∂Ω−
Sl f
at σa.e. point on ∂Ω.
(3.2.7)
Also, item (1) in Theorem 9.52 together with (9.2.34) imply that for every p ∈ (1, ∞) there exists some constant C ∈ (0, ∞) such that ‖N(Sl f)‖L p (∂Ω) + ‖N(dSl f)‖L p (∂Ω) + ‖N(δSl f)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) ,
∀ f ∈ L p (∂Ω, Λ l T M).
(3.2.8)
Also, (9.9.48) gives that in the case when n = dim M ≥ 3 we have if p ∈ (1, n − 1) and 1/q = 1/p − 1/(n − 1) there exists C ∈ (0, ∞) so that ‖N(Sl f)‖L q (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) , ∀ f ∈ L p (∂Ω, Λ l T M), while for each p ∈ (n − 1, ∞) there exists C ∈ (0, ∞) such that N(Sl f)L∞ (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) , ∀ f ∈ L p (∂Ω, Λ l T M), and, corresponding to p = n − 1, for every q ∈ (1, ∞) we have n−1 l N(Sl f)L q (∂Ω) ≤ C q ‖f‖L n−1 (∂Ω,Λ l TM) , ∀ f ∈ L (∂Ω, Λ T M),
(3.2.9)
while corresponding to the critical value p = 1 we have N(Sl f)L(n−1)/(n−2),∞ (∂Ω) ≤ C‖f‖L1 (∂Ω,Λ l TM) , ∀ f ∈ L1 (∂Ω, Λ l T M). In addition, if n = 2 then N(Sl f)L∞ (∂Ω) ≤ C p ‖f‖L p (∂Ω,Λ l TM) for all f ∈ L p (∂Ω, Λ l T M) if p ∈ (1, ∞), and
N(Sl f)L q (∂Ω) ≤ C q ‖f‖L1 (∂Ω,Λ l TM) for all f ∈ L1 (∂Ω, Λ l T M) for all q ∈ (1, ∞).
(3.2.10)
(3.2.11)
Let us also note here that whenever supp (dV) ∩ Ω = ⌀, the operators dSl : L2 (∂Ω, Λ l T M) → H 1/2,2 (Ω, Λ l+1 T M),
(3.2.12)
δSl : L (∂Ω, Λ T M) → H
(3.2.13)
2
l
1/2,2
(Ω, Λ
l−1
T M),
are welldeﬁned, linear, and bounded. This follows from Theorem 9.45, Theorem 9.52 (cf. item (1), and (9.9.54) in particular), formulas (3.2.8), (2.2.50), and the fact that, granted the current assumption on the potential V, for every f ∈ L2 (∂Ω, Λ l T M) we have (∆HL − V)(dSl f) = 0 and (∆HL − V)(δSl f) = 0 in Ω.
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism

119
Going further, given any f ∈ L1 (∂Ω, Λ l T M), on account of the general jumpformula (9.9.80) from Theorem 9.52 and the principal symbol identiﬁcations from Lemma 2.8 we may write for σa.e. x ∈ ∂Ω n.t. dSl f ∂Ω± (x) = ∓ 12 ν(x) ∧ f(x) + P.V. ∫ ⟨d x Γ l (x, y), f(y)⟩ dσ(y),
(3.2.14)
∂Ω
n.t. δSl f ∂Ω± (x) = ± 12 ν(x) ∨ f(x) + P.V. ∫ ⟨δ x Γ l (x, y), f(y)⟩ dσ(y).
(3.2.15)
∂Ω
Here and elsewhere, P.V. indicates that the integral is considered in the principal value sense, by removing small geodesic balls centered at the singularity and passing to limit as the radius shrinks to zero. In the sequel, we shall denote by dS l and δS l the boundary principal value integral operators appearing in the righthand sides of (3.2.14) and (3.2.15), that is, the principal value operators with kernels d x Γ l (x, y) and δ x Γ l (x, y), respectively. In particular, (3.2.14) and (3.2.15) may be rewritten as n.t. dSl ∂Ω± = ∓ 12 ν ∧ I + dS l
and
n.t. δSl ∂Ω± = ± 12 ν ∨ I + δS l
(with I denoting the identity) as operators on L1 (∂Ω, Λ l T M).
(3.2.16)
For future use, let us also note here that, as seen from Proposition 2.41 and (3.2.16), p,δ
ν ∨ S l : L p (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l−1 T M) (3.2.17)
is welldeﬁned, bounded and δ ∂ (ν ∨ S l f ) = −ν ∨ (δS l f), while
∀ f ∈ L (∂Ω, Λ T M), p
l
p,d
ν ∧ S l : L p (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l+1 T M) (3.2.18)
is welldeﬁned, bounded and d ∂ (ν ∧ S l f ) = −ν ∧ (dS l f),
∀ f ∈ L (∂Ω, Λ T M). p
l
In the same geometric setting and with l ∈ {0, 1, . . . , n} arbitrary, we also introduce two pairs of integral operators acting on forms f ∈ L1 (∂Ω, Λ l T M) according to M l f(x) := ν(x) ∨ P.V. ∫ ⟨d x Γ l (x, y), f(y)⟩ dσ(y),
x ∈ ∂Ω,
(3.2.19)
x ∈ ∂Ω,
(3.2.20)
∂Ω
N l f(x) := ν(x) ∧ P.V. ∫ ⟨δ x Γ l (x, y), f(y)⟩ dσ(y), ∂Ω
and ̃ M l f(x) := ν(x) ∨ P.V. ∫ ⟨δ y Γ l+1 (x, y), f(y)⟩ dσ(y),
x ∈ ∂Ω,
(3.2.21)
x ∈ ∂Ω.
(3.2.22)
∂Ω
̃l f(x) := ν(x) ∧ P.V. ∫ ⟨d y Γ l−1 (x, y), f(y)⟩ dσ(y), N ∂Ω
120  3 Potentials for the Hodgede Rham Formalism on UR Domains
Hence, from (3.2.19)–(3.2.22), (3.1.39), and (3.1.51) we obtain ̃ Ml = Ml − ν ∨ Ql
and
̃l = N l − ν ∧ R l−1 . N
(3.2.23)
In particular, from (3.1.44) and (3.1.55) we see that V ≡ constant ⇒ ̃ Ml = Ml
and
̃l = N l . N
(3.2.24)
For further reference, we also wish to note here that the jumprelations (3.2.14) give that, for each f ∈ L1 (∂Ω, Λ l T M), n.t. ν ∨ dSl f ∂Ω± = ∓ 12 ftan + M l f
n.t. ν ∧ δSl f ∂Ω± = ± 12 fnor + N l f.
and
(3.2.25)
Other basic properties of these operators are discussed in the next two propositions. Proposition 3.2. Let Ω be a UR domain in M with outward unit conormal ν, and assume that V is as in (3.1.13). Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} along with an arbitrary integrability exponent p ∈ (1, ∞). Then the operators p M l : L p (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M), Ml , ̃
̃l : L p (∂Ω, Λ l T M) → Nl , N
p Lnor (∂Ω, Λ l T M),
(3.2.26) (3.2.27)
are welldeﬁned, linear, and bounded. Also, M n−l ∗ = − ∗ N l
and
∗ M l = −N n−l ∗ on lforms.
(3.2.28)
Furthermore, the (real ) transposed of M l acting on Ltan (∂Ω, Λ l T M) is the operator M ⊤ l p
acting on Ltan (∂Ω, Λ l T M), with 1/p + 1/p = 1, given by p
̃ M⊤ l = ν ∨ N l+1 (ν ∧ ⋅),
(3.2.29)
p M l )⊤ acting while the (real ) transposed of ̃ M l acting on Ltan (∂Ω, Λ l T M) is the operator (̃ p
on the space Ltan (∂Ω, Λ l T M) by (̃ M l )⊤ = ν ∨ N l+1 (ν ∧ ⋅).
(3.2.30)
Finally, if the potential V is also known to be of class C 1 near ∂Ω (hence, in particular, if supp (dV) ∩ ∂Ω = ⌀) then the operators p,δ p,δ M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M), Ml , ̃
(3.2.31)
p,d p,d ̃l : Lnor Nl , N (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l T M),
(3.2.32)
are welldeﬁned, linear, and bounded.
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism

121
Proof. The claims pertaining to the operators (3.2.26), (3.2.27) are consequences of (3.2.19)–(3.2.22), (3.1.12), (3.1.14), (3.1.15), Lemma 2.2, in view of the brand of CalderónZygmund theory from Theorem 9.52. Also, (3.2.28) follows from (3.1.23) and the various intertwining properties of the Hodge starisomorphism; cf. item (2) of Lemma 2.8. Going further, formula (3.2.29) may be justiﬁed along the lines of the treatment of item (8) in Theorem 9.52. Details are as follows. Let d(⋅, ⋅) denote the geodesic distance on M. For each ε > 0 small and at each x ∈ ∂Ω, consider the following truncated versions of ̃l : the operators M l and N M l, ε f(x) := ν(x) ∨ ∫⟨d x Γ l (x, y), f(y)⟩ dσ(y),
(3.2.33)
y∈∂Ω, d(x,y)>ε
̃l, ε f(x) := ν(x) ∧ ∫⟨d y Γ l−1 (x, y), f(y)⟩ dσ(y), N
(3.2.34)
y∈∂Ω, d(x,y)>ε
where σ is the surface measure on ∂Ω. Fix some f ∈ L p (∂Ω, Λ l T M) with p ∈ (1, ∞). Since M l, ε f(x) → M l f(x) as ε → 0+ ,
for σa.e. x ∈ ∂Ω,
(3.2.35)
̃l f(x) as ε → 0+ , ̃l, ε f(x) → N N
for σa.e. x ∈ ∂Ω,
(3.2.36)
and since the CalderónZygmund theory from [50], cited above, gives the maximal operator bounds p ∫ (sup M l, ε f(x)) dσ(x) < ∞ ∂Ω
̃ p ∫ (sup N l, ε f(x)) dσ(x) < ∞,
and
ε>0
∂Ω
(3.2.37)
ε>0
the Lebesgue Dominated Convergence Theorem applies and yields M l, ε f → M l f
and
̃l, ε f → N ̃l f N
in L p (∂Ω) as ε → 0+ .
(3.2.38) p
Having established this, if we now pick two arbitrary functions f ∈ Ltan (∂Ω, Λ l T M)
and g ∈ Ltan (∂Ω, Λ l T M) with p, p ∈ (1, ∞) satisfying 1/p + 1/p = 1, we may write p
∫ ⟨M l f, g⟩ dσ = lim+ ∫ ⟨M l, ε f, g⟩ dσ ε→0
∂Ω
∂Ω
= lim+ ∫ ε→0
∫ ⟨⟨d x Γ l (x, y), ν(x) ∧ g(x)⟩, f(y)⟩ dσ(x) dσ(y)
x∈∂Ω y∈∂Ω, d(x,y)>ε
= lim+ ∫ ⟨ ∫ ⟨d x Γ l (x, y), ν(x) ∧ g(x)⟩ dσ(x), f(y)⟩ dσ(y) ε→0
y∈∂Ω
x∈∂Ω, d(x,y)>ε
= lim+ ∫ ⟨ν(y) ∨ (ν(y) ∧ ∫ ⟨d x Γ l (x, y), ν(x) ∧ g(x)⟩ dσ(x)), f(y)⟩ dσ(y) ε→0
y∈∂Ω
x∈∂Ω d(x,y)>ε
122  3 Potentials for the Hodgede Rham Formalism on UR Domains ̃l+1, ε (ν ∧ g), f ⟩ dσ = lim+ ∫ ⟨ν ∨ N ε→0
∂Ω
̃l+1 (ν ∧ g), f ⟩ dσ. = ∫ ⟨ν ∨ N
(3.2.39)
∂Ω p
p
∗
Under the identiﬁcation (Ltan (∂Ω, Λ l T M)) = Ltan (∂Ω, Λ l T M), this proves (3.2.29). A similar reasoning applies to (3.2.30). Turning attention to the claims made in the last part of the proposition, we shall ﬁrst derive certain identities of independent interest, namely that in M \ ∂Ω we have² p,δ
δSl f = Sl−1 (δ ∂ f) + R l−1 f,
∀ f ∈ Ltan (∂Ω, Λ l T M),
dSl g = Sl+1 (d ∂ g) + Q l g,
∀ g ∈ Lnor (∂Ω, Λ l T M).
p,d
(3.2.40)
p,δ
To justify the ﬁrst equality in (3.2.40), let f ∈ Ltan (∂Ω, Λ l T M) be arbitrary. For each ﬁxed x ∈ Ω, on account of (3.1.39) and (2.4.20) (used with V := Λ l−1 T x M, O := M \ {x}, and ϕ := Γ l−1 (x, ⋅)), we may then write δSl f(x) = ∫ ⟨δ x (Γ l (x, y)), f(y)⟩ dσ(y) ∂Ω
= ∫ ⟨d y (Γ l−1 (x, y)), f(y)⟩ dσ(y) + ∫ ⟨R l−1 (x, y), f(y)⟩ dσ(y) ∂Ω
∂Ω
= ∫ ⟨Γ l−1 (x, y), (δ ∂ f)(y)⟩ dσ(y) + ∫ ⟨R l−1 (x, y), f(y)⟩ dσ(y) ∂Ω
∂Ω
= Sl−1 (δ ∂ f)(x) + R l−1 f(x),
(3.2.41)
which yields the ﬁrst identity in (3.2.40). The second identity in (3.2.40) follows by a similar argument. Parenthetically, let us also note that by combining (3.2.40) and (3.2.16) we obtain that, on ∂Ω, p,δ
δS l f = S l−1 (δ ∂ f) + R l−1 f,
∀ f ∈ Ltan (∂Ω, Λ l T M),
dS l g = S l+1 (d ∂ g) + Q l g,
∀ g ∈ Lnor (∂Ω, Λ l T M).
p,d
(3.2.42)
We are now in a position to tackle the last part in the statement of Proposition 3.2. This will, in fact, be a direct consequence of the fact that M l and N l almost commute with the boundary derivatives operators δ ∂ and d ∂ , respectively. More concretely, we p,δ shall prove that for any f ∈ Ltan (∂Ω, Λ l T M) we have³ δ ∂ M l f = M l−1 (δ ∂ f) + ν ∨ V S l f + ν ∨ dR l−1 f,
2 we recall the conventions made in Remark 3.1 3 again, we recall the conventions made in Remark 3.1
(3.2.43)
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism

123
p,d
while for any g ∈ Lnor (∂Ω, Λ l T M) we have d ∂ N l g = N l+1 (d ∂ g) + ν ∧ V S l g + ν ∧ δQ l g.
(3.2.44)
Indeed, set u := dSl f ∈ C 1 (Ω, Λ l+1 T M). Clearly, N u ∈ L p (∂Ω) and since by the commutation formula (3.2.40) we have δu = δdSl f = −V Sl f − dδSl f = −V Sl f − dSl−1 (δ ∂ f) − dR l−1 f
(3.2.45)
it follows that N(δu) ∈ L p (∂Ω) as well. In addition, thanks to jumpformula (3.2.25) we n.t. have that ν ∨ u∂Ω = (− 12 I + M l )f . Hence, n.t. δ ∂ M l f = 12 δ ∂ f + δ ∂ (ν ∨ u∂Ω ) n.t. = 12 δ ∂ f − ν ∨ (δu)∂Ω n.t. = 12 δ ∂ f + ν ∨ dSl−1 (δ ∂ f)∂Ω + ν ∨ VS l f + ν ∨ dR l−1 f
= 12 δ ∂ f − 12 δ ∂ f + M l−1 (δ ∂ f) + ν ∨ VS l f + ν ∨ dR l−1 f = M l−1 (δ ∂ f) + ν ∨ VS l f + ν ∨ dR l−1 f,
(3.2.46)
where the second equality uses (2.4.58), the third equality employs (3.2.45), and the fourth equality relies on (3.2.25). This establishes the identity recorded in (3.2.43), and (3.2.44) is proved similarly. Hence, the singular integral operators M l , N l induce wellp,δ p,d deﬁned, bounded mappings from Ltan (∂Ω, Λ l T M) and Lnor (∂Ω, Λ l T M), respectively, ̃l have analogous mapping properties follows from into themselves. That ̃ M l and N what we have just proved, together with (3.2.23), (2.4.58), and (2.4.59). The proof of Proposition 3.2 is therefore complete. The result stated in the next proposition is relevant in the context of the deﬁnition of p,δ p,d the spaces Ltan (∂Ω, Λ l TM) and Lnor (∂Ω, Λ l TM) from § 2.4. Proposition 3.3. Assume Ω ⊂ M is a UR domain satisfying (3.2.1). Fix an arbitrary degree l ∈ {0, 1, . . . , n} along with some integrability exponent p ∈ [1, ∞]. Then, if the differential forms f ∈ L p (∂Ω, Λ l TM) and g ∈ L p (∂Ω, Λ l+1 TM) are such that (3.2.47) ∫ ⟨δϕ, f⟩ dσ = ∫ ⟨ϕ, g⟩ dσ for each ϕ ∈ C 1 (M, Λ l+1 T M), ∂Ω
∂Ω
it follows that, necessarily, f and g are normal to ∂Ω. Consequently, p,d
f ∈ Lnor (∂Ω, Λ l TM),
p,0
g ∈ Lnor (∂Ω, Λ l+1 TM),
and
d ∂ f = g.
(3.2.48)
Similarly, if f ∈ L p (∂Ω, Λ l TM) and g ∈ L p (∂Ω, Λ l−1 TM) are such that ∫ ⟨dϕ, f⟩ dσ = ∫ ⟨ϕ, g⟩ dσ ∂Ω
for each ϕ ∈ C 1 (M, Λ l−1 T M),
(3.2.49)
∂Ω
it follows that, necessarily, p,δ
f ∈ Ltan (∂Ω, Λ l TM),
p,0
g ∈ Ltan (∂Ω, Λ l−1 TM),
and
δ ∂ f = g.
(3.2.50)
124  3 Potentials for the Hodgede Rham Formalism on UR Domains
Proof. Denote by σ the surface measure on ∂Ω and by ν the outward unit conormal to Ω. Recall Ω± from (3.2.2). Then these are UR domains, sharing a common boundary, ∂Ω, and having ±ν as outward unit conormals. To proceed, pick a function V as in (3.1.13) and, associated with this potential, consider fundamental solutions for the operator ∆HL − V as in (3.1.16). Fix an arbitrary point x ∈ M \ ∂Ω along with an arbitrary form η ∈ Λ l+1 T x M. Also, pick some scalar function ψ ∈ C01 (M \ {x}) with the property that ψ ≡ 1 near ∂Ω. Then (cf. (3.1.58)) ϕ(y) := ⟨ψ(y)Γ l+1 (x, y), η⟩x ,
∀ y ∈ M ⇒ ϕ ∈ C 1 (M, Λ l+1 T M),
(3.2.51)
and writing the equality in (3.2.47) for this choice of ϕ yields ⟨ ∫ ⟨δ y (Γ l+1 (x, y)), f(y)⟩ dσ(y), η⟩ = ⟨ ∫ ⟨Γ l+1 (x, y), g(y)⟩ dσ(y), η⟩ . x
∂Ω
(3.2.52)
x
∂Ω
In view of (3.1.51), (3.2.3), and the fact that η ∈ Λ l T x M is arbitrary, from (3.2.52) we conclude that (dSl f)(x) − (Q l f)(x) = Sl+1 g(x), ∀ x ∈ M \ ∂Ω. (3.2.53) Hence, dSl f = Q l f + Sl+1 g in M \ ∂Ω which, in light of (3.2.16) and (3.2.7), gives n.t. n.t. − 12 ν ∧ f + dS l f = dSl f ∂Ω+ = dSl f ∂Ω− = 12 ν ∧ f + dS l f on ∂Ω.
(3.2.54)
In turn, this forces ν ∧ f = 0 on ∂Ω,
(3.2.55)
proving that f is normal to ∂Ω, as wanted. To prove a similar property for g, pick an arbitrary point x ∈ M \ ∂Ω and ﬁx an arbitrary form η ∈ Λ l+2 T x M. As before, let ψ ∈ C01 (M \ {x}) be a scalar function satisfying ψ ≡ 1 near ∂Ω. Then, as seen from (3.1.58), ϕ(y) := ⟨ψ(y)δ y (Γ l+2 (x, y)), η⟩x ,
∀ y ∈ M ⇒ ϕ ∈ C 1 (M, Λ l+1 T M).
(3.2.56)
Writing the equality in (3.2.47) for this choice of ϕ produces 0 = ⟨ ∫ ⟨δ y (Γ l+2 (x, y)), g(y)⟩ dσ(y), η⟩ . ∂Ω
(3.2.57)
x
Collectively, from (3.1.51), (3.2.3), and the fact that η ∈ Λ l+2 T x M is arbitrary, from (3.2.57) we see that (dSl+1 g)(x) = (Q l+1 g)(x),
∀ x ∈ M \ ∂Ω.
(3.2.58)
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism

125
Thus, dSl+1 g = Q l+1 g in M \ ∂Ω which, in concert with (3.2.16), gives n.t. n.t. − 12 ν ∧ g + dS l+1 g = dSl+1 g ∂Ω+ = dSl+1 g ∂Ω− = 12 ν ∧ g + dS l+1 g on ∂Ω
(3.2.59)
hence, ultimately, ν ∧ g = 0 on ∂Ω.
(3.2.60)
This proves that g is normal to ∂Ω, as desired. Having established this, the conclusions in (3.2.48) readily follow from deﬁnitions in § 2.4. Finally, that (3.2.49) implies (3.2.50) is seen from what we have proved already and Hodge duality. Our next result is a companion to Proposition 2.43. Together, they yield a characterip zation of the space Lnor (∂Ω, Λ l+1 T M) when Ω is a UR domain in M, in terms of jumpn.t. n.t. discontinuities across ∂Ω of the type (du+ )∂Ω − (du− )∂Ω , for pairs of ldifferential forms u± deﬁned in Ω± which agree when restricted (in a nontangential fashion) to the common interface ∂Ω. Similar considerations (naturally adapted to the operator δ) p apply to the space Ltan (∂Ω, Λ l−1 T M). Proposition 3.4. Let Ω ⊂ M be a UR domain satisfying (2.4.72). Denote by σ the surface measure on ∂Ω, and introduce Ω+ := Ω and Ω− := M \ Ω. Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} along with some integrability exponent p ∈ (1, ∞). p Then for any f ∈ Lnor (∂Ω, Λ l+1 T M) one can ﬁnd two differential forms, u+ and u− , satisfying u± ∈ C 1 (Ω± , Λ l T M), { { { { p { { { N(u± ), N(du± ) ∈ L (∂Ω), { { { n.t. n.t. u± ∂Ω , (du± )∂Ω exist σa.e. on ∂Ω, { { { { n.t. n.t. { { { u+ ∂Ω = u− ∂Ω on ∂Ω, { { { n.t. n.t. { (du+ )∂Ω − (du− )∂Ω = f.
(3.2.61)
p
Moreover, for any g ∈ Ltan (∂Ω, Λ l−1 T M) one can ﬁnd two differential forms, υ+ and υ− , satisfying υ± ∈ C 1 (Ω± , Λ l T M), { { { { { { N(υ± ), N(δυ± ) ∈ L p (∂Ω), { { { { n.t. n.t. υ± ∂Ω , (δυ± )∂Ω exist σa.e. on ∂Ω, { { { { n.t. n.t. { { υ+ ∂Ω = υ− ∂Ω on ∂Ω, { { { { n.t. n.t. { (δυ+ )∂Ω − (δυ− )∂Ω = g.
(3.2.62)
Proof. Pick a function V as in (3.1.13) and, associated with it and the domain Ω, consider the single layer potential operator as in (3.2.3). If ν denotes the outward unit conormal to Ω, deﬁne u± := −Sl (ν ∨ f) in Ω± . (3.2.63)
126  3 Potentials for the Hodgede Rham Formalism on UR Domains Then u± ∈ C 1 (Ω± , Λ l T M) by (3.1.58). Also, N(u± ), N(du± ) ∈ L p (∂Ω) by (3.2.8). In adn.t. n.t. dition, from (3.2.7) we see that u+ ∂Ω = u− ∂Ω , while (3.2.16) the normality of f permit us to write n.t. n.t. (du+ )∂Ω − (du− )∂Ω = − (− 12 ν ∧ (ν ∨ f) + dS l (ν ∨ f)) + ( 12 ν ∧ (ν ∨ f) + dS l (ν ∨ f)) = ν ∧ (ν ∨ f) = fnor = f on ∂Ω.
(3.2.64)
This establishes the ﬁrst claim in the statement of the proposition. The second claim is seen from this and Hodge duality. In the last part of this section we turn our attention to functional analytic properties of the single layer potential operator associated with the HodgeLaplacian in arbitrary UR domains. The result presented below identiﬁes an optimal context where the said operator is injective and has dense range. Proposition 3.5. Suppose Ω is a proper UR subdomain of M, and assume that the potential V is as in (3.1.13). Fix a degree l ∈ {0, 1, . . . , n} and, associated with these objects, consider the single layer potential S l as in (3.2.6). Then for each p ∈ (1, ∞) the operator S l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)
(3.2.65)
is compact and its (real) transposed is the operator
S l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
1/p + 1/p = 1.
(3.2.66)
If, in addition, Ω satisﬁes (3.2.1) then, whenever n ≥ 3, the operator S l : L2(n−1)/n (∂Ω, Λ l T M) → L2(n−1)/(n−2) (∂Ω, Λ l T M) is welldeﬁned, linear, bounded, injective, and with dense range,
(3.2.67)
while, corresponding to n = 2, ∀ p, q ∈ (1, ∞) ⇒ S l : L p (∂Ω, Λ l T M) → L q (∂Ω, Λ l T M) is welldeﬁned, linear, bounded, injective, and with dense range.
(3.2.68)
As a consequence, if n ≥ 3 then the operator S l in (3.2.65) is injective if p ≥ 2(n − 1)/n, and has dense range if p ≤ 2(n − 1)/(n − 2),
(3.2.69)
while, corresponding to n = 2, the operator S l in (3.2.65) is injective and has dense range for every p ∈ (1, ∞).
(3.2.70)
Proof. We shall treat the case n ≥ 3 (as the situation when n = 2 is handled similarly). That the single layer operator is compact in the context of (3.2.65) for every p ∈ (1, ∞) is
3.2 Layer Potentials for the HodgeLaplacian in the Hodgede Rham Formalism

127
a consequence of Lemma 9.63 (whose applicability in the present setting is guaranteed by (3.1.36)). Also, that the (real) transposed of (3.2.65) is (3.2.66) is seen from (3.1.25). That the operator in (3.2.67) is welldeﬁned, linear, and bounded is a consequence of the Fractional Integration Theorem (in spaces of homogeneous type; cf. [40]). In light of what we have proved already, and given that 2(n − 1)/(n − 2) and 2(n − 1)/n are Hölder conjugate exponents, matters have been reduced to establishing the injectivity of the operator (3.2.67). To this end, let f ∈ L2(n−1)/n (∂Ω, Λ l T M) be such that S l f = 0 on ∂Ω and deﬁne u := Sl f in Ω. Also, denote by ν ∈ T ∗ M the outward unit conormal to the domain Ω and by σ := H n−1 ⌊∂Ω the surface measure on ∂Ω. Then, thanks to (3.2.7), (3.2.9), and (3.2.16), we have u ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, { { { { { { N u ∈ L2(n−1)/(n−2) (∂Ω), { { { { N(δu), N(du) ∈ L2(n−1)/n (∂Ω), (3.2.71) { { { n.t. { { { u∂Ω = S l f = 0 on ∂Ω, { { { { n.t. { { (du)∂Ω = − 12 ν ∧ f + dS l f, { { { { n.t. 1 { (δu)∂Ω = 2 ν ∨ f + δS l f. In concert with the energy identity in Proposition 2.37 this forces du = 0 and δu = 0 in Ω. Next, observe that N u ∈ L2(n−1)/(n−2) (∂Ω) implies u ∈ L2n/(n−2) (Ω, Λ l T M) by (2.2.50). Hence, if we now introduce { u in Ω υ := { (3.2.72) 0 in M \ Ω, { it follows that υ ∈ L1loc (M, Λ l T M). In relation to this, we claim that dυ = 0
and
δυ = 0
in the sense of distributions on M.
(3.2.73)
To justify this claim, ﬁx an arbitrary test form φ ∈ C 1 (M, Λ l+1 T M) and make use of Theorem 2.36 to write D (M) ⟨dυ, φ⟩D(M)
= ∫ ⟨υ, δφ⟩ dVol = ∫⟨u, δφ⟩ dVol M
Ω
n.t. = ∫⟨du, φ⟩ dVol − ∫ ⟨ν ∧ u∂Ω , φ⟩ dσ Ω
∂Ω
= 0 − 0 = 0. This proves dυ = 0 in the sense of distributions on M, and the fact that δυ = 0 is proved similarly. By elliptic regularity (cf. Proposition 2.10), this implies that υ ∈ C 1 (M, Λ l T M). Hence, υ is a harmonic ﬁeld on M which vanishes identically
128  3 Potentials for the Hodgede Rham Formalism on UR Domains in M \ Ω. From Lemma 2.12 we know that M \ Ω is a nonempty open set. Bearing this in mind, and availing ourselves of the unique continuation result from Proposition 2.9, we may conclude that υ = 0 in each connected component of M overlapping with M \ Ω.
(3.2.74)
Since ∂Ω is fully contained in the union of those connected components of M overlapping with M \ Ω, we conclude that υ = 0 near ∂Ω hence n.t. n.t. (du)∂Ω = (dυ)∂Ω+ = 0,
n.t. n.t. (δu)∂Ω = (δυ)∂Ω+ = 0.
(3.2.75)
In view of the last two conditions in (3.2.71) this further entails − 12 ν ∧ f + dS l f = 0 on ∂Ω
1 2ν
and
∨ f + δS l f = 0 on ∂Ω.
(3.2.76)
Now, a similar reasoning carried out in connection with Sl f considered in M \ Ω (which, thanks to (3.2.1), is also a proper UR subdomain of M, with the same boundary as Ω), also shows that 1 2ν
∧ f + dS l f = 0 on ∂Ω
and
− 12 ν ∨ f + δS l f = 0 on ∂Ω.
(3.2.77)
In turn, from (3.2.76) and (3.2.77) we ﬁrst deduce that ν ∧ f = 0 and ν ∨ f = 0 on ∂Ω then, ﬁnally, that f = 0 on ∂Ω by (2.4.3). This shows that the operator S l is indeed injective when acting from L2(n−1)/n (∂Ω, Λ l T M).
3.3 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism The aim of this section is the study of ﬁner functional analytic properties of the op̃l , introduced earlier in § 3.2. In this respect, our main results are Ml , N erators M l , N l , ̃ contained in the next two theorems. Theorem 3.6. Let Ω ⊂ M be a regular SKT domain and assume V is as in (3.1.13). Also, ﬁx p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} arbitrary. Then the following linear operators are compact: p
p
M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M), Nl :
p Lnor (∂Ω, Λ l T M)
̃ Ml :
p Ltan (∂Ω, Λ l T M) p Lnor (∂Ω, Λ l T M)
̃l : N
(3.3.1)
→
p Lnor (∂Ω, Λ l T M),
(3.3.2)
→
p Ltan (∂Ω, Λ l T M), p Lnor (∂Ω, Λ l T M).
(3.3.3)
→
(3.3.4)
3.3 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism

129
Moreover, if the scalar potential V is also of class C 1 near ∂Ω (hence, in particular, if supp (dV) ∩ ∂Ω = ⌀) then the following operators are also compact: p,δ
p,δ
M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M), Nl :
p,d Lnor (∂Ω, Λ l T M)
̃ Ml :
p,δ Ltan (∂Ω, Λ l T M) p,d Lnor (∂Ω, Λ l T M)
̃l : N
(3.3.5)
→
p,d Lnor (∂Ω, Λ l T M),
(3.3.6)
→
p,δ Ltan (∂Ω, Λ l T M), p,d Lnor (∂Ω, Λ l T M).
(3.3.7)
→
(3.3.8)
In fact, given any small positive ε o , one can ensure that any of the operators listed above lies at distance at most ε o from the space of compact operators in that environment by asking that Ω is an εSKT domain for some ε > 0 sufficiently small relative to ε o , the given p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. Let us take a close look at the integral kernel of the operator M l in a small neighborhood U (itself, a local coordinate patch of M) of an arbitrary boundary point x0 ∈ ∂Ω. Recall notation introduced in (3.1.24)–(3.1.26) and abbreviate
e l (x, y) := ∑ e IJ (x, y)dx I ⊗ dy J .
(3.3.9)
I=J=l
Using (3.1.24)–(3.1.26), as well as the conventions introduced in (2.1.3) and (2.1.39), we ﬁrst write
d x Γ l (x, y) = d x e0 (y, x − y) ∧ ( ∑ det G[IJ] (y) dx I ⊗ dy J) I=J=l
+ e0 (y, x − y) d x ( ∑ det G[IJ] (y) dx I ⊗ dy J) + d x e l (x, y) I=J=l
= d x e0 (y, x − y) ∧ ( ∑ det G[IJ] (y) dx I ⊗ dy J) + d x e l (x, y),
(3.3.10)
I=J=l
for x, y ∈ U. Thus, if f is a tangential Λ l T Mvalued differential form on ∂Ω, written in U ∩ ∂Ω as f(y) = ∑K=l f K (y)dy K , the following holds for x, y ∈ U ∩ ∂Ω: ⟨d x Γ l (x, y), f(y)⟩y = d x e0 (y, x − y) ∧ (
∑ det G[IJ] (y)⟨dy J , dy K ⟩f K (y)dx I) + ⟨d x e l (x, y), f(y)⟩y
I=J=K=l
= d x e0 (y, x − y) ∧ ( ∑ δ IK f K (y)dx I) + ⟨d x e l (x, y), f(y)⟩y I=K=l
= d x e0 (y, x − y) ∧ ( ∑ f I (y)dx I) + ⟨d x e l (x, y), f(y)⟩y , I=l
(3.3.11)
130  3 Potentials for the Hodgede Rham Formalism on UR Domains
where for the second equality we have made use of (2.1.24) and (2.1.42) written for A := G (cf. also (2.1.3)–(2.1.5)). Consequently, if ν stands for the outward unit conormal to Ω, the integral kernel of M l acting on f may be written for x, y ∈ U ∩ ∂Ω as ν(x) ∨ ⟨d x Γ l (x, y), f(y)⟩y
= ν(x) ∨ (d x e0 (y, x − y) ∧ ( ∑ f I (y)dx I ) + ⟨d x e l (x, y), f(y)⟩y ) I=l
= − d x e0 (y, x − y) ∧ (ν(x) ∨ ( ∑ f I (y)dx I )) I=l
+ ⟨ν(x), d x e0 (y, x − y)⟩x ∑ f I (y)dx I + ν(x) ∨ ⟨d x e l (x, y), f(y)⟩y ,
(3.3.12)
I=l
where in the last equality in (3.3.12) we have used item (7) in Lemma 2.2. Focusing on the ﬁrst term in the rightmost expression of (3.3.12), observe that by making use of (2.1.37) for x, y ∈ U ∩ ∂Ω we may write n
ν(x) ∨ ( ∑ f I (y)dx I ) = ( ∑ ν i (x)dx i ) ∨ ( ∑ f I (y)dx I ) I=l
I=l
i=1 n
jL
= ∑ ( ∑ g ij (x)ε I ν i (x)f I (y))dx L L=l−1 j=1
= I1 + I2 , where
n
(3.3.13)
jL
I1 := ∑ ( ∑ (g ij (x) − g ij (y))ε I ν i (x)f I (y))dx L ,
(3.3.14)
L=l−1 j=1
and n
jL
I2 := ∑ ( ∑ g ij (y)ε I ν i (x)f I (y))dx L L=l−1 j=1 n
jL
= ∑ ( ∑ g ij (y)ε I (ν i (x) − ν i (y))f I (y))dx L L=l−1 j=1
n
jL
+ ∑ ( ∑ g ij (y)ε I ν i (y)f I (y))dx L L=l−1 j=1
n
jL
= ∑ ( ∑ g ij (y)ε I (ν i (x) − ν i (y))f I (y))dx L ,
(3.3.15)
L=l−1 j=1
where the last equality follows from (2.4.7). The analysis given above shows that, when localized in U ∩ ∂Ω, the operator M l originally deﬁned as in (3.2.19) may be written as a ﬁnite linear combination of principal value singular integral operators with kernels of the following types:
3.3 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism

⟨ν(x), d x e0 (y, x − y)⟩x ,
131
(3.3.16)
ν(x) ∨ d x e l (x, y),
(3.3.17)
∂ x k e0 (y, x − y)(g (x) − g (y))ν i (x),
(3.3.18)
∂ x k e0 (y, x − y)g (y)(ν i (x) − ν i (y)).
(3.3.19)
ij
ij
ij
Owing, respectively, to the smoothness of the tensor metric and (3.1.28), the kernels (3.3.17) and (3.3.18) are weakly singular, hence the corresponding integral operators are compact on L p (∂Ω). See Lemma 9.63 in this regard. Expression (3.3.16) is the integral kernel of the adjoint double layer corresponding to the LaplaceBeltrami operator on M. Its associated integral operator has been studied in [50] where this was shown to be compact on L p (∂Ω), granted the current assumptions on Ω; compare with the results in the ﬁrst part of the statement of Theorem 9.60. Finally, (3.3.19) may be viewed as the integral kernel of the commutator between a CalderónZygmund operator and the operator of pointwise multiplication with components of ν. Since ν ∈ VMO(∂Ω), the version of the CoifmanRochberg commutator theorem proved in § 2.4 of [50] for spaces of homogeneous type applies and yields that the integral operator associated with the kernel (3.3.19) is also compact on L p (∂Ω). See also the last part in the statement of Theorem 9.60. Ultimately this proves that the operator (3.3.1) is compact. That the operator (3.3.2) is also compact follows from what we have just proved and (3.2.28). Next, the claims about the operators (3.3.3) and (3.3.4) are direct consequences of the compactness of (3.3.1) and (3.3.2), formulas (3.2.29) and (3.2.30), and functional analysis. Going further, the compactness of the operators (3.3.5) and (3.3.6) is readily seen from that of (3.3.1) and (3.3.2) and the intertwining identities (3.2.43) and (3.2.44). The compactness of the operators (3.3.7) and (3.3.8) is seen from what we have just proved and (2.4.58) and (2.4.59). Finally, consider the case when some small positive threshold ε o has been given and when Ω is an εSKT domain with ε > 0 sufficiently small relative to ε o , the given p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. In such a case, the mere fact that Ω is an Ahlfors regular domain implies that the integral operators with weakly singular kernels as in (3.3.17) and (3.3.18) continue to be compact on L p (∂Ω). Elucidating he nature of the integral operators with kernels as in (3.3.16) and (3.3.19) requires more care. In this regard, Theorem 9.60 (cf. (9.11.3), and a similar implication for commutator type operators as in (9.11.10)) can be used to ensure that the operators associated with kernels as in (3.3.16) and (3.3.19) are at distance at most ε o from the space of compact operators on L p (∂Ω) by taking ε > 0 sufficiently small to begin with. The second main result alluded to earlier is contained in Theorem 3.7 below. Theorem 3.7. Suppose Ω ⊂ M is a regular SKT domain, and let V be as in (3.1.13). Also, ﬁx some arbitrary degree l ∈ {0, 1, . . . , n}, integrability exponent p ∈ (1, ∞), and complex number z ∈ ℂ \ {0}.
132  3 Potentials for the Hodgede Rham Formalism on UR Domains
Then the following properties hold: (i) The operators p
p
zI + M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M), zI + N l : zI + ̃ Ml : ̃l : zI + N
p Lnor (∂Ω, Λ l T M) p Ltan (∂Ω, Λ l T M) p Lnor (∂Ω, Λ l T M)
→ → →
p Lnor (∂Ω, Λ l T M), p Ltan (∂Ω, Λ l T M), p Lnor (∂Ω, Λ l T M),
(3.3.20) (3.3.21) (3.3.22) (3.3.23)
are Fredholm with index zero. (ii) If, in addition to being as in (3.1.13), the potential V is assumed to be of class C 1 near ∂Ω (hence, in particular, if supp (dV) ∩ ∂Ω = ⌀) then the operators p,δ
p,δ
(3.3.24)
zI + N l : Lnor (∂Ω, Λ l T M) → Lnor (∂Ω, Λ l T M),
p,d
p,d
(3.3.25)
zI + ̃ Ml :
(3.3.26)
zI + M l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M),
̃l : zI + N
p,δ Ltan (∂Ω, Λ l T M) p,d Lnor (∂Ω, Λ l T M)
→ →
p,δ Ltan (∂Ω, Λ l T M), p,d Lnor (∂Ω, Λ l T M),
(3.3.27)
are also Fredholm with index zero, and the following regularity statements hold: p
p,δ
Ker (zI + M l ; Ltan (∂Ω, Λ l T M)) = Ker (zI + M l ; Ltan (∂Ω, Λ l T M)), Ker (zI +
p N l ; Lnor (∂Ω, Λ l T M))
= Ker (zI +
p,d N l ; Lnor (∂Ω, Λ l T M)).
(3.3.28) (3.3.29)
Furthermore, the spaces p
p
Ker (zI + M l ; Ltan (∂Ω, Λ l T M)), p Ker (zI + ̃ M l ; L (∂Ω, Λ l T M)),
Ker (zI + N l ; Lnor (∂Ω, Λ l T M)), p ̃l ; Lnor Ker (zI + N (∂Ω, Λ l T M)),
tan
(3.3.30)
are all independent of p. (iii) If the potential V is a strictly positive constant then for each z ∈ ℂ \ (− 12 , 12 ) the p
p,δ
operator zI + M l is invertible both on Ltan (∂Ω, Λ l T M) and on Ltan (∂Ω, Λ l T M), while p p,d the operator zI + N l is invertible both on Lnor (∂Ω, Λ l T M) and on Lnor (∂Ω, Λ l T M). (iv) As a corollary of item (iii), if the potential V is a strictly positive constant then the spectral radii of the operator M l considered both p p,δ on Ltan (∂Ω, Λ l T M) and on Ltan (∂Ω, Λ l T M) are < 12 ,
(3.3.31)
the spectral radii of the operator N l considered both p p,d on Lnor (∂Ω, Λ l T M) and on Lnor (∂Ω, Λ l T M) are < 12 .
(3.3.32)
and
Consequently, if V is a strictly positive constant then −1
(± 12 I + M l )
∞
j
= − ∑ (∓2)j+1 M l j=0
(3.3.33)
3.3 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism

133
p
with convergence in the operator norm both in the space Ltan (∂Ω, Λ l T M) and in the p,δ space Ltan (∂Ω, Λ l T M), and (± 12 I + N l )
−1
∞
j
= − ∑ (∓2)j+1 N l
(3.3.34)
j=0 p
with convergence in the operator norm both in the space Lnor (∂Ω, Λ l T M) and in the p,d space Lnor (∂Ω, Λ l T M). Finally, all the above results are valid when the hypotheses on the underlying domain are relaxed to just asking that Ω is an εSKT domain for some ε > 0 is sufficiently small relative to the given integrability exponent p, the complex number z, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. In the case when integrability exponent p is near 2, the spectral parameter z is restricted to ℝ \ (− 12 , 12 ), and the set Ω is a Lipschitz subdomain of the manifold M, similar Fredholmness and invertibility results to those claimed in parts (i)–(iii) of Theorem 3.7 have been established in [86], by devising suitable Rellich type identities for differential forms. In the proof of Theorem 3.7, the following abstract index estimate (resembling a monotonicity property of the index on a nested scale of spaces; compare with [102, Lemma 11.9.21, p. 206]) will be useful. Lemma 3.8. Let X j , Y j , j = 0, 1, be two pairs of Banach spaces such that both inclusions X1 → X0 and Y1 → Y0 are continuous with dense range. If T is a linear operator which is Fredholm both from X0 into Y0 and from X1 into Y1 , then index (T : X1 → Y1 ) ≤ index (T : X0 → Y0 ).
(3.3.35)
Furthermore, (3.3.35) holds with equality if and only if Ker (T : X0 → Y0 ) = Ker (T : X1 → Y1 ) and Ker (T ∗ : Y0∗ → X0∗ ) = Ker (T ∗ : Y1∗ → X1∗ ).
(3.3.36)
Proof. Abbreviate A j := Ker (T : X j → Y j )
and
B j := Ker (T ∗ : Y j∗ → X ∗j ),
j = 0, 1.
(3.3.37)
The hypotheses on X j , Y j imply that Y0∗ embeds into Y1∗ , and X0∗ embeds into X1∗ . Hence, we have A1 ⊆ A0 and B0 ⊆ B1 which further entails dim A1 ≤ dim A0 and dim B0 ≤ dim B1 . Consequently, index (T : X1 → Y1 ) = dim A1 − dim B1 ≤ dim A0 − dim B0 = index (T : X0 → Y0 ),
(3.3.38)
proving (3.3.35). The last claim in the statement is also a simple consequence of (3.3.38).
134  3 Potentials for the Hodgede Rham Formalism on UR Domains
Proof of Theorem 3.7. All claims pertaining to Fredholmness with index zero are immediate consequences of Theorem 3.6 and basic functional analysis. Next, (3.3.28) is proved using Lemma 2.42 and Lemma 3.8. A similar argument works in the case of (3.3.29). In fact, this also gives the independence of p for the spaces listed in (3.3.30). To deal with the claims in item (iii) of the statement of the theorem, assume that V is a strictly positive constant. Granted this, we claim that the operator zI + M l is 2,δ injective on Ltan (∂Ω, Λ l T M) whenever z ∈ ℂ \ (− 12 , 12 ). To see that this is the case, let z l be as above and, seeking a contradiction, assume that there exists f ∈ L2,δ tan (∂Ω, Λ T M), such that f ≠ 0, with M l f = −zf . Set u := dSl f in M \ ∂Ω. Then, since [d, V] = 0 in this case, we have (∆HL − V)u = 0 in M \ ∂Ω by (3.2.5) and item (5) in Lemma 2.8. This and the fact that u has appropriate control at the boundary (i.e., (2.3.15) are satisﬁed relative to Ω± , deﬁned as in (3.2.2)) permits us to write the energy identities (2.3.16) corresponding to Ω+ and Ω− . Since du = 0 in M \ ∂Ω, the latter read n.t. n.t. ∫ (δu2 + Vu2 ) dVol = ∓ ∫⟨ν ∨ u∂Ω± , ν ∨ (ν ∧ (δu)∂Ω± )⟩ dσ. Ω±
(3.3.39)
∂Ω±
On the other hand, the jumprelations (3.2.25), (3.2.14) imply n.t. ν ∨ u∂Ω± = (∓ 12 − z)f
and
n.t. n.t. ν ∧ (δu)∂Ω+ = ν ∧ (δu)∂Ω− .
(3.3.40)
At this stage, let us consider the special values z ∈ {− 12 , 12 } in detail. If, say, z = 12 , n.t. then ν ∨ u∂Ω− = 0 which then forces u = 0 in Ω− by the version of (3.3.39) written n.t. n.t. for Ω− . In turn, this further implies ν ∧ (δu)∂Ω+ = ν ∧ (δu)∂Ω− = 0 which, when used back in the version of (3.3.39) written for Ω+ , entails u = 0 in Ω+ . However, in light n.t. of (3.3.40) this would necessarily imply that f = −ν ∨ u∂Ω+ = 0, a possibility ruled out from the start. This contradiction shows that the operator 12 I + M l is injective on 1 l L2,δ tan (∂Ω, Λ T M). Moreover, a similar analysis shows that the operator − 2 I + M l is also 2,δ injective on Ltan (∂Ω, Λ l T M). Having established the desired conclusion in the case when z = ± 12 , let us now consider the scenario when z ∈ ℂ \ [− 12 , 12 ]. Concretely, denote by μ± ∈ [0, ∞), respectively, the solid integrals in the lefthand side of (3.3.39) and observe that identities (3.3.40) may be employed in order to rewrite (3.3.39) as n.t. μ± = ( 12 ∓ z) ∫ ⟨f, ν ∨ (ν ∧ (δu)∂Ω )⟩ dσ.
(3.3.41)
∂Ω
Note that μ+ and μ− cannot vanish simultaneously since, in this case, it is easy to see from (3.3.39), the strict positivity of V, and the jumprelations in (3.2.25), that the function f must be zero, which is something excluded from the outset. Bearing this in mind, it follows from (3.3.41) and simple algebra that z = −(μ+ − μ− )/2(μ+ + μ− ) ∈ ℝ. Hence, on the one hand z ∈ ℝ \ [− 12 , 12 ], while on the other hand z =
1 μ+ − μ− 1 ≤ , 2 μ+ + μ− 2
(3.3.42)
3.3 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism

135
two mutually excluding conclusions. This contradiction goes to show that for each complex number z ∈ ℂ \ (− 12 , 12 ) the operator zI + M l is injective when acting from l L2,δ tan (∂Ω, Λ T M) into itself. Since we already know that in this setting the operator in 2,δ question is also Fredholm with index zero, its invertibility on Ltan (∂Ω, Λ l T M) then follows. Consider next the case when p > 2. Once again, assume z ∈ ℂ \ (− 12 , 12 ). Since p,δ
∂Ω is bounded, the kernel of zI + M l on Ltan (∂Ω, Λ l T M) is a subset of the kernel of l zI + M l on L2,δ tan (∂Ω, Λ T M), which we have just shown to be trivial. Consequently, p,δ the operator zI + M l is injective, hence ultimately invertible, on Ltan (∂Ω, Λ l T M) when p > 2. In summary, so far we know that p,δ
the operator zI + M l is an isomorphism of Ltan (∂Ω, Λ l T M) if p ∈ [2, ∞), z ∈ ℂ \ (− 12 , 12 ), and V is a strictly positive constant.
(3.3.43)
To proceed, retain the assumptions that z ∈ ℂ \ (− 12 , 12 ), and V is a positive constant. By combining (3.3.43), (2.4.67), and (2.4.70) we obtain that zI + M l has dense p range on Ltan (∂Ω, Λ l T M) if p ∈ [2, ∞). Since from the ﬁrst part of the proof we know that this operator is also Fredholm with index zero, we may conclude that p
the operator zI + M l is an isomorphism of Ltan (∂Ω, Λ l T M) if p ∈ [2, ∞), z ∈ ℂ \ (− 12 , 12 ), and V is a strictly positive constant.
(3.3.44)
Based on this, duality, (3.2.29), and (3.2.24) we deduce that zI + N l is an isomorphism p of Lnor (∂Ω, Λ l T M) if p ∈ (1, 2], z ∈ ℂ \ (− 12 , 12 ), and V is a strictly positive constant. Granted this, via Hodge duality (cf. (3.2.28) and (2.4.47)) we arrive at the conclusion p that zI + M l is an isomorphism of Ltan (∂Ω, Λ l T M) if p ∈ (1, 2], z ∈ ℂ \ (− 12 , 12 ), and V is a positive constant. In concert with (3.3.44) this shows that p
the operator zI + M l is an isomorphism of Ltan (∂Ω, Λ l T M) if p ∈ [2, ∞) and z ∈ ℂ \ (− 12 , 12 ), and V is a positive constant.
(3.3.45)
Having proved this, it follows that if V is a strictly positive constant then for each p ∈ p,δ [2, ∞) and z ∈ ℂ \ (− 12 , 12 ) the operator zI + M l is injective on Ltan (∂Ω, Λ l T M), thus invertible, since this is already known to be Fredholm with index zero in this setting. The fact that when V is a strictly positive constant the operator zI + N l is also invertible p p,d on Lnor (∂Ω, Λ l T M) and on Lnor (∂Ω, Λ l T M) for each p ∈ (1, ∞) and z ∈ ℂ \ (− 12 , 12 ) is seen from what we have just proved and Hodge duality (cf. (3.2.28) and (2.4.48)). Finally, assume that Ω is an εSKT domain for some ε > 0 is sufficiently small relative to the given p, the complex number z, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. In this scenario, granted the observation made in the very last part of the statement of Theorem 3.6, the above line of reasoning readily adapts to produce the same type of conclusions as above. Our ﬁnal result in this section improves on the observation that the four nullspaces listed in (3.3.30) are independent of p.
136  3 Potentials for the Hodgede Rham Formalism on UR Domains Theorem 3.9. Let Ω ⊂ M be a regular SKT domain and suppose the potential V is as in (3.1.13). Pick some degree l ∈ {0, 1, . . . , n}, ﬁx an arbitrary complex number z ∈ ℂ \ {0}, and assume that 1 < p ≤ q < ∞. Then the following improved integrability results hold: p
f ∈ Ltan (∂Ω, Λ l T M) and q } ⇒ f ∈ Ltan (∂Ω, Λ l T M), q (zI + M l )f ∈ Ltan (∂Ω, Λ l T M)
(3.3.46)
p
f ∈ Ltan (∂Ω, Λ l T M) and q } ⇒ f ∈ Ltan (∂Ω, Λ l T M), q (zI + ̃ M l )f ∈ Ltan (∂Ω, Λ l T M)
(3.3.47)
p
g ∈ Lnor (∂Ω, Λ l T M) and q } ⇒ g ∈ Lnor (∂Ω, Λ l T M), q (zI + N l )g ∈ Lnor (∂Ω, Λ l T M)
(3.3.48)
p
g ∈ Lnor (∂Ω, Λ l T M) and q } ⇒ g ∈ Lnor (∂Ω, Λ l T M). q ̃l )g ∈ Lnor (∂Ω, Λ l T M) (zI + N
(3.3.49)
If, in addition, the potential V is of class C 1 near ∂Ω (hence, in particular, if supp (dV) ∩ ∂Ω = ⌀) then the following regularity results hold: p
f ∈ Ltan (∂Ω, Λ l T M) and
q,δ
(3.3.50)
} ⇒ g ∈ Lnor (∂Ω, Λ l T M),
q,d
(3.3.51)
f ∈ Ltan (∂Ω, Λ l T M) and q,δ } ⇒ f ∈ Ltan (∂Ω, Λ l T M), q,δ (zI + ̃ M l )f ∈ L (∂Ω, Λ l T M)
(3.3.52)
} ⇒ f ∈ Ltan (∂Ω, Λ l T M),
q,δ
(zI + M l )f ∈ Ltan (∂Ω, Λ l T M) p
g ∈ Lnor (∂Ω, Λ l T M) and q,d
(zI + N l )g ∈ Lnor (∂Ω, Λ l T M) p
tan
p
g ∈ Lnor (∂Ω, Λ l T M) and q,d } ⇒ g ∈ Lnor (∂Ω, Λ l T M). q,d l ̃ (zI + N l )g ∈ Lnor (∂Ω, Λ T M)
(3.3.53)
In fact, all the above results hold when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the targeted exponents p, q, the complex number z, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. In the proof of Theorem 3.9 the following abstract “regularity” result is going to be useful. In its proof, we follow the convention that given a Banach space X, a subspace A of X and a subspace B of X ∗ the annihilators A⊥ and ⊥ B of A, B are deﬁned as A⊥ := {Λ ∈ X ∗ : Λ(x) = 0 for all x ∈ A},
(3.3.54)
⊥
(3.3.55)
B := {x ∈ X : Λ(x) = 0 for all Λ ∈ B}.
Lemma 3.10. Let X0 , X1 , be two Banach spaces such that X0 → X1 continuously and densely. Assume that T is a linear operator which is Fredholm both from X 0 into X0 and
3.3 Fredholm Theory for Layer Potentials in the Hodgede Rham Formalism

137
from X1 into X1 , with the property that index (T : X1 → X1 ) = index (T : X0 → X0 ).
(3.3.56)
Then the following abstract regularity result holds: x1 ∈ X1
Tx1 ∈ X0 ⇒ x1 ∈ X0 .
and
(3.3.57)
As a corollary, X0 ∩ Im (T : X1 → X1 ) = Im (T : X0 → X0 ).
(3.3.58)
Proof. From Lemma 3.8 we know that Ker (T : X0 → X0 ) = Ker (T : X1 → X1 ), ∗
Ker (T :
X0∗
→
X0∗ )
∗
= Ker (T :
X1∗
→
X1∗ ).
(3.3.59) (3.3.60)
Moreover, Im (T : X0 → X0 ) is closed in X0 and Im (T : X1 → X1 ) is closed in X1 . Based on this and (3.3.60), standard functional analysis (cf., e.g., [108, § 4.6, pp. 95–99]) then gives X1 Im (T : X1 → X1 ) = Im (T : X1 → X1 ) = ⊥ (Im (T : X1 → X1 )⊥ ) = ⊥ Ker (T ∗ : X1∗ → X1∗ ) = {y ∈ X1 : Λ(x) = 0 for each Λ ∈ Ker (T ∗ : X1∗ → X1∗ )} = {y ∈ X1 : Λ(x) = 0 for each Λ ∈ Ker (T ∗ : X0∗ → X0∗ )}.
(3.3.61)
Hence, if x1 ∈ X1 is such that y1 := Tx1 ∈ X0 then from (3.3.61) we deduce that y1 ∈ X0 ∩ Im (T : X1 → X1 ) = {y ∈ X0 : Λ(x) = 0 for each Λ ∈ Ker (T ∗ : X0∗ → X0∗ )} = ⊥ Ker (T ∗ : X0∗ → X0∗ ) = ⊥ (Im (T : X0 → X0 )⊥ ) = Im (T : X0 → X0 )
X0
= Im (T : X0 → X0 ).
(3.3.62)
Consequently, there exists x0 ∈ X0 such that Tx0 = y1 . Thus, x1 − x0 ∈ Ker (T : X1 → X1 ) hence, ultimately, x0 − x1 ∈ Ker (T : X0 → X0 ) by (3.3.59). As such, x1 − x0 ∈ X0 hence x1 ∈ X0 , proving (3.3.57). Finally, (3.3.58) is a direct consequence of (3.3.57). We now turn our attention to the Proof of Theorem 3.9. Applying (3.3.57) with T := zI + M l , q
X0 := Ltan (∂Ω, Λ l T M), p
X1 := Ltan (∂Ω, Λ l T M),
(3.3.63)
x1 := f, yields (3.3.46), on account of Theorem 3.6 and (2.4.69). The implications in (3.3.47)– (3.3.49) are proved similarly.
138  3 Potentials for the Hodgede Rham Formalism on UR Domains
As regards the regularity result in (3.3.50), note ﬁrst that thanks to (3.3.46) there is no loss of generality in assuming that q = p. Granted this and keeping in mind Lemma 2.42, another reference to (3.3.57), this time taking T := zI + M l , p,δ
X0 := Ltan (∂Ω, Λ l T M), p
X1 := Ltan (∂Ω, Λ l T M),
(3.3.64)
x1 := f, yields (3.3.50). Moreover, (3.3.51)–(3.3.53) are proved similarly. Finally, the very last claim in the statement of the current theorem may be justiﬁed using the same reasoning, with the help of the very last property recorded in the statement of Theorem 3.6.
4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains In this chapter we set up layer potentials associated with the covariant derivative acting on differential forms (or connection layer potentials for short). We supplement the study of Sl in (3.0.1) with the “double layer” potential Dl , acting on f ∈ L p (∂Ω, Λ l T M), deﬁned by Dl f(x) = ∫ ⟨∇ν♯ (y) Γ l (x, y), f(y)⟩ dσ(y),
x ∈ M \ ∂Ω,
(4.0.1)
∂Ω
ν♯ (y)
where ∈ T y M corresponds to ν(y) ∈ T y∗ M via the metric tensor, and Γ l (x, y) is as in Chapter 3. Results here include nontangential maximal function estimates on Dl f , and also on ∇Sl f , and formulas for nontangential limits of these operators, such as n.t.
Dl f ∂Ω± = (± 12 I + K l )f,
(4.0.2)
where K l is the principal value singular integral operator associated to (4.0.1). Also, ∇ν♯ (Sl f Ω± ) = (∓ 12 I + K ⊤ l )f,
(4.0.3)
where K ⊤ l is the real transpose of K l . We establish boundedness K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), p
p
K l : L1 (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M),
(4.0.4)
the ﬁrst when Ω is a UR domain, and the second if in addition Ω satisﬁes a twosided local John condition. Under this additional condition, we also obtain nontangential maximal function estimates on ∇Dl f , yielding p
∇ν♯ Dl : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), p
∇ν♯ Dl : L p (∂Ω, Λ l T M) → L−1 (∂Ω, Λ l T M),
(4.0.5)
and we establish the useful identities S l (∇ν♯ Dl ) = ( 12 I + K l )(− 12 I + K l ), (∇ν♯ Dl )S l =
( 12 I
+
1 K⊤ l )(− 2 I
+
K⊤ l ),
and
(4.0.6)
on L p (∂Ω, Λ l T M). In § 4.2 we extend the latter boundedness in (4.0.4) to the class of general UR domains, when M carries a C ∞ metric tensor. In § 4.3, we demonstrate compactness in (4.0.4) when Ω is a regular SKT domain. One ingredient in the analysis of Dl and K l is the the Weitzenböck formula, which asserts¹ that on differential forms of arbitrary degree on M the HodgeLaplacian may 1 see the discussion in § 9.3
140  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
be expressed as − ∆HL = ∇∗ ∇ + Ric
(4.0.7)
where Ric is the socalled Weitzenböck operator (cf. Deﬁnition 9.6). More speciﬁcally, Ric is a curvature term of order zero (depending linearly on the Riemann curvature, via real coefficients) that preserves lforms, and is selfadjoint. In addition, Ric vanishes identically on scalar functions and coincides with the classical Ricci tensor when acting on 1forms on M (which explains the present choice of notation).
4.1 The Deﬁnition and Mapping Properties of the Double Layer Based on (9.1.44) and (9.1.38) we compute what we shall refer to as the connection conormal derivative, namely² i Sym (∇∗, ξ)∇u = i Sym (∇∗, ξ)(dx j ⊗ ∇∂ j u) = ⟨ξ, dx j ⟩∇∂ j u = ξ k g kj ∇∂ j u,
(4.1.1)
for every differential formvalued function u and every covector ξ = ξ j dx j ∈ T ∗ M. Here and elsewhere the summation convention over repeated indices is assumed. Since ξ k g kj ∂ j = ξ ♯ , the metric identiﬁcation of ξ ∈ T ∗ M with a tangent vector (cf. (9.1.2), (9.1.3)), it follows from (4.1.1) that i Sym (∇∗, ξ)∇u = ∇ξ ♯ u.
(4.1.2)
Moreover, if u is locally expressed as u = ∑I=l u I dx I and ξ = ξ j dx j , then (4.1.1) and (9.1.49) also give i Sym (∇∗, ξ)∇u = ξ r g rs (∂ s u I + γ s u J ) dx I , IJ
(4.1.3)
where the connection coefficients γ IJ s are the scalar functions deﬁned by the requirement that (cf. (9.1.47))
IJ
∇∂ s dx J = ∑ γ s dx I
for all J and s.
(4.1.4)
I=l
In view of (the ﬁrst line in) (9.2.20), this explicitly gives n
n
k=1
j,k=1
jJ
k k γ IJ s = −( ∑ Γ sk )δ IJ + ∑ Γ js ε kI
for all I, J and s,
k where Γ js are the Christoffel symbols (cf. (9.1.6)).
2 throughout, Sym (P, ξ) denotes the principal symbol of P at the covector ξ
(4.1.5)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
141
We are ready now to deﬁne what we shall call the connection double layer (both as a boundarytodomain and as a boundarytoboundary operator) on a domain Ω ⊂ M and study its properties. We begin by introducing the boundarytodomain version of this double layer. Deﬁnition 4.1. Having picked a function V as in (3.1.13), ﬁx l ∈ {0, 1, . . . , n} and bring in the Schwartz kernel, Γ l (x, y) from (3.1.24), of (∆HL − V)−1 considered on lforms (cf. (3.1.5)). Also, let ∇ stand for the LeviCivita connection on M, whose action is naturally extended to differential forms. In this context, given an Ahlfors regular domain Ω ⊂ M with outward unit conormal ν ∈ T ∗ M and surface measure σ := H n−1 ⌊∂Ω, deﬁne the connection double layer potential operator Dl acting on arbitrary functions f ∈ L1 (∂Ω, Λ l T M) according to Dl f(x) := ∫ ⟨i Sym (∇∗, ν(y))∇y Γ l (x, y), f(y)⟩ dσ(y),
x ∈ M \ ∂Ω.
(4.1.6)
∂Ω
Above, ⟨⋅, ⋅⟩ stands for the real³ pairing in Λ l T M, and the operator i Sym (∇∗, ν(y))∇y acts on the rows of Γ l (x, y) when the latter is identiﬁed with a matrix in local coordinate patches. More speciﬁcally, if Γ l (x, y) is locally expressed as in (3.1.24), then the integral kernel of the connection double layer potential operator (4.1.6) becomes
IJ
∑ dx I ⊗ [i Sym (∇∗, ν(y))∇y ( ∑ Γ l (x, y) dy J )].
I=l
(4.1.7)
J=l
In the context considered in Deﬁnition 4.1, the connection double layer potential operator satisﬁes 1 Dl f ∈ Cloc (M \ ∂Ω, Λ l T M),
∀ f ∈ L1 (∂Ω, Λ l T M),
(4.1.8)
∀ f ∈ L1 (∂Ω, Λ l T M).
(4.1.9)
and, in the sense of distributions, (∆HL − V)Dl f = 0 in M \ ∂Ω,
Thus, from this point of view, the connection double layer potential is a mechanism for generating lots of nullsolutions for the differential operator ∆HL − V, away from ∂Ω. Going further, inspired by the structure of the integral kernel of the connection double layer potential operator, we also make the following deﬁnition. Deﬁnition 4.2. Let Ω ⊂ M be an Ahlfors regular domain with outward unit conormal ν ∈ T ∗ M and let ∇ stand for the LeviCivita connection on M, whose action is naturally extended to differential forms. For a given form u ∈ C 1 (Ω, Λ l T M), where
3 as opposed to sesquilinear (hence, there is no complex conjugation involved)
142  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains l ∈ {0, 1, . . . , n}, then deﬁne the boundary conormal derivative as⁴ n.t. ∇ν♯ u := i Sym (∇∗, ν)(∇u)∂Ω on ∂Ω,
(4.1.10)
whenever this is meaningful in a pointwise, a.e. sense (with respect to the surface measure). That these objects interact naturally with one another is apparent from the integral identities established in our next two proposition. Proposition 4.3. Let the potential V be as in (3.1.13) and assume that Ω ⊂ M is an Ahlfors regular domain, with surface measure σ := H n−1 ⌊∂Ω and outward unit conormal ν ∈ L∞ (∂Ω, T ∗ M). Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n}. With Γ l (x, y) associated with V as in (3.1.24), let Dl be the connection double layer from Deﬁnition 4.1 and let Sl be the single layer potential operator from (3.2.3). Then each differential form u satisfying 1,1 u ∈ Hloc (Ω, Λ l T M),
N u, N(∇u) ∈ L1 (∂Ω),
(∆HL − V)u ∈ L p (Ω, Λ l T M) for some p > n/2, n.t. n.t. u∂Ω and (∇u)∂Ω exist at σa.e. point on ∂Ω,
(4.1.11)
has the integral representation formula n.t. u(x) = Dl (u∂Ω )(x) − Sl (∇ν♯ u)(x)
(4.1.12)
+ ∫⟨Γ l (x, y), ((∆HL − V)u)(y)⟩ dVol(y),
for each x ∈ Ω.
Ω
Furthermore, n.t. 0 = Dl (u∂Ω )(x) − Sl (∇ν♯ u)(x)
(4.1.13)
+ ∫⟨Γ l (x, y), ((∆HL − V)u)(y)⟩ dVol(y),
for each x ∈ M \ Ω.
Ω
As a corollary, each differential form satisfying u ∈ C 1 (Ω, Λ l T M),
(∆HL − V)u = 0 in Ω,
N u, N(∇u) ∈ L (∂Ω), and such that there 1
(4.1.14)
n.t. n.t. exist u∂Ω and (∇u)∂Ω at σa.e. point on ∂Ω, admits the layer potential integral representation formula n.t. u = Dl (u∂Ω ) − Sl (∇ν♯ u) in Ω.
(4.1.15)
n.t. 0 = Dl (u∂Ω ) − Sl (∇ν♯ u) in M \ Ω.
(4.1.16)
In addition,
4 note that ν being the outward unit conormal to Ω makes ν♯ the outward unit normal vector to Ω
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
143
Below we isolate a useful ingredient in the proof of this proposition, itself a particular case of Lemma 2.7. Lemma 4.4. Consider integrability exponents 1 ≤ p, q, p , q ≤ ∞ such that q ≤ p and 1/p + 1/p = 1 = 1/q + 1/q , and ﬁx an arbitrary degree l ∈ {0, 1, . . . , n}. Then for each open set O ⊆ M the integration by parts formula ∫⟨∇u, υ⟩ dVol = ∫⟨u, ∇∗ υ⟩ dVol O
(4.1.17)
O
holds whenever p
u ∈ Lloc (O, Λ l T M), q
∇u ∈ Lloc (O, T ∗ M ⊗ Λ l T M),
υ ∈ Lloc (O, T ∗ M ⊗ Λ l T M),
q
p
∇∗ υ ∈ Lloc (O, Λ l T M),
(4.1.18)
and either u, or υ, is compactly supported in O. Having recorded the above integration by parts formula, we now turn our attention to presenting the proof of Proposition 4.3. Proof of Proposition 4.3. As regards the integral representation formula (4.1.12), consider a differential form u enjoying the properties listed in (4.1.11). Then, thanks to (2.2.45), (2.1.113), and the hypotheses on u, we have 2,q
1,∞ u ∈ Hloc (Ω, Λ l T M) ∩ Hloc (Ω, Λ l T M),
for some q > n/2.
(4.1.19)
To proceed, ﬁx an arbitrary point x ∈ Ω and consider the mapping F⃗ : Ω → T M ⊗ Λ l T x M
(4.1.20)
deﬁned uniquely at each y ∈ Ω by the requirement that for every ξ ∈ T y∗ M we have ⃗
T y∗ M (ξ, F(y))T y M
= ⟨i Sym (∇∗, ξ)∇y Γ l (x, y), u(y)⟩y − ⟨Γ l (x, y), i Sym (∇∗, ξ)(∇u)(y)⟩y
(4.1.21)
where, in the present context, ⟨⋅, ⋅⟩y denotes the pairing in Λ l T y M. Note that F⃗ is unambiguously deﬁned since the righthand side of (4.1.21) is linear in ξ . Upon recalling 1,∞ that u ∈ Hloc (Ω, Λ l T M), it follows from (3.1.18) that F⃗ ∈ L1loc (Ω, Λ l T x M ⊗ T M). In fact, we locally have F⃗ = F j ∂ j with F j (y) = ⟨i Sym(∇∗, dx j )∇y Γ l (x, y), u(y)⟩y − ⟨Γ l (x, y), i Sym(∇∗, dx j )(∇u)(y)⟩y .
(4.1.22)
144  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains For any given scalar function ψ ∈ C 02 (Ω) we may then compute ⃗
D (Ω) (div F, ψ)D(Ω)
= − D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T y M ⟨(grad ψ)(y), F(y)⟩ T y M dVol(y) Ω
⃗ = − ∫ T y∗ M ((grad ψ)♭ (y), F(y)) T y M dVol(y) Ω
⃗ = − ∫ T y∗ M (dψ(y), F(y)) T y M dVol(y) Ω
= − ∫⟨i Sym (∇∗, dψ)∇Γ l (x, ⋅), u⟩ dVol Ω
+ ∫⟨Γ l (x, ⋅), i Sym (∇∗, dψ)∇u⟩ dVol,
(4.1.23)
Ω
where the last equality uses (2.3.5) with ξ := dψ(y) ∈ T y∗ M. Let us work on the penultimate integral in (4.1.23). Making use of the transposition law (2.1.65) and the commutator formula (2.1.69), thanks to (4.1.19), (3.1.21), and Lemma 4.4, we may successively transform the aforementioned integral as follows: − ∫⟨i Sym (∇∗, dψ)∇Γ l (x, ⋅), u⟩ dVol = ∫⟨∇Γ l (x, ⋅), i Sym(∇, dψ)u⟩ dVol Ω
Ω
= ∫⟨Γ l (x, ⋅), ∇∗ (i Sym(∇, dψ)u)⟩ dVol Ω
= − ∫⟨Γ l (x, ⋅), ∇∗ [∇, ψ]u⟩ dVol.
(4.1.24)
Ω
The commutator formula (2.1.69) also permits us to rewrite the last integral in (4.1.23) as ∫⟨Γ l (x, ⋅), i Sym (∇∗, dψ)∇u⟩ dVol = − ∫⟨Γ l (x, ⋅), [∇∗, ψ]∇u⟩ dVol. Ω
(4.1.25)
Ω
At this stage, the idea is to avail ourselves of a general commutator identity to the effect that − A[B, C] − [A, C]B = −[AB, C] (4.1.26) for A := ∇∗ , B := ∇, C := ψ, and to observe that Weitzenböck’s formula (cf. (4.0.7)) implies − [∇∗ ∇, ψ] = [(∆HL − V) + (V + Ric), ψ] = [∆HL − V, ψ]. (4.1.27)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
145
Mindful of the fact that Γ l (x, ⋅) is a fundamental solution with mass at x for the operator ∆HL − V, we then deduce from (4.1.23)–(4.1.27) that ⃗
D (Ω) (div F, ψ)D(Ω)
= ∫⟨Γ l (x, ⋅), [∆HL − V, ψ]u⟩ dVol Ω
= ∫⟨Γ l (x, ⋅), (∆HL − V)(ψu)⟩ dVol Ω
− ∫ ψ⟨Γ l (x, ⋅), (∆HL − V)u⟩ dVol Ω
= (ψu)(x) − ∫ ψ⟨Γ l (x, ⋅), (∆HL − V)u⟩ dVol Ω
= D (Ω) (u(x)δ x − ⟨Γ l (x, ⋅), (∆HL − V)u⟩, ψ)
D(Ω)
(4.1.28)
where δ x is the Dirac distribution with mass at x. From this we ultimately conclude that div F⃗ = u(x)δ x − ⟨Γ l (x, ⋅), (∆HL − V)u⟩ in D (Ω). (4.1.29) In particular, thanks to (3.1.37) and the fact that (∆HL − V)u ∈ L p (Ω, Λ l T M) for some p > n/2, we have div F⃗ ∈ V ⊗ E (Ω) + V ⊗ L1 (Ω), (4.1.30) where V stands for the ﬁnite dimensional vector space Λ l T x M. Moreover, from (4.1.21), n.t. (3.1.18), and the last line in (4.1.14) we see that the nontangential trace F⃗ ∂Ω exists σa.e. on ∂Ω and that for σa.e. y ∈ ∂Ω we have ⃗ n.t. T y∗ M (ν(y), ( F ∂Ω )(y))
Ty M
n.t. = ⟨i Sym (∇∗, ν(y))∇y Γ l (x, y), (u∂Ω )(y)⟩
y
n.t. − ⟨Γ l (x, y), i Sym (∇ , ν(y))(∇u)∂Ω (y)⟩ y ∗
(4.1.31)
as vectors in V. In addition, by choosing the truncation parameter ρ > 0 smaller than dist (x, ∂Ω) and relying on (3.1.18), we may estimate N ρ (F)⃗ ≤ C N u + C N(∇u) pointwise on ∂Ω which, by the second line in (4.1.14), places N ρ (F)⃗ in L1 (∂Ω). Having also checked this, Theorem 9.67 applies (for the ﬁnite dimensional vector ﬁeld V := Λ l T x M) and gives that u(x) −∫⟨Γ l (x, ⋅), (∆HL − V)u⟩ dVol = (C 1 (Ω))∗ (div F,⃗ 1)C 1 (Ω) = ∫ b
b
Ω
⃗ n.t. T ∗ M (ν, F ∂Ω )TM dσ
∂Ω
n.t. = ∫ ⟨i Sym (∇ , ν(y))∇y Γ l (x, y), (u∂Ω )(y)⟩y dσ(y) ∗
∂Ω
n.t. − ∫ ⟨Γ l (x, y), i Sym (∇∗, ν(y))(∇u)∂Ω (y)⟩y dσ(y) ∂Ω
n.t. = Dl (u∂Ω )(x) − Sl (∇ν♯ u)(x),
(4.1.32)
146  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
where the last equality follows upon recalling the deﬁnitions of the connection double layer from (4.1.6) and the connection conormal from (4.1.10). This establishes (4.1.12). To prove formula (4.1.13), ﬁx now an arbitrary point x ∈ M \ Ω then deﬁne F⃗ as in (4.1.20) and (4.1.21) for this choice of x. In this scenario, Γ l (x, ⋅) ∈ C 1 (Ω) and we have (∆HL − V)[Γ l (x, ⋅)] = 0 in Ω. Consequently, in lieu of (4.1.29) we now obtain div F⃗ = −⟨Γ l (x, ⋅), (∆HL − V)u⟩ in D (Ω).
(4.1.33)
Bearing this in mind and relying on Theorem 9.67 as before, in place of (4.1.12) we now arrive at (4.1.13). At this stage, there remains to observe that, granted (4.1.14), formulas (4.1.15), (4.1.16) become consequences of (4.1.12) and (4.1.13), respectively. This concludes the proof of the proposition. Here is the second proposition mentioned earlier, containing a natural Greentype formula involving the conormal derivative from (4.1.10) proved under minimal regularity assumptions. As usual, ∇ stands for the LeviCivita connection on M, whose action is naturally extended to differential forms. Proposition 4.5. Assume Ω ⊂ M is an Ahlfors regular domain with outward unit conormal ν and surface measure σ. Pick an arbitrary scalarvalued potential V ∈ L n (Ω). Finally, select an arbitrary degree l ∈ {0, 1, . . . , n}, along with two integrability exponents p, p ∈ [1, ∞] such that 1/p + 1/p = 1. Then for any two differential forms u, υ satisfying u ∈ C 1 (Ω, Λ l T M), υ ∈ C 1 (Ω, Λ l T M),
(∆HL − V)u ∈ L np /(np −n+1) (Ω, Λ l T M), (∆HL − V)υ ∈ L np/(np−n+1) (Ω, Λ l T M),
(4.1.34)
N u, N(∇u) ∈ L p (∂Ω), N υ, N(∇υ) ∈ L p (∂Ω),
n.t. n.t. n.t. n.t. u∂Ω , (∇u)∂Ω , υ∂Ω , (∇υ)∂Ω exist σa.e. on ∂Ω, one has ∫⟨∆HL u, υ⟩ dVol − ∫⟨u, ∆HL υ⟩ dVol Ω
Ω
= ∫⟨(∆HL − V)u, υ⟩ dVol − ∫⟨u, (∆HL − V)υ⟩ dVol Ω
Ω
n.t. n.t. = ∫ ⟨∇ν♯ u, υ∂Ω ⟩ dσ − ∫ ⟨u∂Ω , ∇ν♯ υ⟩ dσ, ∂Ω
∂Ω
where all integrals are absolutely convergent.
(4.1.35)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
147
In particular, if u, υ are two differential such that u, υ ∈ C 1 (Ω, Λ l T M), (∆HL − V)u = (∆HL − V)υ = 0 in Ω,
N u, N(∇u) ∈ L p (∂Ω), N υ, N(∇υ) ∈ L p (∂Ω), and there exist
(4.1.36)
n.t. n.t. n.t. n.t. u∂Ω , (∇u)∂Ω , υ∂Ω , (∇υ)∂Ω at σa.e. point on ∂Ω, then
n.t. n.t. ∫ ⟨∇ν♯ u, υ∂Ω ⟩ dσ = ∫ ⟨u∂Ω , ∇ν♯ υ⟩ dσ. ∂Ω
(4.1.37)
∂Ω
l Proof. Since from assumptions we have u, υ ∈ L∞ loc (Ω, Λ T M) and np /(np −n+1)
∆HL u = (∆HL − V)u + Vu ∈ Lloc
np/(np−n+1)
∆HL υ = (∆HL − V)υ + Vυ ∈ Lloc
(Ω, Λ l T M),
(Ω, Λ l T M),
(4.1.38)
from (2.1.113) (used with V = 0) we conclude that 2, np /(np −n+1)
u ∈ Hloc
(Ω, Λ l T M)
and
2, np/(np−n+1)
υ ∈ Hloc
(Ω, Λ l T M).
(4.1.39)
Consider next the vector ﬁeld F⃗ : Ω → T M uniquely determined by the requirement that for each x ∈ Ω and each ξ ∈ T x∗ M we have ⃗
T x∗ M (ξ, F(x))T x M
= ⟨i Sym (∇∗, ξ)(∇u)(x), υ(x)⟩x − ⟨u(x), i Sym (∇∗, ξ)(∇υ)(x)⟩x .
(4.1.40)
Then the conditions on u, υ ensure that F⃗ ∈ L1loc (Ω, T M). Also, N(F)⃗ ∈ L1 (∂Ω), and n.t. F⃗ ∂Ω exists σa.e. on ∂Ω. In fact, we may locally express F⃗ = F j ∂ j with F j (x) = ⟨i Sym(∇∗, dx j )(∇u)(x), υ(x)⟩x − ⟨u(x), i Sym(∇∗, dx j )(∇υ)(x)⟩x .
(4.1.41)
Given an arbitrary scalar function ψ ∈ C 02 (Ω) we next compute ⃗
D (Ω) (div F, ψ)D(Ω)
= − D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T y M ⟨(grad ψ)(y), F(y)⟩ T y M dVol(y) Ω
⃗ = − ∫ T y∗ M ((grad ψ)♭ (y), F(y)) T y M dVol(y) Ω
⃗ = − ∫ T y∗ M (dψ(y), F(y)) T y M dVol(y) Ω
= − ∫⟨i Sym (∇∗, dψ)∇u, υ⟩ dVol Ω
+ ∫⟨u, i Sym (∇∗, dψ)∇υ⟩ dVol, Ω
(4.1.42)
148  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains where the last equality employes (2.3.5) for the choice ξ := dψ(y) ∈ T y∗ M. To proceed, observe that Weitzenböck’s formula (4.0.7) entails − ∇∗ ∇ = (∆HL − V) + (V + Ric)
(4.1.43)
Bearing this in mind together with the commutator formula (2.1.69), and by relying on Lemma 4.4 (whose applicability in the present setting is justiﬁed with the help of (4.1.39)) we may successively transform the penultimate integral in (4.1.42) as follows: − ∫⟨i Sym (∇∗, dψ)∇u, υ⟩ dVol = ∫⟨[∇∗, ψ]∇u, υ⟩ dVol Ω
Ω
= ∫⟨∇∗ (ψ∇u), υ⟩ dVol − ∫ ψ⟨∇∗ ∇u, υ⟩ dVol Ω
Ω
= ∫ ψ⟨∇u, ∇υ⟩ dVol + ∫ ψ⟨(∆HL − V)u, υ⟩ dVol Ω
Ω
+ ∫ ψ⟨(V + Ric)u, υ⟩ dVol
(4.1.44)
Ω
Together with (4.1.43), the commutator formula (2.1.69) also permits us to rewrite the last integral in (4.1.42) as ∫⟨u, i Sym (∇∗, dψ)∇υ⟩ dVol = − ∫⟨u, [∇∗, ψ]∇υ⟩ dVol Ω
Ω
= − ∫⟨u, ∇∗ (ψ∇υ)⟩ dVol + ∫ ψ⟨u, ∇∗ ∇υ⟩ dVol Ω
Ω
= − ∫ ψ⟨∇u, ∇υ⟩ dVol − ∫ ψ⟨u, (∆HL − V)υ⟩ dVol Ω
Ω
− ∫ ψ⟨u, (V + Ric)υ⟩ dVol.
(4.1.45)
Ω
In view of the symmetry of Ric, by combining (4.1.42), (4.1.44), and (4.1.45) we arrive at div F⃗ = ⟨(∆HL − V)u, υ⟩ − ⟨u, (∆HL − V)υ⟩ in D (Ω). (4.1.46) Since from (2.2.50) and the assumptions on u, υ we have u ∈ L np/(n−1) (Ω, Λ l T M)
and
υ ∈ L np /(n−1) (Ω, Λ l T M),
(4.1.47)
it follows from (4.1.46), (4.1.47), as well as the integrability conditions imposed on (∆HL − V)u and (∆HL − V)υ in (4.1.34) that div F⃗ ∈ L1 (Ω).
(4.1.48)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
149
Moreover, given that V ∈ L n (Ω), the memberships in (4.1.47) also give V⟨u, υ⟩ ∈ L1 (Ω).
(4.1.49)
div F⃗ = ⟨∆HL u, υ⟩ − ⟨u, ∆HL υ⟩ in D (Ω).
(4.1.50)
In turn, this further implies that
Upon noting from (4.1.40) that at σa.e. point on ∂Ω we have ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
n.t. n.t. = ⟨∇ν♯ u, υ∂Ω ⟩ − ⟨u∂Ω , ∇ν♯ υ⟩,
(4.1.51)
Theorem 9.67 applies and yields (4.1.35). We now make a deﬁnition which is somewhat akin to decomposing the gradient of a function into normal and tangential components relative to a given surface of codimension one in the ambient. Deﬁnition 4.6. Let Ω ⊂ M be an Ahlfors regular domain with outward unit conormal ν ∈ T ∗ M and let ∇ stand for the LeviCivita connection on M, whose action is naturally extended to differential forms. Fix a degree l ∈ {0, 1, . . . , n} and consider a vector bundle E → M. In this setting, given a ﬁrstorder differential operator P : C 1 (M, Λ l T M) → C 0 (M, E)
(4.1.52)
with continuous coefficients deﬁne ∂ τ P , the tangential part of P on ∂Ω, according to ∂ τ P := P + i Sym (P, ν)∇ν♯ . (4.1.53) A word of clariﬁcation is in order here. If in local coordinates P is expressed as P = ∑ A j ∇∂ j + B
(4.1.54)
j
then since
♯
∑ ν k ν k = ∑ g jk ν k ν j = ⟨ν, ν⟩ = ν2 = 1, k
(4.1.55)
j,k
we may write ♯
∂ τ P = ∑ A j ∇∂ j + B − ∑ A j ν j ν k ∇∂ k j
j,k
=
♯ ∑ A j ν k ν k ∇∂ j j,k
=
♯ ∑ A j ν k (ν k ∇∂ j j,k ♯
♯
+ B − ∑ A j ν j ν k ∇∂ k j,k
− ν j ∇∂ k ) + B
= − ∑ A j ν k ∂ τ jk + B, j,k
(4.1.56)
150  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
where the last line uses the deﬁnition of ∂ τ jk , ﬁrst considered on smooth functions restricted to ∂Ω as in (9.5.47), then subsequently extended as a linear and bounded p mapping from L1 (∂Ω, Λ l T M) into L p (∂Ω, Λ l T M) as in (9.6.77). It is the latter interpretation that allows us to consider p
∂ τ P : L1 (∂Ω, Λ l T M) → L p (∂Ω, E),
1 < p < ∞,
as a welldeﬁned, linear and bounded operator.
(4.1.57)
In relation to this, we propose to prove the following decomposition formula. Proposition 4.7. Let E be a C 1 Hermitian vector bundle over M and let Ω ⊂ M be an Ahlfors regular domain with outward unit conormal ν and surface measure σ. Also, ﬁx a degree l ∈ {0, 1, . . . , n} along with an exponent p ∈ (1, ∞) and consider a ﬁrstorder differential operator P : C 1 (M, Λ l T M) → C 0 (M, E) with continuous coefficients. Then for any differential form u satisfying 2,1
u ∈ Hloc (Ω, Λ l T M), N u, N(∇u) ∈ L p (∂Ω), n.t. n.t. and u∂Ω , (∇u)∂Ω exist σa.e. on ∂Ω,
(4.1.58)
p n.t. one has u∂Ω ∈ L1 (∂Ω, Λ l T M) and
n.t. n.t. (Pu)∂Ω = ∂ τ P (u∂Ω ) − i Sym (P, ν)∇ν♯ u on ∂Ω.
(4.1.59)
p n.t. Proof. That under the present assumptions u∂Ω belongs to L1 (∂Ω, Λ l T M) follows from Proposition 9.36. Moreover, if we now write P locally as in (4.1.54) and interpret ∂ τ P as in (4.1.56) and (4.1.57), formula (9.6.85) permits us to compute ♯ n.t. n.t. n.t. ∂ τ P (u∂Ω ) = − ∑ A j ν k ∂ τ jk (u∂Ω ) + B(u∂Ω ) j,k
♯ n.t. n.t. n.t. = − ∑ A j ν k (ν j (∇∂ k u)∂Ω − ν k (∇∂ j u)∂Ω ) + B(u∂Ω ) j,k
n.t. ♯ ♯ n.t. = (∑ ν k ν k )(∑ A j ∇∂ j u + Bu) − ∑ A j ν k ν j (∇∂ k u)∂Ω ∂Ω j
k
j,k
n.t. = (Pu)∂Ω − ∑ A j ν j ∇ν♯ u j
n.t. = (Pu)∂Ω + i Sym (P, ν)∇ν♯ u,
(4.1.60)
making use of (4.1.55) and deﬁnitions (cf. (2.1.63)). From this, (4.1.59) readily follows, ﬁnishing the proof of the proposition. We now return to the study of our connection double layer by also considering its boundarytoboundary version. In this vein, the reader is advised to recall the deﬁnition and properties of Sobolev spaces considered on boundaries of Ahlfors regular domains in § 9.5.
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
151
Theorem 4.8. Let the potential V be as in (3.1.13), ﬁx a degree l ∈ {0, 1, . . . , n} and bring in the Schwartz kernel, Γ l (x, y) from (3.1.24), of (∆HL − V)−1 considered on lforms (cf. (3.1.5)). Also, let ∇ stand for the LeviCivita connection on M, whose action is naturally extended to differential forms. Next, assume Ω ⊂ M is a UR domain satisfying ∂(Ω) = ∂Ω, and denote by σ := H n−1 ⌊∂Ω its surface measure and by ν ∈ T ∗ M its outward unit conormal. In this setting, consider the principal value boundary version of the connection double layer operator Dl from (4.1.6), i.e., K l f(x) := P.V. ∫ ⟨i Sym (∇∗, ν(y))∇y Γ l (x, y), f(y)⟩ dσ(y)
(4.1.61)
∂Ω
= P.V. ∫ ⟨∇ν♯ (y) Γ l (x, y), f(y)⟩ dσ(y),
x ∈ ∂Ω.
∂Ω
In other words, x ∈ ∂Ω,
K l f(x) := lim+ K l,ε f(x), ε→0
(4.1.62)
where, for each ε > 0, the truncated singular integral operator K l,ε is given by⁵ K l,ε f(x) := ∫⟨i Sym (∇∗, ν(y))∇y Γ l (x, y), f(y)⟩ dσ(y),
(4.1.63)
y∈∂Ω, d(x,y)>ε
for each x ∈ ∂Ω. In relation to this, also introduce the maximal operator K l,∗ f(x) := sup K l,ε f(x), ε>0
x ∈ ∂Ω.
(4.1.64)
Then the following conclusions are valid. (1) For each p ∈ (1, ∞) there exists C ∈ (0, ∞) such that ‖N(Dl f)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) ,
∀ f ∈ L p (∂Ω, Λ l T M),
(4.1.65)
‖N(∇Sl f)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) ,
∀ f ∈ L (∂Ω, Λ T M).
(4.1.66)
p
l
Also, corresponding to p = 1, one has⁶ ‖N(Dl f)‖L1,∞ (∂Ω) ≤ C‖f‖L1 (∂Ω,Λ l TM) ,
∀ f ∈ L1 (∂Ω, Λ l T M),
(4.1.67)
‖N(∇Sl f)‖L1,∞ (∂Ω) ≤ C‖f‖L1 (∂Ω,Λ l TM) ,
∀ f ∈ L (∂Ω, Λ T M).
(4.1.68)
1
l
(2) For each f ∈ L1 (∂Ω, Λ l T M) the principal value integral in (4.1.61) exists σa.e. on ∂Ω and the operators K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
1 < p < ∞,
K l : L1 (∂Ω, Λ l T M) → L1,∞ (∂Ω, Λ l T M), are welldeﬁned, linear, and bounded. 5 where the integral kernel is interpreted as in (4.1.7) with x, y ∈ ∂Ω 6 here and elsewhere, L1,∞ (∂Ω) stands for the standard weakL1 space on ∂Ω
(4.1.69) (4.1.70)
152  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
(3) The maximal operators K l,∗ : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), K l,∗ : L (∂Ω, Λ T M) → L 1
l
1 < p < ∞,
(∂Ω, Λ T M),
1,∞
l
(4.1.71) (4.1.72)
are welldeﬁned, sublinear, and bounded. Hence, as a consequence of (4.1.62), (4.1.71), and Lebesgue’s Dominated Convergence Theorem, for every differential form f ∈ L p (∂Ω, Λ l T M) with p ∈ (1, ∞) one therefore has lim K l,ε f = K l f in L p (∂Ω, Λ l T M).
ε→0+
(4.1.73)
(4) The (real) transposed of K l , considered in the context of (4.1.69), is the operator ∗ K⊤ l f(x) = P.V. ∫ ⟨i Sym(∇ , ν(x))∇x Γ l (x, y), f(y)⟩ dσ(y)
(4.1.74)
∂Ω
= P.V. ∫ ⟨∇ν♯ (x) Γ l (x, y), f(y)⟩ dσ(y),
x ∈ ∂Ω,
∂Ω
acting on L p (∂Ω, Λ l T M) with 1/p + 1/p = 1. Moreover, K ⊤ l enjoys similar properties to those listed for K l in (4.1.69)–(4.1.73). (5) There exists C ∈ (0, ∞) with the property that for every f ∈ L2 (∂Ω, Λ l T M) one has the L2 square function estimates 2 ∫ ∇(Dl f)(x) dist (x, ∂Ω) dVol(x) ≤ C ∫ f2 dσ, M\∂Ω
(4.1.75)
∂Ω
and 2 ∫ ∇2 (Sl f)(x) dist (x, ∂Ω) dVol(x) ≤ C ∫ f2 dσ. M\∂Ω
(4.1.76)
∂Ω
In addition, the operators Dl : L2 (∂Ω, Λ l T M) → H 1/2,2 (Ω, Λ l T M), Sl :
L2−1 (∂Ω, Λ l T M)
→ H
1/2,2
(Ω, Λ T M), l
(4.1.77) (4.1.78)
are welldeﬁned, linear, and bounded. (6) For each f ∈ L1 (∂Ω, Λ l T M), n.t. (Dl f)∂Ω± = (± 12 I + K l )f
at σa.e. point on ∂Ω,
(4.1.79)
Ω− := M \ Ω.
(4.1.80)
where I is the identity, and Ω+ := Ω
and
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
153
(7) For each f ∈ L1 (∂Ω, Λ l T M) the nontangential limits n.t. (∇Sl f)∂Ω± exist at σa.e. point on ∂Ω, and
∇ν♯ (Sl f Ω± ) = (∓ 12 I + K ⊤ l )f
at σa.e. point on ∂Ω,
(4.1.81)
(4.1.82)
where I is the identity, and K ⊤ l is the (real ) transposed of K l . (8) With the boundary single layer S l deﬁned as in (3.2.6), for each p ∈ (1, ∞) one has the intertwining formula p l Sl K⊤ l = K l S l on L (∂Ω, Λ T M).
(4.1.83)
(9) Whenever 2 < p ≤ ∞ there exists a ﬁnite constant C = C(Ω, p) > 0 with the property that for each f ∈ L p (∂Ω, Λ l T M) the following L p square function estimate holds 1/2 sup (r1−n ∫ ∇Dl f 2 dist (⋅, ∂Ω) dVol) ≤ C‖f‖L p (∂Ω,Λ l TM) . p r>0 L z (∂Ω) B (z)\∂Ω
(4.1.84)
r
In particular, corresponding to the endpoint case p = ∞, for each differential form f ∈ L∞ (∂Ω, Λ l T M) one has the following Carleson measure estimate: 2 sup (r1−n ∫ ∇Dl f dist (⋅, ∂Ω) dVol) ≤ C‖f‖2L∞ (∂Ω,Λ l TM) .
z∈∂Ω, r>0
(4.1.85)
B r (z)\∂Ω
(10) In the case when p ∈ (2, ∞), there exists a ﬁnite constant C > 0 such that the single layer operator Sl (deﬁned as in (3.2.3)) satisﬁes the L p square function estimate 1/2 p sup (r1−n ∫ ∇Sl f 2 dist (⋅, ∂Ω) dVol) ≤ C‖f‖L−1 (∂Ω,Λ l TM) p r>0 L z (∂Ω) B (z)\∂Ω
(4.1.86)
r
p
for every differential form f ∈ L−1 (∂Ω, Λ l T M). (11) Given a vector bundle E → M and a ﬁrstorder differential operator P : C 1 (M, Λ l T M) → C 0 (M, E)
(4.1.87)
with continuous coefficients, recall that ∂ τ P stands for the tangential part of P on ∂Ω in the sense of Deﬁnition 4.6. Then for every form f ∈ L p (∂Ω, Λ l T M) with p ∈ (1, ∞), at σa.e. point on ∂Ω one has n.t. = −i Sym (P, ν)(∓ 12 I + K ⊤ (PSl f ) l )f + ∂ τ P (S l f). ∂Ω±
(4.1.88)
As a consequence, for each f ∈ L p (∂Ω, Λ l T M) with 1 < p < ∞, at σa.e. point on ∂Ω one has n.t. n.t. (PSl f)∂Ω+ − (PSl f)∂Ω− = i Sym (P, ν)f. (4.1.89)
154  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
Proof. The claims in items (1)–(5) are direct consequence of corresponding properties in the more general setting of Theorem 9.52. To deal with the jumpformula from item (6), ﬁx a function f ∈ L1 (∂Ω, Λ l T M) and, for each x ∈ M \ ∂Ω, write ⊤
Dl f(x) = ∫ ⟨i ∇y Γ l (x, y), Sym (∇∗, ν(y)) f(y)⟩ dσ(y).
(4.1.90)
∂Ω
Our intention is to apply Theorem 9.52 when we take L := ∆HL − V, with ∆HL denoting the HodgeLaplacian on M, the ﬁrstorder differential operator P := i ∇, and consider the function Sym (∇∗, ν)⊤ f in place of f . From the general jumpformula (9.9.74) we then obtain n.t.
Dl f ∂Ω± = ± 12 i Sym(L, ν)−1 [i Sym (∇, ν)⊤ Sym (∇∗, ν) f ] + K l f ⊤
⊤
= ± 12 i Sym (∆HL , ν)−1 [i (Sym (∇∗, ν) Sym (∇, ν)) f ] + K l f = ± 12 i Sym (∆HL , ν)−1 [i Sym (∇∗ ∇, ν)⊤ f ] + K l f = ± 12 i Sym (∆HL , ν)−1 [i Sym (−∆HL , ν)⊤ f ] + K l f = ± 12 i Sym (∆HL , ν)−1 [(−i) Sym (∆HL , ν)f ] + K l f = (± 12 I + K l )f
at σa.e. point on ∂Ω,
(4.1.91)
as wanted. Consider now the claims made in (4.1.81) and (4.1.82). Start by ﬁxing an arbitrary differential formvalued function f ∈ L1 (∂Ω, Λ l T M) and writing locally ∇Sl f = dx j ⊗ ∇∂ j Sl f in Ω.
(4.1.92)
Next, for each j ∈ {1, . . . , n}, express ∇∂ j Sl f(x) = ∫ ⟨∇∂ xj Γ l (x, y), f(y)⟩ dσ(y),
x ∈ M \ ∂Ω.
(4.1.93)
∂Ω
For each ﬁxed j ∈ {1, . . . , n}, the idea is to apply Theorem 9.52 with L := ∆HL − V and ̃ := ∇∂ . Speciﬁcally, upon observing that by for the ﬁrstorder differential operator P j (9.1.45) and (9.1.3) we have Sym (∇∂ j , ν) = iν j
and
Sym (∆HL , ν) = −1,
(4.1.94)
the general jumpformula (9.9.80) gives that, at σa.e. point on ∂Ω, n.t. ∇∂ j Sl f ∂Ω± = ∓ 12 i Sym (∇∂ j , ν) Sym (L, ν)−1 f + T j f = ∓ 12 ν j f + T j f,
(4.1.95)
where T j f(x) := P.V. ∫ ⟨∇∂ xj Γ l (x, y), f(y)⟩y dσ(y),
x ∈ ∂Ω.
(4.1.96)
∂Ω
Consequently, at σa.e. point on ∂Ω, n.t. n.t. (∇Sl f)∂Ω± = dx j ⊗ (∇∂ j Sl f)∂Ω± = dx j ⊗ (∓ 12 ν j f + T j f) = ∓ 12 (ν j dx j ) ⊗ f + dx j ⊗ T j f = ∓ 12 ν ⊗ f + Tf,
(4.1.97)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
155
where Tf(x) := P.V. ∫ ⟨dx j ⊗ ∇∂ xj Γ l (x, y), f(y)⟩y dσ(y) ∂Ω
= P.V. ∫ ⟨∇x Γ l (x, y), f(y)⟩y dσ(y),
x ∈ ∂Ω.
(4.1.98)
∂Ω
Since by Theorem 9.52 this is a meaningfully deﬁned object at σa.e. point x ∈ ∂Ω, the claim in (4.1.81) is established. Moreover, all ingredients are now in place to compute, making use of (4.1.10), (9.1.44), n.t. ∇ν♯ (Sl f Ω± ) = i Sym (∇∗, ν)(∇Sl f)∂Ω± = ∓ 12 i Sym (∇∗, ν)(ν ⊗ f) + i Sym (∇∗, ν)Tf = ∓ 12 f + K ⊤ l f,
(4.1.99)
after observing from (4.1.98) and (4.1.74) that i Sym (∇∗, ν)Tf = K ⊤ l f . The proof of (4.1.82) is therefore complete. Hence, item (7) has been fully dealt with. As regards the intertwining formula (4.1.83) from item (8), pick an arbitrary differential form f ∈ L p (∂Ω, Λ l T M) and set u := Sl f in Ω. Then the differential form u satisﬁes the properties listed in (4.1.14). As such, the integral representation formula (4.1.15) and the jumprelations established earlier give Sl f = Dl (S l f) − Sl ((− 12 I + K ⊤ l )f ) in Ω.
(4.1.100)
By taking the nontangential traces of both sides we therefore arrive at S l f = ( 12 I + K l )(S l f) − S l ((− 12 I + K ⊤ l )f ) on ∂Ω,
(4.1.101)
from which one readily deduces that S l K ⊤ l = Kl Sl . Next, the claims in items (9) and (10) are direct consequences of similar estimates in the more general setting described in items (11) and (12) of Theorem 9.52. Finally, formula (4.1.88) in item (11) is a consequence of Proposition 4.7, (3.2.7), and (4.1.82). The proof of Theorem 4.8 is therefore complete. Moving on, we turn to the study of harmonic layer potentials associated with LeviCivita connection that involve L p based Sobolev spaces of order one on the boundary of the underlying domain. For a review of this brand of Sobolev spaces see the discussion in § 9.5. Theorem 4.9. Retain the same background hypotheses as in Theorem 4.8 and recall the double layer potential operator Dl from (4.1.6), its principal value boundary version K l from (4.1.61), as well as the single layer Sl from (3.2.3) together with its boundary version S l from (3.2.6). Also, ﬁx some p ∈ (1, ∞) arbitrary. Then the operators p
S l : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M), Sl :
p L−1 (∂Ω, Λ l T M)
→ L (∂Ω, Λ T M), p
l
(4.1.102) (4.1.103)
156  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains are welldeﬁned, linear, and bounded, and there exists a constant C ∈ (0, ∞) such that p
p ‖N(Sl f)‖L p (∂Ω) ≤ C‖f‖L−1 (∂Ω,Λ l TM) ,
∀ f ∈ L−1 (∂Ω, Λ l T M),
(4.1.104)
p
Also, for every f ∈ L−1 (∂Ω, Λ l T M) one has Sl f ∈ C 0 (M \ ∂Ω, Λ l T M)
and (∆HL − V)Sl f = 0 in M \ ∂Ω,
(4.1.105)
and n.t.
n.t.
Sl f ∂Ω+ = S l f = Sl f ∂Ω− σa.e. on ∂Ω,
p
∀ f ∈ L−1 (∂Ω, Λ l T M).
(4.1.106)
In addition, the single layer satisﬁes the L2 square function estimate 2 ( ∫ ∇(Sl f)(x) dist (x, ∂Ω) dVol(x))
1/2
≤ C‖f‖L2−1 (∂Ω,Λ l TM) .
(4.1.107)
M\∂Ω
Finally, if S l,V1 and S l,V2 are the boundary single layers associated with two potentials V1 , V2 as in (3.1.13), then the difference p
S l,V1 − S l,V1 : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M) is compact.
(4.1.108)
Moreover, under the additional assumption that Ω satisﬁes a twosided local John condition the following conclusions are valid. p (1) There exists C ∈ (0, ∞) such that for every form f ∈ L1 (∂Ω, Λ l T M) one has ‖N(∇Dl f)‖L p (∂Ω) ≤ C‖f‖L1p (∂Ω,Λ l TM) ,
(4.1.109)
‖Dl f‖W 1,np/(n−1) (Ω,Λ l TM) ≤ C‖f‖L1p (∂Ω,Λ l TM) .
(4.1.110)
and (2) The operators p
p
K l : L1 (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M), K⊤ l :
p L−1 (∂Ω, Λ l T M)
→
p L−1 (∂Ω, Λ l T M),
(4.1.111) (4.1.112)
are welldeﬁned, linear, and bounded. p (3) If Ω± are as in (4.1.80), then for each form f ∈ L1 (∂Ω, Λ l T M) the pointwise nontangential boundary traces n.t. (∇Dl f)∂Ω± exist σa.e. on ∂Ω
(4.1.113)
and, with the boundary conormal derivative ∇ν♯ deﬁned as in (4.1.10), one has ∇ν♯ (Dl f Ω+ ) = ∇ν♯ (Dl f Ω− ) at σa.e. point on ∂Ω.
(4.1.114)
p
Granted this, for each given f ∈ L1 (∂Ω, Λ l T M) it is meaningful to simply abbreviate ∇ν♯ Dl f := ∇ν♯ (Dl f Ω± ).
(4.1.115)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
157
(4) With convention (4.1.115) in mind, the operator p
∇ν♯ Dl : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)
(4.1.116)
is welldeﬁned, linear, and bounded. In addition, if the exponent p ∈ (1, ∞) is such that 1/p + 1/p = 1, then ∫ ⟨∇ν♯ Dl f, g⟩ dσ = ∫ ⟨f, ∇ν♯ Dl g⟩ dσ ∂Ω
(4.1.117)
∂Ω p
p
for every f ∈ L1 (∂Ω, Λ l T M) and g ∈ L1 (∂Ω, Λ l T M). As a consequence, the operator (4.1.116) further extends to a welldeﬁned, linear, and bounded mapping p
∇ν♯ Dl : L p (∂Ω, Λ l T M) → L−1 (∂Ω, Λ l T M)
(4.1.118)
which is the transposed of the operator in (4.1.116) (corresponding to the Hölder conjugate exponent p of p). (5) The following operator identities hold: ( 12 I + K l )(− 12 I + K l ) = S l (∇ν♯ Dl ) on L p (∂Ω, Λ l T M),
(4.1.119)
⊤ p l 1 ♯ ( 12 I + K ⊤ l )(− 2 I + K l ) = (∇ν Dl )S l on L (∂Ω, Λ T M),
(4.1.120)
♯ K⊤ l (∇ν Dl )
= (∇ν♯ Dl )K l on L (∂Ω, Λ T M). p
l
(4.1.121)
Moreover, the intertwining identity (4.1.83) extends to l Sl K⊤ l = K l S l on L −1 (∂Ω, Λ T M), p
(4.1.122)
and one also has ⊤ l 1 ♯ ( 12 I + K ⊤ l )(− 2 I + K l ) = (∇ν Dl )S l on L −1 (∂Ω, Λ T M). p
(4.1.123)
(6) Given a vector bundle E → M and a ﬁrstorder differential operator P : C 1 (M, Λ l T M) → C 0 (M, E)
(4.1.124)
with continuous coefficients, recall that ∂ τ P stands for the tangential part of P on ∂Ω p in the sense of Deﬁnition 4.6. Then for every f ∈ L1 (∂Ω, Λ l T M) one has, at σa.e. point on ∂Ω, n.t. (PDl f)∂Ω± = ∂ τ P (± 12 I + K l )f − i Sym (P, ν)∇ν♯ Dl f.
(4.1.125)
p
As a consequence, for every f ∈ L1 (∂Ω, Λ l T M) one has n.t. n.t. (PDl f)∂Ω+ − (PDl f)∂Ω− = ∂ τ P f at σa.e. point on ∂Ω.
(4.1.126)
158  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
Proof. For starters, the claims pertaining to the single layer in (4.1.102)–(4.1.107) are direct consequences of the general theory presented in Theorem 9.52. Concerning (4.1.108), the key observation is that the choice of the potential V does not affect the top singularity in the fundamental solution Γ l (x, y). Indeed, an inspection of (3.1.26) and (3.1.27) reveals that, given any two potentials V1 , V2 as in (3.1.13) then (with a selfexplanatory piece of notation) for each j ∈ {1, . . . , n} we have ∂ x j Γ l,V1 (x, y) − ∂ x j Γ l,V2 (x, y) = O(x − y−(n−2+ε) ),
∀ ε > 0.
(4.1.127)
Granted this, the compactness of S l,V1 − S l,V1 in the context of (4.1.108) may be seen with the help of Lemma 9.63. Moving on, in order to get a better insight of the nature of the operator (4.1.6), ﬁx an arbitrary point x o ∈ ∂Ω and suppose U is a coordinate patch in M containing x o . A few conventions about notation used throughout the proof are as follows. Whenever convenient, we shall identify the portion of Ω contained in U with its Euclidean image under the coordinate chart. In such a scenario, we let νE = (νEj )1≤j≤n be the outward unit normal to ∂Ω with respect to the Euclidean metric in ℝn , and let σE := Hℝn−1 ⌊∂Ω n be the surface measure induced by the ﬂatspace Euclidean metric δ jk on ∂Ω. These are related to the manifold conormal ν and surface measure σ (associated with the original Riemannian metric g on M) via explicit formulas given in (9.5.4), (9.5.5), and (9.5.11). p Returning to the mainstream discussion, let f ∈ L1 (∂Ω, Λ l T M) be a differential N form supported in U ∩ ∂Ω with fU∩∂Ω = ∑N=l f N dx . In Euclidean coordinates we p shall simply identify f with the collection of its components f N ∈ L1 (∂Ω, σE ). Deﬁne for every two ordered arrays I, J of equal length g IJ := det ((g ij (y))i∈I,j∈J ),
(4.1.128)
and recall from (3.1.26) that g IJ (y) = det ((g ij (y))i∈I,j∈J ). In particular, Lemma 2.5 implies that (4.1.129) g IJ g JL = δ IL , ∀ I, L. Writing ν = ν r dx r in U ∩ ∂Ω, we then see from (4.1.6), (4.1.7) and (4.1.3) that, for each x ∈ Ω ∩ U, IJ
JH
Dl f(x) = ∫ ⟨ν r (y)g rs (y)(∂ y s Γ l (x, y) + γ s (y)Γ lIH (x, y))dx I⊗ dy J , f(y)⟩ dσ(y) ∂Ω IJ
= ( ∫ ν r (y)g rs (y)∂ y s Γ l (x, y)g JN (y)f N (y) dσ(y))dx I ∂Ω JH
+ ( ∫ ν r (y)g rs (y)γ s (y)Γ lIH (x, y)g JN (y)f N (y) dσ(y))dx I .
(4.1.130)
∂Ω
To be able to focus on individual components, express
Dl f(x) = ∑ (Dl f)I (x)dx I , I=l
x ∈ Ω ∩ U.
(4.1.131)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
159
Then for each ordered array I with I = l we have (Dl f)I (x) = ∫ νEr (y)g rs (y)∂ y s Γ l (x, y)g JN (y)f N (y) √ g(y) dσE (y) IJ
(4.1.132)
∂Ω
+ ∫ νEr (y)g rs (y)γ s (y)Γ l (x, y)g JN (y)f N (y) √ g(y) dσE (y). JH
IJ
∂Ω
Fix now an arbitrary point z ∈ ∂Ω ∩ U and some ordered array I of length l. Then for each x ∈ Ω ∩ U write (Dl f)I (x) = A1 + A2 + A3 , (4.1.133) where IJ A1 := ∫ νEr (y)g rs (y)∂ y s Γ l (x, y)g JN (y)[f N (y) − f N (z)]√ g(y) dσE (y),
(4.1.134)
∂Ω
A2 := f N (z) ∫ νEr (y)g rs (y)∂ y s Γ lIJ (x, y)g JN (y)√ g(y) dσE (y),
(4.1.135)
∂Ω IJ JN E A3 := ∫ νEr (y)g rs (y)γ JH s (y)Γ l (x, y)g (y)f N (y) √ g(y) dσ (y).
(4.1.136)
∂Ω
For each ﬁxed ordered array N of length l, use the Divergence Theorem to transform the integral appearing in A2 above as ∫ νEr (y)g rs (y)∂ y s Γ l (x, y)g JN (y)√ g(y) dσE (y) IJ
∂Ω
= ∫ ∂ y r {g rs (y)∂ y s Γ l (x, y)g JN (y)√ g(y)} dy IJ
Ω IJ = ∫ ∂ y r {g rs (y)∂ y s Γ l (x, y)}g JN (y)√ g(y) dy Ω
+ ∫ g rs (y)∂ y s Γ l (x, y)∂ y r {g JN (y)√ g(y)} dy IJ
Ω
=: B1 + B2 .
(4.1.137)
From (2.1.96) and (9.9.96) we see that, schematically, IJ ∂ y r {g rs (y)∂ y s Γ l (x, y)}
(4.1.138)
y
IJ IJML = ∆HL Γ l (x, y) + a t (y)∂ y t Γ lML (x, y) + b IJML (y)Γ lML (x, y) −1
= (√ g(x) ) δ IJ δ x (y) + a t
IJML
(y)∂ y t Γ lML (x, y) + b IJML (y)Γ lML (x, y),
160  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
so, after an integration by parts, (y)∂ y t Γ lML (x, y)g JN (y)√ g(y) dy B1 = g IN (x) + ∫ a IJML t Ω
+ ∫ b IJML (y)Γ lML (x, y)g JN (y)√ g(y) dy Ω IJML = g (x) + ∫ Γ lML (x, y){b IJML (y)g JN (y)√ g(y) − ∂ y t {a t (y)g JN (y)√ g(y) }} dy IN
Ω
+∫
IJML a t (y)νEt (y)Γ lML (x, y)g JN (y)√ g(y) dσE .
(4.1.139)
∂Ω
To treat B2 integrate by parts IJ B2 = ∫ g rs (y)∂ y s Γ l (x, y)∂ y r {g JN (y)√ g(y)} dy Ω IJ = − ∫ Γ l (x, y)∂ y s {g rs (y)∂ y r {g JN (y)√ g(y) }} dy Ω
+ ∫ g rs (y)νEs (y)Γ lIJ (x, y)∂ y r {g JN (y)√ g(y)} dσE (y).
(4.1.140)
∂Ω
Given j ∈ {1, . . . , n}, at each x ∈ Ω write (∂ x j Dl f(x))I = ∂ x j A1 + ∂ x j A2 + ∂ x j A3 , i.e., (∂ x j Dl f(x))I = II, j (x, z) + III, j (x, z) + IIII, j (x, z) + IVI, j (x, z) − VI, j (x, z) + VII, j (x, z) + VIII, j (x),
(4.1.141)
where II, j (x, z) := ∫ νEr (y)g rs (y)∂ x j ∂ y s Γ l (x, y)g JN (y)[f , N (y) − f N (z)]√ g(y) dσE (y) IJ
∂Ω
III, j (x, z) := f N (z)(∂ j g IN )(x), IIII, j (x, z) := f N (z) ∫ ∂ x j Γ lML (x, y){ b IJML (y)g JN (y)√ g(y) Ω
− ∂ y t {a IJML (y)g JN (y)√ g(y) }} dy t
(y)νEt (y)∂ x j Γ lML (x, y)g JN (y)√ g(y) dσE (y), IVI, j (x, z) := f N (z) ∫ a IJML t ∂Ω
VI, j (x, z) := f N (z) ∫ ∂ x j Γ lIJ (x, y)∂ y s {g rs (y)∂ y r {g JN (y)√ g(y) }} dy, Ω
VII, j (x, z) := f N (z) ∫ g rs (y)νEs (y)∂ x j Γ lIJ (x, y)∂ y r {g JN (y)√ g(y) } dσE (y), ∂Ω
VIII, j (x) := ∫ ∂Ω
IJ JN E νEr (y)g rs (y)γ JH s (y)∂ x j Γ l (x, y)g (y)f N (y) √ g(y) dσ (y).
(4.1.142)
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
161
To handle II, j (x, z), ﬁrst observe from (3.1.24)–(3.1.28) and (3.1.33) that for each ordered array J with J = l and each j, s ∈ {1, . . . , n} we may write IJ IJ ∂ x j ∂ y s [Γ l (x, y)] = ∂ x j ∂ y s [e0 (y, x − y)]g IJ (y) + R js (x, y)
= −∂ y j ∂ y s [e0 (y, x − y)]g IJ (y) + R IJ js (x, y),
(4.1.143)
where R IJ js (x, y) are generic residual terms (changing from line to line) satisfying R js (x, y) = O(x − y−(n−1+ε) ), IJ
∀ ε > 0.
(4.1.144)
Use (4.1.143) and (4.1.129) to rewrite II, j (x, z) as II, j (x, z) = − ∫ g rs (y)νEr (y)∂ y j ∂ y s [e0 (y, x − y)][f I (y) − f I (z)]√ g(y) dσE (y) ∂Ω JN E √ + ∫ g rs (y)νEr (y)R IJ js (x, y)g (y)[f N (y) − f N (z)] g(y) dσ (y)
=:
∂Ω (1) II, j (x, z)
(2)
+ II, j (x, z).
(4.1.145) (1)
Note that since −νEr (y)∂ y j = ∂ τ jr (y) − νEj (y)∂ y r we may further express II, j (x, z) as (1) II, j (x, z) = ∫ g rs (y)∂ τ jr (y) ∂ y s [e0 (y, x − y)][f I (y) − f I (z)]√ g(y) dσE (y) ∂Ω
− ∫ g rs (y)νEj (y)∂ y r ∂ y s [e0 (y, x − y)][f I (y) − f I (z)]√ g(y) dσE (y) ∂Ω
= ∫ g rs (y)∂ τ jr (y) ∂ y s [e0 (y, x − y)][f I (y) − f I (z)]√ g(y) dσE (y)
(4.1.146)
∂Ω
since, thanks to the cancellation property recorded in (3.1.34), the second integral above vanishes. In the last integral in (4.1.145) integrate by parts on ∂Ω in order to relocate the boundary tangential derivative operator ∂ τ jr (y) away from ∂ y s [e0 (y, x − y)], thus obtaining (1) II, j (x, z) = ∫ ∂ y s [e0 (y, x − y)]∂ τ rj (y) {g rs (y)[f I (y) − f I (z)]√ g(y) } dσE (y) ∂Ω
= ∫ ∂ y s [e0 (y, x − y)]∂ τ rj (g rs √g )(y)[f I (y) − f I (z)] dσE (y) ∂Ω
+ ∫ g rs (y)∂ y s [e0 (y, x − y)]√ g(y) (∂ τ rj f I )(y) dσE (y) ∂Ω
=:
(11) II, j (x, z)
(12)
+ II, j (x).
(4.1.147)
162  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains (12)
The idea for further handling the term II, j (x) above is to reverseengineer Γ lIJ (x, y) starting from e0 (y, x − y). Indeed, invoking (3.1.26) and (3.1.28), for each s ∈ {1, . . . , n} and each ordered array J with J = l we see that IJ
IJ
∂ y s [e0 (y, x − y)]g IJ (y) = ∂ y s [Γ l (x, y)] + Q s (x, y),
(4.1.148)
where the residual term Q IJ s (x, y) satisﬁes Q s (x, y) = O(x − y−(n−2+ε) ), IJ
∀ ε > 0.
(4.1.149)
Mindful of the fact that ∂ τ rj f I = g IJ g JN ∂ τ rj f N (cf. (4.1.129)), we may then use (4.1.148) to (12)
rewrite II, j (x) as (12)
II, j (x) = ∫ g rs (y)∂ y s Γ lIJ (x, y)√ g(y) g JN (y)(∂ τ rj f N )(y) dσE (y) ∂Ω JN E + ∫ g rs (y)Q IJ s (x, y)√ g(y) g (y)(∂ τ rj f N )(y) dσ (y) ∂Ω (121)
(122)
=: II, j (x) + II, j (x).
(4.1.150)
Upon observing that ∂ τ rj f N = νEr (∇Etan f N )j − νEj (∇Etan f N )r
(4.1.151)
(121)
(cf. (9.5.37)), we may further recast II, j (x) in the form (121)
(1211)
II, j (x) = II, j
(1212)
(x) − II, j
(x),
(4.1.152)
where (1211)
II, j
(x) := ∫ g rs (y)∂ y s Γ lIJ (x, y)g JN (y)√ g(y) νEr (y)(∇Etan f N )j (y) dσE (y),
(4.1.153)
∂Ω (1212)
II, j
(x) := ∫ g rs (y)∂ y s Γ lIJ (x, y)g JN (y)√ g(y) νEj (y)(∇Etan f N )r (y) dσE (y).
(4.1.154)
∂Ω
Finally, from (4.1.153) and (4.1.130)–(4.1.132) we recognize that (1211)
II, j
(last)
(x) = (Dl (∇Etan f)j )I (x) − II, j (x),
(4.1.155)
where (last)
II, j (x) := ∫ νEr (y)g rs (y)γ s (y)Γ l (x, y)g JN (y)(∇Etan f)j (y) √ g(y) dσE (y). JH
IJ
(4.1.156)
∂Ω
At this stage, we turn to the task of estimating the nontangential maximal functions of various pieces in which (∂ x j Dl f(x))I has been broken up, starting in (4.1.141). To get started, recall that Γ(z) ⊂ Ω stands for the nontangential approach region with
4.1 The Deﬁnition and Mapping Properties of the Double Layer 
163
vertex at z ∈ ∂Ω, and note that the piece (Dl (∇Etan f)j )I (x), appearing in (4.1.155), falls directly under the scope of the CalderónZygmund theory presented in Theorem 9.52 which ultimately gives 1/p p E E ≤ C‖f‖L1p (∂Ω,σE ) . ( ∫ sup (Dl (∇tan f)j )I (x) dσ (z)) x∈Γ(z)
(4.1.157)
∂Ω
(1212)
The piece II, j (x) from (4.1.154), as well as the piece VIII, j (x) from (4.1.142), may also be treated via CalderónZygmund theory. For these, Theorem 9.52 applies directly and gives, respectively, 1/p (1212) p ≤ C‖f‖L1p (∂Ω,σE ) , ( ∫ sup II, j (x) dσE (z)) x∈Γ(z)
(4.1.158)
1/p p ≤ C‖f‖L p (∂Ω,σE ) . ( ∫ sup VIII, j (x) dσE (z)) x∈Γ(z)
(4.1.159)
∂Ω
and
∂Ω
(last)
Next, the contribution from the piece II, j (x), appearing in (4.1.156), may be handled, purely based on the size of its kernel (which is weakly singular; cf. (3.1.26) and (3.1.27)). Indeed, making use of Lemma 9.54 we obtain 1/p
(last) p ( ∫ sup II, j (x) dσE (z)) x∈Γ(z)
≤ C‖f‖L1p (∂Ω,σE ) .
(4.1.160)
∂Ω
(122)
In fact, granted (4.1.149), similar considerations apply to II, j (x) for which we also conclude 1/p (122) p ≤ C‖f‖L1p (∂Ω,σE ) . (4.1.161) ( ∫ sup II, j (x) dσE (z)) x∈Γ(z) ∂Ω
(11)
By way of contrast, while the integral kernel in II, j (x, z), appearing in (4.1.147), has a strong singularity (of order n − 1; cf. (3.1.26)), the special algebraic structure of this piece makes it possible for us to invoke the estimate established in Lemma 9.55 (used here with ε = 0) which then gives 1/p (11) p ≤ C‖f‖L1p (∂Ω,σE ) . ( ∫ sup II, j (x, z) dσE (z)) x∈Γ(z)
(4.1.162)
∂Ω
IJ
Given the hypersingular nature of the kernels R js (cf. (4.1.144)), the full force of (2) Lemma 9.55 is required when dealing with II, j (x, z). As before, we arrive at 1/p
(2) p ( ∫ sup II, j (x, z) dσE (z)) x∈Γ(z) ∂Ω
≤ C‖f‖L1p (∂Ω,σE ) .
(4.1.163)
164  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains This completes the analysis of II, j (x, z) from (4.1.142). For the piece III, j (x, z) appearing in (4.1.142) we trivially have 1/p p ( ∫ sup III, j (x, z) dσE (z)) ≤ C‖f‖L p (∂Ω,σE ) x∈Γ(z)
(4.1.164)
∂Ω
and, in fact, the same type of estimate holds for IIII,j (x, z) and VI,j (x, z) from (4.1.142) given that sup ∫∂ x j Γ lML (x, y) b IJML (y)g JN (y)√ g(y) x∈Ω Ω − ∂ y t {a IJML (y)g JN (y)√ g(y) } dy < ∞ (4.1.165) t and IJ (4.1.166) ∫∂ x j Γ l (x, y) ∂ y s {g rs (y)∂ y r {g JN (y)√ g(y) }} dy < ∞, Ω
as may be seen with the help of (3.1.26)–(3.1.28), (aware of the identiﬁcations of the coefficients in (4.1.138) with those in (2.1.96), and keeping in mind that the metric is of class C 2 and that Ω is relatively compact). At this stage, there remains to estimate the contribution from IVI,j (x, z) and VII,j (x, z) in (4.1.142). Structurally, they are both of the form f N (z)Θ I,N,j (x) where Θ I,N,j (x) := ∫ ∂ x j Γ lAB (x, y)h INAB (y) dσE (y),
(4.1.167)
∂Ω
with IJAB E JN ν t g √ g(y)
h INAB := a t
∈ L∞ (∂Ω, σE ) ⊂ ⋂ L q (∂Ω, σE )
(4.1.168)
1 0, k l (x, y) dσ(y) = [⟨νE (y), x − y⟩b(y, x − y) + O(x − y−(n−2+ε) )] dσE (y) where
1
(4.3.8)
−n/2
. (4.3.9) (g jk (y)z j z k ) ω n−1 Since the function b(z, y) satisﬁes the properties listed in (9.11.1), Theorem 9.60 applies and, in concert with Lemma 9.63 (which takes care of the weak singularity in (4.3.8)), ultimately gives that the operator (4.3.1) is compact. From this and duality we then conclude that the operator (4.3.2) is compact. In dealing with the compactness of K l on Sobolev spaces, we shall recycle a large portion of the argument developed in the proof of Theorem 4.9. In particular, we shall retain notation introduced on that occasion. Again, work locally in a small neighborp hood U of an arbitrary ﬁxed boundary point. As before, we let f ∈ L1 (∂Ω, Λ l T M) be a N differential form supported in U ∩ ∂Ω with fU∩∂Ω = ∑N=l f N dx . Working in Euclidean coordinates, for any given j, k ∈ {1, . . . , n}, ﬁrst write b(y, z) := −
∂ τ jk (K l f)I (z) = ∂ τ jk ( 12 f + K l f)I (z) − 12 (∂ τ jk f I )(z)
(4.3.10)
n.t. n.t. = νEj (z)(∂ k Dl f )I ∂Ω (z) − νEk (z)(∂ j Dl f )I ∂Ω (z) − 12 (∂ τ jk f I )(z), where we have made use of (4.1.65), (4.1.79), (4.1.109), (4.1.113), and Proposition 9.16. Second, observe that from (4.1.79) and (4.1.151) we have n.t. n.t. νEj (z) (Dl (∇Etan f)k )I ∂Ω (z) − νEk (z) (Dl (∇Etan f)j )I ∂Ω (z) = νEj (z)( 12 (∇Etan f)k + K l (∇Etan f)k )I (z) − νEk (z)( 12 (∇Etan f)j + K l (∇Etan f)j )I (z) = 12 {νEj (∇Etan f I )k − νEk (∇Etan f I )j }(z) + νEj (z)(K l (∇Etan f)k )I (z) − νEk (z)(K l (∇Etan f)j )I (z) = 12 (∂ τ jk f I )(z) + (K l (νEj (∇Etan f)k ))I (z) − (K l (νEk (∇Etan f)j ))I (z) + ([M νE , K l ](∇Etan f)k ) (z) − ([M νE , K l ](∇Etan f)j ) (z) j
I
k
I
4.3 Compactness of the Double Layer on Regular SKT Domains  175
= 12 (∂ τ jk f I )(z) + (K l (∂ τ jk f))I (z) + ([M νE , K l ](∇Etan f)k ) (z) − ([M νE , K l ](∇Etan f)j ) (z). I
j
(4.3.11)
I
k
(1211)
The above formula is relevant when assessing the contribution from II, j (x), written n.t. n.t. as in (4.1.155), in the context of νEj (z)(∂ k Dl f )I ∂Ω (z) − νEk (z)(∂ j Dl f )I ∂Ω (z). Speciﬁcally, starting with (4.3.10), then recalling the decomposition (4.1.141) and the subsequent analysis of the intervening pieces, for any given j, k ∈ {1, . . . , n} we then obtain ∂ τ jk (K l f)I (z) = (K l (∂ τ jk f))I (z) + ([M νE , K l ](∇Etan f)k ) (z) − ([M νE , K l ](∇Etan f)j ) (z) I
j
−
(last) n.t. νEj (z) II, k ∂Ω (z)
+
(1212) n.t. ∂Ω (z)
− νEj (z) II, k
k
I
(last) n.t. νEk (z) II, j ∂Ω (z) (1212) n.t. ∂Ω (z)
+ νEk (z) II, j
(122) n.t. (122) n.t. + νEj (z) II, k ∂Ω (z) − νEk (z) II, j ∂Ω (z) (11) (11) n.t. n.t. + νEj (z) II, k (⋅, z)∂Ω (z) − νEk (z) II, j (⋅, z)∂Ω (z) (2) (2) n.t. n.t. + νEj (z) II, k (⋅, z)∂Ω (z) − νEk (z) II, j (⋅, z)∂Ω (z) n.t. n.t. + νEj (z) III, k (⋅, z)∂Ω (z) − νEk (z) III, j (⋅, z)∂Ω (z) n.t. n.t. + νEj (z) IIII, k (⋅, z)∂Ω (z) − νEk (z) IIII, j (⋅, z)∂Ω (z) n.t. n.t. + νEj (z) IVI, k (⋅, z)∂Ω (z) − νEk (z) IVI, j (⋅, z)∂Ω (z) n.t. n.t. − νEj (z) VI, k (⋅, z)∂Ω (z) + νEk (z) VI, j (⋅, z)∂Ω (z) n.t. n.t. + νEj (z) VII, k (⋅, z)∂Ω (z) − νEk (z) VII, j (⋅, z)∂Ω (z) n.t. n.t. + νEj (z) VIII, k ∂Ω (z) − νEk (z) VIII, j ∂Ω (z),
(4.3.12)
at σa.e. point z ∈ ∂Ω. We plan to read the compactness of K l on the Sobolev space p L1 (∂Ω, Λ l T M) off the above decomposition, considering each line individually. In the p process, we shall tacitly identify a form f ∈ L1 (∂Ω, Λ l T M) with the collection of its p E components f N ∈ L1 (∂Ω, σ ) in the local writing f = ∑N=l f N dx N . To get started, observe that the assignment p
L1 (∂Ω) ∋ f → (K l (∂ τ jk f))I ∈ L p (∂Ω)
(4.3.13)
is compact, thanks to (4.3.1) and the fact that p
∂ τ jk : L1 (∂Ω) → L p (∂Ω) is bounded,
∀ j, k ∈ {1, . . . , n}.
(4.3.14)
Next, from (4.3.14) and the portion of Theorem 9.60 dealing with commutators it follows that for each j, k ∈ {1, . . . , n} the mapping p
L1 (∂Ω) ∋ f → ([M νE , K l ](∇Etan f)k ) ∈ L p (∂Ω) j
I
(4.3.15)
176  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains is compact. This takes care of the second line of (4.3.12). Next, for each j, k ∈ {1, . . . , n} the mapping⁸ p p (last) n.t. L1 (∂Ω) ∋ f → νEj (z) II, k ∂Ω (z) ∈ L z (∂Ω) (4.3.16) is compact, by virtue of (4.1.156), (4.3.14), and Lemma 9.63. As such, the third line of (4.3.12) is accounted for. As regards the fourth line in (4.3.12), observe ﬁrst that from (4.1.154) we obtain (keeping in mind that the jumpterms produced according to Theorem 9.52 by taking nontangential boundary traces actually cancel in the combination considered below) (1212) n.t. ∂Ω (z)
νEj (z) II, k
(1212) n.t. ∂Ω (z)
− νEk (z) II, j
(4.3.17)
= P.V. ∫ g rs (y)∂ y s Γ l (z, y)g JN (y)√ g(y) νEj (z)νEk (y)(∇Etan f N )r (y) dσE (y) IJ
∂Ω
− P.V. ∫ g rs (y)∂ y s Γ l (z, y)g JN (y)√ g(y) νEk (z)νEj (y)(∇Etan f N )r (y) dσE (y) IJ
∂Ω
= P.V. ∫ g rs (y)∂ y s Γ l (z, y)g JN (y)√ g(y) νEj (z)(νEk (y) − νEk (z))(∇Etan f N )r (y) dσE (y) IJ
∂Ω
− P.V. ∫ g rs (y)∂ y s Γ l (z, y)g JN (y)√ g(y) νEk (z)(νEj (y) − νEj (z))(∇Etan f N )r (y) dσE (y). IJ
∂Ω
Individually, each of the principal value singular integral operators in the rightmost side of (4.3.17) are of commutator type, amenable to treatment via (the last part in) Theorem 9.60. Mindful of (4.3.14), we conclude from this that the fourth line in (4.3.12) is also of a nature which suits our present purposes. (122) Recall next that II, j (x) has been deﬁned in (4.1.150). Given the weak singular (122) n.t. nature of the integral kernel in II, j ∂Ω (cf. (4.1.149)), it follows that for each indices j, k ∈ {1, . . . , n} the mapping p p (122) n.t. L1 (∂Ω) ∋ f → νEj (z) II, k ∂Ω (z) ∈ L z (∂Ω)
(4.3.18)
is compact, by Lemma 9.63. Thus, the ﬁfth line in (4.3.12) is also of the right nature. (11) Going further, recall II, j (x, z) from (4.1.147). In relation to this, Lemma 9.62 applies (with ε = 0) and gives that, for each j, k ∈ {1, . . . , n}, the assignment p p (11) n.t. L1 (∂Ω) ∋ f → νEj (z) II, k (⋅, z)∂Ω (z) ∈ L z (∂Ω)
(4.3.19)
(2)
is compact, taking care of the sixth line in (4.3.12). Recall now II, j (x, z) from (4.1.145). Aware of (4.1.144) and making use of the full strength of Lemma 9.62, we also have that for each j, k ∈ {1, . . . , n} the operator p p (2) n.t. L1 (∂Ω) ∋ f → νEj (z) II, k (⋅, z)∂Ω (z) ∈ L z (∂Ω)
p
(4.3.20)
8 here and elsewhere we are writing F ∈ L z (∂Ω) simply to indicate that F belongs to L p (∂Ω) as a function of the variable z
4.3 Compactness of the Double Layer on Regular SKT Domains  177
is compact. This clariﬁes matters, as far as the seventh line in (4.3.12) is concerned. The nature of the eighth line in (4.3.12), involving III, j (x, z) originally deﬁned in (4.1.142), is easily elucidated by invoking the compact embeddings in (9.5.41) and (9.5.42). In fact, by (4.1.165) and (4.1.166) the same is true for for the contributions of IIII, j (x, z) and VI, j (x, z) (originally deﬁned in (4.1.142)) in the context of the ninth and eleventh line in (4.3.12). All these contributions are therefore under control. Moving on, we shall simultaneously treat the contributions that IVI, j (x, z) and VI, j (x, z) (originally deﬁned in (4.1.142)) make, respectively, in the tenth and twelfth lines of (4.3.12). The idea is that there exists p o ∈ (p, ∞) with the property that the embedding in (4.1.171) is actually compact (as seen from (9.5.41) and (9.5.42)). Granted this, essentially the same argument based on CalderónZygmund theory and Hölder inequality as in (4.1.167)–(4.1.172) applies and gives that each j, k ∈ {1, . . . , n} the operators p p n.t. L1 (∂Ω) ∋ f → νEj (z) IVI, k (⋅, z)∂Ω (z) ∈ L z (∂Ω), (4.3.21) p p n.t. L1 (∂Ω) ∋ f → νEj (z) VII, k (⋅, z)∂Ω (z) ∈ L z (∂Ω), are compact. Hence, the tenth and twelfth lines in (4.3.12) behave in the desired fashion as well. Finally, the contribution VIII, j (x, z) (ﬁrst introduced in (4.1.142)) makes in the thirteenth line of (4.3.12) can be seen to have the right nature given that for each indices j, k ∈ {1, . . . , n} the operator p p n.t. L1 (∂Ω) ∋ f → νEj (z) VIII, k ∂Ω (z) ∈ L z (∂Ω)
(4.3.22)
is compact, thanks to the CalderónZygmund theory from Theorem 9.50 bearing in mind that, as seen from (9.5.41)(9.5.42), the embedding p
L1 (∂Ω, σE ) → L p (∂Ω, σE ) is compact, for all p ∈ (1, ∞).
(4.3.23)
With all lines of (4.3.12) accounted in the desired manner, the ﬁnal conclusion is that p K l is compact on L1 (∂Ω, Λ l T M). This takes care of the claim in (4.3.3), and the claim in (4.3.4) follows from this and duality. To complete the proof of the theorem there remains to observe that, given any small positive ε o , the same reasoning as above gives that any of the operators (4.3.1)– (4.3.4) lie at distance at most ε o from the space of compact operators in their respective environment, provided Ω is an εSKT domain for some ε > 0 sufficiently small relative to the threshold ε o , the intervening integrability exponent p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. In the second part of this section we wish to establish Fredholm properties involving the principal value connection double layer K l associated with a larger class of domains, which do not necessarily satisfy a two sided local John condition and, as such, may not be regular SKT domains. To proceed in this regard, we ﬁrst recall a piece of notation, originally introduced in (1.4.68). Speciﬁcally, given an Ahlfors regular domain
178  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains Ω ⊂ M and a vector ﬁeld X ∈ T M∂Ω locally expressed as X = X j ∂ j , for a differential form 1 l u ∈ C (Ω, Λ T M) deﬁne its covariant derivative along X on ∂Ω in local coordinates as n.t. ∇X u := X j (∇∂ j u)∂Ω on ∂Ω,
(4.3.24)
assuming that the intervening pointwise nontangential traces exist. Theorem 4.13. Let V be a potential as in (3.1.13) and ﬁx a degree l ∈ {0, 1, . . . , n}. Assume Ω ⊂ M is a UR domain with outward unit conormal ν ∈ T ∗ M and surface measure σ = H n−1 ⌊∂Ω. In this context, let Γ l (x, y) be the fundamental solution for the Schrödinger operator ∆HL − V as in (3.1.16), and use it to deﬁne the single layer operator Sl as in (3.2.3). Also, ﬁx a vector ﬁeld⁹ X ∈ L∞ (∂Ω, T M).
(4.3.25)
Then for every f ∈ L1 (∂Ω, Λ l T M) one has the jumpformula¹⁰ ∇X Sl f = (− 12 ⟨X, ν♯ ⟩I + R)f at σa.e. point on ∂Ω,
(4.3.26)
where R is the principal value singular integral operator given by Rf(x) := P.V. ∫ ⟨∇X(x) Γ l (x, y), f(y)⟩y dσ(y),
x ∈ ∂Ω.
(4.3.27)
1 < p < ∞,
(4.3.28)
1 < p < ∞,
(4.3.29)
∂Ω
The latter operator induces a linear and bounded mapping R : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), whose (real ) transposed is the operator R⊤ : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), acting on each f ∈ L p (∂Ω, Λ l T M) according to R⊤ f(x) = P.V. ∫ ⟨∇X(y) Γ l (x, y), f(y)⟩y dσ(y),
x ∈ ∂Ω.
(4.3.30)
∂Ω
Moreover, if (4.3.25) is strengthened to X ∈ L∞ (∂Ω, T M) ∩ VMO(∂Ω, T M) then
R + R⊤ : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) is a compact operator for each p ∈ (1, ∞).
9 with real components 10 with the left side understood in the sense of (4.3.24)
(4.3.31)
(4.3.32)
4.3 Compactness of the Double Layer on Regular SKT Domains  179
If in addition to (4.3.31) one also assumes that X is transversal to ∂Ω, in the sense that there exists some c > 0 such that ⟨X, ν♯ ⟩ ≥ c at σa.e. point on ∂Ω,
(4.3.33)
then there exist two Hölder conjugate integrability exponents p o , po ∈ [1, ∞] satisfying 1 ≤ p o < 2 < po ≤ ∞
(4.3.34)
and which depend on the above framework, with the property that the operators ± 12 ⟨X, ν♯ ⟩I + R : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), ± 12 ⟨X, ν♯ ⟩I + R⊤ : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), are all Fredholm with index zero for every p ∈
(4.3.35)
(p o , po ).
In fact, a result like (4.3.35) holds assuming only that X ∈ L∞ (∂Ω, T M) is transversal to ∂Ω and has the property that dist (X, VMO(∂Ω, T M)) is sufficiently small,
(4.3.36)
relative to the relevant background structures, where the distance is measured in BMO. Proof. The jumpformula recorded in (4.3.26) is a direct consequence of (4.3.24) and (4.1.95), (4.1.96). That for each p ∈ (1, ∞) the operator R, hence also R⊤ , maps L p (∂Ω, Λ l T M) linearly and boundedly into itself is seen from item (7) in Theorem 9.52. Also, item (8) in Theorem 9.52 yields the representation (4.3.30) for R⊤ (bearing in mind that, at the global level, the HodgeLaplacian is symmetric). Next, strengthen (4.3.25) to (4.3.31), with the goal of proving (4.3.32). To this end, consider the kernel k R (x, y) of the integral operator R from (4.3.27). Working in local coordinates and writing X = X j ∂ j , it follows from (4.1.4), (4.1.5), and (3.1.24)–(3.1.30) that k R (x, y) := ∇X(x) Γ l (x, y) = X j (x)∇∂ xj Γ l (x, y) = X j (x)∇∂ xj (Γ lIJ (x, y)dx I ) ⊗ dy J NJ I J = X j (x)(∂ x j Γ lIJ (x, y) + γ IN j (x)Γ l (x, y))dx ⊗ dy
= X j (x)∂ x j Γ lIJ (x, y)dx I ⊗ dy J + O(x − y−(n−2) ) = X j (x)∂ x j [e0 (y, x − y)g IJ (y) + e IJ (x, y)]dx I ⊗ dy J + O(x − y−(n−2) ) = X j (x)(∂ IIj e0 )(y, x − y)g IJ (y) dx I ⊗ dy J + O(x − y−(n−2+ε) ),
∀ ε > 0,
(4.3.37)
where ∂IIj e0 denotes the partial derivative of e0 (⋅, ⋅) with respect to the jcomponent of its second argument. Likewise, the kernel k R⊤ (x, y) of the integral operator R⊤ from
180  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
(4.3.30) may be expressed in local coordinates as k R⊤ (x, y) := ∇X(y) Γ l (x, y) = X j (y)∇∂ yj Γ l (x, y) = dx I ⊗ X j (y)∇∂ yj (Γ lIJ (x, y)dy J ) IN I J = X j (y)(∂ y j Γ lIJ (x, y) + γ JN j (y)Γ l (x, y))dx ⊗ dy
= X j (y)∂ y j Γ lIJ (x, y)dx I ⊗ dy J + O(x − y−(n−2) ) = X j (y)∂ y j [e0 (y, x − y)g IJ (y) + e IJ (x, y)]dx I ⊗ dy J + O(x − y−(n−2) ) = − X j (y)(∂ IIj e0 )(y, x − y)g IJ (y) dx I ⊗ dy J + O(x − y−(n−2+ε) ),
∀ ε > 0,
(4.3.38)
Consequently, the integral kernel ksym (x, y) of the operator R + R⊤ may be locally written as ksym (x, y) := k R (x, y) + k R⊤ (y, x) = (X j (x) − X j (y))(∂ IIj e0 )(y, x − y)g IJ (y) dx I ⊗ dy J + O(x − y−(n−2+ε) ),
∀ ε > 0.
(4.3.39)
At this stage, Theorem 9.61 applies to the ﬁrst term in the right side of (4.3.39) upon observing that (∂IIj e0 )(y, x − y) is a kernel of CalderónZygmund type of an operator bounded on L2 (∂Ω) (since Ω is a UR domain; cf. [50]), and keeping in mind that we are currently assuming (4.3.31). Also, Lemma 9.63 applies to the residual term in the right side of (4.3.39). In concert, these show that ksym (x, y) is the integral kernel of an integral operator which is compact on every L p (∂Ω, Λ l T M), 1 < p < ∞. This establishes (4.3.32). In fact, an inspection of the proof reveals that merely assuming that the vector ﬁeld X ∈ L∞ (∂Ω, T M) satisﬁes (4.3.36) implies that, for each exponent p ∈ (1, ∞), the distance from R + R⊤ to the space of linear and compact operators on L p (∂Ω, Λ l T M) is (correspondingly) small.
(4.3.40)
Moving on, assume that X is as in (4.3.31) and, in addition, is transversal to ∂Ω, in the sense described in (4.3.33). Then the operators A± := ± 12 ⟨X, ν♯ ⟩I + 12 (R − R⊤ ) : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) are welldeﬁned, linear and bounded for every p ∈ (1, ∞).
(4.3.41)
Furthermore, corresponding to p = 2, condition (4.3.33) and the deﬁnition of A± imply that for every f ∈ L2 (∂Ω, Λ l T M) we have c‖f‖2L2 (∂Ω,Λ l TM) ≤ ∫ ⟨X, ν♯ ⟩f2 dσ = (±2) ∫ ⟨A± f, f ⟩ dσ ∂Ω
∂Ω
≤ 2‖A± f‖L2 (∂Ω,Λ l TM) ‖f‖L2 (∂Ω,Λ l TM) ,
(4.3.42)
4.3 Compactness of the Double Layer on Regular SKT Domains  181
from which we ultimately deduce that (c/2)‖f‖L2 (∂Ω,Λ l TM) ≤ ‖A± f‖L2 (∂Ω,Λ l TM) ,
∀ f ∈ L2 (∂Ω, Λ l T M).
(4.3.43)
In turn, this implies that the operators A± : L2 (∂Ω, Λ l T M) → L2 (∂Ω, Λ l T M) are injective and have closed ranges.
(4.3.44)
1 1 ♯ ⊤ In fact, a similar argument applies to A⊤ ± = ± 2 ⟨X, ν ⟩I + 2 (R − R) and yields 2 l 2 l A⊤ ± : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
are injective with closed ranges.
(4.3.45)
From (4.3.45) and duality we then conclude that A± : L2 (∂Ω, Λ l T M) → L2 (∂Ω, Λ l T M) are surjective.
(4.3.46)
In concert with (4.3.44) this shows that actually A± : L2 (∂Ω, Λ l T M) → L2 (∂Ω, Λ l T M) are invertible.
(4.3.47)
Finally, from (4.3.41), (4.3.47), and perturbation theory (cf. [62]) it follows that there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞, with the property that ± 12 ⟨X, ν♯ ⟩I + 12 (R − R⊤ ) : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) are invertible operators for every p ∈ (p o , po ).
(4.3.48)
Combining (4.3.48) with (4.3.32) then readily yields (4.3.35). Finally, the very last claim in the statement of the theorem is a consequence of (4.3.40) and (4.3.48). In turn, Theorem 4.13 is a key ingredient in establishing the Fredholm properties for the principal value connection double layer recorded in the corollary below. Once again, the class of domains considered is larger than in the past, though the range of spaces in which the aforementioned results are proved is smaller (speciﬁcally, the integrability exponent is assumed to be near 2; compare with Theorem 4.12). Corollary 4.14. Let V be a potential as in (3.1.13) and ﬁx a degree l ∈ {0, 1, . . . , n}. Assume Ω ⊂ M is a UR domain whose outward unit conormal has the property that ν ∈ VMO(∂Ω, T ∗ M).
(4.3.49)
In this context, consider the principal value double layer operator K l as in (4.1.61), and recall that K ⊤ l stands for its (real ) transposed.
182  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains Then there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞, which depend on the above framework, with the property that for each p ∈ (p o , po ) and each z ∈ ℂ \ {0} the following operators are Fredholm with index zero: zI + K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), zI +
K⊤ l :
L (∂Ω, Λ T M) → L (∂Ω, Λ T M). p
l
p
l
(4.3.50) (4.3.51)
In fact, the results above continue to hold if (4.3.49) is relaxed to merely demanding that dist (ν, VMO(∂Ω, T ∗ M)) is sufficiently small,
(4.3.52)
relative to the background structures, where the distance is measured in BMO. Proof. Under the current hypotheses, we may invoke the full force of Theorem 4.13 with (4.3.53) X := ν♯ ∈ L∞ (∂Ω, T M) ∩ VMO(∂Ω, T M) upon noting that, for this choice, ⟨X, ν♯ ⟩ = ⟨ν♯ , ν♯ ⟩ = ⟨ν, ν⟩ = ν2 = 1 at σa.e. point on ∂Ω,
(4.3.54)
thanks to (9.1.3). In this case, comparing the jumpformula (4.3.26) with (4.1.82) reveals that R is actually K ⊤ l . As such, (4.3.32) presently becomes p l p l Kl + K⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
is a compact operator for each p ∈ (1, ∞).
(4.3.55)
For some z ∈ ℂ \ {0}, consider next the operator A := zI + 12 (K l − K ⊤ l ) and run the same program which, starting with (4.3.41), has produced (4.3.48). This now yields two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞, with the property that p l p l zI + 12 (K l − K ⊤ l ) : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
is an invertible operator for every p ∈ (p o , po ).
(4.3.56)
Moreover, a close inspection of this argument shows that p o may be chosen as to not depend on z (for z large using a Neumann series, and for z in a compact set using the results in [62]). Combing (4.3.55) with (4.3.56) then shows that the operators (4.3.50) and (4.3.51) are Fredholm with index zero for each p ∈ (p o , po ). Finally, the case when (4.3.49) is relaxed to (4.3.52) is handled analogously, making use of (9.11.13). Similar Fredholm properties hold for the operators zI + K l , zI + K ⊤ l on Sobolev spaces (of positive and negative order of smoothness, respectively) provided the scale of Sobolev spaces on ∂Ω exhibits suitable embedding properties into the scale of Lebesgue spaces on ∂Ω, as indicated below. Again, compared with Theorem 4.12, the novelty here is the fact that the underlying set is not necessarily a regular SKT domain.
4.3 Compactness of the Double Layer on Regular SKT Domains  183
Corollary 4.15. Suppose V is a potential as in (3.1.13) and pick a degree l ∈ {0, 1, . . . , n}. Let Ω ⊂ M be a UR domain whose outward unit conormal has the property that ν ∈ VMO(∂Ω, T ∗ M).
(4.3.57)
In addition, assume that for every p near 2 there exists p∗ > p with the property that p L1 (∂Ω) → L p∗ (∂Ω) compactly.
(4.3.58)
In this setting, deﬁne the principal value double layer operator K l as in (4.1.61), and recall that K ⊤ l stands for its (real) transposed. Then there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞ such that for each p ∈ (p o , po ) and each z ∈ ℂ \ {0} the following operators are Fredholm with index zero: p
p
zI + K l : L1 (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M), zI +
K⊤ l
:
p L−1 (∂Ω, Λ l T M)
→
p L−1 (∂Ω, Λ l T M).
(4.3.59) (4.3.60)
Moreover, the results above continue to hold if (4.3.57) is relaxed to asking that dist (ν, VMO(∂Ω, T ∗ M)) is sufficiently small,
(4.3.61)
relative to the background structures, where the distance is measured in BMO. Proof. Given that we are presently assuming (4.3.49), by relying on Theorem 9.61 (in place of Theorem 9.60), from the identity derived in (4.3.12) and the ensuing analysis (which remains valid for the current class of domains, as long as (4.3.58) is assumed), we conclude that for any given j, k ∈ {1, . . . , n} there exists p
C jk : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) linear and compact operator for each p ∈ (1, ∞),
(4.3.62)
with the property that for each p ∈ (1, ∞) we have ∂ τ jk (K l f) = K l (∂ τ jk f) + C jk f,
p
∀ f ∈ L1 (∂Ω, Λ l T M).
(4.3.63)
In particular, this entails that for each p ∈ (1, ∞) and z ∈ ℂ \ {0} we have ∂ τ jk [(zI + K l )f ] = (zI + K l )(∂ τ jk f) + C jk f,
p
∀ f ∈ L1 (∂Ω, Λ l T M).
(4.3.64)
Next, ﬁx p ∈ (p o , po ). The fact that the operator (4.3.50) is Fredholm implies that this has a quasiinverse on L p (∂Ω, Λ l T M). That is, there exist Q : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) linear, bounded, C : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) linear, compact, such that Q(zI + K l ) = I + C on L (∂Ω, Λ T M). p
l
(4.3.65)
184  4 Harmonic Layer Potentials Associated with the LeviCivita Connection on UR Domains
In what follows, we agree to ̃ generic linear compact operators from denote by C p l L1 (∂Ω, Λ T M) into L p (∂Ω, Λ l T M), whose speciﬁc nature may vary from one occurrence to another.
(4.3.66)
p
With this in mind, for every f ∈ L1 (∂Ω, Λ l T M) we may then estimate ‖∂ τ jk f‖L p (∂Ω,Λ l TM) ≤ ‖(I + C)∂ τ jk f‖L p (∂Ω,Λ l TM) + ‖C(∂ τ jk f)‖L p (∂Ω,Λ l TM) ̃ L p (∂Ω,Λ l TM) = ‖Q(zI + K l )(∂ τ jk f)‖L p (∂Ω,Λ l TM) + ‖Cf‖ ̃ L p (∂Ω,Λ l TM) ≤ c‖(zI + K l )(∂ τ jk f)‖L p (∂Ω,Λ l TM) + ‖Cf‖ ̃ L p (∂Ω,Λ l TM) ≤ c‖∂ τ jk (zI + K l )f‖L p (∂Ω,Λ l TM) + ‖Cf‖ ̃ L p (∂Ω,Λ l TM) , ≤ c‖(zI + K l )f‖L1p (∂Ω,Λ l TM) + ‖Cf‖
(4.3.67)
where c ∈ (0, ∞) is independent of f . Similarly, for every f ∈ L p (∂Ω, Λ l T M) we may estimate ‖f‖L p (∂Ω,Λ l TM) ≤ ‖(I + C)f‖L p (∂Ω,Λ l TM) + ‖Cf‖L p (∂Ω,Λ l TM) = ‖Q(zI + K l )f‖L p (∂Ω,Λ l TM) + ‖Cf‖L p (∂Ω,Λ l TM) ≤ c‖(zI + K l )f‖L p (∂Ω,Λ l TM) + ‖Cf‖L p (∂Ω,Λ l TM)
(4.3.68)
where, once again, c ∈ (0, ∞) is independent of f . Together, (4.3.67) and (4.3.68) prove that if p ∈ (p o , po ) and z ∈ ℂ \ {0} then there exists a constant c ∈ (0, ∞) and an operator ̃ as in (4.3.66) with the property that C ̃ L p (∂Ω,Λ l TM) , ‖f‖L1p (∂Ω,Λ l TM) ≤ c‖(zI + K l )f‖L1p (∂Ω,Λ l TM) + ‖Cf‖ p
for every differential form f ∈ L1 (∂Ω, Λ l T M).
(4.3.69)
In turn, (4.3.69) permits us to conclude that for every z ∈ ℂ \ {0} the operator in (4.3.59) has closed range and ﬁnite dimensional kernel. In particular, {zI + K l }z is a family of p semiFredholm operators on L1 (∂Ω, Λ l T M) continuously parametrized by z ∈ ℂ \ {0}. From the homotopy invariance of the index, and the fact that zI + K l is actually invertp ible on L1 (∂Ω, Λ l T M) if z is large (via a Neumann series), we ultimately conclude that the operator in (4.3.59) is Fredholm with index zero. From this and duality, a similar claim about the operator in (4.3.60) then follows. The following observation is going to be relevant in the last part of § 5.2. Remark 4.16. In the case when M = ℝn , equipped with the standard Euclidean metric, identity (4.3.63) with C jk as in (4.3.62) holds without having to assume that condition (4.3.58) holds. See [50, (6.2.6), (6.2.7), p. 2793].
5 Dirichlet and Neumann Boundary Value Problems for the HodgeLaplacian on Regular SKT Domains Our goal in this chapter is to provide the proofs of Theorems 1.8–1.16. This requires having a good understanding of key functional analytic properties of our brand of harmonic layer potentials introduced in Chapter 4, such as injectivity, dense range, compactness and, eventually, invertibility. Among the results established in § 5.1 is that, if Ω is a UR domain satisfying V + Ric ≥ 0 on M \ Ω and V + Ric > 0 on a subset of positive measure of each connected component of M \ Ω,
(5.0.1)
then there exists p n ∈ (1, ∞) (where n := dim M) such that 1 2I
pn l pn l + K⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
(5.0.2)
is injective. Given this and compactness results on (4.0.4) from Chapter 4, together with a regularity result following from such compactness, it follows that if Ω is a regular SKT domain, then ⊤ p l p l 1 (5.0.3) 2 I + K l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M) is invertible for all p ∈ (1, ∞). Such invertibility then follows also for 12 I + K l . Details can be found in § 5.2, along with a proof that the operators p
S l : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M), p
S l : L−1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)
(5.0.4)
are invertible, whenever Ω is a regular SKT domain, and that, if (5.0.1) also holds, then 1 2I
p
p
+ K l : L1 (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M)
(5.0.5)
l is invertible, and also 12 I + K ⊤ l is invertible on L −1 (∂Ω, Λ T M), for all p ∈ (1, ∞). Section 5.2 also has a Fatou type result (which one can regard as a warmup for the material in Chapter 6), pertaining to solutions of (∆HL − V)u = 0 in Ω with N u and N(∇u) in L p (∂Ω). Results of §§ 5.1 and 5.2 are applied to the proofs of Theorems 1.8–1.16 in § 5.3. For the existence part of Theorem 1.8, one arranges (5.0.1), with V ≡ 0 in Ω, and sets p
−1
u = Dl (( 12 I + K l ) f ) in Ω,
(5.0.6)
u = Sl (S−1 l f) in Ω.
(5.0.7)
or, alternatively,
Uniqueness (to the extent applicable) requires a further argument. We refer to § 5.3 for details, as well as for the proofs of Theorems 1.9–1.16.
186  5 Dirichlet and Neumann Boundary Value Problems
5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains Throughout, Ric denotes the Weitzenböck operator of M introduced in Deﬁnition 9.6, and n stands for the dimension of the manifold M. We debut by establishing an energy type identity involving the LeviCivita connection and the Weitzenböck operator on M. Proposition 5.1. Let Ω ⊂ M be an Ahlfors regular domain. As in the past, denote by ν the outward unit conormal to Ω and let σ := H n−1 ⌊∂Ω (where, as usual, n = dim M ≥ 2) be the surface measure on ∂Ω. Select an arbitrary degree l ∈ {0, 1, . . . , n} together with an integrability exponent p satisfying 2(n − 1)/n < p < ∞,
(5.1.1)
and ﬁx a realvalued function V ∈ L r (Ω)
r := np/(np − 2n + 2) ∈ (1, ∞).
where
(5.1.2)
In this setting, suppose the differential form u satisﬁes (with p denoting the Hölder conjugate exponent of p): l (hence u ∈ L∞ loc (Ω, Λ T M) by (2.2.45)),
N u ∈ L p (∂Ω)
2,r (∆HL − V)u = 0 in Ω (hence u ∈ Hloc (Ω, Λ l T M) by (2.1.118)), n.t. n.t. there exist u∂Ω and (∇u)∂Ω at σa.e. point on ∂Ω, N(∇u) ∈ L q (∂Ω) with q := max{p , 2(n − 1)/n}.
(5.1.3)
Then, with Ric denoting the Weitzenböck operator on M (cf. Deﬁnition 9.6), the following energy identity holds¹: n.t. ∫{∇u2 + Vu2 + ⟨Ric u, u ⟩} dVol = ∫ ⟨u∂Ω , ∇ν♯ u⟩ dσ. Ω
(5.1.4)
∂Ω
Proof. For starters, observe that thanks to (2.2.45), (2.1.118), and the hypotheses on u, 2,r l l u ∈ L∞ loc (Ω, Λ T M) ∩ H loc (Ω, Λ T M)
l and ∇u ∈ L∞ loc (Ω, Λ T M).
(5.1.5)
In addition, (2.2.50) and the nontangential maximal function properties of u give u ∈ L np/(n−1) (Ω, Λ l T M) ∇u ∈ L
nq/(n−1)
and
(5.1.6)
(Ω, Λ T M) ⊂ L2 (Ω, Λ l T M). l
To proceed, introduce the vector ﬁeld F⃗ : Ω → T M in an unequivocal manner by asking that for every x ∈ Ω and ξ ∈ T x∗ M we have ⃗
T x∗ M (ξ, F(x))T x M
= ⟨u(x), i Sym (∇∗, ξ)(∇u(x))⟩ .
1 all integrals involved being absolutely convergent
x
(5.1.7)
5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains 
187
It is then clear from (5.1.7) that F⃗ ≤ Cu∇u pointwise in Ω. Thanks to (5.1.5) this 1 ⃗ implies that F⃗ ∈ L∞ loc (Ω, T M) ⊂ L loc (Ω, T M). In fact, we may locally express F = F j ∂ j with (5.1.8) F j (x) = ⟨u(x), i Sym(∇∗, dx j )(∇u(x))⟩ . x
For any scalar function ψ ∈ ⃗
D (Ω) (div F, ψ)D(Ω)
C 01 (Ω),
we may then compute
= −D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T x M ⟨(grad ψ)(x), F(x)⟩ T x M dVol(x) Ω
⃗ = − ∫ T x∗ M ((grad ψ)♭ (x), F(x)) T x M dVol(x) Ω
⃗ = − ∫ T x∗ M (dψ(x), F(x)) T x M dVol(x) Ω
= − ∫⟨u, i Sym (∇∗, dψ)(∇u)⟩ dVol,
(5.1.9)
Ω
where the last equality above is based on (5.1.7) used here with ξ := dψ(x) ∈ T x∗ M. Next, (2.1.69) and the symmetry of the Weitzenböck operator Ric (cf. (9.3.26)) permit us to express (in a pointwise sense in Ω) ⟨u, i Sym (∇∗, dψ)(∇u)⟩ = −⟨u, [∇∗, ψ](∇u)⟩ = −⟨u, ∇∗ (ψ∇u)⟩ + ⟨u, ψ∇∗ ∇u⟩ = −⟨u, ∇∗ (ψ∇u)⟩ + ⟨u, ψ(−∆HL − Ric)u⟩ = −⟨u, ∇∗ (ψ∇u)⟩ − ψ{Vu2 + ⟨Ric u, u⟩}.
(5.1.10)
In addition, thanks to (5.1.5) and Lemma 4.4, ∫⟨u, ∇∗ (ψ∇u)⟩ dVol = ∫⟨∇u, ψ∇u⟩ dVol Ω
Ω
= ∫ ψ∇u2 dVol.
(5.1.11)
Ω
Collectively, (5.1.9)–(5.1.11) prove that div F⃗ = ∇u2 + Vu2 + ⟨Ric u, u ⟩ in D (Ω).
(5.1.12)
In concert with (5.1.6) and (5.1.2), this implies that div F⃗ ∈ L1 (Ω). Moreover, it is clear n.t. from (5.1.7) and the assumptions on u that the nontangential trace F⃗ ∂Ω exists σa.e. on ∂Ω and ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
n.t. n.t. = ⟨u∂Ω , i Sym (∇∗, ν)(∇u)∂Ω ⟩
n.t. = ⟨u∂Ω , ∇ν♯ u⟩,
(5.1.13)
188  5 Dirichlet and Neumann Boundary Value Problems where the last equality is based on (4.1.10). Upon noting that N(F)⃗ ≤ C N u ⋅ N(∇u) pointwise on ∂Ω (as seen from (2.3.19)) we conclude (from the ﬁrst and last lines in (5.1.3)) that N(F)⃗ ∈ L1 (∂Ω). At this stage, formula (5.1.4) follows from (5.1.12) and (5.1.13), by invoking Theorem 9.68. Two useful consequences of Proposition 5.1 are recorded next. The reader is reminded that n stands for the dimension of the manifold M. Proposition 5.2. Let Ω ⊂ M be an Ahlfors regular domain with outward unit conormal ν and surface measure σ = H n−1 ⌊∂Ω. Fix an arbitrary degree l ∈ {0, 1, . . . , n}, and suppose V ∈ L n/2 (Ω) is a realvalued function with the property that² Ric + V ≥ 0 at the level of lforms in Ω.
(5.1.14)
Then any differential form u satisfying u ∈ C 1 (Ω, Λ l T M), Nu ∈ L
2(n−1)/(n−2)
(∂Ω),
(∆HL − V)u = 0 in Ω, N(∇u) ∈ L 2(n−1)/n (∂Ω),
(5.1.15)
n.t. n.t. there exist u∂Ω and (∇u)∂Ω at σa.e. point on ∂Ω, as well as ∇ν♯ u = 0 at σa.e. point on ∂Ω,
(5.1.16)
is parallel in Ω, i.e., it satisﬁes ∇u = 0 in Ω. Moreover, if in addition to (5.1.14) one also has³ Ric + V > 0 at the level of lforms in a subset of positive measure in each connected component of Ω,
(5.1.17)
then any differential form u satisfying (5.1.15) and (5.1.16) necessarily vanishes identically in Ω. Proof. This is an immediate consequence of the energy identity from Proposition 5.1 and the unique continuation result from Proposition 9.4. Here is the other consequence of Proposition 5.1, alluded to earlier. Proposition 5.3. Suppose Ω ⊂ M is an Ahlfors regular domain with outward unit conormal ν and surface measure σ = H n−1 ⌊∂Ω. Pick an arbitrary degree l ∈ {0, 1, . . . , n}, and assume V ∈ L n/2 (Ω) is a realvalued function with the property that, at the level of lforms, V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of (5.1.18) positive measure of every connected component of Ω.
2 that is, for a.e. x ∈ Ω one has ⟨(Ric + V)η, η ⟩x ≥ 0 for every η ∈ Λ l T x M, where ⟨⋅, ⋅⟩x is the (real) pairing in Λ l T x M 3 i.e., there exists a set O ⊆ Ω of positive measure with the property that for each x ∈ O one has ⟨(Ric + V)η, η ⟩x > 0 for every η ∈ Λ l T x M, where ⟨⋅, ⋅⟩x is the (real) pairing in Λ l T x M
5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains 
189
Then any differential form u satisfying u ∈ C 1 (Ω, Λ l T M), Nu ∈ L
n.t. u∂Ω
(∆HL − V)u = 0 in Ω,
(∂Ω), N(∇u) ∈ L2(n−1)/n (∂Ω), n.t. = 0 and (∇u)∂Ω exists at σa.e. point on ∂Ω, 2(n−1)/(n−2)
(5.1.19)
necessarily vanishes identically in Ω. Proof. This is seen from the energy identity established in Proposition 5.1 and the unique continuation result from Proposition 9.4. The result discussed next identiﬁes an optimal context where the principal value double layer gives rise to operators that are injective or have dense range in arbitrary UR domains. Proposition 5.4. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω and ﬁx an arbitrary degree l ∈ {0, 1, . . . , n}. Suppose V is a potential as in (3.1.13) with the additional property that, at the level of lforms, V + Ric ≥ 0 in M \ Ω, and V + Ric > 0 in a subset of positive measure of each connected component of M \ Ω.
(5.1.20)
Associated with the domain Ω and potential V consider the principal value double layer potential operator K l deﬁned as in (4.1.61). Then, in the case when n ≥ 3, 1 2I
+ K l : L2(n−1)/(n−2) (∂Ω, Λ l T M) → L2(n−1)/(n−2) (∂Ω, Λ l T M)
is a welldeﬁned, linear and bounded operator, with dense range,
(5.1.21)
and, with K ⊤ l denoting the (real ) transposed of K l , 1 2I
2(n−1)/n + K⊤ (∂Ω, Λ l T M) → L2(n−1)/n (∂Ω, Λ l T M) l : L
is a welldeﬁned, linear, bounded, and injective operator.
(5.1.22)
Moreover, corresponding to the case when n = 2, for every p ∈ (1, ∞) one has that ⊤ 1 2 I + Kl : 1 2 I + Kl :
L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) is injective, while L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) has dense range.
(5.1.23)
Proof. Denote by ν ∈ T ∗ M the outward unit conormal to Ω and by σ := H n−1 ⌊∂Ω the surface measure on ∂Ω. Abbreviate Ω+ := Ω and Ω− := M \ Ω. Also, let ∇ stand for the LeviCivita connection on M, acting on differential forms. Consider ﬁrst the case when n ≥ 3. With the goal of establishing the injectivity of the operator (5.1.22), let f ∈ L2(n−1)/n (∂Ω, Λ l T M) be such that ( 12 I + K ⊤ l )f = 0
(5.1.24)
190  5 Dirichlet and Neumann Boundary Value Problems
and deﬁne u± := Sl f in Ω± .
(5.1.25)
Then from (5.1.25), (3.2.5), (3.2.9), (3.2.7), and (4.1.81) we deduce that { (∆HL − V)u± = 0 in Ω± , { { { N(u± ) ∈ L2(n−1)/(n−2) (∂Ω), N(∇u± ) ∈ L2(n−1)/n (∂Ω), { { { { n.t. n.t. ∃ u± ∂Ω± , ∃ (∇u± )∂Ω± at σa.e. point on ∂Ω. {
(5.1.26)
In addition, (5.1.24) and (4.1.82) imply ∇ν♯ u− = 0 at σa.e. point on ∂Ω.
(5.1.27)
In light of Proposition 5.2 (which is applicable for u− in Ω− , given (5.1.20)), we see that n.t. n.t. (5.1.27) forces u− = 0 in Ω− . In turn, this implies u+ ∂Ω+ = u− ∂Ω− = 0 on ∂Ω. Having es+ tablished this, we may use the energy identity (2.3.16) for u in Ω+ in order to conclude (mindful of the fact that V ≥ 0 on M) that du+ = 0 and δu+ = 0 in Ω+ . At this stage, the same reasoning which, starting with υ in (3.2.72) has produced (3.2.75) now yields ∇ν♯ u+ = 0 at σa.e. point on ∂Ω. (5.1.28) Having proved this, we now make use of the jumpformula (4.1.82) to conclude that f = ∇ν♯ u− − ∇ν♯ u+ = 0 at σa.e. point on ∂Ω.
(5.1.29)
2(n−1)/n (∂Ω, Λ l T M). Of course, (5.1.21) This shows that 12 I + K ⊤ l is indeed injective on L then readily follows from what we have just proved and duality. The claims about the operators in (5.1.23) are justiﬁed in a similar manner. Finally, when n = 2, the proof proceeds much as before, bearing in mind that, this time, (3.2.10) and (3.2.11) imply N(u± ) ∈ L∞ (∂Ω) and N(∇u± ) ∈ L p (∂Ω), a scenario in which Proposition 5.2 with n = 2 continues to apply.
Here is a companion result to Proposition 5.4. Proposition 5.5. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω and ﬁx an arbitrary degree l ∈ {0, 1, . . . , n}. Suppose V is a potential as in (3.1.13) with the additional property that, at the level of lforms, V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of positive measure of every connected component of Ω.
(5.1.30)
Associated with the domain Ω and potential V, consider the principal value double layer potential operator K l deﬁned as in (4.1.61). Then when n ≥ 3, it follows that − 12 I + K l : L2(n−1)/(n−2) (∂Ω, Λ l T M) → L2(n−1)/(n−2) (∂Ω, Λ l T M) is a welldeﬁned, linear and bounded operator, with dense range,
(5.1.31)
5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains 
and
2(n−1)/n − 12 I + K ⊤ (∂Ω, Λ l T M) → L2(n−1)/n (∂Ω, Λ l T M) l : L
is a welldeﬁned, linear, bounded, and injective operator.
191
(5.1.32)
Furthermore, corresponding to the case when n = 2, for each p ∈ (1, ∞) one has that p l p l − 12 I + K ⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M) is injective, while
− 12 I + K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) has dense range.
(5.1.33)
Proof. This follows directly from Proposition 5.4 with the roles of Ω and M \ Ω interchanged (mindful of the fact that the outward unit conormal for the latter domain is −ν). We now consider the action of the principal value double layer potential operator on Sobolev spaces on the boundary. Proposition 5.6. Let Ω ⊂ M be an Ahlfors regular domain satisfying a twosided local John condition. Denote by ν ∈ T ∗ M the outward unit conormal to Ω and by σ := H n−1 ⌊∂Ω the surface measure on ∂Ω. Fix an arbitrary degree l ∈ {0, 1, . . . , n} and let V be a potential as in (3.1.13) with the additional property that, at the level of lforms, V + Ric ≥ 0 in M, and V + Ric > 0 in a subset of positive measure of each connected component of Ω and of M \ Ω.
(5.1.34)
Associated with the domain Ω and potential V consider the principal value double layer potential operator K l deﬁned as in (4.1.61). Then, in the case when n ≥ 3, 2(n−1)/n
± 12 I + K l : L1
2(n−1)/n
(∂Ω, Λ l T M) → L1
(∂Ω, Λ l T M)
(5.1.35)
are welldeﬁned, linear, bounded, and injective operators, and, with K ⊤ l denoting the (real ) transposed of K l , ± 12 I + K ⊤ l : L −1
2(n−1)/(n−2)
2(n−1)/(n−2)
(∂Ω, Λ l T M) → L−1
(∂Ω, Λ l T M)
(5.1.36)
are a welldeﬁned, linear and bounded operators, with dense ranges. Moreover, corresponding to the case when n = 2, for every p ∈ (1, ∞) one has that p
p
± 12 I + K l : L1 (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M) are injective, while l l ± 12 I + K ⊤ l : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M) have dense ranges. p
p
(5.1.37)
Proof. We shall treat the case n ≥ 3 (as the situation when n = 2 is handled similarly). Under the current assumptions on Ω, we have ∂(Ω) = ∂Ω and the embeddings in (9.5.39) imply 2(n−1)/n
L1
(∂Ω, Λ l T M) → L2(n−1)/(n−2) (∂Ω, Λ l T M).
(5.1.38)
192  5 Dirichlet and Neumann Boundary Value Problems As in the past, abbreviate Ω+ := Ω and Ω− := M \ Ω. With the goal of establish2(n−1)/n ing the injectivity of the operator 12 I + K l acting on L1 (∂Ω, Λ l T M), pick some 2(n−1)/n f ∈ L1 (∂Ω, Λ l T M) such that ( 12 I + K l )f = 0 on ∂Ω.
(5.1.39)
Thanks to (5.1.38), we have f ∈ L2(n−1)/(n−2) (∂Ω, Λ l T M), so if we now deﬁne u± := Dl f in Ω±
(5.1.40)
then (4.1.9) and Theorems 4.8, 4.9 ensure that { (∆HL − V)u± = 0 in Ω± , { { { N(u± ) ∈ L2(n−1)/(n−2) (∂Ω), N(∇u± ) ∈ L2(n−1)/n (∂Ω), { { { { n.t. n.t. ∃ u± ∂Ω± , ∃ (∇u± )∂Ω± at σa.e. point on ∂Ω. {
(5.1.41)
Furthermore, (5.1.39) implies n.t. u+ ∂Ω+ = 0
at σa.e. point on ∂Ω,
(5.1.42)
while (4.1.114) gives ∇ν♯ u + = ∇ν♯ u −
at σa.e. point on ∂Ω.
(5.1.43)
Granted (5.1.34), (5.1.41), and (5.1.42), Proposition 5.3 applies and gives that u+ = 0 in Ω+ . Together with concert with (5.1.43), this further implies ∇ν♯ u− = 0 at σa.e. point on ∂Ω− . With this in hand, Proposition 5.2 applied to u− in Ω− gives u− = 0 in Ω− . n.t. n.t. Hence, ultimately, f = u+ ∂Ω+ − u− ∂Ω− = 0 on ∂Ω (cf. (4.1.79)). This proves that 12 I + K l
is injective on L1 (∂Ω, Λ l T M). Via duality, we also obtain that 12 I + K ⊤ l has dense 2(n−1)/(n−2) range when acting on L−1 (∂Ω, Λ l T M). Finally, that similar properties are also enjoyed by the operators − 12 I + K l , acting 2(n−1)/n 2(n−1)/(n−2) on L1 (∂Ω, Λ l T M), and − 12 I + K ⊤ (∂Ω, Λ l T M), is proved l , acting on L −1 in an analogous fashion, swapping the roles of Ω+ and Ω− (mindful of the fact that (5.1.34) is unaffected by such a change). 2(n−1)/n
Regarding functional analytic properties of the conormal derivative of the connection double layer (as deﬁned in item (3) of Theorem 4.9) we have the following result. Proposition 5.7. Suppose Ω ⊂ M is an Ahlfors regular domain satisfying a twosided local John condition. Denote by ν ∈ T ∗ M the outward unit conormal to Ω and by σ := H n−1 ⌊∂Ω the surface measure on ∂Ω. Fix an arbitrary degree l ∈ {0, 1, . . . , n} and let V be a potential as in (3.1.13) with the additional property that, at the level of lforms, V + Ric ≥ 0 in M, and V + Ric > 0 in a subset of positive measure of each connected component of Ω and of M \ Ω.
(5.1.44)
5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains 
193
Associated with the domain Ω and potential V consider the principal value double layer potential operator Dl introduced in Deﬁnition 4.1. Also, let ∇ stand for the LeviCivita connection on M, acting on differential forms. In relation to these, recall (4.1.115). Then, in the case when n ≥ 3, 2(n−1)/n
∇ν♯ Dl : L 1
(∂Ω, Λ l T M) → L2(n−1)/n (∂Ω, Λ l T M)
(5.1.45)
is a welldeﬁned, linear, bounded and injective operator, while 2(n−1)/(n−2)
∇ν♯ Dl : L2(n−1)/(n−2) (∂Ω, Λ l T M) → L−1
(∂Ω, Λ l T M)
(5.1.46)
is a welldeﬁned, linear, and bounded operator, with dense range. Finally, corresponding to the case when n = 2 for each p ∈ (1, ∞) one has p
∇ν♯ Dl : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) is injective, while p
∇ν♯ Dl : L p (∂Ω, Λ l T M) → L−1 (∂Ω, Λ l T M) has dense range.
(5.1.47)
Proof. The argument is similar to the proof of Proposition 5.6. Assume that n ≥ 3 (the case n = 2 is dealt with analogously). To establish the injectivity of the operator (5.1.45), 2(n−1)/n let f ∈ L1 (∂Ω, Λ l T M) be such that ∇ν♯ Dl f = 0 on ∂Ω.
(5.1.48)
Thanks to (5.1.38), we have f ∈ L2(n−1)/(n−2) (∂Ω, Λ l T M), so if we now deﬁne u± := Dl f in Ω± then from (4.1.9) and Theorems 4.8, 4.9 we conclude that { (∆HL − V)u± = 0 in Ω± , { { { N(u± ) ∈ L2(n−1)/(n−2) (∂Ω), N(∇u± ) ∈ L2(n−1)/n (∂Ω), { { { { n.t. n.t. ∃ u± ∂Ω± , ∃ (∇u± )∂Ω± at σa.e. point on ∂Ω. {
(5.1.49)
In addition, (5.1.48) implies ∇ν♯ u± = 0 at σa.e. point on ∂Ω.
(5.1.50)
Bring in Proposition 5.2, which is presently applicable both in Ω+ and in Ω− , given (5.1.44). In light of (5.1.50) this forces u± = 0 in Ω± . In concert with (4.1.79), this further n.t. n.t. implies f = u+ ∂Ω+ − u− ∂Ω− = 0 on ∂Ω. With this in hand, the claims made in (5.1.45) readily follow. Finally, the claims made in (5.1.46) are consequences of (5.1.45), item (4) of Theorem 4.9, and duality. We continue by recording the following useful consequence of Theorem 4.12. Corollary 5.8. Let Ω ⊂ M be a regular SKT domain and suppose the potential V is as in (3.1.13). Pick some degree l ∈ {0, 1, . . . , n}, ﬁx an arbitrary complex number z ∈ ℂ \ {0}, and select two integrability exponents p, q ∈ (1, ∞).
194  5 Dirichlet and Neumann Boundary Value Problems
Then one has the following improved integrability result: f ∈ L p (∂Ω, Λ l T M) and q
q
(5.1.51)
q
(5.1.52)
} ⇒ f ∈ L1 (∂Ω, Λ l T M).
(zI + K l )f ∈ L1 (∂Ω, Λ l T M) In particular,
Ker (zI + K l : L p (∂Ω, Λ l T M)) = Ker (zI + K l : L1 (∂Ω, Λ l T M)).
Moreover, similar results are valid when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the targeted exponents p, q, the complex number z, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. This follows from Theorem 4.12 by arguing as in the proof of Theorem 3.9 based on the abstract regularity result from Lemma 3.10. In our last result in this section we take a look at the point spectrum of the transpose of the principal value connection double layer operator. Proposition 5.9. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω and ﬁx some degree l ∈ {0, 1, . . . , n}. Also, let V be a potential as in (3.1.13) with the additional property that V + Ric ≥ 0 at the level of lforms in M.
(5.1.53)
Associated with the above domain Ω and potential V consider the principal value boundary version the connection double layer operator K l deﬁned as in (4.1.61). Then, assuming n ≥ 3, it follows that the point spectrum of the operator K ⊤ l acting from the space L2(n−1)/n (∂Ω, Λ l T M) into itself is a subset of the interval [− 12 , 12 ]. Moreover, corresponding to the case when n = 2, for every p ∈ (1, ∞) it follows that p l the point spectrum of the operator K ⊤ l acting from the space L (∂Ω, Λ T M) into itself is 1 1 a subset of the interval [− 2 , 2 ]. Proof. Consider ﬁrst the case when n ≥ 3. Let λ ∈ ℂ be an arbitrary eigenvalue of the 2(n−1)/n (∂Ω, Λ l T M) into itself. Then there exists operator K ⊤ l acting from the space L f ∈ L2(n−1)/n (∂Ω, Λ l T M)
such that f ≢ 0 and λf = K ⊤ l f.
(5.1.54)
Associated with the given domain Ω and potential V, consider the single layer operator Sl as in (3.2.3). Introduce
u± := Sl f in Ω± ,
(5.1.55)
where, as before, Ω+ := Ω and Ω− := M \ Ω. Notice that (3.2.9) implies N(u± ) ∈ L2(n−1)/(n−2) (∂Ω).
(5.1.56)
Also, as a consequence of (3.2.6), (3.2.8), (3.2.7), and (3.2.16), there holds (∆HL − V)u± = 0 in Ω± , N(∇u± ) ∈ L2(n−1)/n (∂Ω) and n.t. n.t. there exist u± ∂Ω and (∇u± )∂Ω at σa.e. point on ∂Ω,
(5.1.57)
5.1 Functional Analytic Properties for Harmonic Layer Potentials in UR Domains 
where σ := H we may write
195
n−1 ⌊∂Ω is the surface measure on ∂Ω. Next, based on (4.1.82) and (5.1.54)
1 ∇ν♯ u± = (∓ 12 I + K ⊤ l ) f = (λ ∓ 2 )f
at σa.e. point on ∂Ω,
(5.1.58)
where ν : ∂Ω → T ∗ M is the outward unit conormal to Ω. If we now set I± := ∫ {∇u± 2 + Vu± 2 + ⟨Ric u± , u± ⟩} dVol,
(5.1.59)
Ω±
it follows from (5.1.56), (5.1.57), (5.1.58), and the energy identity (5.1.4) from Proposition 5.1 (written for both Ω+ and Ω− ) that n.t. I± = ∫ ⟨u± ∂Ω± , ∇ν♯ u± ⟩ dσ = ∫ ⟨S l f, (− 12 I ± K ⊤ l )f ⟩ dσ ±
∂Ω
∂Ω
= (±λ − 12 ) ∫ ⟨S l f, f ⟩ dσ,
(5.1.60)
∂Ω
where we have denoted by ν± = ±ν the outward unit conormals to Ω± . Therefore, 2
(λ + 12 )I+ = (λ − 14 ) ∫ ⟨S l f, f ⟩ dσ = (−λ + 12 )I− ,
(5.1.61)
∂Ω
where the ﬁrst identity in (5.1.61) has been obtained from the version of (5.1.60) corresponding to the choice of sign plus, by multiplication of the leftmost and rightmost terms of the sequence of identities by (λ + 12 ), while the second identity in (5.1.61) follows from the version of (5.1.60) corresponding to the choice of sign minus, and multiplication of the leftmost and rightmost terms of the sequence of identities by (−λ + 12 ). To proceed, recall that the Ricci tensor is real and symmetric (cf. (9.3.26)). Granted these, the expressions ⟨Ric u± , u± ⟩ are actually real at every point in Ω± . As such, (5.1.59) and the hypothesis (5.1.53) allow us to conclude that I± ∈ [0, ∞).
(5.1.62)
I+ + I− > 0.
(5.1.63)
Going further we claim that, in fact,
To justify this claim, argue by contradiction. Assuming that I+ + I− = 0 then forces both I+ and I− to vanish, in light of (5.1.62). In such a scenario, based on the deﬁnition of the expressions I± from (5.1.59) and the hypothesis (5.1.53) we obtain that ∫Ω ∇u± 2 dVol = 0. Consequently, ∇u± = 0 in Ω± . In concert with the jump relations ± (4.1.82) this ﬁnally yields f = ∇ν♯ u− − ∇ν♯ u+ = 0, (5.1.64)
196  5 Dirichlet and Neumann Boundary Value Problems contradicting the fact that f ≢ 0. This contradiction completes the proof of the claim made in (5.1.63). With (5.1.63) in hand, we now return to (5.1.61) and note that the equality between its ﬁrst and third terms allows us to express λ as λ=
1 I− − I+ ⋅( ). 2 I− + I+
(5.1.65)
From this and (5.1.62), (5.1.63) we then conclude that λ ∈ [− 12 , 12 ], as desired. Finally, the proof of the very last claim in the statement of the proposition, pertaining to the case when n = 2, may be carried out using the same ingredients as above.
5.2 Invertibility Results for Layer Potentials Associated with the LeviCivita Connection As a culmination of the work carried out in Chapter 4 and Section 5.1 we are now well equipped to formulate and prove the following basic invertibility results. Theorem 5.10. Suppose V is a potential as in (3.1.13) and let Ω ⊂ M be a regular SKT domain. Select an arbitrary l ∈ {0, 1, . . . , n} and, associated with the domain Ω and potential V, consider the single layer potential S l deﬁned as in (3.2.6), as well as the principal value double layer potential operator K l deﬁned as in (4.1.61). Also, ﬁx some p ∈ (1, ∞) arbitrary. Then the following conclusions hold: (1) The operators p
S l : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M), Sl :
p L−1 (∂Ω, Λ l T M)
→ L (∂Ω, Λ T M), p
l
(5.2.1) (5.2.2)
are invertible. (2) The operators p
∇ν♯ Dl : L p (∂Ω, Λ l T M) → L−1 (∂Ω, Λ l T M), ∇ν♯ Dl :
p L1 (∂Ω, Λ l T M)
→ L (∂Ω, Λ T M), p
l
(5.2.3) (5.2.4)
are Fredholm with index zero. They are actually invertible if, at the level of lforms, V + Ric ≥ 0 on M, and V + Ric > 0 in a subset of positive measure of each connected component of Ω and of M \ Ω.
(5.2.5)
(3) Under the assumption that at the level of lforms one has V + Ric ≥ 0 in M \ Ω, and V + Ric > 0 in a subset of positive measure of each connected component of M \ Ω
(5.2.6)
5.2 Invertibility Results for Layer Potentials Associated with the LeviCivita Connection  197
the following operators are invertible: 1 2 I + Kl : ⊤ 1 2 I + Kl : 1 2 I + Kl : 1 2I
+
K⊤ l :
L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
(5.2.7)
L (∂Ω, Λ T M) → L (∂Ω, Λ T M),
(5.2.8)
p
l
p L1 (∂Ω, Λ l T M) p L−1 (∂Ω, Λ l T M)
p
→ →
l
p L1 (∂Ω, Λ l T M), p L−1 (∂Ω, Λ l T M).
(5.2.9) (5.2.10)
(4) Under the additional assumption that at the level of lforms one has V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of positive measure of each connected component of Ω
(5.2.11)
the following operators are invertible: − 12 I + K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
(5.2.12)
p l p l − 12 I + K ⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M),
(5.2.13)
p − 12 I + K l : L1 (∂Ω, Λ l T M) p l − 12 I + K ⊤ l : L −1 (∂Ω, Λ T M)
→ →
p L1 (∂Ω, Λ l T M), p L−1 (∂Ω, Λ l T M).
(5.2.14) (5.2.15)
Finally, the same results are valid when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the exponents p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. The claims in items (3) and (4) are direct consequences of Theorem 4.12, Propositions 5.45.5, as well as Fredholm theory. Assume now that V is a potential as in (3.1.13) and ﬁx an arbitrary p ∈ (1, ∞). For this portion of the proof, we agree to write S l,V , K l,V in order to emphasize the dependence on the potential function V. From Theorem 4.12 it follows that the operators ± 12 I + K l,V are Fredholm with index zero both on L p (∂Ω, Λ l T M) and on p p l L1 (∂Ω, Λ l T M), while ± 12 I + K ⊤ l,V are Fredholm with index zero both on L (∂Ω, Λ T M) p and on L−1 (∂Ω, Λ l T M). Granted these, we may conclude with the help of (4.1.119) that p S l,V : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M) has a ﬁnite codimensional (hence also closed) range. In concert with the last part in Proposition 3.5, this implies that this operator is semiFredholm. On the other hand, if we now associate boundary layer potentials with a potential function V o which is a sufficiently large positive constant (anything larger that supx∈M ‖Ricx ‖Λ l T x M→Λ l T x M will do), and run the same argument as above, we arrive at the conclusion that S l,V o is invertible as an operator from L p (∂Ω, Λ l T M) p onto L1 (∂Ω, Λ l T M) (given that, from what we have already proved in items (3) and (4), p the operators ± 12 I + K l,V o are invertible both on L p (∂Ω, Λ l T M) and on L1 (∂Ω, Λ l T M), p p l l while ± 12 I + K ⊤ l,V o are invertible both on L (∂Ω, Λ T M) and on L −1 (∂Ω, Λ T M)). Since from (4.1.108) we know that p
S l,V − S l,V o : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M)
(5.2.16)
198  5 Dirichlet and Neumann Boundary Value Problems is a compact operator, writing S l,V = S l,V o + (S l,V − S l,V o ) leads to the conclusion that p the operator S l,V : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M) is Fredholm with index zero. Having established this, another reference to the last claim in Proposition 3.5 ﬁnally allows to conclude that the operator (5.2.1) is invertible. Then the invertibility of (5.2.2) follows from this and transposition (cf. Proposition 3.5). Turning attention to the claims in item (2), we may use (4.1.119) and what we have proved in item (1) to express ∇ν♯ Dl , both as an operator from L p (∂Ω, Λ l T M) to p p L−1 (∂Ω, Λ l T M) and from L1 (∂Ω, Λ l T M) to L p (∂Ω, Λ l T M), as the composition 1 1 S−1 l ( 2 I + K l )(− 2 I + K l ).
(5.2.17)
In either context, the latter is Fredholm with index zero, proving the ﬁrst claim in item (2). Moreover, when (5.2.5) holds, then all factors in the composition (5.2.17) are invertible (from what we have shown already), so the operators (5.2.3) and (5.2.4) become invertible in this case as well. That the very ﬁnal claim in the statement of the theorem holds is a consequence of the fact that all ingredients used in the proof thus far continue to hold in the case described there. Generally speaking, given a linear and bounded operator T acting on a Banach space X, we agree to denote by Spec (T; X) := {z ∈ ℂ : zI − T is not invertible on X}
(5.2.18)
the spectrum of the operator T on X. Concerning the spectral properties of the harmonic double layer operator in regular SKT domains we record the following result. Proposition 5.11. Let V be a potential as in (3.1.13) and assume that Ω ⊂ M is a regular SKT domain. Pick an arbitrary degree l ∈ {0, 1, . . . , n} and, associated with the domain Ω and potential V, consider the principal value double layer potential operator K l deﬁned as in (4.1.61). Finally, ﬁx an arbitrary integrability exponent p ∈ (1, ∞). Then the following conclusions hold: (1) If at the level of lforms one has V + Ric ≥ 0 on M
(5.2.19)
it follows that p
Spec (K l ; L p (∂Ω, Λ l T M)) = Spec (K l ; L1 (∂Ω, Λ l T M)) ⊆ [− 12 , 12 ]
(5.2.20)
and the two spectra are actually independent of p. (2) If at the level of lforms one has V + Ric ≥ 0 on M, and V + Ric > 0 in a subset of positive measure of each connected component of Ω and of M \ Ω,
(5.2.21)
the spectral radii of the operator K l considered both p on L p (∂Ω, Λ l T M) and on L1 (∂Ω, Λ l T M) are < 12 .
(5.2.22)
then
5.2 Invertibility Results for Layer Potentials Associated with the LeviCivita Connection  199
In particular, whenever (5.2.21) holds, one may expand (± 12 I + K l )
−1
∞
j
= − ∑ (∓2)j+1 K l
(5.2.23)
j=0
with convergence in the operator norm both in the space L p (∂Ω, Λ l T M) and in the p space L1 (∂Ω, Λ l T M). p
Proof. Since the Banach spaces L p (∂Ω, Λ l T M) and L1 (∂Ω, Λ l T M) are inﬁnite dimensional and since the operators (4.3.1)(4.3.3) are compact, Riesz’s theorem implies that, on the one hand, 0 ∈ Spec (K l ; L p (∂Ω, Λ l T M)) and (5.2.24) p 0 ∈ Spec (K l ; L1 (∂Ω, Λ l T M)). On the other hand, for every z ∈ ℂ \ {0}, the operator zI − K l is Fredholm with index p zero both on L p (∂Ω, Λ l T M) and on L1 (∂Ω, Λ l T M). In concert with (5.1.52) and (5.2.24), this implies that p
Spec (K l ; L p (∂Ω, Λ l T M)) = Spec (K l ; L1 (∂Ω, Λ l T M))
(5.2.25)
and the two spectra are in fact independent of p. With this in hand and granted (5.2.21), the last inclusion in (5.2.20) then follows from the point spectrum result established in Proposition 5.9. This completes the treatment of item (1). Finally, the claims in item (2) are seen from what we have just proved with the help of items (3) and (4) in Theorem 5.10. We shall also ﬁnd it useful to have a Fatou type result of the sort described in the proposition below. Proposition 5.12. Assume that Ω ⊂ M is a regular SKT domain with outward unit conormal ν ∈ T ∗ M and surface measure σ := H n−1 ⌊∂Ω. Let the function V be as in (3.1.13), and ﬁx l ∈ {0, 1, . . . , n} along with some p ∈ (1, ∞). As usual, let ∇ stand for the LeviCivita connection on M acting on differential forms. Then any differential form u satisfying u ∈ C 1 (Ω, Λ l T M), and
(∆HL − V)u = 0 in Ω,
N(∇u) ∈ L p (∂Ω),
(5.2.26)
has the property that N u ∈ L p (∂Ω),
(5.2.27)
and the pointwise nontangential traces n.t. n.t. u∂Ω , (∇u)∂Ω exist at σa.e. point on ∂Ω. Moreover,
p n.t. u∂Ω belongs to L1 (∂Ω, Λ l T M),
n.t. (∇u)∂Ω belongs to L p (∂Ω, T ∗ M ⊗ Λ l T M),
(5.2.28)
(5.2.29)
200  5 Dirichlet and Neumann Boundary Value Problems and there exists a constant C ∈ (0, ∞), independent of u, such that n.t. n.t. p u∂Ω L (∂Ω,Λ l TM) + (∇u)∂Ω L p (∂Ω,T ∗ M⊗Λ l TM) 1 ≤ C(‖N u‖L p (∂Ω) + ‖N(∇u)‖L p (∂Ω) ).
(5.2.30)
In fact, all results above continue to hold when Ω is assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the given p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. From Proposition 9.19 (cf. (9.5.56) in this regard) we know that assumptions (5.2.26) enure that the membership in (5.2.27) holds. To proceed, let {Ω j }j∈ℕ approximate Ω in the manner described in Proposition 2.35. For each ﬁxed integer j, we have u ∈ C 1 (Ω j , Λ l T M), and from what we proved so far it follows that uΩ j does satisfy the conditions similar to (4.1.14) relative to Ω j . As such, (4.1.15) gives the integral representation formula n.t. n.t. u = Dl,j (u∂Ω j ) − Sl,j (∇ν♯j u∂Ω j ) in Ω j ,
(5.2.31)
where the subscript j indicates that the objects in question are associated with Ω j . Fix n.t. now a point x ∈ Ω and consider limj→∞ Dl,j (u∂Ω j )(x). Making use of the change of variable formula (2.2.85) and keeping in mind item (viii) of Proposition 2.35, we have n.t. lim Dl,j (u∂Ω j )(x)
(5.2.32)
j→∞
= lim ∫ ⟨i Sym (∇∗, (ν j ∘ 𝛶j )(y))(∇2 Γ l )(x, 𝛶j (y)), (u ∘ 𝛶j )(y)⟩𝛶j (y) J j (y) dσ(y), j→∞
∂Ω
where composition of forms with 𝛶j is interpreted as pullback. Note that since by (2.2.86) for each j we have u ∘ 𝛶j  ≤ N u pointwise on ∂Ω, it follows from (5.2.27) that the sequence {u ∘ 𝛶j }j∈ℕ is bounded in L p (∂Ω, Λ l T M). Thus, by eventually passing to a subsequence, there is no loss of generality in assuming that there exists a differential form f ∈ L p (∂Ω, Λ l T M), independent of x, with the property that u ∘ 𝛶j → f weakly in L p (∂Ω, Λ l T M), as j → ∞ .
(5.2.33)
Keeping this in mind and availing ourselves of (2.2.80)–(2.2.85), we deduce from (5.2.32) that n.t. Dl,j (u∂Ω j )(x) → Dl f(x) as j → ∞, (5.2.34) where Dl is the connection double layer associated with the original domain Ω. A similar reasoning also shows that there exists some function g ∈ L p (∂Ω, Λ l T M) (again, independent of x) with the property that n.t.
Sl,j (∇ν♯j u∂Ω j )(x) → Sl g(x)
as j → ∞,
(5.2.35)
5.2 Invertibility Results for Layer Potentials Associated with the LeviCivita Connection  201
where Sl is the single layer associated with Ω. In concert, (5.2.31), (5.2.34), (5.2.35) prove that there exist f, g ∈ L p (∂Ω, Λ l T M) (5.2.36) such that u = Dl f − Sl g in Ω. n.t. As a consequence of this, (4.1.79), and (3.2.7) we therefore conclude that u∂Ω exists at σa.e. point on ∂Ω and, in fact, n.t. u∂Ω = ( 12 I + K l )f − S l g on ∂Ω.
(5.2.37)
When used in combination with the information in the second line of (5.2.26), this ensures that Proposition 9.17 may be currently invoked. Based on this, we then conclude p n.t. that u∂Ω ∈ L1 (∂Ω, Λ l T M) (plus a natural estimate). In turn, this membership forces p n.t. ( 12 I + K l )f = u∂Ω + S l g ∈ L1 (∂Ω, Λ l T M),
(5.2.38)
thanks to (4.1.102). Next, make use of Corollary 5.8 to deduce from (5.2.38) that actup ally f ∈ L1 (∂Ω, Λ l T M). Having established this, we may invoke (4.1.113) and (4.1.81) to ﬁnally conclude that n.t. n.t. n.t. (∇u)∂Ω = (∇Dl f)∂Ω − (∇Sl g)∂Ω
exists σa.e. on ∂Ω .
(5.2.39)
All other claims are readily seen at this point. In the last part of this section we revisit the issue of the invertibility of boundary layer operators of the sort considered in Theorem 5.10. The novelty here is the consideration of a class of domains larger than in the past, though the range of spaces in which the aforementioned invertibility results are proved is smaller (speciﬁcally, the integrability exponent is assumed to be closed to 2). Theorem 5.13. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω and whose outward unit conormal has the property that ν ∈ VMO(∂Ω, T ∗ M).
(5.2.40)
Also, suppose V is a potential as in (3.1.13) and ﬁx an arbitrary degree l ∈ {0, 1, . . . , n}. Associated with the domain Ω and potential V consider the principal value double layer potential operator K l deﬁned as in (4.1.61). Then there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞ making the following statements true. (1) Under the assumption that, at the level of lforms, V + Ric ≥ 0 in M \ Ω, and V + Ric > 0 in a subset of positive measure of each connected component of M \ Ω,
(5.2.41)
it follows that the operators 1 2I
+ Kl ,
1 2I
p l p l + K⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
are isomorphisms for each p ∈ (p o , po ).
(5.2.42)
202  5 Dirichlet and Neumann Boundary Value Problems
(2) If at the level of lforms one has V + Ric ≥ 0 in Ω, and V + Ric > 0 in a subset of positive measure of each connected component of Ω,
(5.2.43)
it follows that the operators p l p l − 12 I + K l , − 12 I + K ⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
are isomorphisms for each p ∈ (p o , po ).
(5.2.44)
(3) Under the assumption that V + Ric ≥ 0 at the level of lforms in M,
(5.2.45)
it follows that for every complex number z ∈ ℂ \ [− 12 , 12 ] the operators p l p l zI + K l , zI + K ⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M)
are isomorphisms for each p ∈ (p o , po ).
(5.2.46)
Moreover, under the additional assumption that for every p near 2 there exists p∗ > p with the property p that L1 (∂Ω) → L p∗ (∂Ω) compactly,
(5.2.47)
the following claims are also valid. (4) Whenever (5.2.41) holds, the operators 1 2I 1 2I
+ K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
l l + K⊤ l : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M), p
p
(5.2.48)
are isomorphisms for each p ∈ (p o , po ). (5) Whenever (5.2.43) holds, the operators − 12 I + K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), l l − 12 I + K ⊤ l : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M), p
p
(5.2.49)
are isomorphisms for each p ∈ (p o , po ). (6) Whenever (5.2.45) holds, for every complex number z ∈ ℂ \ [− 12 , 12 ] the operators zI + K l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M), l l zI + K ⊤ l : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M), p
p
(5.2.50)
are isomorphisms for each p ∈ (p o , po ). Finally, all results in items (1)–(6) above continue to hold if (5.2.40) is relaxed to merely demanding that dist (ν, VMO(∂Ω, T ∗ M)) is sufficiently small, relative to the background structures, where the distance is measured in BMO.
(5.2.51)
5.2 Invertibility Results for Layer Potentials Associated with the LeviCivita Connection  203
Proof. The claims in item (1) are direct consequences of Corollary 4.14 and Proposition 5.4, the claims in item (2) readily follow from Corollary 4.14 and Proposition 5.5, while the claims in item (3) are immediate from Corollary 4.14 and Proposition 5.9. Finally, under the additional assumption made in (5.2.47), the claims in items (4)–(6) are seen from the corresponding claims in items (1)–(3) and Corollary 4.15. The following corollary augments Theorem 5.13 by considering invertibility properties of the single layer in the class of UR domains whose outward unit normal is in VMO and whose boundary Sobolev spaces satisfy suitable embedding properties (cf. (5.2.47)). Corollary 5.14. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω and whose outward unit conormal satisﬁes⁴ ν ∈ VMO(∂Ω, T ∗ M). Also, assume that for every p near 2 there exists p∗ > p with the property p that L1 (∂Ω) → L p∗ (∂Ω) compactly.
(5.2.52)
In this context, pick a potential V as in (3.1.13) and ﬁx a degree l ∈ {0, 1, . . . , n}. Associated with the domain Ω and potential V then consider the single layer potential operator S l deﬁned as in (3.2.6). Then there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞ such that the operators p
S l : L p (∂Ω, Λ l T M) → L1 (∂Ω, Λ l T M), p
S l : L−1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
(5.2.53)
are isomorphisms for each p ∈ (p o , po ). Proof. This is proved by arguing as in the proof of item (1) in Theorem 5.10, making use of Theorem 5.13 in place of Theorem 4.12. We conclude by making two observations pertaining to the nature of Theorem 5.13 and Corollary 5.14. Remark 5.15. A posteriori, condition (5.2.52) may be shown to hold in the class of UR domains Ω ⊂ M with outward unit conormal ν ∈ VMO(∂Ω, T ∗ M), and for which ∂( Ω) = ∂Ω. Indeed, it suffices to work in the context when such a bounded domain Ω has been selected in ℝn . In such a scenario, one runs the same program that has culminated in Corollary 5.14 and, thanks to Remark 4.16, conclude that in the Euclidean setting, the single layer potential S associated with the (scalar) Helmholtz operator ∆ − 1 induces isomorphisms in the context of (5.2.53) for p near 2. For such a range of p’s, this allows p us to describe L1 (∂Ω) as the image of S acting on L p (∂Ω) and then (5.2.52) follows as soon as we show that for each p near 2 there exists p∗ > p with the property that (5.2.54) S : L p (∂Ω) → L p∗ (∂Ω) is a compact operator. 4 all results remain true if instead one assumes that dist(ν, VMO(∂Ω, T ∗ M)) is sufficiently small relative to the relevant background structures, where the distance is measured in BMO
204  5 Dirichlet and Neumann Boundary Value Problems
This, however, is a consequence of Lemma 9.63, (3.2.67), (3.2.68), and Krasnoselski’s interpolation theorem (see, e.g., [4, Theorem 2.9, p. 203]). Remark 5.16. As a consequence of the observation recorded in Remark 5.15 it follows that the claims in items (4)–(6) of Theorem 5.13, as well as Corollary 5.14 continue to hold without imposing the conditions stipulated in (5.2.47) and (5.2.52).
5.3 Solving the Dirichlet, Neumann, Transmission, Poincaré, and Robin Boundary Value Problems The stage has now been set for us to tackle the proof of Theorem 1.8. Proof of Theorem 1.8. Consider ﬁrst the issue of existence for (1.4.2). Given the nature of the conclusion we seek, we may harmlessly discard all connected components of M which are disjoint from ∂Ω. As such, there is no loss of generality in assuming that ∂Ω intersects each connected component of M.
(5.3.1)
Granted this, we may then pick a function V as in (3.1.13) with the additional property that V vanishes identically in Ω. (5.3.2) If now, associated with this, we introduce the single layer operators Sl and S l as in § 3, then (5.2.2) in Theorem 5.10 ensures that u := Sl (S−1 l f ) in Ω,
(5.3.3)
is meaningfully deﬁned. Note that ∆HL u = 0 in Ω, thanks to (5.3.2) and (4.1.105), hence u ∈ C 1 (Ω, Λ l T M) by elliptic regularity (cf. Proposition 2.10). In addition, (4.1.104) in Theorem 4.9 gives the third condition in (1.4.2), as well as the estimate in (1.4.3). In turn, (1.4.3) and (2.2.50) imply (1.4.4). Next, (4.1.106) ensures that the boundary condition in (1.4.2) is satisﬁed, while (4.1.107) provides (1.4.5). A combination of Theorem 9.45, (1.4.5), and (1.4.4) then justiﬁes (1.4.6). Finally, item (12) of Theorem 9.52 yields the L p square function estimate (1.4.7) for p ∈ (2, ∞). All things considered, this proves that u in (5.3.3) is a solution of (1.4.2) satisfying all additional properties listed in item (i) in the statement of the theorem. Alternatively, in place of (5.3.3) we could have expressed the solution u using the harmonic double layer, namely as −1
u := Dl (( 12 I + K l ) f ) in Ω,
(5.3.4)
with the same effectiveness as before. For this to work, we need to ensure that the inverse appearing in (5.3.4) is meaningful. A sufficient condition to this effect has been identiﬁed in (5.2.6). In the present case, when (5.3.1) is assumed, a simple choice is V := λ 1M\Ω ,
where λ is a constant > sup ‖Ricx ‖Λ l T x M→Λ l T x M . x∈M\Ω
(5.3.5)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  205
This choice also ensures that (5.3.2) holds which, in turn, guarantees that u given by (5.3.4) is a harmonic form in Ω (cf. (4.1.9)). The integral representation in (5.3.4) is going to be useful in the proof of the Fatou result presented later in Theorem 6.6. Here we wish to note that for the solution u of (1.4.2) considered in (5.3.4) the L p square function estimate (1.4.7) for p ∈ (2, ∞) is a consequence of item (11) in Theorem 9.52. Next, we shall show that one has genuine uniqueness for the Dirichlet problem (1.4.2) if the working hypothesis (5.3.1) is retained. Our proof is based on Theorem 5.10 and follows along the lines of the argument in [50, § 7.1] with a number of adaptations. The starting pint is the construction of a suitable Green function with pole at an arbitrary ﬁxed point x ∈ Ω. In this regard, we take G l (x, y) := Γ l (x, y) − Sl (S−1 l (Γ l (x, ⋅)∂Ω ))(y),
∀ y ∈ Ω,
(5.3.6)
and observe that G l (x, y) is a meaningfully deﬁned double form of bidegree (l, l). To describe its main properties, pick some positive number ρ o such that ρ o < 14 dist (x, ∂Ω). Then, if N ρ o stands for the truncated nontangential maximal function at height ρ o (cf. (9.12.9)), 1+θ G l (x, ⋅) ∈ Cloc (Ω \ {x}, Λ l T x M ⊗ Λ l T M) for some θ > 0, n.t. G(x, ⋅) = 0 at σa.e. point on ∂Ω, and ∂Ω N ρ o (G l (x, ⋅)), N ρ o (∇y G l (x, ⋅)) ∈ L p (∂Ω) where 1p + p1 = 1.
(5.3.7) (5.3.8) (5.3.9)
In addition, G l (x, ⋅) is a fundamental solution for the HodgeLaplacian ∆HL in Ω with pole at x, in the sense that υ(x) = ∫⟨G l (x, ⋅), ∆HL υ⟩ dVol,
∀ υ ∈ C 01 (Ω, Λ l T M).
(5.3.10)
Ω
To proceed for each ρ ∈ (0, diam (Ω)), consider the (onesided) ρcollar of the boundary Oρ := {z ∈ Ω : dist (z, ∂Ω) ≤ ρ}, and pick a family of functions ψ ρ , indexed by ρ > 0, with the following properties: ψ ρ ∈ C 02 (Ω),
0 ≤ ψ ρ ≤ 1, ∇ψ ρ  ≤ Cρ
ψ ρ ≡ 1 on Ω \ Oρ , −1
and
ψ ρ ≡ 0 on Oρ/2 , −2
∇ ψ ρ  ≤ Cρ . 2
(5.3.11) (5.3.12)
Let now u be a solution for the homogeneous version of (1.4.2). Then Leibniz’s formulas from item (4) in Lemma 2.8 permit us to compute δd(ψ ρ u) = δ[dψ ρ ∧ u + ψ ρ du] = δ[dψ ρ ∧ u] + ψ ρ δdu − (dψ ρ ) ∨ du,
(5.3.13)
and dδ(ψ ρ u) = d[ψ ρ δu − (dψ ρ ) ∨ u] = ψ ρ dδu + dψ ρ ∧ δu − d[(dψ ρ ) ∨ u].
(5.3.14)
206  5 Dirichlet and Neumann Boundary Value Problems Hence, in view of the fact that ∆HL u = 0 in Ω , ∆HL (ψ ρ u) = −δd(ψ ρ u) − dδ(ψ ρ u)
(5.3.15)
= −δ[dψ ρ ∧ u] + (dψ ρ ) ∨ du − dψ ρ ∧ δu + d[(dψ ρ ) ∨ u]. Consequently, bearing in mind that ψ ρ u ∈ C 01 (Ω, Λ l T M) since and ψ ρ ≡ 0 near ∂Ω, and that x ∈ Ω \ Oρ for ρ ∈ (0, ρ o ), for each ρ small we may write u(x) = (ψ ρ u)(x) = ∫⟨G l (x, ⋅), ∆HL (ψ ρ u)⟩ dVol Ω
= −I + II − III + IV,
(5.3.16)
where I := ∫⟨G l (x, ⋅), δ[dψ ρ ∧ u]⟩ dVol = ∫⟨dψ ρ ∨ dG l (x, ⋅), u⟩ dVol, Ω
(5.3.17)
Ω
II := ∫⟨G l (x, ⋅), (dψ ρ ) ∨ du⟩ dVol = ∫⟨δ[dψ ρ ∧ G l (x, ⋅)], u⟩ dVol, Ω
(5.3.18)
Ω
III := ∫⟨G l (x, ⋅), dψ ρ ∧ δu⟩ dVol = ∫⟨d[dψ ρ ∨ G l (x, ⋅)], u⟩ dVol, Ω
(5.3.19)
Ω
IV := ∫⟨G l (x, ⋅), d[(ψ ρ ) ∨ u]⟩ dVol = ∫⟨dψ ρ ∧ δG l (x, ⋅), u⟩ dVol. Ω
(5.3.20)
Ω
From (5.3.16)–(5.3.20), (5.3.13), and (5.3.14) we then conclude that u(x) ≤ ∫{∇2 ψ ρ G l (x, ⋅) + ∇ψ ρ ∇G l (x, ⋅)}u dVol Ω
≤ Cρ −2 ∫ G l (x, y)u(y) dVol(y) Oρ \Oρ/2
+ Cρ−1 ∫ ∇y G l (x, y)u(y) dVol(y) Oρ \Oρ/2
=: A ρ + B ρ .
(5.3.21)
As regards the ﬁrst term in the rightmost side above, based on (5.3.11), (5.3.12), (9.12.11), and Hölder’s inequality we may estimate (as usual, σ denotes the surface measure on ∂Ω) A ρ ≤ C ∫ N ρ (∇y G l (x, ⋅)u) dσ ∂Ω
≤ C ∫ N ρ (∇y G l (x, ⋅))N ρ u dσ ∂Ω
≤ CN ρ (∇y G l (x, ⋅))L p (∂Ω) ‖N ρ u‖L p (∂Ω) .
(5.3.22)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  207
Now, on the one hand, (5.3.9) ensures that ρ (5.3.23) N (∇y G l (x, ⋅))L p (∂Ω) = O(1) as ρ → 0+ . n.t. On the other hand, upon recalling that u∂Ω = 0 and N u ∈ L p (∂Ω), Lebesgue’s Dominated Convergence Theorem gives that as ρ → 0+ ,
‖N ρ u‖L p (∂Ω) = o(1)
(5.3.24)
From (5.3.22)–(5.3.24) we then conclude that, for each ﬁxed x ∈ Ω, lim A ρ = 0.
(5.3.25)
ρ→0+
Let us now turn our attention to treating B ρ in (5.3.16). In preparation to this, consider a decomposition of Ω into nonoverlapping Whitney cubes {I k }k and, for each ﬁxed ρ > 0, set ρ Jρ := {k : I k := I k ∩ Oρ ≠ ⌀}. (5.3.26) ρ
It follows that the sidelength of each I k is comparable with ρ. Going further, since ∂Ω (equipped with the measure σ and the geodesic distance) is a space of homogeneous type, there exists a decomposition of ∂Ω into a grid of dyadic boundary “cubes” Q ρ , of sidelength comparable with ρ. For each k ∈ Jρ , select one such boundary dyadic ρ cube Q k with the property that ρ
ρ
ρ
dist (I k , ∂Ω) = dist (I k , Q k ).
(5.3.27)
Matters can be arranged so that the concentric dilates of these boundary dyadic cubes have bounded overlap. That is, for every λ ≥ 1 there exists a ﬁnite constant C > 0 such that (5.3.28) ∑ 1λ Q ρ ≤ C on ∂Ω, k∈Jρ ρ
k
ρ
where λ Q k := {x : dist(x, Q k ) ≤ λρ}. Next, ﬁx x ∈ Ω, and assume that ρ > 0 is much smaller than dist (x, ∂Ω). We may then write 1 ρ2
G l (x, y) u(y) dVol(y)
∫
ρ 2 ≤dist(y,∂Ω)≤ρ
y∈Ω
≤ ∑ k∈Jρ
1 ∫ G l (x, y) u(y) dVol(y) ρ2
(5.3.29)
ρ
Ik
1/p
1 G l (x, y) p ≤ ∑ ( ∫( ) dVol(y)) ρ ρ k∈J ρ
ρ Ik
1/p
(
1 ∫ (1I ρ u)(y)p dVol(y)) k ρ
.
Oρ
n.t. Using G(x, ⋅)∂Ω = 0, the fact that there exists λ ≥ 1 such that ρ
supp N ρ (1I ρ u) ⊂ λ Q k , k
(5.3.30)
208  5 Dirichlet and Neumann Boundary Value Problems
and (9.12.11), we then obtain 1 ρ2
G l (x, y) u(y) dVol(y)
∫
ρ 2 ≤dist(y,∂Ω)≤ρ
y∈Ω
1 ∫ G l (x, y) u(y) dVol(y) ρ2
≤ ∑ k∈Jρ
(5.3.31)
ρ
Ik
1/p 1/p p 1 1 − G(x, ⋅) dσ dVol(y)) ( ∫ N ρ up dσ) , ≤ ∑ ( ∫ G l (x, y) − ∫ ρ ρ k∈J ρ
ρ
ρ
Ik
ρ
Qk
λ Qk
ρ
with the barred integral denoting integral average. Note that every y ∈ I k is a corkscrew ρ point relative to any point z ∈ Q k . Then Lemma 2.21 (whose applicability is justiﬁed by the fact that any regular SKT domain is a twosided NTA domain) guarantees the existence of a rectiﬁable path γ(z, y) from z to y, of length bounded by Cρ and such that for each point w ∈ γ(z, y) we have dist (w, z) ≈ dist (w, ∂Ω). In particular, using a possibly bigger aperture parameter in the deﬁnition of the nontangential approach regions (without affecting the ﬁnal outcome) we may ensure that γ(z, y) ⊂ Γ(z).
(5.3.32)
ρ Ik
Fix now y ∈ and, based on the Fundamental Theorem of Calculus and (5.3.32), write 1 1 ≤ ∫ G l (x, y) − ∫ − − G l (x, y) − G l (x, z) dσ(z) G (x, z) dσ(z) l ρ ρ ρ ρ Qk
Qk
1 d −( G l (x, w) ds(w)) dσ(z) ≤ ∫ ∫ ds ρ ρ γ(z,y) Qk
− N(∇G l (x, ⋅))(z) dσ(z), ≤∫
(5.3.33)
ρ
Qk
where ds and d/ds denote, respectively, the arclength measure and tangential derivative along γ(z, w) (considered in the second variable of the function G l (⋅, ⋅)). Returning with this back in (5.3.31) allows us to estimate 1 ρ2
G l (x, y) u(y) dVol(y)
∫ −1
2 ρ≤dist(y,∂Ω)≤ρ y∈Ω
p ≤ ∑ (∫ N(∇G l (x, ⋅)) dσ) k∈Jρ
1/p
1/p
( ∫ N ρ up dσ)
ρ
ρ
Qk
λ Qk 1/p
p ≤ C( ∫ N(∇G l (x, ⋅)) dσ) ∂Ω
1/p
( ∫ N ρ up dσ) ∂Ω
,
(5.3.34)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  209
by Hölder’s inequality and (5.3.28). Replacing ρ by 2−j ρ, with j ∈ ℕ0 , in (5.3.34) then yields 1 ρ2
G l (x, y) u(y) dVol(y)
∫ 2
−j−1
−j
ρ≤dist(y,∂Ω)≤2 ρ y∈Ω
1/p
p ≤ C4 ( ∫ N(∇G l (x, ⋅)) dσ) −j
∂Ω
1/p
( ∫ N ρ up dσ)
.
(5.3.35)
∂Ω
At this stage, by summing up estimates like (5.3.35) over j ∈ ℕ0 we arrive at 1 ∫ G l (x, y) u(y) dVol(y) ρ2 Oρ
1/p
p ≤ C( ∫ N(∇G l (x, ⋅)) dσ) ∂Ω
1/p
( ∫ N ρ up dσ)
,
(5.3.36)
∂Ω
which further entails Bρ ≤
C ∫ G l (x, y) u(y) dVol(y), ρ2 Oρ
1/p
p ≤ C( ∫ N(∇G l (x, ⋅)) dσ) ∂Ω
1/p
( ∫ N ρ up dσ)
.
(5.3.37)
∂Ω
Thus, thanks to (5.3.23) and (5.3.24), we also have lim B ρ = 0.
ρ→0+
(5.3.38)
Collectively, (5.3.16), (5.3.23), and (5.3.38), prove that u(x) = 0. Since x ∈ Ω has been chosen arbitrarily, this shows that the problem (1.4.2) has a unique solution, under the assumption (5.3.1). The statement regarding the uniqueness issue for the Dirichlet boundary value problem (1.4.2) in the general case, as recorded in the ﬁrst part of item (iv) of the theorem, then readily follows from this. Indeed, if u solves the homogeneous version of (1.4.2) and if M is an arbitrary connected component of M contained in Ω, then the energy identity (2.3.16) written in the domain M forces du = 0 and δu = 0 in M . In this vein, let us also note that (1.4.15) implies (1.4.16), thanks to the energy identity (5.1.4) (written for ω and M in place of u and Ω), and (9.2.35). In light of the unique continuation result described in Proposition 9.4 it is also clear that (1.4.17) implies genuine uniqueness for the Dirichlet problem (1.4.2). Finally, the fact that we have dim NΩl = b l (Ω, M) < +∞ is a consequence of classical de Rham theory. This fully justiﬁes all claims made in item (iv). At this stage, from the treatment of item (i) and the conclusions in item (iv) it follows that any solution of the Dirichlet boundary value problem (1.4.2) is of the form u = Sl (S−1 l f ) + ω in Ω,
for some ω ∈ NΩl .
(5.3.39)
210  5 Dirichlet and Neumann Boundary Value Problems
Having established this, the leftpointing implication in (1.4.8) is clear from Proposition 9.36, while the rightpointing implication in (1.4.8) is seen from (5.3.3), the invertibility of the operator (5.2.1), and (4.1.66). Also, the L2 square function estimate in (1.4.9) is seen from (5.3.3) and (4.1.76). This justiﬁes the claims made in item (ii). Consider now the case when the HodgeLaplacian ∆HL is replaced by the Schrödinger operator ∆HL − V for a realvalued, nonnegative potential V ∈ L r (Ω), with r > n. That is, consider the Dirichlet boundary value problem (1.4.10). In this scenario, pick a real number λ > supx∈M ‖Ricx ‖Λ l T x M→Λ l T x M and, in place of (5.3.5), choose the potential { V in Ω, ̃ := V (5.3.40) { λ in M \ Ω. { ̃ = 0 in Ω. Then the same type of argument Note that we continue to have (∆HL − V)u as before may be carried out in the present case as well, with a solution representable either as in (5.3.3), or as in (5.3.4) (with the understanding that the layer potentials iñ This justiﬁes the claims volved are associated with the Schrödinger operator ∆HL − V). made in items (iii) and (v). Finally, the claim made in item (vi) is implicit in what we have proved so far, while the claim made in item (vii) is dealt with analogously. This ﬁnishes the proof of Theorem 1.8. Going further, we present the proof of Theorem 1.9. Proof of Theorem 1.9. For starters, it is clear that the space consisting of nullsolutions of the Neumann problem (1.4.38) restricted to the union of all components of M conl l , where NΩ,V is the space deﬁned in (1.4.19). tained in Ω has dimension dim NΩ,V Keeping this in mind, in order to address the ﬁrst claim in the statement of the theorem it suffices to discard those connected components of M which do not intersect ∂Ω and show that, when viewed in this smaller ambient, the corresponding Neumann problem is Fredholm solvable with index zero. With this convention in mind, we may henceforth assume that ∂Ω intersects each connected component of M.
(5.3.41)
̃ as in (3.1.13) with the In particular, (5.3.41) makes it possible to select a function V property that ̃ = V. (5.3.42) V Ω ̃ as in (5.3.40). For the sake of concreteness, take V To proceed, let us abbreviate Bpl (Ω; V) := {u ∈ C 1 (Ω, Λ l T M) : (∆HL − V)u = 0 in Ω, and N u, N(∇u) ∈ L p (∂Ω)}.
(5.3.43)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  211
Then the Fatou result proved in Proposition 5.12 ensures that, for each u ∈ Bpl (Ω; V), p n.t. u∂Ω exists σa.e. on ∂Ω and belongs to L1 (∂Ω, Λ l T M),
n.t. (∇u)∂Ω exists σa.e. on ∂Ω and belongs to L p (∂Ω, Λ l T M).
(5.3.44)
Hence, ∇ν♯ u also exists σa.e. on ∂Ω and belongs to L p (∂Ω, Λ l T M) for each differential form u ∈ Bpl (Ω; V). Thus, it is meaningful to deﬁne Npl (Ω; V) := {u ∈ Bpl (Ω; V) : ∇ν♯ u = 0 on ∂Ω}.
(5.3.45)
̃ and the domain Ω, consider the single Associate with the potential function V layer operators Sl , S l , and connection double layer Dl , as well as its principal value l version K l , with transposed K ⊤ l . For each u ∈ Bp (Ω; V), it follows from (5.3.44) and Proposition 4.3 that the integral representation formula (4.1.15) holds. When further specialized to forms u in Npl (Ω; V) this reduces to n.t. u = Dl (u∂Ω ) in Ω,
(5.3.46)
which, after taking nontangential boundary traces, further entails n.t. (− 12 I + K l )(u∂Ω ) = 0.
(5.3.47)
This proves that the linear assignment n.t. Npl (Ω; V) ∋ u → u∂Ω ∈ Ker (− 12 I + K l ; L p (∂Ω, Λ l T M))
(5.3.48)
is welldeﬁned. In fact, in view of formula (5.3.46), this assignment is also injective, which goes to show that, on the one hand, dim Npl (Ω; V) ≤ dim Ker (− 12 I + K l ; L p (∂Ω, Λ l T M)).
(5.3.49)
On the other hand, the linear assignment p l l Ker (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)) ∋ f → Sl f ∈ Np (Ω; V)
(5.3.50)
is welldeﬁned, thanks to (4.1.82), and we claim that this assignment is also injective. p l Indeed, if f ∈ Ker (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)) is such that Sl f = 0 in Ω then we have n.t. S l f = Sl f ∂Ω = 0 on ∂Ω, which forces f = 0 given that the boundary single layer is invertible in the context of (5.2.1). Consequently, p l l dim Ker (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)) ≤ dim Np (Ω; V).
(5.3.51)
Upon observing that p l dim Ker (− 12 I + K l ; L p (∂Ω, Λ l T M)) = dim Ker (− 12 I + K ⊤ l ; L (∂Ω, Λ T M))
(5.3.52)
212  5 Dirichlet and Neumann Boundary Value Problems
thanks to the fact that the operators involved are Fredholm with index zero on the entire Lebesgue scale and the regularity result from Corollary 5.8, we deduce from (5.3.49) and (5.3.51) that, in fact, p l dim Npl (Ω; V) = dim Ker (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)).
(5.3.53)
Next, let us deﬁne Rpl (∂Ω; V) := {∇ν♯ u : u ∈ Bpl (Ω; V)} ⊆ L p (∂Ω, Λ l T M).
(5.3.54)
In relation to this, observe ﬁrst that since the mapping, Sl : L p (∂Ω, Λ l T M) → Bpl (Ω; V)
(5.3.55)
is welldeﬁned, the jumpformula (4.1.82) gives that p l l Im (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)) ⊆ Rp (∂Ω; V).
(5.3.56)
We claim that the opposite inclusion is also true. To see that this is the case, start with an arbitrary f ∈ Rpl (∂Ω; V). Then there exists u ∈ Bpl (Ω; V) such that f = ∇ν♯ u. To proceed, observe that our working assumption (5.3.41) implies that the L p Regularity problem for the Schrödinger operator ∆HL − V in Ω discussed in Theorem 1.8 is wellposed in the present setting (cf. (1.4.20)). Moreover, solutions of the said problem have the structure described in (5.3.3). Bearing this in mind, the mere membership of u to Bpl (Ω; V), coupled with the Fatou result in (5.3.44), yields the representation formula n.t. u = Sl (S−1 l (u ∂Ω )) in Ω.
(5.3.57)
As such, the original function f may be expressed as n.t. n.t. ⊤ 1 f = ∇ν♯ u = ∇ν♯ Sl (S−1 l (u ∂Ω )) = (− 2 I + K l )(u ∂Ω ),
(5.3.58)
thanks to the jumpformula (4.1.82). In turn, the identity just established shows that f p l belongs to Im (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)), ﬁnishing the proof of the opposite inclusion in (5.3.56). Thus, at this stage we have shown p l Rpl (∂Ω; V) = Im (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)),
(5.3.59)
from which we further conclude that dim (
L p (∂Ω, Λ l T M) Rpl (∂Ω; V)
) = dim (
L p (∂Ω, Λ l T M) ) p l Im (− 12 I + K ⊤ l ; L (∂Ω, Λ T M))
p l = dim Ker (− 12 I + K ⊤ l ; L (∂Ω, Λ T M)),
(5.3.60)
p l since − 12 I + K ⊤ l is Fredholm with index zero on L (∂Ω, Λ T M). Aware of the signifl l icance of the spaces Rp (∂Ω; V) and Np (Ω; V) in relation to the Neumann problem
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  213
(1.4.38), from (5.3.60) and (5.3.53) we deduce that this boundary value problem is Fredholm solvable with index dim Npl (Ω; V) − dim (
L p (∂Ω, Λ l T M) Rpl (∂Ω; V)
) = 0,
(5.3.61)
which is precisely the kind of conclusion we wanted (in view of the conventions made at the beginning of the proof). To summarize, the above argument shows that the Neumann boundary value problem (1.4.31) for the Schrödinger operator ∆HL − V is Fredholm solvable with index l dim NΩ,V , the dimension of the space deﬁned in (1.4.19). This concludes the treatment of item (i) in the statement of the theorem. Turning our attention to the claim made in item (ii), assume ﬁrst that some differential form f ∈ L p (∂Ω, Λ l T M) has been given with the property that the Neumann boundary value problem (1.4.31) has a solution u for this boundary datum. Consider an arbitrary differential form υ ∈ Ker(SCHNp )l . Bearing in mind the Fatou type result of the sort described in Proposition 5.12, it follows that
υ ∈ C 1 (Ω, Λ l T M), (∆HL − V)υ = 0 in Ω, N υ, N(∇υ) ∈ L p (∂Ω), n.t. n.t. the traces υ∂Ω , (∇υ)∂Ω exist at σa.e. point on ∂Ω, and ∇ν♯ υ = 0.
(5.3.62)
Base on these and Green’s formula (4.1.37) established in Proposition 4.5 we may then compute n.t. n.t. ∫ ⟨f, υ∂Ω ⟩ dσ = ∫ ⟨∇ν♯ u, υ∂Ω ⟩ dσ ∂Ω
∂Ω
n.t. = ∫ ⟨u∂Ω , ∇ν♯ υ⟩ dσ = 0,
(5.3.63)
∂Ω
as wanted. For the converse implication, assume f ∈ L p (∂Ω, Λ l T M) is such that (1.4.33) holds. Since the goal is to show that the Neumann boundary value problem (1.4.31) has a solution u for the boundary datum f , there is no loss of generality in assuming that ∂Ω intersects each connected component of M. Granted this, extend the given V to a ̃ as in (3.1.13) and consider boundary layer potentials associated with the function V ̃ and the domain Ω. We claim that potential function V n.t. {υ∂Ω : υ ∈ Ker (SCHNp )l } = Ker (− 12 I + K l : L p (∂Ω, Λ l T M)).
(5.3.64)
Indeed, if υ ∈ Ker (SCHNp )l then the layer potential integral representation formula (4.1.15) yields (keeping (5.3.62) in mind) n.t. υ = Dl (υ∂Ω ) in Ω.
(5.3.65)
In view of (4.1.79), going nontangentially to the boundary in (5.3.65) then produces n.t. n.t. υ∂Ω = ( 12 I + K l )(υ∂Ω )
(5.3.66)
214  5 Dirichlet and Neumann Boundary Value Problems n.t. from which we ultimately deduce that υ∂Ω ∈ Ker (− 12 I + K l : L p (∂Ω, Λ l T M)). This establishes the lefttoright inclusion in (5.3.64). In the opposite direction, consider an arbitrary form g ∈ Ker (− 12 I + K l : L p (∂Ω, Λ l T M)). Corollary 5.8 then places g in
p
the Sobolev space L1 (∂Ω, Λ l T M). As such, if we deﬁne υ := Dl g in Ω it follows from (4.1.8), (4.1.9), (4.1.65), (4.1.109), (4.1.113), and (4.1.79) that
υ ∈ C 1 (Ω, Λ l T M),
(∆HL − V)υ = 0 in Ω, N υ, N(∇υ) ∈ L p (∂Ω), n.t. n.t. the nontangential traces υ∂Ω , (∇υ)∂Ω exist at σa.e. point on ∂Ω, p n.t. υ∂Ω belongs to L1 (∂Ω, Λ l T M), ∇ν♯ υ belongs to L p (∂Ω, Λ l T M), n.t. and in fact υ∂Ω = ( 12 I + K l )g = 12 g + 12 g = g at σa.e. point on ∂Ω.
(5.3.67)
In addition, from (4.1.119) and the assumption on g we see that S l (∇ν♯ υ) = S l (∇ν♯ Dl g) = ( 12 I + K l )(− 12 I + K l )g = 0
(5.3.68)
which, in light of item (1) in Theorem 5.10, forces ∇ν♯ υ = 0 on ∂Ω.
(5.3.69)
n.t. Ultimately, (5.3.69) and (5.3.67) place υ in Ker(SCHNp )l . Hence, g = υ∂Ω belongs to the space in the left side of (5.3.64). This establishes the righttoleft inclusion in (5.3.64), thus concluding the proof of (5.3.64). p l On the other hand, the fact that − 21 I + K ⊤ l is a Fredholm operator on L (∂Ω, Λ T M) implies that this operator has closed range. This permits us to write
⊥
p l {Ker (− 12 I + K l : L p (∂Ω, Λ l T M))} = Im (− 12 I + K ⊤ l : L (∂Ω, Λ T M)),
(5.3.70)
∗
where {. . .}⊥ refers to the annihilator of {. . .} in (L p (∂Ω, Λ l T M)) = L p (∂Ω, Λ l T M). In concert with (5.3.64), this proves that ⊥ n.t. p l {υ∂Ω : υ ∈ Ker (SCHNp )l } = Im (− 12 I + K ⊤ l : L (∂Ω, Λ T M)).
(5.3.71)
Recall that we are presently assuming that f ∈ L p (∂Ω, Λ l T M) is such that (1.4.33) holds. It follows from (5.3.71) that there exists g ∈ L p (∂Ω, Λ l T M) such that f = (− 12 I + K ⊤ l )g.
(5.3.72)
Granted this, it follows from (3.2.4), (3.2.5), (3.2.8), (4.1.66), and (4.1.82) that the function u := Sl g solves the Neumann boundary value problem (1.4.31) for the given boundary datum f . This ﬁnishes the treatment of the claim made in item (ii) in the statement of the theorem.
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  215
Next, suppose that some nullsolution u of the Neumann problem (1.4.31) has been given. From the welldeﬁniteness of the assignment (5.3.48) and the regularity result n.t. in Corollary 5.8 we then conclude that u∂Ω ∈ L21 (∂Ω, Λ l T M). When used back in the integral representation formula (5.3.46), this implies (cf. (4.1.109)) that u satisﬁes u ∈ C 1 (Ω, Λ l T M), ∆HL u = 0 in Ω, N u, N(∇u) ∈ L2 (∂Ω), n.t. n.t. and there exist u∂Ω as well as (∇u)∂Ω at σa.e. point on ∂Ω.
(5.3.73)
Granted this, we may invoke the energy identity (5.1.4) which, under the assumption that (1.4.34) holds, forces ∇u = 0 in Ω. If (1.4.34) is strengthened to (1.4.36), then the same reasoning produces u = 0 in Ω, by virtue of Proposition 9.4. Moving on, observe that when (1.4.36) holds, then (5.2.11) holds in which case item p l (4) in Theorem 5.10 guarantees the invertibility of − 12 I + K ⊤ l on L (∂Ω, Λ T M). Granted this, we may solve the Neumann problem (1.4.31) taking −1
u := Sl ((− 12 I + K ⊤ l ) f ) in Ω,
(5.3.74)
which also produces the estimate recorded in (1.4.37). All tools are now in place for proving Theorem 1.10, a task to which we turn next. Proof of Theorem 1.10. For starters, we propose to show that u ∈ C 1 (Ω, Λ l T M)
} } } u = ∇ν♯ u. (∆HL − V)u = 0 in Ω } ⇒ ∇weak ν♯ } } p N u, N(∇u) ∈ L (∂Ω)}
(5.3.75)
Given the nature of the conclusion we seek, there is no loss of generality in assuming that (5.3.76) ∂Ω intersects each connected component of M. ̃ as in (3.1.13). Throughout, Granted this, we may extend the potential V to a function V ̃ and the domain Ω. all boundary layer potentials are associated with this potential V Turning to the task of proving (5.3.75) in the earnest, pick an arbitrary differential p p ∗ form f ∈ L1 (∂Ω, Λ l T M) = (L−1 (∂Ω, Λ l T M)) . According to the solvability of the L p Regularity problem presented in item (vi) of Theorem 1.8 there exists a differential form υ satisfying υ ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)υ = 0 in Ω, { { N υ, N(∇υ) ∈ L p (∂Ω), { { { { n.t. { υ∂Ω = f on ∂Ω.
(5.3.77)
216  5 Dirichlet and Neumann Boundary Value Problems
With the help of (1.4.46), (1.4.43), and Green’s formula from Proposition 4.5 may then compute p
(L1 (∂Ω,Λ l TM))∗
u, f )L p (∂Ω,Λ l TM) (∇weak ν♯ 1
n.t. ⊤ = (L p (∂Ω,Λ l TM))∗ (ΨDN (u∂Ω ), f )L p (∂Ω,Λ l TM) 1
1
n.t. = (L p (∂Ω,Λ l TM))∗ (u∂Ω , ΨDN f )L p (∂Ω,Λ l TM) n.t. = ∫ ⟨u∂Ω , ΨDN f ⟩ dσ ∂Ω
n.t. n.t. = ∫ ⟨u∂Ω , ∇ν♯ υ⟩ dσ = ∫ ⟨∇ν♯ u, υ∂Ω ⟩ dσ ∂Ω
∂Ω
= (L p (∂Ω,Λ l TM))∗ (∇ν♯ u, f )L p (∂Ω,Λ l TM) . 1
(5.3.78)
1
p
In view of the arbitrariness of f ∈ L1 (∂Ω, Λ l T M), this establishes (5.3.75). In turn, (5.3.75) readily gives the righttoleft inclusion in (1.4.49). To prove the opposite inclusion, consider u ∈ Ker (SCHweakNp )l and pick an arbitrary differential p
form f ∈ L1 (∂Ω, Λ l T M). Then, much as in the ﬁrst part of (5.3.78), we have n.t. n.t. ∫ ⟨u∂Ω , ΨDN f ⟩ dσ = (L p (∂Ω,Λ l TM))∗ (u∂Ω , ΨDN f )L p (∂Ω,Λ l TM) n.t. ⊤ = (L p (∂Ω,Λ l TM))∗ (ΨDN (u∂Ω ), f )L p (∂Ω,Λ l TM)
∂Ω
1
=
1
p (L1 (∂Ω,Λ l TM))∗
u, f )L p (∂Ω,Λ l TM) (∇weak ν♯ 1
= 0.
(5.3.79)
Let us specialize this formula to the case when f = S l g, for g ∈ L p (∂Ω, Λ l T M) arbitrary. In particular, such a choice entails ΨDN f = ΨDN (S l g) = ∇ν♯ (Sl g) = (− 12 I + K ⊤ l )g,
(5.3.80)
by (1.4.43), the fact that Sl g ∈ C 1 (Ω, Λ l T M),
N(Sl g), N(∇Sl g) ∈ L p (∂Ω),
(∆HL − V)Sl g = 0 in Ω, n.t. Sl g ∂Ω = S l g on ∂Ω,
(5.3.81)
and the jumpformula (4.1.82). Utilizing (5.3.80) back in (5.3.79) then gives n.t. 0 = ∫ ⟨u∂Ω , ΨDN f ⟩ dσ ∂Ω
n.t. = ∫ ⟨u∂Ω , (− 12 I + K ⊤ l )g⟩ dσ ∂Ω
n.t. = ∫ ⟨(− 21 I + K l )(u∂Ω ), g⟩ dσ ∂Ω
(5.3.82)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  217
which, in view of the arbitrariness of g ∈ L p (∂Ω, Λ l T M), forces n.t. u∂Ω ∈ Ker (− 12 I + K l : L p (∂Ω, Λ l T M)).
(5.3.83)
Availing ourselves of this and Corollary 5.8 we conclude that p n.t. u∂Ω ∈ L1 (∂Ω, Λ l T M).
(5.3.84)
Bearing in mind that u ∈ Ker (SCHweakNp )l , it follows from (5.3.84) and the regularity statement recorded in (1.4.8) that N(∇u) ∈ L p (∂Ω). Hence, further, u ∈ Ker (SCHNp )l , completing the proof of the lefttoright inclusion in (1.4.49). This ﬁnishes the proof of (1.4.49). Consider next the claim made in item (ii) of the statement of Theorem 1.10. Supp p ∗ pose ﬁrst that some differential form f ∈ L−1 (∂Ω, Λ l T M) = (L1 (∂Ω, Λ l T M)) has been given which satisﬁes the compatibility conditions recorded in (1.4.51). The goal is to show that the weak Neumann boundary value problem (1.4.48) is solvable for the boundary datum f . With this goal in mind, we ﬁrst note that (5.3.64) and Corollary 5.8 imply p n.t. {υ∂Ω : υ ∈ Ker (SCHNp )l } = Ker (− 12 I + K l : L1 (∂Ω, Λ l T M)).
(5.3.85)
l Second, since − 12 I + K ⊤ l has closed range on L −1 (∂Ω, Λ T M) (given that this is a Fredholm operator on the space in question), we have p
p
⊥
l {Ker (− 12 I + K l : L1 (∂Ω, Λ l T M))} = Im (− 12 I + K ⊤ l : L −1 (∂Ω, Λ T M)), p
(5.3.86)
where, in the present context, {. . .}⊥ refers to the annihilator of {. . .} taken in the space p p ∗ (L1 (∂Ω, Λ l T M)) = L−1 (∂Ω, Λ l T M). Together with (5.3.85), this proves that p ⊥ n.t. l {υ∂Ω : υ ∈ Ker (SCHNp )l } = Im (− 12 I + K ⊤ l : L −1 (∂Ω, Λ T M)) p
∗
(5.3.87)
p
where the annihilator is taken in (L1 (∂Ω, Λ l T M)) = L−1 (∂Ω, Λ l T M). Given that f belongs to the latter space, condition (1.4.51) then implies l f ∈ Im (− 12 I + K ⊤ l : L −1 (∂Ω, Λ T M)).
(5.3.88)
g ∈ L−1 (∂Ω, Λ l T M) such that f = (− 12 I + K ⊤ l )g.
(5.3.89)
u := Sl g in Ω,
(5.3.90)
p
Hence, there exists p
If we now set it follows from (4.1.105) and (4.1.104) that u ∈ C 0 (Ω, Λ l T M),
(∆HL − V)u = 0 in Ω,
N u ∈ L p (∂Ω).
(5.3.91)
218  5 Dirichlet and Neumann Boundary Value Problems
Moreover, (3.2.6) gives
n.t. u∂Ω = S l g on ∂Ω.
(5.3.92)
In relation to u from (5.3.90) we also claim that p
∇weak u = f in L−1 (∂Ω, Λ l T M). ν♯
(5.3.93)
p
To justify this, pick an arbitrary form h ∈ L1 (∂Ω, Λ l T M) then use the solvability of the L p Regularity problem from item (vi) of Theorem 1.8 in order to ﬁnd some form w satisfying w ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)w = 0 in Ω, { { { N w, N(∇w) ∈ L p (∂Ω), { { { n.t. { w∂Ω = h on ∂Ω.
(5.3.94)
In fact, as indicated in (5.3.3), we may take w := Sl (S−1 l h) in Ω,
(5.3.95)
where S−1 l is the inverse of the operator (5.2.1). In light of (1.4.43), (5.3.94), and (4.1.82), such a choice entails ⊤ −1 1 ΨDN h = ∇ν♯ w = ∇ν♯ Sl (S−1 l h) = (− 2 I + K l )(S l h).
(5.3.96)
At this stage, based on (1.4.46), (5.3.92), (5.3.96), Proposition 3.5, item (8) in Theorem 4.8, and (5.3.89) we may compute p
(L1 (∂Ω,Λ l TM))∗
u, h)L p (∂Ω,Λ l TM) (∇weak ν♯ 1
n.t. ⊤ = (L p (∂Ω,Λ l TM))∗ (ΨDN (u∂Ω ), h)L p (∂Ω,Λ l TM) 1
1
n.t. = (L p (∂Ω,Λ l TM))∗ (u∂Ω , ΨDN h)L p (∂Ω,Λ l TM) n.t. = ∫ ⟨u∂Ω , ΨDN h⟩ dσ ∂Ω −1 = ∫ ⟨S l g, (− 12 I + K ⊤ l )(S l h)⟩ dσ ∂Ω −1 = ∫ ⟨g, S l (− 12 I + K ⊤ l )(S l h)⟩ dσ ∂Ω
= ∫ ⟨g, (− 12 I + K l )h⟩ dσ = ∫ ⟨(− 12 I + K ⊤ l )g, h⟩ dσ ∂Ω
∂Ω
= ∫ ⟨f, h⟩ dσ = (L p (∂Ω,Λ l TM))∗ (f, h)L p (∂Ω,Λ l TM) . 1
1
∂Ω p
In view of the arbitrariness of h ∈ L1 (∂Ω, Λ l T M), this establishes (5.3.93).
(5.3.97)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  219
p
In summary, given any f ∈ L−1 (∂Ω, Λ l T M) satisfying (1.4.51), the differential form u deﬁned in (5.3.90) (with g as in (5.3.89)) enjoys the properties listed in (5.3.91) and (5.3.93). In particular, u solves the weak Neumann boundary value problem (1.4.48) for the boundary datum f . In the converse direction, suppose u solves the weak Neumann boundary value problem (1.4.48) and pick some arbitrary υ ∈ Ker (SCHNp )l . Then p
(L1 (∂Ω,Λ l TM))∗
n.t. (f, υ∂Ω )L p (∂Ω,Λ l TM)
= =
1
p (L1 (∂Ω,Λ l TM))∗
n.t. u, υ∂Ω )L p (∂Ω,Λ l TM) (∇weak ν♯ 1
n.t. n.t. ⊤ (ΨDN (u∂Ω ), υ∂Ω )L p (∂Ω,Λ l TM) p (L1 (∂Ω,Λ l TM))∗ 1
n.t. n.t. = (L p (∂Ω,Λ l TM))∗ (u∂Ω , ΨDN (υ∂Ω ))L p (∂Ω,Λ l TM) n.t. = ∫ ⟨u∂Ω , ∇ν♯ υ⟩ dσ ∂Ω
= 0,
(5.3.98)
thanks to the deﬁnition of the weak conormal derivative, as well as the nature of u and υ. This proves that (1.4.51) is a necessary condition for the solvability of the weak Neumann boundary value problem (1.4.48). The justiﬁcation of the claim made in item (ii) of the statement of Theorem 1.10 is therefore complete. We are now prepared to ﬁnish the proof of item (i) in Theorem 1.10. Concretely, bearing in mind (1.4.32) (both for p and p ) it follows from (1.4.49) and item (ii) in Theorem 1.10 that the weak Neumann boundary value problem (1.4.48) is Fredholm solvable. By deﬁnition, its index is the dimension of its nullspace minus the number of linearly independent compatibility conditions the boundary data must satisfy to ensure its solvability. In view of (1.4.49) and item (ii) in Theorem 1.10, this translates into index of (1.4.48) = dim Ker (SCHweakNp )l − dim Ker (SCHweakNp )l l = index of (1.4.31) = dim NΩ,V ,
(5.3.99)
where the penultimate equality is based on item (ii) in Theorem 1.9, while the last equality is implied by item (i) in Theorem 1.9. This ﬁnishes the treatment of item (i) in the statement of Theorem 1.10. Consider next the regularity property (1.4.52) in item (iii). In one direction, if u is a solution of the problem (1.4.48) with the additional property that N(∇u) ∈ L p (∂Ω) then on account of (5.3.75) and (1.4.30) we may write n.t. u = ∇ν♯ u = ν k g kj (∇∂ j u)∂Ω ∈ L p (∂Ω, Λ l T M) f = ∇weak ν♯
(5.3.100)
where the last membership is implied by Proposition 5.12. Conversely, assume u solves the weak Neumann boundary value problem (1.4.48) for some f ∈ L p (∂Ω, Λ l T M)
220  5 Dirichlet and Neumann Boundary Value Problems p
(viewed as a subspace of L−1 (∂Ω, Λ l T M); cf. § § 9.49.5). Given the more regular nature of f , the compatibility conditions (1.4.51) presently become n.t. 0 = (L p (∂Ω,Λ l TM))∗ (f, υ∂Ω )L p (∂Ω,Λ l TM) 1
1
n.t. = ∫ ⟨f, υ∂Ω ⟩ dσ,
∀ υ ∈ Ker (SCHNp )l .
(5.3.101)
∂Ω
Thus, (1.4.33) holds which then implies that one may ﬁnd a differential form w such that w ∈ C 1 (Ω, Λ l T M), { { { { { { (∆HL − V)w = 0 in Ω, (5.3.102) { { { N w, N(∇w) ∈ L p (∂Ω), { { { { ∇ν♯ w = f on ∂Ω. Note that, as a consequence of this and (5.3.75), we also have w = f. ∇weak ν♯
(5.3.103)
Consequently, the difference ω := u − w belongs to the space Ker (SCHweakNp )l hence also to Ker (SCHNp )l , thanks to (1.4.49). Since one of the attributes of the latter membership is the fact that N(∇ω) ∈ L p (∂Ω), we deduce that N(∇u) ∈ L p (∂Ω), as wanted. The proof of (1.4.52) is therefore complete. Since (1.4.53) is already contained in (5.3.75), all claims in item (iii) have been justiﬁed at this point. As regards item (iv) in the statement of Theorem 1.10, suppose that the conditions in (1.4.54) hold. From item (iv) in Theorem 1.9 (used both for p and p ) it follows that both Ker (SCHNp )l and Ker (SCHNp )l are trivial. In light of (1.4.49) and item (ii) in Theorem 1.10 we may conclude that the weak Neumann boundary value problem (1.4.48) is uniquely solvable. In such a scenario, estimate (1.4.55) is implicit in the speciﬁc manner in which a solution u has been produced in the treatment of item (ii) (see (5.3.90)). Finally, a similar line of reasoning works for the more general class of domains considered in item (v), ﬁnishing the proof of Theorem 1.10. We continue by presenting the proof of Theorem 1.11. Proof of Theorem 1.11. In a ﬁrst stage, we agree to discard all connected components of M which are disjoint from ∂Ω. This guarantees that ∂Ω intersects each connected component of M,
(5.3.104)
a property which is going to be relevant shortly. We proceed by associating with the domain Ω and potential V layer potential operators in the fashion described in Chapter 4. Introducing υ± := u± − (Dl f Ω± ) (5.3.105)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  221
and recalling (4.1.8), (4.1.9), (4.1.79), (4.1.65), (4.1.109), then transforms (1.4.59) into the reduced transmission problem υ± ∈ C 1 (Ω± , Λ l T M), { { { { { { (∆HL − V)υ± = 0 in Ω± , { { { N υ± , N(∇υ± ) ∈ L p (∂Ω), { { { { n.t. n.t. { { υ+ ∂Ω = υ− ∂Ω on ∂Ω, { { { + − p l { ∇ν♯ υ − μ∇ν♯ υ = g̃ ∈ L (∂Ω, Λ T M),
(5.3.106)
where we have set g̃ := g − (1 − μ)∇ν♯ Dl f ∈ L p (∂Ω, Λ l T M).
(5.3.107)
In view of (5.3.104) and the ﬁrst three lines in (5.3.106), Corollary 6.11 ensures that there exist two differential forms h± ∈ L p (∂Ω, Λ l T M) uniquely determined by the property that υ± = Sl h± in Ω± . Bearing this in mind as well as (3.2.7), the fourth line in (5.3.106) then becomes S l h+ = S l h− on ∂Ω which, given that the operator (5.2.1) is an isomorphism, forces h+ = h− . At this stage, abbreviating h := h± ∈ L p (∂Ω, Λ l T M) yields the single layer representation formulas υ± = Sl h in Ω± .
(5.3.108)
Moreover, h ∈ L p (∂Ω, Λ l T M) is unique with this property. In this notation, thanks to the conormal jump formulas (4.1.82), the last line in (5.3.106) then becomes g̃ = ∇ν♯ υ+ − μ∇ν♯ υ− = ∇ν♯ (Sl hΩ+ ) − μ∇ν♯ (Sl hΩ− ) ⊤ 1 = (− 12 I + K ⊤ l )h − μ( 2 I + K l )h
= (− 12 (1 + μ)I + (1 − μ)K ⊤ l )h,
(5.3.109)
or, equivalently, (λI + K ⊤ l )h =
1 g̃ 1−μ
where λ :=
μ+1 ∈ ℂ \ [− 12 , 12 ], 2(μ − 1)
(5.3.110)
where the last membership relies on (1.4.58). Since the spectral result from Proposition 5.11 implies that, if 1/p + 1/p = 1, we have
p l p l 1 1 Spec (K ⊤ l ; L (∂Ω, Λ T M)) = Spec (K l ; L (∂Ω, Λ T M)) ⊆ [− 2 , 2 ],
(5.3.111)
we conclude that the operator p l p l λI + K ⊤ l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M) is invertible.
(5.3.112)
As such, condition (5.3.110) determines h uniquely, namely h=
1 −1 ̃ (λI + K ⊤ l ) g. 1−μ
(5.3.113)
222  5 Dirichlet and Neumann Boundary Value Problems
The above argument goes to show that the reduced transmission problem (5.3.106) has a unique solution, namely υ± =
1 −1 ̃ Sl (λI + K ⊤ l ) g in Ω ± . 1−μ
(5.3.114)
Upon recalling (5.3.105), we conclude that, under the assumption (5.3.104), the original transmission problem (1.4.59) has a unique solution, namely u± =
1 −1 ̃ Sl (λI + K ⊤ l ) g + (Dl f Ω± ) in Ω ± , 1−μ
(5.3.115)
where λ ∈ ℂ \ [− 12 , 12 ] is as in (5.3.110) and g̃ ∈ L p (∂Ω, Λ l T M) is as in (5.3.107). Having established this, all other claims in the statement of Theorem 1.11 pertaining to the Transmission boundary value problem (1.4.59) now follow easily, by making use of the energy identities (2.3.16) and (5.1.4) (as well as the unique continuation results from Proposition 9.4 and Proposition 2.9) in order to clarify the structure of smooth nullsolutions of the Schrödinger operator ∆HL − V in those connected components of M which are disjoint from ∂Ω. Moving on, the treatment of the weak Transmission boundary value problem formulated in (1.4.66) is completely analogous, working with one unit of regularity less than before. Both item (1) in Theorem 5.10 and Corollary 6.11 are accommodating in this regard, and we only wish to clarify the manner in which a couple of key aspects of our earlier approach manifest themselves in the present setting. Since, currently, the single layer representation formulas (5.3.108) hold for some h belonging to the Sobolev p space of negative order L−1 (∂Ω, Λ l T M), the ﬁrst aspect we wish to elaborate on is the validity of the weak trace formula ⊤ 1 ∇weak ν♯ (Sl h) = (− 2 I + K l )h,
p
∀ h ∈ L−1 (∂Ω, Λ l T M).
(5.3.116)
This, however, follows by a density argument, observing ﬁrst that the above equality is true when h ∈ L p (∂Ω, Λ l T M) (thanks to (5.3.75) and (4.1.82)), then noting that both operators p p l l ∇weak ν♯ Sl : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M), (5.3.117) p p l l − 12 I + K ⊤ l : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M), are welldeﬁned, linear and bounded (by (1.4.47) and (4.1.104) for the ﬁrst operator, and by (4.1.112) for the second operator), and ﬁnally invoking the fact that p L p (∂Ω, Λ l T M) is dense in L−1 (∂Ω, Λ l T M) (cf. (9.4.15) and the discussion in § 9.5). The weak trace formula (5.3.116) is relevant in the context of (5.3.109). The second aspect worth considering is that Proposition 5.11 implies that, whenever 1/p + 1/p = 1, we have p
l l 1 1 Spec (K ⊤ l ; L −1 (∂Ω, Λ T M)) = Spec (K l ; L 1 (∂Ω, Λ T M)) ⊆ [− 2 , 2 ]. p
(5.3.118)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  223
In particular, if λ ∈ ℂ \ [− 12 , 12 ] is as in (5.3.110) then the operator l l λI + K ⊤ l : L −1 (∂Ω, Λ T M) → L −1 (∂Ω, Λ T M) is invertible. p
p
(5.3.119)
Based on these, much as before we may conclude that, under the assumption (5.3.104), the weak Transmission problem (1.4.66) has a unique solution, given by u± = where (λI + K ⊤ l )
−1
1 −1 ̂ Sl (λI + K ⊤ l ) g + (Dl f Ω± ) in Ω ± , 1−μ
(5.3.120)
is now the inverse of (5.3.119) and where
p weak l ĝ := g − ∇weak ν♯ (Dl f Ω+ ) + μ∇ν♯ (Dl f Ω− ) ∈ L −1 (∂Ω, Λ T M).
(5.3.121)
p
From the structure of this solution is then clear that whenever f ∈ L1 (∂Ω, Λ l T M) and g ∈ L p (∂Ω, Λ l T M) we necessarily have N(∇u± ) ∈ L p (∂Ω). The converse implication follows from Proposition 5.12. This establishes the regularity property recorded in (1.4.67) and ﬁnishes the proof of Theorem 1.11. We are ready to discuss the proof of Theorem 1.12, dealing with the L p PoincaréRobin boundary value problem for the Schrödinger operator ∆HL − V. Proof of Theorem 1.12. As in the proof of Theorem 1.9 it suffices to treat the case when ∂Ω intersects each connected component of M,
(5.3.122)
the goal now being to show that, in such a scenario, the corresponding PoincaréRobin problem is Fredholm solvable with index zero. To proceed, observe that assumption (5.3.122) makes it possible to extend the given ̃ as in (3.1.13). Bring in layer potentials associated with V ̃ potential V to a function V and Ω of the usual sort. Granted (5.3.122) it follows from Corollary 6.11 that any differential form u satisfying the conditions in the ﬁrst three lines of (1.4.72) for some p ∈ (1, ∞) may be represented as u = Sl g in Ω for some (uniquely determined) form g ∈ L p (∂Ω, Λ l T M). Bearing in mind this integral representation and recalling the jumpformula established in (4.3.26), the boundary condition in (1.4.72) then becomes (− 12 ⟨X, ν♯ ⟩I + R + Θ ∘ S l )g = f on ∂Ω,
(5.3.123)
where R is the principal value singular integral operator from (4.3.27). In turn, this argument shows two things, namely that given any exponent p ∈ (1, ∞), the L p PoincaréRobin problem (1.4.72) is solvable if and only if the boundary datum f belongs to the image of − 12 ⟨X, ν♯ ⟩I + R + Θ ∘ S l on L p (∂Ω, Λ l T M),
(5.3.124)
224  5 Dirichlet and Neumann Boundary Value Problems
and that for any p ∈ (1, ∞), the nullspace of the L p PoincaréRobin problem (1.4.72) is in a onetoone correspondence (via Sl ) with the kernel of the operator − 12 ⟨X, ν♯ ⟩I + R + Θ ∘ S l acting on the space L p (∂Ω, Λ l T M).
(5.3.125)
Let us now restrict p to the interval (p o , po ) where 1 ≤ p o < 2 < po ≤ ∞ are the two Hölder conjugate exponents as in Theorem 4.13. In particular, this guarantees that (4.3.35) holds. Hence, in light of the fact that the single layer operator is welldeﬁned, linear and bounded in the context of (4.1.102), having Θ as in (1.4.71) implies that − 12 ⟨X, ν♯ ⟩I + R + Θ ∘ S l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) is a Fredholm operator with index zero.
(5.3.126)
At this stage, from (5.3.124)–(5.3.126) we conclude that for each integrability exponent p ∈ (p o , po ) the L p PoincaréRobin boundary value problem (1.4.72) is Fredholm solvable with index, whenever (5.3.122) holds. Having established this, all other conclusions readily follow, ﬁnishing the proof of the theorem. The companion result to Theorem 1.12 is Theorem 1.13, dealing with the L p Robin boundary value problem for the Schrödinger operator ∆HL − V. This is proved next. Proof of Theorem 1.13. The argument mirrors the proof of Theorem 1.12 in the special case when X := ν♯ . Such a choice entails ⟨X, ν♯ ⟩ = 1 at σ a.e. point on ∂Ω. Also, in view of (4.3.26) and (4.1.82), we now have R = K ⊤ l . The upshot of this is that the Fredholmness result from (4.3.35) is presently valid with p o = 1 (i.e., for every p ∈ (1, ∞)), thanks to the compactness of the operator (4.3.2) from Theorem 4.12. This takes care of all claims pertaining to the L p Robin boundary value problem (1.4.73). The treatment of the weak Robin problem formulated in (1.4.76) proceeds along similar lines. First, Corollary 6.11 ensures that any form u as in the ﬁrst three lines of (1.4.76) may be represented as u = Sl g in Ω for some (uniquely determined) p differential form g ∈ L−1 (∂Ω, Λ l T M).
(5.3.127)
Next, from this integral representation and the jumpformula established in (5.3.116), the boundary condition in (1.4.76) then becomes (− 12 I + K ⊤ l + Θ ∘ S l )g = f on ∂Ω.
(5.3.128)
Having noticed this, we conclude that the weakRobin problem (1.4.76) is solvable if and only if the boundary datum f belongs to the image of the operator p l − 12 I + K ⊤ l + Θ ∘ S l acting on the space L −1 (∂Ω, Λ T M),
(5.3.129)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  225
and the nullspace of the weakRobin problem (1.4.76) is in a onetoone correspondence (via the action of Sl ) with the kernel of the operator − 12 I + K ⊤ l + Θ ∘ S l acting on p l the space L−1 (∂Ω, Λ T M).
(5.3.130)
Making now use of the compactness of the operator (4.3.4) in Theorem 4.12, the same argument as in the endgame of the proof of Theorem 1.12 gives that the weak Robin l . problem (1.4.76) is also Fredholm solvable with index dim NΩ,V Let us now turn our attention to the regularity property recorded in (1.4.78) when the operator Θ is both as in (1.4.74) and (1.4.75). In such a scenario, from Theorem 4.12, the boundedness of the operators (4.1.102) and (4.1.103), and the assumptions on Θ, we deduce that the operator − 12 I + K ⊤ l + Θ ∘ S l is Fredholm with index zero p both on the space L p (∂Ω, Λ l T M) and on L−1 (∂Ω, Λ l T M).
(5.3.131)
Granted this, from the abstract regularity result from Lemma 3.10, employed here with X0 := L p (∂Ω, Λ l T M), p
X1 := L−1 (∂Ω, Λ l T M), T :=
− 12 I
+
K⊤ l
(5.3.132)
+ Θ ∘ Sl ,
and whose applicability is ensured by (9.5.45), it follows that p
if h ∈ L−1 (∂Ω, Λ l T M) is such that p l (− 12 I + K ⊤ l + Θ ∘ S l )h ∈ L (∂Ω, Λ T M)
(5.3.133)
then necessarily h ∈ L (∂Ω, Λ T M). p
l
Pick now a solution u of the weak Robin problem (1.4.76) for some boundary datum f belonging to L p (∂Ω, Λ l T M). Then (5.3.127) and (5.3.129) imply that there exists some p g ∈ L−1 (∂Ω, Λ l T M) such that u = Sl g in Ω and (− 12 I + K ⊤ l + Θ ∘ S l )g = f . In concert with (5.3.133), the latter condition then places g in L p (∂Ω, Λ l T M) which further permits us to write N(∇u) = N(∇Sl g) ∈ L p (∂Ω) by (4.1.66). This establishes the rightpointing implication in (1.4.78). In the converse direction, assume that u is a solution of the weak Robin problem (1.4.76) with the property that N(∇u) ∈ L p (∂Ω). Then Proposition 5.12 gives that, for each vector ﬁeld X with continuous components, n.t. n.t. u∂Ω and (∇X u)∂Ω exist at σa.e. point on ∂Ω, and
(5.3.134)
p n.t. n.t. u∂Ω ∈ L1 (∂Ω, Λ l T M), (∇X u)∂Ω ∈ L p (∂Ω, Λ l T M). (5.3.135) n.t. Thus, on the one hand, Θ(u∂Ω ) ∈ L p (∂Ω, Λ l T M) by (1.4.74). On the other hand, the interior regularity result from (2.1.118) guarantees that any solution u of the weak Robin
226  5 Dirichlet and Neumann Boundary Value Problems problem (1.4.76) belongs to C 1 (Ω, Λ l T M). Granted this, from (5.3.75) and (1.4.30), locally we obtain n.t. ∇weak u = ∇ν♯ u = ν k g kj (∇∂ j u)∂Ω . (5.3.136) ν♯ n.t. All together, this proves that f = ∇weak u + Θ(u∂Ω ) ∈ L p (∂Ω, Λ l T M), concluding the ν♯ proof of (1.4.78). Finally, (1.4.77) is a direct consequence of (1.4.78), so the proof of Theorem 1.13 is ﬁnished. Here is the proof of Theorem 1.14, dealing with Dirichlet and Neumann problems in a class of UR domains. Proof of Theorem 1.14. By harmlessly discarding all connected components of M which are disjoint from ∂Ω, we see that there is no loss of generality in assuming that ∂Ω intersects each connected component of M. (5.3.137) In this scenario, pick a real number λ > supx∈M ‖Ricx ‖Λ l T x M→Λ l T x M and deﬁne the potential { V in Ω, ̃ := V (5.3.138) { λ in M \ Ω. { Associated with this potential and the domain Ω, consider boundary layer potentials as in Chapter 4. In particular, (1) in Theorem 5.13 implies that there exist two Hölder conjugate integrability exponents 1 ≤ p o < 2 < po ≤ ∞ with the property that the operator p l p l 1 2 I + K l : L (∂Ω, Λ T M) → L (∂Ω, Λ T M) (5.3.139) is an isomorphism for each p ∈ (p o , po ). Granted this, whenever p ∈ (p o , po ) a solution to the Dirichlet problem (1.4.79) may be constructed taking −1 u := Dl (( 12 I + K ⊤ (5.3.140) l ) f ) in Ω. Similar considerations apply to the Neumann problem (1.4.31) when p ∈ (p o , po ), taking −1 u := Sl ((− 12 I + K ⊤ (5.3.141) l ) f ) in Ω, where, this time, the intervening boundary layer potential operators are associated ̃ as in (3.1.13). with any extension of the original potential V to a function V Next, we turn our attention to the proof of Theorem 1.15. Proof of Theorem 1.15. Given the current goals, we may harmlessly discarding all connected components of M which are disjoint from ∂Ω. Hence, there is no loss of generality in assuming that ∂Ω intersects each connected component of M, a scenario ̃ as in which makes it possible for us to extend the given function V to a potential V ̃ (3.1.13) (for example, V from (5.3.138) will do). Assuming that this has been done, let ̃ constructed Γ l (x, y) be the fundamental solution for the Schrödinger operator ∆HL − V,
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  227
as in (3.1.16). Also, recall the principal value singular integral operator R introduced in (4.3.27). Having ﬁxed some p ∈ (1, ∞), we look for a solution of the Dirichlet problem (1.4.79) expressed as u(x) := ∫ ⟨∇X(y) Γ l (x, y), g(y)⟩y dσ(y) ∂Ω
+ ∫ ⟨Γ l (x, y), h(y)⟩y dσ(y),
∀ x ∈ Ω,
(5.3.142)
∂Ω
where g, h ∈ L p (∂Ω, Λ l T M)
(5.3.143)
are two differential forms yet to be speciﬁed. For now we note that, regardless of the choice of g, h in (5.3.143), the candidate deﬁned as in (5.3.142) satisﬁes u ∈ C 0 (Ω, Λ l T M), { { { { { { (∆HL − V)u = 0 in Ω, { { N u ∈ L p (∂Ω), { { { { n.t. 1 ♯ ⊤ { u∂Ω = ( 2 ⟨X, ν ⟩I + R )g + S l h on ∂Ω,
(5.3.144)
thanks to items (3) and (5) in Theorem 9.52, along with (9.1.45), (2.1.95), (4.3.30), and (3.2.7). On the other hand, from Theorem 4.13 we know that there exist two Hölder conjugate exponents 1 ≤ p o < 2 < po ≤ ∞ with the property that the operator ♯ 1 2 ⟨X, ν
⟩I + R⊤ : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M),
is Fredholm with index zero for every p ∈ (p o , po ).
(5.3.145)
In light of Proposition 3.5, matters may also be arranged (by taking p o closer to 2 if necessary) so that the operator S l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M) has dense range whenever p ∈ (p o , po ).
(5.3.146)
Consequently, whenever p ∈ (p o , po ), Vp := Im ( 12 ⟨X, ν♯ ⟩I + R⊤ : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)) is a closed, ﬁnite codimensional subspace of L p (∂Ω, Λ l T M), while
Wp := Im (S l : L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)) is a dense subspace of L p (∂Ω, Λ l T M).
(5.3.147)
(5.3.148)
228  5 Dirichlet and Neumann Boundary Value Problems If we now ﬁx p ∈ (p o , po ) and consider T : L p (∂Ω, Λ l T M) ⊕ L p (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M, T(g, h) := ( 12 ⟨X, ν♯ ⟩I + R⊤ )g + S l h,
∀ g, h ∈ L p (∂Ω, Λ l T M),
(5.3.149)
it follows that T is a welldeﬁned, linear, and bounded operator, whose image is precisely Vp + Wp . The latter sum is a linear subspace of L p (∂Ω, Λ l T M) which, thanks to (5.3.147) and (5.3.148), is simultaneously dense and of ﬁnite codimension in the space L p (∂Ω, Λ l T M). Some standard functional analysis (cf., e.g., [105, Lemma 2, p. 156]) then forces Vp + Wp = L p (∂Ω, Λ l T M), ultimately proving that the operator T in (5.3.149) is surjective for p ∈ (p o , po ).
(5.3.150)
Granted this, the Open Mapping Theorem may be invoked in order to quantify the ontoness of T in the following manner: for each p ∈ (p o , po ) there exists a constant C ∈ (0, ∞) with the property that for every f ∈ L p (∂Ω, Λ l T M) one can ﬁnd two forms g, h ∈ L p (∂Ω, Λ l T M) such that T(g, h) = f and ‖g‖L p (∂Ω,Λ l TM) + ‖h‖L p (∂Ω,Λ l TM) ≤ C‖f‖L p (∂Ω,Λ l TM) .
(5.3.151)
Having established this, it follows that whenever the boundary datum f for the Dirichlet problem (1.4.79) belongs to L p (∂Ω, Λ l T M) with p ∈ (p o , po ) we may ﬁnd g, h as in (5.3.143) so that and T(g, h) = f
and
‖g‖L p (∂Ω,Λ l TM) + ‖h‖L p (∂Ω,Λ l TM) ≤ C‖f‖L p (∂Ω,Λ l TM) ,
(5.3.152)
for some constant C ∈ (0, ∞) independent of f . In light of (5.3.144), we conclude that u deﬁned as in (5.3.142) for this choice of g, h is a solution of the Dirichlet problem (1.4.79). Moreover, the fact that the solution u constructed as in (5.3.142) with g, h as in (5.3.152) also enjoys the properties speciﬁed in (1.4.86)(1.4.88) may be seen as in the proof of Theorem 1.8. Finally, the very last claim in the statement of theorem is a consequence of the property recorded in the last part of Theorem 4.13. We conclude this section by proving Theorem 1.16. p
p
Proof of Theorem 1.16. Given f ∈ Ltan (∂Ω, Λ l−1 T M) and g ∈ Lnor (∂Ω, Λ l+1 T M), introduce h := ν ∧ f + ν ∨ g ∈ L p (∂Ω, Λ l T M), (5.3.153) then consider a solution u of the Dirichlet boundary value problem for the HodgeLaplacian formulated as in (1.4.2) for this boundary datum h. Since, by (2.4.6), we have n.t. ν ∨ u∂Ω = ν ∨ h = ν ∨ (ν ∧ f) = f,
n.t. ν ∧ u∂Ω = ν ∧ h = ν ∧ (ν ∨ g) = g,
(5.3.154)
5.3 Solving Dirichlet, Neumann, and Other Boundary Problems  229
it follows that u solves the boundary value problem (1.4.92). Moreover, a differential form u is a nullsolution of (1.4.92) if and only if u is a nullsolution of the Dirichlet problem (1.4.2) since, as seen from (2.4.3), n.t. n.t. n.t. u∂Ω = ν ∧ (ν ∨ u∂Ω ) + ν ∨ (ν ∧ u∂Ω ).
(5.3.155)
Next, for an arbitrary solution u of (1.4.92), the equivalence in (1.4.93) follows from (1.4.8), Proposition 9.17, and (5.3.155) which gives n.t. u∂Ω = ν ∧ f + ν ∨ g.
(5.3.156)
Also, the last inclusion in (1.4.93) is seen from (9.6.129) and (9.6.131). All other remaining claims are direct consequences of Theorem 1.8.
6 Fatou Theorems and Integral Representations for the HodgeLaplacian on Regular SKT Domains The goal in this chapter is to prove a Fatou type result and an integral representation formula for nullsolutions of the HodgeLaplacian whose nontangential maximal function is pth power integrable in a regular SKT domain. Our main result implies that if Ω is a regular SKT domain, u is an lform on Ω such that ∆HL u = 0 on Ω and n.t. N u ∈ L p (∂Ω) for some p ∈ (1, ∞), then u∂Ω exists and belongs to L p (∂Ω, Λ l T M), and u = Dl f,
n.t. f = ( 12 I + K l )−1 (u∂Ω ).
(6.0.1)
Roughly speaking, the strategy is to take approximating domains Ω j ↗ Ω as in Proposition 2.35, in which case u = Dl,j f j in Ω j with f j = ( 12 I + K l,j )−1 (u∂Ω j ) (where Dl,j and K l,j are the analogues of Dl and K l , with Ω replaced by Ω j ), and show that it is possible to pass to the limit. The latter step involves some extensive technical work, carried out in § 6.1. In particular, we obtain uniform operator bounds on ( 12 I + K l,j )−1 and convergence results as Ω j ↗ Ω, under the hypothesis (5.0.1). Using this, in § 6.2 we establish the basic Fatou result in Theorem 6.6. In § 6.3 we deduce from Theorem 6.6 that elements of H l,p (Ω) have nontangential boundary values in L p (∂Ω, Λ l T M), when Ω is a regular SKT domain. We also establish that the subspaces of H l,p (Ω) denoted H∨l (Ω) and H∧l (Ω) are ﬁnite dimensional and independent of the integrability exponent p ∈ (1, ∞). These results extend to εSKT domains, for p in any given compact subset of (1, ∞), if ε is small enough.
6.1 Convergence of Families of Singular Integral Operators The initial aim here is to prove the following convergence result for sequences of singular integral operators considered on the boundaries of a family of domains as in the approximation scheme from Proposition 2.35. Theorem 6.1. There exists a positive integer N = N(n) with the following signiﬁcance. Suppose Ω ⊆ ℝn is a bounded regular SKT domain and denote by σ := H n−1 ⌊∂Ω the surface measure on ∂Ω and by ν the geometric measure theoretic outward unit normal to Ω. Fix some open neighborhood U ⊂ ℝn of Ω and consider a function U × (ℝn \ {0}) ∋ (x, z) → b(x, z) ∈ ℝ, odd and (positively)
homogeneous of degree 1 − n in the variable z ∈ ℝn \ {0}, γ and such that ∂ z b(x, z) is continuous and bounded on n−1 U×S whenever the multiindex γ ∈ ℕ0n satisﬁes γ ≤ N.
(6.1.1)
232  6 Fatou Theorems and Integral Representations Associate to the function in (6.1.1) and to some ﬁxed multiindices α, β ∈ ℕ0n the variable kernel singular integral operator acting on functions f deﬁned on ∂Ω according to T αβ f(x) := P.V. ∫ ν(x)α ν(y)β b(x, x − y)f(y) dσ(y),
x ∈ ∂Ω.
(6.1.2)
∂Ω
Next, let Ω j ↗ Ω as j → ∞ in the sense of Proposition 2.35 and, in a manner analogous to (6.1.2), for each j ∈ ℕ associate the singular integral operator acting on functions f deﬁned at each x ∈ ∂Ω j according to αβ
T j f(x) := P.V. ∫ ν j (x)α ν j (y)β b(x, x − y)f(y) dσ j (y),
(6.1.3)
∂Ω j
where, for each j ∈ ℕ, the surface measure on ∂Ω j has been denoted by σ j := H n−1 ⌊∂Ω j , while the geometric measure theoretic outward unit normal to Ω j has been denoted by ν j . Finally, recall that 𝛶j : ∂Ω → ∂Ω j , j ∈ ℕ, denote the biLipschitz homeomorphisms from item (iv) of Proposition 2.35, and ﬁx some p ∈ (1, ∞). Then one has αβ p limj→∞ [T j (f j ∘ 𝛶−1 j )] ∘ 𝛶j = T f in L (∂Ω), αβ
whenever the functions f, f j ∈ L p (∂Ω), j ∈ ℕ,
(6.1.4)
are such that lim f j = f in L (∂Ω), p
j→∞
plus a similar convergence result for operators associated as in (6.1.2) and (6.1.3) with the function b(y, x − y) in place of b(x, x − y). Moreover, all results above continue to hold when Ω is merely assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the given p, the size of the function b, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. We shall proceed in a series of steps, starting with: Step I. For each integrability exponent p ∈ (1, ∞), the operators T αβ : L p (∂Ω) → L p (∂Ω) αβ and T j : L p (∂Ω) → L p (∂Ω) for j ∈ ℕ are welldeﬁned, linear and bounded. In addition, αβ
sup ‖T j ‖L p (∂Ω j )→L p (∂Ω j ) < ∞.
(6.1.5)
j∈ℕ
As far as T αβ is concerned, the CalderónZygmund theory for variable coefficient kernel singular integral operators in UR domains from [50] applies and yields that for each given p ∈ (1, ∞) the operator T αβ : L p (∂Ω) → L p (∂Ω) is welldeﬁned, linear and bounded, with γ
‖T αβ ‖L p (∂Ω)→L p (∂Ω) ≤ C(Ω, p) ∑ ‖∂ z b(x, z)‖L∞ (U×S n−1 ) ,
(6.1.6)
γ≤N o
where C(Ω, p) ∈ (0, ∞) depends only on p and the UR character of Ω, while N o is a purely dimensional constant. Thus, taking N ≥ N o ensures that T αβ has all desired
6.1 Convergence of Families of Singular Integral Operators

233
αβ
qualities. Finally, the same type of reasoning applies to the operators T j . In particαβ ular, (6.1.5) follows from the analogue of (6.1.6) written for T j , bearing in mind the properties of the approximating sequence {Ω j }j∈ℕ from Proposition 2.35. Step II. If for some integrability exponent p ∈ (1, ∞) one has αβ p lim [T j (f ∘ 𝛶−1 j )] ∘ 𝛶j = T f in L (∂Ω) αβ
j→∞
(6.1.7)
for every function f ∈ L p (∂Ω), then the claim in (6.1.4) holds for that p. To justify this, ﬁrst observe that the change of variable formula from ∂Ω j to ∂Ω described in item (vii) of Proposition 2.35 gives ‖g ∘ 𝛶j ‖L p (∂Ω) ≈ ‖g‖L p (∂Ω j ) , uniformly for g ∈ L p (∂Ω j ) and j ∈ ℕ, f∘
p 𝛶−1 j ‖L (∂Ω j )
≈ ‖f‖L p (∂Ω) , uniformly for f ∈ L (∂Ω) and j ∈ ℕ. p
(6.1.8) (6.1.9)
Given f ∈ L p (∂Ω) and f j ∈ L p (∂Ω) for j ∈ ℕ, such that limj→∞ f j = f in L p (∂Ω), making use of (6.1.8)(6.1.9) and the uniform operator norm bounds established in Step I we may compute αβ αβ −1 lim sup [T j ((f j − f) ∘ 𝛶−1 L p (∂Ω) ≤ C lim sup T j ((f j − f) ∘ 𝛶j )L p (∂Ω j ) j )] ∘ 𝛶j j→∞
j→∞
≤ C lim sup (f j − f) ∘ 𝛶−1 L p (∂Ω j ) j j→∞
≤ C lim sup ‖f j − f‖L p (∂Ω) = 0.
(6.1.10)
j→∞
In turn, this goes to show that −1 lim [T j (f j ∘ 𝛶−1 j )] ∘ 𝛶j = lim [T j (f ∘ 𝛶j )] ∘ 𝛶j αβ
αβ
j→∞
j→∞
= T αβ f in L p (∂Ω),
(6.1.11)
as wanted. Step III. If for some p ∈ (1, ∞) the convergence in (6.1.7) holds for arbitrary Lipschitz functions f ∈ Lip (∂Ω), then (6.1.4) holds for that p. Indeed, this is an immediate consequence of Step II, the estimates in (6.1.8) and (6.1.9), the uniform operator norm bounds established in Step I, and the density of Lip (∂Ω) in L p (∂Ω). Step IV. The claim in (6.1.4) holds for a given p ∈ (1, ∞) provided αβ
lim [T j 1] ∘ 𝛶j = T αβ 1 in L p (∂Ω).
j→∞
(6.1.12)
234  6 Fatou Theorems and Integral Representations
To see that this is true, recall from Step II and Step III that it suffices to establish the convergence in (6.1.7) for arbitrary functions f ∈ Lip (∂Ω). With this goal in mind, pick some f ∈ Lip (∂Ω) and, for σa.e. x ∈ ∂Ω, decompose ([T j (f ∘ 𝛶−1 j )] ∘ 𝛶j )(x) = (Ij f)(x) + (IIj f)(x), αβ
(6.1.13)
where α
(Ij f)(x) := P.V. ∫ ν j (𝛶j (x)) ν j (y)β
(6.1.14)
× b(𝛶j (x), 𝛶j (x) − y)[(f ∘
∂Ω j
𝛶−1 j )(y)
− f(x)] dσ j (y),
α
(IIj f)(x) := f(x) P.V. ∫ ν j (𝛶j (x)) ν j (y)β b(𝛶j (x), 𝛶j (x) − y) dσ j (y).
(6.1.15)
∂Ω j
−1 Then since (f ∘ 𝛶−1 j )(y) − f(x) ≤ C𝛶j (y) − x ≈ Cy − 𝛶j (x) and, generally speaking, n for every x ∈ U and z ∈ ℝ \ {0} we have b(x, z) ≤ ‖b‖L∞ (U×S n−1 ) z1−n , it follows that the integrand in (6.1.14) is absolutely integrable. Keeping this in mind and making a change of variables as in item (vii) of Proposition 2.35, we may rewrite (Ij f)(x) as α
(Ij f)(x) = ∫ ν j (𝛶j (x)) ν j (𝛶j (y))
β
(6.1.16)
× b(𝛶j (x), 𝛶j (x) − 𝛶j (y))[f(y) − f(x)]J j (y) dσ(y).
∂Ω
As before, the integrand in (6.1.16) satisﬁes, for some C ∈ (0, ∞) independent of j, α β ν j (𝛶j (x)) ν j (𝛶j (y)) b(𝛶j (x), 𝛶j (x) − 𝛶j (y)) × [f(y) − f(x)]J j (y) ≤ Cx − y2−n for σa.e. x, y ∈ ∂Ω.
(6.1.17)
Since by (2.2.82), (2.2.84), and the last condition in (2.2.80), for σa.e. x, y ∈ ∂Ω we also have α β ν j (𝛶j (x)) ν j (𝛶j (y)) b(𝛶j (x), 𝛶j (x) − 𝛶j (y))[f(y) − f(x)]J j (y) (6.1.18) converges to ν(x)α ν(y)β b(x, x − y)[f(y) − f(x)] as j → ∞, we may invoke Lebesgue’s Dominated Convergence Theorem in order to conclude that lim (Ij f)(x) = ∫ ν(x)α ν(y)β b(x, x − y)[f(y) − f(x)] dσ(y)
j→∞
∂Ω
= (T αβ f)(x) − f(x)(T αβ 1)(x) for σa.e. x ∈ ∂Ω.
(6.1.19)
Upon observing from (6.1.16) and (6.1.17) that there exists a suitable ﬁnite constant C = C(∂Ω, f) > 0, independent of j, with the property that (Ij f)(x) ≤ C for x ∈ ∂Ω, we deduce from (6.1.19) and Lebesgue’s Dominated Convergence Theorem that lim Ij f = T αβ f − f T αβ 1 in L p (∂Ω).
j→∞
(6.1.20)
As regards IIj f , directly from deﬁnitions we have αβ
IIj f = f [T j 1] ∘ 𝛶j .
(6.1.21)
6.1 Convergence of Families of Singular Integral Operators

235
Given that (6.1.12) implies αβ
‖f [T j 1] ∘ 𝛶j − f T αβ 1‖L p (∂Ω)
(6.1.22)
αβ ≤ ‖f‖L∞ (∂Ω) [T j 1] ∘ 𝛶j − T αβ 1L p (∂Ω) → 0 in L p (∂Ω) as j → ∞, it follows that lim IIj f = f T αβ 1 in L p (∂Ω).
(6.1.23)
j→∞
At this point, (6.1.7) is clear from (6.1.13), (6.1.20) and (6.1.23). Step V. If (6.1.4) holds for every p ∈ (1, ∞) then a similar convergence property holds αβ for every p ∈ (1, ∞) with α, β replaced (in the deﬁnitions of T αβ , T j from (6.1.2) and ̃ , β̃ ∈ ℕn such that α ̃ ≥ α and β̃ ≥ β. (6.1.3)) by any other pair of multiindices α 0
To justify this, observe that by Step II and Step III it suffices to establish that ̃ ̃β α
̃
̃β α p lim [T j (f ∘ 𝛶−1 j )] ∘ 𝛶j = T f in L (∂Ω)
(6.1.24)
j→∞
for every p ∈ (1, ∞) and every f ∈ Lip (∂Ω). To this end, pick p ∈ (1, ∞) and f ∈ Lip (∂Ω) arbitrary and, for σa.e. x ∈ ∂Ω, decompose ̃ ̃β α
̃ ̃β α
̃ ̃β α
̃ ̃β α
([T j (f ∘ 𝛶−1 j )] ∘ 𝛶j )(x) = (Ij f )(x) + (IIj f )(x) + (IIIj f )(x),
(6.1.25)
where ̃ ̃β α
(Ij f )(x) := [ν j (𝛶j (x))
̃ −α α
̃
− ν(x)α̃ −α ](P.V. ∫ ν j (𝛶j (x)) ν j (y)β α
(6.1.26)
∂Ω j
× b(𝛶j (x), 𝛶j (x) − y)(f ∘ 𝛶−1 j )(y) dσ j (y)), ̃ ̃β α
(IIj f )(x) := ν(x)α̃ −α (P.V. ∫ ν j (𝛶j (x)) ν j (y)β b(𝛶j (x), 𝛶j (x) − y) α
(6.1.27)
∂Ω j ̃
× [ν j (y)β−β − ν(𝛶−1 j (y)) ̃ ̃β α
̃ β−β
](f ∘ 𝛶−1 j )(y) dσ j (y)),
(IIIj f )(x) := ν(x)α̃ −α (P.V. ∫ ν j (𝛶j (x)) ν j (y)β b(𝛶j (x), 𝛶j (x) − y) α
(6.1.28)
̃
∂Ω j
× ((ν β−β f) ∘ 𝛶−1 j )(y) dσ j (y)).
Observe that ̃ ̃β α
Ij f = [(ν j ∘ 𝛶j ) ̃ ̃β α IIj f ̃ ̃β α
̃ −α α
̃ αβ
− ν α̃ −α ][T j (f ∘ 𝛶−1 j )] ∘ 𝛶j , ̃
̃
= ν α̃ −α [T j ((((ν j ∘ 𝛶j )β−β − ν β−β )f ) ∘ 𝛶−1 j )] ∘ 𝛶j , αβ
̃
IIIj f = ν α̃ −α [T j ((ν β−β f) ∘ 𝛶−1 j )] ∘ 𝛶j . αβ
(6.1.29) (6.1.30) (6.1.31)
236  6 Fatou Theorems and Integral Representations
Using (6.1.5), (6.1.8), (6.1.9), and Hölder’s inequality, we see that for some ﬁnite constant C > 0 independent of j, ̃ −α α α̃ β̃ − ν α̃ −α L2p (∂Ω) f L2p (∂Ω) , Ij f L p (∂Ω) ≤ C (ν j ∘ 𝛶j )
(6.1.32)
̃ ̃ β−β α̃ β̃ − ν β−β L2p (∂Ω) f L2p (∂Ω) . IIj f L p (∂Ω) ≤ C (ν j ∘ 𝛶j )
(6.1.33)
and
The fact that f ∈ Lip (∂Ω) makes ‖f‖L2p (∂Ω) ﬁnite. Given that, as seen from (vi) of Proposition 2.35 and Lebesgue’s Dominated Convergence Theorem, for every multiindex γ ∈ ℕ0n we have γ
(ν j ∘ 𝛶j ) → ν γ as j → ∞
in any L q (∂Ω) with q < ∞,
(6.1.34)
we conclude that, on the one hand, ̃ ̃β α
Ij f → 0
̃ ̃β α
IIj f → 0 in L p (∂Ω) as j → ∞.
and
(6.1.35)
̃
On the other hand, since ν β−β f ∈ L p (∂Ω), the present working hypotheses permit us to conclude that ̃ ̃β α
̃
lim IIIj f = lim ν α̃ −α [T j ((ν β−β f) ∘ 𝛶−1 j )] ∘ 𝛶j
j→∞
αβ
j→∞
̃
̃
= ν α̃ −α T αβ (ν β−β f) = T α̃ β f in L p (∂Ω).
(6.1.36)
At this stage, (6.1.24) is seen from (6.1.25), (6.1.35), and (6.1.36). Step VI. It suffices to prove that (6.1.7) holds for every for every function f ∈ L p (∂Ω) with p ∈ (1, ∞), in the special case when α = β = (0, . . . , 0) ∈ ℕ0n and when b(x, z) depends only on z, in the following speciﬁc fashion: b(z) =
P(z) zn−1+ℓ
for every z ∈ ℝn \ {0},
(6.1.37)
where P is an odd, homogeneous, harmonic polynomial in ℝn of degree ℓ. To justify the claim made in Step VI, we begin by considering the following orthonormal set {Ψ iℓ : ℓ ∈ 2ℕ + 1, 1 ≤ i ≤ Hℓ }, spanning the set of all odd spherical harmonics in ℝ. Here, n+ℓ−1 n+ℓ−3 )−( ) ℓ ℓ−2
if ℓ ≥ 3.
(6.1.38)
Hℓ ≤ C n (ℓ − 1) ⋅ ℓ ⋅ ⋅ ⋅ (n + ℓ − 2) ⋅ (n + ℓ − 3) ≤ C n ℓn−1
(6.1.39)
H1 := n,
and
Hℓ := (
In particular,
and, whenever ℓ ∈ 2ℕ + 1 and 1 ≤ i ≤ Hℓ , ∆ S n−1 Ψ iℓ = −ℓ(n + ℓ − 2)Ψ iℓ
and
Ψ iℓ (
P iℓ (x) x )= x xℓ
(6.1.40)
6.1 Convergence of Families of Singular Integral Operators

237
for some homogeneous, odd, harmonic polynomial P iℓ of degree ℓ in ℝn . Thus, if for x ∈ U we now set a iℓ (x) := ∫ b(x, ω)Ψ iℓ (ω) dω,
ℓ ∈ 2ℕ + 1, 1 ≤ i ≤ Hℓ ,
(6.1.41)
S n−1
a familiar argument involving successive integration by parts on the unit sphere gives that γ a iℓ ‖L∞ (U) ≤ C n ℓ−N ∑ ‖∂ z b(x, z)‖L∞ (U×S n−1 ) (6.1.42) γ≤N
for all indices ℓ ∈ 2ℕ + 1 and 1 ≤ i ≤ Hℓ . Let us now separate the variables x ∈ U and z ∈ ℝn \ {0} by expanding b(x, z) = =
Hℓ z 1 z 1 b a iℓ (x) n−1 Ψ iℓ ( ) = (x, ) ∑ ∑ n−1 z z z z ℓ∈2ℕ+1 i=1
∑
Hℓ
∑ a iℓ (x)b iℓ (z),
(6.1.43)
ℓ∈2ℕ+1 i=1
where b iℓ (z) :=
1 P iℓ (z) z Ψ iℓ ( ) = n−1+ℓ , n−1 z z z
z ∈ ℝn \ {0},
(6.1.44)
is a kernel which satisﬁes (6.1.37). Let us also point out here that once an even integer d > N o + (n − 1)/2 has been ﬁxed (where N o has appeared earlier, in connection with (6.1.6)), then for each ℓ ∈ 2ℕ + 1 and 1 ≤ i ≤ Hℓ , ‖b iℓ S n−1 ‖C No ≤ C n (I − ∆ S n−1 )d/2 (b iℓ S n−1 )L2 (S n−1 ) ≤ C n ℓd ,
(6.1.45)
thanks to (6.1.40), Sobolev’s Embedding Theorem, and the fact that b iℓ = Ψ iℓ on S n−1 . Moving on, ﬁx p ∈ (1, ∞) along with some f ∈ L p (∂Ω). Let us agree to temporarily write T b in place of T (0,...,0)(0,...,0) in order to stress the dependence on the integral (0,...,0)(0,...,0) kernel. A similar convention will apply to each T j in place of which we now write T b,j . With this in mind, for each j ∈ ℕ we make use of (6.1.43) in order to decompose [T b,j (f ∘ 𝛶−1 j )] ∘ 𝛶j = ∑
Hℓ
−1 ∑ [a iℓ T b iℓ ,j (f ∘ 𝛶−1 j )] ∘ 𝛶j + [R M,j (f ∘ 𝛶j )] ∘ 𝛶j
(6.1.46)
1≤ℓ≤M i=1 ℓ odd
where M ∈ ℕ will be chosen later, and where the residual operator R M,j is given by R M,j := ∑
Hℓ
∑ a iℓ T b iℓ ,j .
ℓ≥M+1 i=1 ℓ odd
(6.1.47)
238  6 Fatou Theorems and Integral Representations
It follows from (6.1.47), the version of (6.1.6) written for T b iℓ ,j , (6.1.45), (6.1.42), and (6.1.39) that Hℓ
R M,j L p (∂Ω j )→L p (∂Ω j ) ≤ C(Ω, p) ∑
∑ ‖a iℓ ‖L∞ (∂Ω j ) ‖b iℓ S n−1 ‖C No
ℓ≥M+1 i=1 ℓ odd
γ
≤ C(Ω, p)( ∑ ‖∂ z b(x, z)‖L∞ (U×S n−1 ) ) ∑
Hℓ
∑ ℓ−N+d
ℓ≥M+1 i=1 ℓ odd
γ≤N
≤ C(Ω, p, N, b)( ∑ ℓ−N+(n−1)+d ).
(6.1.48)
ℓ≥M+1 ℓ odd
Thus, if N = N(n) is large enough relative to N o to begin with, it follows that given ε > 0, one may ensure that R M,j L p (∂Ω j )→L p (∂Ω j ) ≤ ε for every (6.1.49) j ∈ ℕ by taking M = M(Ω, p, N, b, ε) ∈ ℕ sufficiently large. Parallel arguments apply to T b , which we expand as Tb = ∑
Hℓ
∑ a iℓ T b iℓ + R M .
(6.1.50)
1≤ℓ≤M i=1 ℓ odd
Here R M abbreviates R M := ∑
Hℓ
∑ a iℓ T b iℓ
(6.1.51)
ℓ≥M+1 i=1 ℓ odd
and, much as in (6.1.49), given ε > 0, one may ensure that R M L p (∂Ω)→L p (∂Ω) ≤ ε by taking M = M(Ω, p, N, b, ε) ∈ ℕ sufficiently large.
(6.1.52)
In light of the current goals, given that the operators T b iℓ ,j , T b iℓ are associated with kernels as in (6.1.37) it is permissible to assume that for each p ∈ (1, ∞), and each ℓ ∈ 2ℕ + 1 and i ∈ {1, . . . , Hℓ }, p p lim [T b iℓ ,j (f ∘ 𝛶−1 j )] ∘ 𝛶j = T b iℓ f in L (∂Ω), ∀ f ∈ L (∂Ω).
(6.1.53)
j→∞
Granted this, assume some f ∈ L p (∂Ω) with p ∈ (1, ∞) and, for some arbitrary M ∈ ℕ, estimate lim sup [T b,j (f ∘ 𝛶−1 L p (∂Ω) j )] ∘ 𝛶j − T b f j→∞
Hℓ Hℓ ≤ lim sup ∑ ∑ [a iℓ T b iℓ ,j (f ∘ 𝛶−1 )] ∘ 𝛶 − a T f ∑ ∑ j iℓ b iℓ j p j→∞ 1≤ℓ≤M i=1 L (∂Ω) 1≤ℓ≤M i=1 ℓ odd
ℓ odd
+ lim sup [R M,j (f ∘ 𝛶−1 L p (∂Ω) + R M f L p (∂Ω) . j )] ∘ 𝛶j j→∞
(6.1.54)
6.1 Convergence of Families of Singular Integral Operators

239
Thanks to (6.1.49), (6.1.52), and (6.1.53), for any given ε > 0 we may then conclude (by taking M large enough) that lim sup [T b,j (f ∘ 𝛶−1 L p (∂Ω) ≤ 2ε j )] ∘ 𝛶j − T b f
(6.1.55)
j→∞
from which (6.1.7) follows when α = β = (0, . . . , 0) ∈ ℕ0n . The latter constraint may then be eliminated by invoking Step V. Interlude. As a consequence of Step VI, in all subsequent steps we may (and will) assume that the kernel b(x, z) depends only on the variable z alone. Thanks to Step V, we may (and typically, though not always, will) also assume that α = β = (0, . . . , 0) ∈ ℕ0n , αβ
in which scenario we agree to drop the superscripts αβ when writing T αβ and T j . Step VII. The claim in (6.1.7) holds in the special case when, for some i ∈ {1, . . . , n}, b(z) = −
1 ω n−1
zi zn
for every z = (z1 , . . . , z n ) ∈ ℝn \ {0}.
(6.1.56)
In such a scenario, observe that b(x − y) = ∂ y i [E∆ (x − y)]
for all x, y ∈ ℝ with x ≠ y,
(6.1.57)
where E∆ stands for the standard radial fundamental solution for the ﬂatspace Laplacian ∆ = ∂21 + ⋅ ⋅ ⋅ + ∂2n in ℝn , given by 1 1 { { { ω n−1 (2 − n) xn−2 E∆ (x) := { { 1 { ln x { 2π
if n ≥ 3, ∀ x ∈ ℝn \ {0}.
(6.1.58)
if n = 2,
To proceed, denote by ν = (ν1 , . . . , ν n ) the outward unit normal to Ω and decompose (using the conventions adopted in the interlude) Tf(x) = P.V. ∫ ∂Ω
xi − yi f(y) dσ(y) x − yn
= P.V. ∫ ∂ y i [E∆ (x − y)]f(y) dσ(y) ∂Ω n
= ∑ P.V. ∫ ν k (y)ν k (y)∂ y i [E∆ (x − y)]f(y) dσ(y) k=1
∂Ω
n
= ∑ P.V. ∫ ν k (y)∂ τ ki (y) [E∆ (x − y)]f(y) dσ(y) k=1
∂Ω n
+ ∑ P.V. ∫ ν k (y)ν i (y)∂ y k [E∆ (x − y)]f(y) dσ(y) k=1
∂Ω
= Af(x) + Bf(x),
for σa.e. x ∈ ∂Ω,
(6.1.59)
240  6 Fatou Theorems and Integral Representations where for σa.e. x ∈ ∂Ω we have set n
Af(x) := ∑ P.V. ∫ ν k (y)∂ τ ki (y) [E∆ (x − y)]f(y) dσ(y), k=1
(6.1.60)
∂Ω
Bf(x) := P.V. ∫ ν i (y)∂ ν(y) [E∆ (x − y)]f(y) dσ(y).
(6.1.61)
∂Ω
Analogously, if for each j ∈ ℕ we let ν j = ((ν j )1 , . . . , (ν j )n ) be the outward unit normal to Ω j , we may decompose xi − yi f(y) dσ j (y) = A j (x) + B j f(x) x − yn
T j f(x) = P.V. ∫ ∂Ω j
(6.1.62)
where for σ j a.e. x ∈ ∂Ω j we now take (with ∂ τ j := (ν j )k ∂ i − (ν j )i ∂ k on ∂Ω j ) ki
n
A j f(x) := ∑ P.V. ∫ (ν j )k (y)∂ τ j
ki (y)
k=1
[E∆ (x − y)]f(y) dσ j (y),
(6.1.63)
∂Ω j
B j f(x) := P.V. ∫ (ν j )i (y)∂ ν j (y) [E∆ (x − y)]f(y) dσ j (y).
(6.1.64)
∂Ω j
Let us also consider the following versions of the operators A, B, A j , B j : ̃ A k f(x) := P.V. ∫ ∂ τ ki (y) [E∆ (x − y)]f(y) dσ(y),
1 ≤ k ≤ n,
(6.1.65)
∂Ω
̃ Bf(x) := P.V. ∫ ∂ ν(y) [E∆ (x − y)]f(y) dσ(y),
(6.1.66)
∂Ω
for σa.e. x ∈ ∂Ω, and for each j ∈ ℕ, ̃ A jk f(x) := P.V. ∫ ∂ τ j
ki (y)
[E∆ (x − y)]f(y) dσ j (y),
1 ≤ k ≤ n,
(6.1.67)
∂Ω j
̃ j f(x) := P.V. ∫ ∂ ν (y) [E (x − y)]f(y) dσ j (y). B j ∆
(6.1.68)
∂Ω j
Then Proposition 9.21 implies that, for each k ∈ {1, . . . , n} and each j ∈ ℕ, ̃ A k 1 = 0 at σa.e. point on ∂Ω, and ̃ A jk 1 = 0 at σ j a.e. point on ∂Ω j ,
(6.1.69)
while formula (9.8.13) (used here both for Ω and Ω j ) gives that, for each j ∈ ℕ, ̃ = B1 ̃ j1 = B
1 2 1 2
at σa.e. point on ∂Ω, and at σ j a.e. point on ∂Ω j .
(6.1.70)
6.1 Convergence of Families of Singular Integral Operators

241
Granted these, it follows from Step IV that for every function f ∈ L p (∂Ω) we have p ̃ A jk (f ∘ 𝛶−1 lim [̃ j )] ∘ 𝛶j = A k f in L (∂Ω),
j→∞
∀ k ∈ {1, . . . , n},
(6.1.71)
and ̃ in L p (∂Ω). ̃ j (f ∘ 𝛶−1 )] ∘ 𝛶j = Bf lim [B j
(6.1.72)
j→∞
In turn, from (6.1.71) and (6.1.72), we deduce on account of Step V that for every function f ∈ L p (∂Ω) we have p lim [A j (f ∘ 𝛶−1 j )] ∘ 𝛶j = Af in L (∂Ω),
j→∞
p lim [B j (f ∘ 𝛶−1 j )] ∘ 𝛶j = Bf in L (∂Ω).
(6.1.73)
j→∞
In view of (6.1.59) and (6.1.62), these imply that for every function f ∈ L p (∂Ω) we have p lim [T j (f ∘ 𝛶−1 j )] ∘ 𝛶j = Tf in L (∂Ω),
j→∞
(6.1.74)
as wanted. Step VIII. Assume that for some ﬁxed odd integer l ≥ 3, the claim in (6.1.7) holds for every f ∈ L p (∂Ω) with p ∈ (1, ∞) whenever T and T j are associated as in (6.1.2) and (6.1.3) with a function of the form b(z) = P(z)/zn−1+l for z ∈ ℝn \ {0} where P is an odd harmonic homogeneous polynomial of degree ≤ l − 2 in ℝn .
(6.1.75)
Then (6.1.75) holds with l replaced by l + 2. In the proof of this inductive claim we shall make use of the Clifford algebra formalism from § 9.13, which the reader is invited to review ﬁrst. To get started, pick an arbitrary odd harmonic homogeneous polynomial P(x) of degree l in ℝn , and let the principal value singular integral operators T and T j be associated as indicated in (6.1.2) and (6.1.3) with the function b(z) := P(z)/zn−1+l for z ∈ ℝn \ {0}. Next, consider the family P rs (x), 1 ≤ r, s ≤ n, of odd harmonic homogeneous polynomials of degree l − 2, as well as the family of odd, C ∞ functions b rs : ℝn \ {0} → ℝn → Cℓn , associated with P as in Lemma 9.70. For each 1 ≤ i, j ≤ n set b rs (z) := P rs (z)/zn+l−3
for z ∈ ℝn \ {0}.
(6.1.76)
Fix some p ∈ (1, ∞) and introduce the integral operator acting on Clifford algebravalued functions, f = ∑I f I e I with scalar components f I ∈ L p (∂Ω) according to T rs f(x) := P.V. ∫ b rs (x − y)f(y) dσ(y) ∂Ω
= ∑ ( ∫ b rs (x − y)f I (y) dσ(y))e I , I
∂Ω
x ∈ ∂Ω.
(6.1.77)
242  6 Fatou Theorems and Integral Representations Likewise, for each j ∈ ℕ, consider the integral operator acting on Clifford algebravalued functions, f ∈ L p (∂Ω j ) ⊗ Cℓn according to T jrs f(x) := P.V. ∫ b rs (x − y)f(y) dσ j (y),
x ∈ ∂Ω j .
(6.1.78)
∂Ω j
Then from the properties of the P rs ’s and the induction hypothesis (6.1.75) we conclude that for each 1 ≤ r, s ≤ n we have rs p lim [T jrs (f j ∘ 𝛶−1 j )] ∘ 𝛶j = T f in L (∂Ω) ⊗ Cℓn
j→∞
whenever
f, f j ∈ L p (∂Ω) ⊗ Cℓn are such that lim f j = f in L p (∂Ω) ⊗ Cℓn .
(6.1.79)
j→∞
Moving on, for every r, s ∈ {1, . . . , n} consider the singular integral operator acting on a Cℓn valued function f deﬁned in ∂Ω with L p scalar components by T rs f(x) := P.V. ∫ b rs (x − y) ⊙ f(y) dσ(y),
x ∈ ∂Ω.
(6.1.80)
∂Ω
Likewise, for each j ∈ ℕ, consider the integral operator acting on Clifford algebravalued functions, f ∈ L p (∂Ω j ) ⊗ Cℓn according to T rs,j f(x) := P.V. ∫ b rs (x − y) ⊙ f(y) dσ j (y),
x ∈ ∂Ω j .
(6.1.81)
∂Ω j
Thanks to formula (9.13.26), whenever the function f is deﬁned on ∂Ω and takes values in ℝ → Cℓn , one may recover Tf (where T is the operator introduced earlier in the current proof) from (6.1.80) by means of the identity n
Tf(x) = ∑ [T rs f(x)]s
for σa.e. x ∈ ∂Ω.
(6.1.82)
r,s=1
Similarly, if f : ∂Ω j → ℝ → Cℓn , one may recover T j f (again, with T j as deﬁned earlier in the current proof) from (6.1.81) by means of the identity n
T j f(x) = ∑ [T rs,j f(x)]s
for σ j a.e. x ∈ ∂Ω j .
(6.1.83)
r,s=1
To proceed, ﬁx an arbitrary point x ∈ ℝn \ Ω. As a consequence of the integration by parts formula (9.13.18) (used here with u = b rs (x − ⋅) ∈ C ∞ (Ω), and υ ≡ 1) and (9.13.27), for each r, s ∈ {1, . . . , n} we then obtain ∫ b rs (x − y) ⊙ ν(y) dσ(y) = − ∫(D R b rs )(x − y) dy Ω
∂Ω
=
∂ P rs (x − y) l−1 ( ) dy ∫ n + l − 3 ∂y r x − yn+l−3 Ω
l−1 = ∫ b rs (x − y)ν r (y) dσ(y), n+l−3 ∂Ω
(6.1.84)
6.1 Convergence of Families of Singular Integral Operators

243
where ν r is the rth component of ν, the outward unit normal to Ω. Upon letting the point x ∈ ℝn \ Ω approach ∂Ω in a nontangential fashion we obtain from (6.1.84) and the jumpformulas in UR domains from [50] that¹ −
1 ̂ b rs (ν(x)) ⊙ ν(x) + P.V. ∫ b rs (x − y) ⊙ ν(y) dσ(y) √ 2 −1 ∂Ω
l−1
̂ rs (ν(x))ν r (x) b 2√−1(n + l − 3) l−1 P.V. ∫ b rs (x − y)ν r (y) dσ(y) + n+l−3
= −
(6.1.85) for σa.e. x ∈ ∂Ω.
∂Ω
In light of (6.1.80) and (6.1.77), this formula translates into −
1 ̂ b rs (ν(x)) ⊙ ν(x) + (T rs ν)(x) 2√−1 l−1 ̂ rs (ν(x))ν r (x) b = − 2√−1(n + l − 3) l−1 (T rs ν r )(x) for σa.e. x ∈ ∂Ω. + n+l−3
(6.1.86)
In this connection, let us remark that the functions b rs belong to C ∞ (ℝn \ {0}) and ̂ are homogeneous of degree 1 − n. Hence, b rs is a tempered distribution in ℝn and b rs , n originally considered in the class of tempered distributions in ℝ , satisﬁes (cf. [79, Exercise 4.60, p. 133], or [118, Chapter 3, Proposition 8.1]) ∞ n ̂ b rs ∈ C (ℝ \ {0}).
(6.1.87)
̂ rs ∈ C ∞ (ℝn \ {0}). b
(6.1.88)
Similarly,
In addition, in a completely similar fashion to (6.1.86), for each j ∈ ℕ we obtain −
1 ̂ b rs (ν j (x)) ⊙ ν j (x) + (T rs,j ν j )(x) √ 2 −1 l−1 ̂ rs (ν (x))(ν )r (x) b = − j j 2√−1(n + l − 3) l−1 + (T rs (ν j )r )(x) for σa.e. x ∈ ∂Ω j . n+l−3 j
(6.1.89)
̃ rs and T ̃ rs,j be the versions of T rs from (6.1.80) and T rs,j from Going further, let T (6.1.81) in which the original integral kernel has been Clifford multiplied from the right
1 with “hat” denoting the Fourier transform, and √−1 denoting the unique complex number z with positive imaginary part satisfying z2 = −1
244  6 Fatou Theorems and Integral Representations with the outward unit normal at y. Speciﬁcally, given r, s ∈ {1, . . . , n}, if f : ∂Ω → Cℓn set ̃ rs f(x) := P.V. ∫ (b rs (x − y) ⊙ ν(y)) ⊙ f(y) dσ(y), T
x ∈ ∂Ω,
(6.1.90)
∂Ω
and, for each j ∈ ℕ, if f : ∂Ω j → Cℓn set ̃ rs,j f(x) := P.V. ∫ (b rs (x − y) ⊙ ν j (y)) ⊙ f(y) dσ j (y), T
x ∈ ∂Ω j .
(6.1.91)
∂Ω j
Since by design ̃ rs 1 = T rs ν T ̃ rs,j 1 = T rs,j ν j T
for each r, s ∈ {1, . . . , n},
whereas
for each r, s ∈ {1, . . . , n}
and j ∈ ℕ,
(6.1.92)
from (6.1.92), (6.1.89), (2.2.82), (6.1.87), (6.1.88), (6.1.34), (6.1.79), (6.1.89), and (6.1.86), we deduce that, for each r, s ∈ {1, . . . , n}, we have ̃ rs,j 1] ∘ 𝛶j = lim [T rs,j ν j ] ∘ 𝛶j lim [T
j→∞
j→∞
1 ̂ lim b rs (ν j ∘ 𝛶j ) ⊙ (ν j ∘ 𝛶j ) 2√−1 j→∞ l−1 ̂ rs (ν ∘ 𝛶 )(ν ∘ 𝛶 )r lim b − j j j j j→∞ √ 2 −1(n + l − 3) l−1 + lim [T rs (ν j )r ] ∘ 𝛶j n + l − 3 j→∞ j 1 ̂ = b rs (ν) ⊙ ν 2√−1 l−1 ̂ rs (ν)ν r − b √ 2 −1(n + l − 3) l−1 + T rs (ν r ) n+l−3 ̃ rs 1 in L p (∂Ω) ⊗ Cℓn . = T rs ν = T =
Thus,
̃ rs 1 in L p (∂Ω) ⊗ Cℓn . ̃ rs,j 1] ∘ 𝛶j = T lim [T
j→∞
(6.1.93)
(6.1.94)
With this in hand, Step IV applies and, for each 1 ≤ r, s ≤ n, yields the conclusion that ̃ rs f in L p (∂Ω) ⊗ Cℓn ̃ rs,j (f j ∘ 𝛶−1 )] ∘ 𝛶j = T lim [T j
j→∞
whenever
f, f j ∈ L p (∂Ω) ⊗ Cℓn are such that lim f j = f in L p (∂Ω) ⊗ Cℓn .
(6.1.95)
j→∞
L p (∂Ω)
⊗ Cℓn . If for each j ∈ ℕ we deﬁne f j := (ν j ∘ 𝛶j ) ⊙ f , Fix now an arbitrary f ∈ it follows that f j ∈ L p (∂Ω) ⊗ Cℓn and from (2.2.82) and Lebesgue’s Dominated Convergence Theorem we deduce that limj→∞ f j = ν ⊙ f in L p (∂Ω) ⊗ Cℓn . The idea is to
6.1 Convergence of Families of Singular Integral Operators
 245
write (6.1.95) for this convergent sequence {f j }j∈ℕ . Upon recalling (6.1.90) and (6.1.91) as well as (6.1.80) and (6.1.81), and mindful of the fact that ν ⊙ ν = −1 (cf. (9.13.1)), for each 1 ≤ r, s ≤ n this produces p lim [T rs,j (f ∘ 𝛶−1 j )] ∘ 𝛶j = T rs f in L (∂Ω) ⊗ Cℓn
j→∞
(6.1.96)
for every function f ∈ L p (∂Ω) ⊗ Cℓn . In particular, taking f = 1 leads to the conclusion that lim [T rs,j 1] ∘ 𝛶j = T rs 1 in L p (∂Ω) ⊗ Cℓn
j→∞
for all 1 ≤ r, s ≤ n.
(6.1.97)
Thanks to (6.1.82) and (6.1.83), this entails lim [T j 1] ∘ 𝛶j = T1 in L p (∂Ω),
j→∞
(6.1.98)
and the desired conclusion follows from this by invoking Step IV. The endgame in the proof of Theorem 6.1. From Step VII, Step VIII and induction on l, we are able to conclude that the claim in (6.1.4) holds for any p ∈ (1, ∞) whenever the singular integral operators in question are associated with a function of the form b(x, z) = P(z)/zn−1+l , z ∈ ℝn \ {0}, for any odd homogeneous harmonic polynomial P of degree l in ℝn , in the case when α = β = (0, . . . , 0) ∈ ℕ0n . In concert with Step VI, this ﬁnishes the proof of the claim made in (6.1.4) in full generality. Finally, for the very last claim in the statement of the theorem, pertaining to singular integral operators associated with b(y, x − y) in place of b(x, x − y), the same line of reasoning applies almost verbatim. The only notable difference originates in (6.1.41), where we now have a iℓ (y) := ∫ b(y, ω)Ψ iℓ (ω) dω,
ℓ ∈ 2ℕ + 1, 1 ≤ i ≤ Hℓ ,
(6.1.99)
S n−1
with the estimate (6.1.42) still valid. Given that in place of (6.1.43) we presently have Hℓ a iℓ (y)b iℓ (z) with b iℓ (z) as in (6.1.44), in place of (6.1.46) we may b(y, z) = ∑ℓ∈2ℕ+1 ∑i=1 now decompose [T b,j (f ∘ 𝛶−1 j )] ∘ 𝛶j = ∑
Hℓ
−1 ∑ [T b iℓ ,j (f jiℓ ∘ 𝛶−1 j ))] ∘ 𝛶j + [R M,j (f ∘ 𝛶j )] ∘ 𝛶j
(6.1.100)
1≤ℓ≤M i=1 ℓ odd
where we have set f jiℓ := (a iℓ ∘ 𝛶j )f ∈ L p (∂Ω) (hence, in particular, f jiℓ → a iℓ f in L p (∂Ω) as j → ∞), and where the residual operator R M,j is now given by R M,j := ∑
Hℓ
∑ T b iℓ ,j a iℓ ,
ℓ≥M+1 i=1 ℓ odd
(6.1.101)
246  6 Fatou Theorems and Integral Representations
regarding T b iℓ ,j a iℓ as the composition between T b iℓ ,j and the operator of pointwise multiplication by the function a iℓ . With this interpretation estimate (6.1.48) continues to hold and, from this point on, the same reasoning as in Step VI applies. This concludes the proof of Theorem 6.1. It turns out that Theorem 6.1 has a natural counterpart with strong convergence in Lebesgue spaces replaced by weak convergence. Speciﬁcally, we have the following result. Theorem 6.2. Let Ω ⊆ ℝn be a bounded regular SKT domain. Fix some open neighborhood U ⊂ ℝn of Ω and consider a function b(x, z) as in (6.1.1). Associate to this and to some ﬁxed multiindices α, β ∈ ℕ0n the variable kernel singular integral operators T αβ as in (6.1.2). Also, if Ω j ↗ Ω as j → ∞ in the sense of Proposition 2.35 for each j ∈ ℕ conαβ
sider T j deﬁned as in (6.1.3). Recall that 𝛶j : ∂Ω → ∂Ω j , j ∈ ℕ, denote the biLipschitz homeomorphisms from item (iv) of Proposition 2.35, and ﬁx some p ∈ (1, ∞). Then one has αβ p lim [T j (f j ∘ 𝛶−1 j )] ∘ 𝛶j = T f weakly in L (∂Ω), αβ
j→∞
whenever the functions f, f j ∈ L p (∂Ω), for j ∈ ℕ, are such that lim f j = f weakly in L p (∂Ω).
(6.1.102)
j→∞
Moreover, a similar weak convergence result holds for the integral operators associated as in (6.1.2) and (6.1.3) with the function b(y, x − y) in place of b(x, x − y). In fact, all results above remain true in the case when the set Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the given p, the size of the function b, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. Recall that Theorem 6.1 also works when b(x, x − y) is replaced by b(y, x − y) αβ in (6.1.2) and (6.1.3). Thus one can replace the operators T j and T αβ by their adjoints and have the same conclusions. In view of this, Theorem 6.2 follows by duality. We ﬁnd it useful to complement Theorems 6.1 and 6.2 with a convergence result for weakly singular integral operators of the sort described in our next lemma. Lemma 6.3. Assume the kernel k : (ℝn × ℝn ) \ diag → ℝ is continuous and there exists α > 0 with the property that k(x, y) ≤ Cx − yn−1−α for all x ≠ y, x, y ∈ ℝn . Also, suppose Ω ⊂ ℝn is a compact εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. Let 𝛶j be the mappings introduced in Proposition 2.35 and ﬁx some p ∈ (1, ∞). Then, if {f j }j∈ℕ is a sequence of functions in L p (∂Ω) which converges to weakly in p L (∂Ω) to some f ∈ L p (∂Ω), it follows that, as functions of x ∈ ∂Ω, ∫ k(𝛶j (x), 𝛶j (y))f j (y) dσ(y) → ∫ k(x, y)f(y) dσ(y) ∂Ω
∂Ω
in L (∂Ω) as j → ∞. p
(6.1.103)
6.1 Convergence of Families of Singular Integral Operators 
247
Proof. Fix an arbitrary number r > 0 and, for each point x ∈ ∂Ω, decompose the domain of integration into {y ∈ ∂Ω : x − y ≥ r} and {y ∈ ∂Ω : x − y ≤ r}. For the ﬁrst resulting term, Lebesgue’s Dominated Convergence Theorem applies and gives that, as functions of x ∈ ∂Ω, ∫ k(𝛶j (x), 𝛶j (y))f j (y) dσ(y) → ∫ k(x, y)f(y) dσ(y) ∂Ω\B r (x)
(6.1.104)
∂Ω\B r (x)
in L (∂Ω) as j → ∞. p
As regards the second term, Schur’s test (or interpolation between L1 (∂Ω) and L∞ (∂Ω)) yields ∫ k(𝛶j (x), 𝛶j (y))f j (y)1x−y≤r dσ(y) p ∂Ω L x (∂Ω) ≤ C( sup x∈∂Ω
∫ y∈∂Ω∩B r (x)
dσ(y) ) sup ‖f j ‖L p (∂Ω) . x − yn−1−α j∈ℕ
(6.1.105)
Bearing in mind that ∂Ω is an Ahlfors regular set, from this we may conclude that the lefthand side of (6.1.105) is ≤ Cr α . Since Schur’s test may also be used to show that, as functions of x ∈ ∂Ω, we have ∫
k(x, y)f(y) dσ(y) → ∫ k(x, y)f(y) dσ(y)
∂Ω\B r (x)
(6.1.106)
∂Ω
in L (∂Ω) p
the lemma follows by sending r →
as
+
r→0 ,
0+ .
Remark 6.4. Thanks to (3.1.24)–(3.1.29) and Lemma 6.3, it follows that Theorem 6.1 and Theorem 6.2 apply to principal value singular integral operators on the boundaries of εSKT domains (assuming ε > 0 small relative to the potential V, the exponent p, and the Ahlfors and John constants of the underlying domain) in a manifold M, whose integral kernels are locally representable as linear combinations of ﬁrstorder derivatives of the fundamental solution Γ l (x, y) from (3.1.16) multiplied by (powers of the) components of the outward unit conormal to the underlying domain. This basically covers all principal value singular integral operators considered in this work. Having established the general convergence results discussed earlier in this section, we are now in a position to prove some useful uniform invertibility results for the principal value connection double layer operator, of the sort presented in the lemma below. Lemma 6.5. Let Ω ⊂ M be a regular SKT domain and ﬁx a degree l ∈ {0, 1, . . . , n} and some exponent p ∈ (1, ∞). Also, consider a potential V ∈ L r (M), r > n, which is real, nonnegative, and satisﬁes V + Ric ≥ 0 in M \ Ω, and V + Ric > 0 in a subset of positive measure of each connected component of M \ Ω.
(6.1.107)
248  6 Fatou Theorems and Integral Representations Then one can ﬁnd a family {Ω j }j∈ℕ approximating Ω in the manner described in Proposition 2.35, along with a constant C ∈ (0, ∞) independent of j ∈ ℕ, such that if for each j ∈ ℕ the potential V also satisﬁes V + Ric ≥ 0 in M \ Ω j , and V + Ric > 0 in a subset of positive measure of each connected component of M \ Ω j , then
‖f‖L p (∂Ω j ,Λ l TM) ≤ C( 12 I + K l,j )f L p (∂Ω j ,Λ l TM)
(6.1.108)
(6.1.109)
for every f ∈ L p (∂Ω j , Λ l T M), where K l,j denotes the principal value connection double layer operator, acting on lforms, deﬁned as in (4.1.61) in relation to Ω j and the potential V. In addition, whenever f ∈ L p (∂Ω, Λ l T M) and f j ∈ L p (∂Ω j , Λ l T M), j ∈ ℕ, satisfy lim f j ∘ 𝛶j = f
j→∞
weakly in L p (∂Ω, Λ l T M),
(6.1.110)
where 𝛶j : ∂Ω → ∂Ω j , j ∈ ℕ, denote the biLipschitz homeomorphisms from item (iv) of Proposition 2.35 and where the composition is interpreted as pullback, it follows that −1
−1
lim [( 12 I + K l,j ) f j ] ∘ 𝛶j = ( 12 I + K l ) f
j→∞
(6.1.111)
weakly in L p (∂Ω, Λ l T M), where K l denotes the principal value connection double layer operator deﬁned in (4.1.61) for the potential V. Moreover, all these results remain valid in the case when the set Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the given exponent p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. If (6.1.109) fails, there exist forms f j ∈ L p (∂Ω j , Λ l T M) for j ∈ ℕ such that f j ‖L p (∂Ω j ,Λ l TM) = 1 and 1 ( 2 I + K l,j )f j L p (∂Ω j ,Λ l TM) → 0 as j → ∞.
(6.1.112)
With composition interpreted as pullback, set g j := f j ∘ 𝛶j ∈ L p (∂Ω, Λ l T M) for every j ∈ ℕ. Then, thanks to (6.1.112), (6.1.8), and (6.1.9), we have {g j }j∈ℕ is a bounded sequence in L p (∂Ω, Λ l T M).
(6.1.113)
By eventually passing to a subsequence, we may therefore ensure that {g j }j∈ℕ converges weakly in L p (∂Ω, Λ l T M) to some form g ∈ L p (∂Ω, Λ l T M). We claim that g must be zero. To justify this claim, observe that the last condition in (6.1.112) implies lim [( 12 I + K l,j )(g j ∘ 𝛶−1 j )] ∘ 𝛶j = 0
j→∞
strongly in L p (∂Ω, Λ l T M).
(6.1.114)
6.1 Convergence of Families of Singular Integral Operators
 249
At the same time, lim g j = g
j→∞
weakly in L p (∂Ω, Λ l T M).
(6.1.115)
In the present context, when composition is regarded as pullback, Theorem 6.2 continues to be applicable thanks to item (viii) in Proposition 2.35 (which ensures that the effect of working with differential forms in place of scalarvalued functions is negligible for the current purposes). Bearing this in mind, we then conclude from (6.1.115) that 1 lim [( 1 I + K l,j )(g j ∘ 𝛶−1 j )] ∘ 𝛶j = ( 2 I + K l )g j→∞ 2 (6.1.116) weakly in L p (∂Ω, Λ l T M). Collectively, (6.1.114)–(6.1.116) then imply that ( 12 I + K l )g = 0 and since 12 I + K l is an isomorphism of L p (∂Ω, Λ l T M), we see that g must be zero, as claimed. We shall show that this leads to a contradiction. To this end, we ﬁrst note that for each j ∈ ℕ, the proof of Theorem 4.12 in concert with Proposition 9.64 permit us to decompose K l,j = R l,j + Q l,j where ‖Q l,j ‖L p (∂Ω j ,Λ l TM)→L p (∂Ω j ,Λ l TM) < 1/4
(6.1.117)
and R l,j f(x) = ∫ ⟨k l,j (x, y), f(y)⟩ dσ j (y),
x ∈ ∂Ω j ,
(6.1.118)
∂Ω j
with an integral kernel k l,j (x, y) a ﬁnite linear combination of terms of the form η(x)ζ(y) Sym (∇∗, ν j (y))∇y Γ l (x, y),
x, y ∈ ∂Ω j ,
(6.1.119)
where η, ζ ∈ C01 (ℝn ) are scalarvalued functions (independent of j) with the property that η ≡ 0 near supp ζ . Then, on the one hand, Lemma 6.3 gives ‖R l,j f j ‖L p (∂Ω j ,Λ l TM) → 0
as j → ∞.
(6.1.120)
On the other hand, (6.1.117) implies that for every j ∈ ℕ we have 1 2 ‖f j ‖L p (∂Ω j ,Λ l TM)
≤ ( 12 I + K l,j )f j L p (∂Ω j ,Λ l TM) + ‖K l,j f j ‖L p (∂Ω j ,Λ l TM) ≤ ( 12 I + K l,j )f j L p (∂Ω j ,Λ l TM) + ‖R l,j f j ‖L p (∂Ω j ) + ‖Q l,j f j ‖L p (∂Ω j ,Λ l TM) 1 ≤ ( 2 I + K l,j )f j L p (∂Ω j ,Λ l TM) + ‖R l,j f j ‖L p (∂Ω j ) + 14 ‖f j ‖L p (∂Ω j ,Λ l TM) ,
(6.1.121)
‖f j ‖L p (∂Ω j ) ≤ 4( 12 I + K l,j )f j L p (∂Ω j ,Λ l TM) + 4‖R l,j f j ‖L p (∂Ω j ,Λ l TM) .
(6.1.122)
hence,
250  6 Fatou Theorems and Integral Representations
Passing to the limit in (6.1.122) leads to an obvious contradiction on account of (6.1.112) and (6.1.120). This completes the proof of the claim made in the ﬁrst part of the statement of Lemma 6.5. To treat the claim made in the last part of the statement of Lemma 6.5 recall that, as a consequence of the uniform boundedness principle, every weakly convergent sequence in a Banach space is bounded. In our case, this implies that sup ‖f j ∘ 𝛶j ‖L p (∂Ω,Λ l TM) < ∞.
(6.1.123)
j∈ℕ
To proceed, introduce −1
g j := [( 12 I + K l,j ) f j ] ∘ 𝛶j
for each j ∈ ℕ,
(6.1.124)
and note that thanks to (6.1.8) and (6.1.9) for each j ∈ ℕ we have −1 ‖g j ‖L p (∂Ω,Λ l TM) ≈ ( 12 I + K l,j ) f j L p (∂Ω j ,Λ l TM) −1 ≤ C( 12 I + K l,j )( 12 I + K j ) f j L p (∂Ω j ,Λ l TM) = C‖f j ‖L p (∂Ω j ,Λ l TM) ≈ ‖f j ∘ 𝛶j ‖L p (∂Ω,Λ l TM) ≤ C,
(6.1.125)
for some ﬁnite constant C > 0 independent of j. Above, we have also used Lemma 6.5 in the second step, and (6.1.123) in the last step. Hence, {g j }j∈ℕ is a bounded sequence in L p (∂Ω, Λ l T M). As such, it contains a subsequence which converges weakly in L p (∂Ω, Λ l T M) to some g ∈ L p (∂Ω, Λ l T M). In fact, for reasons which are clear a posteriori, there is no loss of generality in assuming that the entire sequence {g j }j∈ℕ converges weakly in L p (∂Ω, Λ l T M) to this g.
(6.1.126)
Assuming that this is the case, we may once again rely on Theorem 6.2, together with (6.1.124) and (6.1.110), in order to write ( 12 I + K l )g = lim [( 12 I + K l,j )(g j ∘ 𝛶−1 j )] ∘ 𝛶j j→∞
= lim f j ∘ 𝛶j = f j→∞
As a consequence,
weakly in L p (∂Ω, Λ l T M). −1
g = ( 12 I + K l ) f.
(6.1.127)
(6.1.128)
Bearing this in mind, (6.1.111) now follows from (6.1.126) and (6.1.124), ﬁnishing the proof of the lemma.
6.2 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains In this section we are adequately prepared to prove the main result in this chapter, dealing with a Fatou theorem for the HodgeLaplacian in regular SKT domains and a naturally accompanying double layer representation formula, as stated below.
6.2 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains
 251
Theorem 6.6. Suppose Ω ⊂ M is a regular SKT domain, with surface measure σ and outward unit conormal ν. Fix l ∈ {0, 1, . . . , n} along with p ∈ (1, ∞), and consider some potential² V ∈ L r (Ω) with r > n. (6.2.1) Then any differential form u satisfying u ∈ C 0 (Ω, Λ l T M),
(∆HL − V)u = 0 in Ω,
N u ∈ L p (∂Ω),
(6.2.2)
has the property that n.t. the nontangential trace u∂Ω exists at σa.e. point on ∂Ω. (6.2.3) n.t. Also, the differential form u∂Ω belongs to the space L p (∂Ω, Λ l T M) and there exists a ﬁnite constant C > 0 independent of u such that n.t. u∂Ω L p (∂Ω,Λ l TM) ≤ C‖N u‖L p (∂Ω) .
(6.2.4)
Moreover, if actually the potential V is real and nonnegative, and the domain Ω does not contain any connected component of M,
(6.2.5)
then the potential { V in Ω, ̃ := V { λ in M \ Ω, {
with λ > sup ‖Ricx ‖Λ l T x M→Λ l T x M ,
(6.2.6)
x∈M\Ω
satisﬁes all properties listed in (3.1.13). It is therefore meaningful to consider the fundã associate with the potential mental solution Γ l (x, y) for the Schrödinger operator ∆HL − V ̃ V as in (3.1.16). In relation to this, the conclusion is that there exists a unique differential form f ∈ L p (∂Ω, Λ l T M) such that u(x) = ∫ ⟨i Sym (∇∗, ν(y))∇y Γ l (x, y), f(y)⟩ dσ(y),
∀ x ∈ Ω.
(6.2.7)
∂Ω
In fact, u = Dl f in Ω,
(6.2.8)
̃ on ∂Ω, where Dl is the connection double layer potential operator associated with V and −1 n.t. f = ( 12 I + K l ) (u∂Ω ), (6.2.9) where K l is the principal value double layer on ∂Ω associated as in (4.1.61) with the ̃ for the potential V ̃ given in (6.2.6) (in which scenario, the existence of operator ∆HL − V
2 not necessarily nonnegative (or even realvalued)
252  6 Fatou Theorems and Integral Representations −1
the inverse ( 12 I + K l ) on L p (∂Ω, Λ l T M) is guaranteed by item (3) of Theorem 5.10). As a corollary, whenever (6.2.5) holds one has n.t. u∂Ω L p (∂Ω,Λ l TM) ≈ ‖N u‖L p (∂Ω) ,
(6.2.10)
uniformly for u as in (6.2.2). Finally, all results above continue to hold when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the given exponent p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. Prior to presenting the proof of Theorem 6.6, in the lemma below we study the nature of the composition between a volumelike potential operator and a boundarylike potential operator. Lemma 6.7. Let Ω be a nonempty proper open subset of M, and let σ be a ﬁnite Borel measure on ∂Ω. Fix a function W ∈ L r (Ω)
with r > n.
(6.2.11)
Also, pick α > 0 and consider a measurable function k α : Ω × Ω → ℝ satisfying k α (x, y) ≤ C dist (x, y)−(n−α) ,
∀ x, y ∈ Ω.
(6.2.12)
Using this as an integral kernel, deﬁne the integral operator T α u(x) := ∫ k α (x, y)W(y)u(y) dVol(y),
x ∈ Ω,
(6.2.13)
Ω
acting on measurable functions u in Ω with the property that ∫ Ω
W(y) u(y) dVol(y) < ∞ dist (x, y)n−α
for a.e. x ∈ Ω.
(6.2.14)
Next, for each β > 0, denote by R(Ω, β) the class of all integral operators R β mapping functions f ∈ L1 (∂Ω) into functions deﬁned in Ω according to R β f(x) := ∫ h β (x, y)f(y) dσ(y),
x ∈ Ω,
(6.2.15)
∂Ω
where the integral kernel h β : Ω × ∂Ω → ℝ is a measurable function satisfying h β (x, y) ≤ C dist (x, y)−(n−β) ,
∀ x ∈ Ω, ∀ y ∈ ∂Ω.
(6.2.16)
Then, if n 1 n , β > , α + β < n(1 + ), r r r it follows that for every operator R β ∈ R(Ω, β) the composition α>
T α ∘ R β is welldeﬁned and belongs to R(Ω, γ) n where γ := α + β − . r
(6.2.17)
(6.2.18)
6.2 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains
 253
As a consequence corresponding to α = 2 and β = 1, there exists some γ > 1 with the property that, given any R1 ∈ R(Ω, 1), for each number N ∈ ℕ the composition ⋅ ⋅⏟⏟⏟⏟⏟⏟⏟⏟ ∘ T⏟2⏟ ∘ R1 is welldeﬁned and belongs to R(Ω, γ). ⏟⏟T⏟⏟⏟⏟⏟⏟⏟⏟ 2 ∘ ⋅⏟⏟⏟
(6.2.19)
N times
Proof. Suppose α, β are as in (6.2.17) and select an arbitrary function f ∈ L1 (∂Ω). Then, at least formally, Fubini’s theorem gives T α (R β f)(x) = ∫ h γ (x, z)f(z) dσ(z),
for x ∈ Ω,
(6.2.20)
x ∈ Ω, z ∈ ∂Ω.
(6.2.21)
∂Ω
where h γ (x, z) := ∫ k α (x, y)W(y)h β (y, z) dVol(y), Ω
To justify the applicability of Fubini’s theorem, let r := r/(r − 1) be the Hölder conjugate exponent of r. Then, having ﬁxed x ∈ Ω and z ∈ ∂Ω, by working in local coordinates we may estimate ∫k α (x, y) W(y) h β (y, z) dVol(y) Ω
1/r
r r ≤ ‖W‖L r (Ω) (∫ k α (x, y) h β (y, z) dVol(y)) Ω
≤ C‖W‖
L r (Ω)
1 1 dy) (∫ (n−α)r y − z(n−β)r x − y n
1/r
ℝ
= C‖W‖L r (Ω) ( ∫ ℝn
= C‖W‖L r (Ω) ( ≤
1 y(n−α)r
1 dy) y − υ(n−β)r
1
υ(2n−α−β)r −n
∫ ℝn
1 y(n−α)r
1/r
1 dy) y − ω(n−β)r
C‖W‖L r (Ω) C‖W‖L r (Ω) = , 2n−α−β−n/r υ x − z2n−α−β−n/r
1/r
(6.2.22)
where we have made the change of variables y → y + x, redenoted υ := z − x, then changed variables again according to y → υy, and redenoted ω := υ/υ ∈ S n−1 , and where we have used the fact that³ since α, β are as in (6.2.17) we have sup ( ∫
ω∈S n−1
ℝn
1 y(n−α)r
1 dy) y − ω(n−β)r
1/r
∈ (0, ∞).
3 in fact, the integral in the right side of (6.2.23) is independent of ω ∈ S n−1
(6.2.23)
254  6 Fatou Theorems and Integral Representations This proves that the integral deﬁning h γ (x, z) in (6.2.21) is absolutely convergent and h γ (x, z) ≤ C dist (x, z)−(n−γ) ,
x ∈ Ω, z ∈ ∂Ω,
(6.2.24)
where γ := α + β − n/r. As a byproduct of (6.2.22), we also have that the composition (T α ∘ R β )f is meaningful and given by (6.2.20). From (6.2.20) and (6.2.24) we ultimately conclude that the claim made in (6.2.18) holds. Finally, the claim in (6.2.19) follows by iterating (6.2.18) (bearing in mind that the class R(Ω, β) is nested with respect to the parameter β). The results presented in Lemma 6.7 are going to be useful in the proof of Theorem 6.6 which we discuss next. Proof of Theorem 6.6. For the claim made in the ﬁrst part of the theorem there is no loss of generality in assuming that Ω does not contain any connected component of M (otherwise we simply discard those). Assume that this is the case and ﬁx a real number λ0 with the property that λ0 > sup ‖Ricx ‖Λ l T x M→Λ l T x M .
(6.2.25)
x∈M
Viewing this as a constant potential, deﬁne the Newtonian volume potential Π l0 associated with ∆HL − λ0 as the operator acting on some given lform υ in Ω according to Π l0 υ(x) := ∫⟨Γ l0 (x, y), υ(y)⟩ dVol(y),
x ∈ M,
(6.2.26)
Ω
where Γ l0 (x, y) is the Schwartz kernel of the operator (∆HL − λ0 )−1 (cf. (3.1.16)). Then for each q ∈ (1, ∞) the operator Π l0 : L q (Ω, Λ l T M) → H 2,q (M, Λ l T M)
(6.2.27)
is welldeﬁned, linear and bounded (cf. [97]). Observe that (2.2.50) implies u ∈ L np/(n−1) (Ω, Λ l T M)
and
‖u‖L np/(n−1) (Ω,Λ l TM) ≤ C‖N u‖L p (∂Ω) ,
(6.2.28)
for some constant C ∈ (0, ∞) independent of u. In concert with (6.2.1) this implies (V − λ0 )u ∈ L p1 (Ω, Λ l T M) where p1 ∈ (1, ∞) 1 n−1 1 = + . and satisﬁes p1 r pn
(6.2.29)
Next, let Ω j ↗ Ω be an approximating sequence as in Proposition 2.35 and, for each j ∈ ℕ, consider the lform
then set
̃j := Π l0 ((V − λ0 )1Ω j u) in M, w
(6.2.30)
̃j Ω in Ω j . w j := w j
(6.2.31)
6.2 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains
 255
Then, thanks to (6.2.29) and (6.2.27), we have ̃j ‖H 2,p1 (M,Λ l TM) ≤ C‖(V − λ0 )u‖L p1 (Ω,Λ l TM) < ∞. sup ‖w
(6.2.32)
j∈ℕ
Moreover, since (2.2.45) and (6.2.1) entail (V − λ0 )1Ω j u ∈ L r (Ω, Λ l T M) for each j ∈ ℕ and since r > n/2, we deduce from (6.2.27) and standard embedding results that 0 ̃j ∈ H 2,r (M, Λ l T M) ⊂ Cloc (M, Λ l T M) w
for each j ∈ ℕ.
(6.2.33)
Hence, w j ∈ C 0 ( Ω j , Λ l T M)
for each j ∈ ℕ.
(6.2.34)
Recall from item (ii) in Proposition 2.35 that each ∂Ω j is an Ahlfors regular set with all constants involved independent of j. Bearing this in mind, known quantitative trace results on Ahlfors regular sets (such as those established by A. Jonsson and H. Wallin in [61, Theorem 1, p. 182]⁴, and H. Triebel in [123, Theorem 18.6, p. 139]; cf. also [10]) used in concert with (6.2.30)–(6.2.34) then imply n.t. w j ∂Ω j exists for each j ∈ ℕ and (6.2.35) n.t. M0 := sup w j ∂Ω j L p1 (∂Ω j ,Λ l TM) < ∞. j∈ℕ
Going further, for each j ∈ ℕ consider u j := uΩ j − w j ∈ C 0 (Ω j , Λ l T M)
(6.2.36)
and notice that (∆HL − λ0 )u j = 0 in Ω j ,
u j ∈ L∞ (Ω j ),
n.t. f j := u j ∂Ω j exists.
(6.2.37)
In particular, if Nj denotes the nontangential maximal operator associated with the domain Ω j , we have ‖Nj u j ‖L∞ (∂Ω j ) < ∞. (6.2.38) In addition, with p∗ := min {p, p1 } ∈ (1, ∞),
(6.2.39)
for each j ∈ ℕ we have n.t. ‖f j ‖L p∗ (∂Ω j ,Λ l TM) ≤ u∂Ω j L p∗ (∂Ω j ,Λ l TM) + w j ∂Ω j L p∗ (∂Ω j ,Λ l TM) n.t. ≤ C((u∂Ω j ) ∘ 𝛶j ) ∘ 𝛶−1 L p (∂Ω j ,Λ l TM) + Cw j ∂Ω j L p1 (∂Ω j ,Λ l TM) j ≤ C(u∂Ω j ) ∘ 𝛶j L p (∂Ω,Λ l TM) + CM0 ≤ C‖N u‖L p (∂Ω,Λ l TM) + CM0 ,
(6.2.40)
4 regarding traces of functions belonging to Sobolev spaces into Besov spaces on Ahlfors regular subsets of the ambient; here we do not require the full force of this result, as we are only interested in ordinary Lebesgue spaces in place of Besov spaces (in a context in which the latter imbed continuously into the former)
256  6 Fatou Theorems and Integral Representations
hence sup ‖f j ‖L p∗ (∂Ω j ,Λ l TM) < ∞.
(6.2.41)
j∈ℕ
Moving on, for each j ∈ ℕ we let K 0l,j denote the principal value double layer on ∂Ω j associated as in (4.1.61) with ∆HL − λ0 . Given that, by design, for every j ∈ ℕ we have λ0 + Ric > 0 in M \ Ω j , from item (3) of Theorem 5.10 it follows that the inverse ( 12 I + K 0l,j ) L p∗ (∂Ω
Λ l T M).
(6.2.42) −1
exists on the
space For each j ∈ ℕ, let us also denote by the version of the j, connection double layer potential operator from Deﬁnition 4.1 associated as in (4.1.6) with the constant potential λ0 in the domain Ω j . Observe now that for each ﬁxed j, the function u j satisﬁes conditions similar to those in the ﬁrst three listed of the Dirichlet problem (1.4.2), relative to the domain Ω j , the constant potential λ0 , and the integrability exponent p∗ . Granted these, (5.3.4) applies and, for each j ∈ ℕ, gives the integral representation formula D0l,j
−1
u j = D0l,j (( 12 I + K 0l,j ) f j ) in Ω j .
(6.2.43)
Since (6.2.41) implies that the sequence {f j ∘ 𝛶j }j∈ℕ is bounded in L p∗ (∂Ω, Λ l T M), by eventually passing to a subsequence, there is no loss of generality in assuming that there exists f ∈ L p∗ (∂Ω, Λ l T M) (6.2.44) with the property that f j ∘ 𝛶j → f weakly in L p∗ (∂Ω, Λ l T M) as j → ∞.
(6.2.45)
Making use of the change of variable formula (2.2.85), while bearing in mind item (viii) of Proposition 2.35, as well as (6.2.30), (6.2.31), (6.2.36), and (6.2.29), for almost every x ∈ Ω we may then write u(x) = lim Π l0 ((V − λ0 )1Ω j u)(x) + lim u j (x) j→∞
=
j→∞
Π l0 ((V
− λ0 )u)(x)
+ lim ∫ ⟨i Sym(∇∗, (ν j ∘ 𝛶j )(y))(∇2 Γ l0 )(x, 𝛶j (y)), j→∞
∂Ω
((( 12 I + K 0l,j )−1 f j ) ∘ 𝛶j )(y)⟩
𝛶j (y)
J j (y) dσ(y).
(6.2.46)
Thanks to (6.1.111), and (2.2.80)–(2.2.84), it follows from (6.2.46) that u = Π l0 ((V − λ0 )u)Ω + D0l f in Ω,
(6.2.47)
where D0l is the connection double layer associated with the original domain Ω and the constant potential λ0 .
6.2 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains
 257
In terms of the notation introduced in Lemma 6.7, we have D0l ∈ R(Ω, 1)
(6.2.48)
and identity (6.2.47) may be recast as u = T2 u + D0l f in Ω,
(6.2.49)
where, for each differential form w ∈ L n/(n−1) (Ω, Λ l T M), we have set T2 w := Π l0 ((V − λ0 )w)Ω in Ω.
(6.2.50)
Let us note here that from (6.2.50), (6.2.27), and standard embedding results we have the implications w ∈ L q (Ω, Λ l T M) implies T2 w ∈ L (Ω, Λ T M) q∗
l
n 2 1 −1 ≤q q, q r n
with
(6.2.51)
and w ∈ L q (Ω, Λ l T M)
with q > (
2 1 −1 − ) n r
(6.2.52)
implies T2 w ∈ C 0 ( Ω, Λ l T M). The idea is now to use the righthand side of (6.2.49) to replace u appearing in the righthand side of (6.2.49), thus producing u = T2 ∘ (T2 u) + T2 ∘ D0l f + D0l f in Ω.
(6.2.53)
In light of (6.2.19) (cf. also (6.2.48)), this translates into u = T2 ∘ (T2 u) + R γ f + D0l f in Ω,
(6.2.54)
for some γ > 1. Inductively, this procedure shows that for each number N ∈ ℕ we have u = ⏟⏟T⏟⏟⏟⏟⏟⏟⏟⏟ ⋅ ⋅⏟⏟⏟⏟⏟⏟⏟⏟ ∘ T⏟2⏟ u + R γ f + D0l f in Ω 2 ∘ ⋅⏟⏟⏟ N times
for some operator R γ ∈ R(Ω, γ) with γ > 1, whose integral kernel h γ (x, y) has the property that for σa.e. y ∈ ∂Ω the function h γ (⋅, y) extends continuously to Ω \ {y},
(6.2.55)
where the last property above makes use of (3.1.17). If N is sufficiently large, it follows from (6.2.28), (6.2.51), and (6.2.52) that ⋅ ⋅⏟⏟⏟⏟⏟⏟⏟⏟ ∘ T⏟2⏟ u ∈ C 0 ( Ω, Λ l T M). ⏟⏟T⏟⏟⏟⏟⏟⏟⏟⏟ 2 ∘ ⋅⏟⏟⏟
(6.2.56)
N times
In turn, from (6.2.55), (6.2.56), Lemma 9.54, and item (6) of Theorem 4.8 we ﬁnally n.t. conclude that the nontangential trace u∂Ω exists at σa.e. point on ∂Ω. This proves
258  6 Fatou Theorems and Integral Representations n.t. (6.2.3). Moreover, it is implicit in the above reasoning that u∂Ω is measurable. With n.t. this in hand, the fact that u∂Ω actually belongs to the space L p (∂Ω, Λ l T M) is a consequence of (2.2.39). Furthermore, estimate (6.2.4) is a direct consequence of what we have proved so far and (2.2.39). Moving on, strengthen the assumptions on V and Ω by also assuming (6.2.5), then ̃ as in (6.2.6). In particular, deﬁne V ̃ + Ric > 0 in M \ Ω. V
(6.2.57)
Next, if u is as in (6.2.2), it follows from what we have just proved that n.t. g := u∂Ω is a welldeﬁned form, belonging to L p (∂Ω, Λ l T M).
(6.2.58)
̃ ∈ L r (M) with r > n, V
̃ ≥ 0, V
Hence, by design, u solves the L p Dirichlet problem u ∈ C 0 (Ω, Λ l T M), (∆HL − V)u = 0 in Ω, n.t. N u ∈ L p (∂Ω), u∂Ω = g ∈ L p (∂Ω, Λ l T M).
(6.2.59)
Having noticed this, from Theorem 1.8 and its proof (see the discussion pertaining to (5.3.4)) we then conclude that u may be represented as in (6.2.7) with f given by (6.2.9). From this, all other conclusions follow, ﬁnishing the proof of Theorem 6.6. In concert with the CalderónZygmund theory from § 9.9, Theorem 6.6 implies the following global regularity result and L2 square function estimate. Corollary 6.8. Let Ω ⊂ M be an εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. Denote by ∇ the LeviCivita connection acting on differential forms on M, and ﬁx a degree l ∈ {0, 1, . . . , n}. Then any differential form satisfying u ∈ C 0 (Ω, Λ l T M),
∆HL u = 0 in Ω,
and
N u ∈ L2 (∂Ω),
(6.2.60)
has the property that u ∈ H 1/2,2 (Ω, Λ l T M).
(6.2.61)
Moreover, under the additional assumption that Ω does not contain any connected component of M, there exists a constant C ∈ (0, ∞) such that the following L2 square function estimate holds: 1/2
2 (∫ (∇u)(x) dist (x, ∂Ω) dVol(x))
≤ C‖N u‖L2 (∂Ω) .
(6.2.62)
Ω
Proof. We begin by noting that, under the assumption that Ω does not contain any connected component of M, the L2 square function estimate in (6.2.62) is a consequence of (6.2.7)–(6.2.10) and (9.9.77). Having established this, the regularity result in (6.2.61) follows with the help of Theorem 9.45, bearing in mind that, as proved in Theorem 1.8, any two solutions of the L2 Dirichlet problem for the HodgeLaplacian in Ω (as formulated in (1.4.2)) differ by a form belonging to the space NΩl deﬁned in (1.4.11).
6.2 A Fatou Theorem for the HodgeLaplacian in Regular SKT Domains
 259
We augment Corollary 6.8 with an L p square function estimate in the range 2 < p < ∞. Before stating it, the reader is reminded that, as usual, ∇ denotes the LeviCivita connection acting on differential forms, and that n denotes the dimension of the ambient manifold M. Corollary 6.9. Fix an exponent p ∈ (2, ∞) along with a degree l ∈ {0, 1, . . . , n}. Assume that Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Denote by σ the surface measure on ∂Ω. Then any differential form satisfying u ∈ C 0 (Ω, Λ l T M),
∆HL u = 0 in Ω,
and N u ∈ L p (∂Ω),
(6.2.63)
has the property that 2 ∫ sup (r1−n ∫ ∇u(x) dist (x, ∂Ω) dVol(x)) ∂Ω
r>0
p/2
dσ(z) < ∞.
(6.2.64)
B r (z)∩Ω
Furthermore, under the additional assumption that Ω does not contain any connected component of M, there exists a constant C ∈ (0, ∞), independent of u, such that the following L p square function estimate holds: 1/2 sup (r1−n ∫ ∇u(x)2 dist (x, ∂Ω) dVol(x)) ≤ C‖N u‖L p (∂Ω) . (6.2.65) p r>0 L z (∂Ω) B (z)∩Ω r
Proof. This is readily seen with the help of Theorem 6.6 and Theorem 1.8. The Fatou type result and double layer representation formula from Theorem 6.6 also has the following consequence, which will be useful in future endeavors. Lemma 6.10. Fix l ∈ {0, 1, . . . , n} along with p ∈ (1, ∞), and let Ω be an εSKT domain for some ε > 0 sufficiently small relative to the given exponent p, as well as the Ahlfors regularity constants and local John constants of Ω. In this context, for some ﬁxed number α ∈ (0, n − 1) consider the fractional integration operator (α)
(IΩ u)(x) := ∫ Ω
u(y) dVol(y), dist (x, y)n−α
x ∈ Ω.
(6.2.66)
N u ∈ L p (∂Ω),
(6.2.67)
Then any differential form u satisfying u ∈ C 0 (Ω, Λ l T M), has the property that
∆HL u = 0 in Ω, (α)
N(IΩ u) ∈ L p α (∂Ω)
where
−1 { (1/p − α/(n − 1)) { { { p α := { any q ∈ (1, ∞) { { { ∞ {
(6.2.68)
if p ∈ (1, (n − 1)/α), if p = (n − 1)/α, if p ∈ ((n − 1)/α, ∞).
(6.2.69)
260  6 Fatou Theorems and Integral Representations
Proof. Given the nature of the conclusion we seek, there is no loss of generality in assuming that Ω does not contain any connected component of M. Recall from Theorem 6.6 (used here with V = 0) that, in such a context, any differential form u satisfying (6.2.67) may be represented as in (6.2.7) for some f ∈ L p (∂Ω, Λ l T M). Then, if σ and ν are, respectively, the surface measure and the outward unit conormal of Ω, for each x ∈ Ω we may estimate dVol(y) (α) (IΩ u)(x) = ∫ ∫ ⟨i Sym (∇∗, ν(z))∇z Γ l (y, z), f(z)⟩ dσ(z) dist (x, y)n−α Ω ∂Ω
≤ C ∫ (∫ ∂Ω
Ω
∇z Γ l (y, z) dVol(y))f(z) dσ(z). dist (x, y)n−α
(6.2.70)
∇z Γ l (y, z) dVol(y), dist (x, y)n−α
(6.2.71)
Let us abbreviate k(x, z) := ∫ Ω
x ∈ Ω, z ∈ ∂Ω.
Working in local coordinates and arguing as in (6.2.22)(6.2.24), we see that k(x, z) ≤ C dist (x, z)−(n−1−α) ,
x ∈ Ω, z ∈ ∂Ω.
(6.2.72)
Now (6.2.68), (6.2.69) follows from this, (6.2.70), (6.2.71), and Lemma 9.54. Recall that, among other things, Theorem 6.6 establishes a double layer integral representation of certain nullsolutions of the Schrödinger operator in a regular SKT domain. We conclude this section by discussion a single layer integral representation formula which is useful in the proof of the transmission problem formulated in Theorem 1.11. Corollary 6.11. Fix l ∈ {0, 1, . . . , n} along with p ∈ (1, ∞). Also, let V be a potential as in (3.1.13). Consider a regular εSKT domain Ω ⊂ M with the property that Ω does not contain any connected component of M,
(6.2.73)
and such that ε > 0 is sufficiently small relative to the given exponent p, the potential V, as well as the Ahlfors regularity constants and local John constants of Ω. In this context, consider the single layer potential Sl associated with the potential V and the domain Ω. Then the operators p
Sl : L−1 (∂Ω, Λ l T M) → {u ∈ C 0 (Ω, Λ l T M) : (∆HL − V)u = 0 in Ω
and N u ∈ L p (∂Ω)}
(6.2.74)
and Sl : L p (∂Ω, Λ l T M) → {u ∈ C 1 (Ω, Λ l T M) : (∆HL − V)u = 0 in Ω
and N u, N(∇u) ∈ L p (∂Ω)} are isomorphisms.
(6.2.75)
6.3 Spaces of Harmonic Fields and Green Type Formulas
 261
Proof. From (4.1.105) and (4.1.104) it follows that the single layer operator induces a welldeﬁned mapping in the context of (6.2.74). To prove that this is surjective, pick an arbitrary form u ∈ C 0 (Ω, Λ l T M) satisfying (∆HL − V)u = 0 in Ω as well as N u ∈ L p (∂Ω). n.t. Theorem 6.6 ensures that f := u∂Ω exists and belongs to L p (∂Ω, Λ l T M). From (5.2.2) we know that the operator p
S l : L−1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)
(6.2.76)
l is an isomorphism. Thus, g := S−1 l f is welldeﬁned and belongs to L −1 (∂Ω, Λ T M). We now observe that Sl g = u in Ω. Indeed, granted (6.2.73), this is a consequence of the uniqueness in the Dirichlet problem discussed in Theorem 1.8. All together, this shows that (6.2.74) is a welldeﬁned and surjective mapping. The fact that this mapping is also injective is a consequence of the invertibility of the operator in (6.2.76). This ﬁnishes the proof of the fact that (6.2.74) is a welldeﬁned isomorphism. A similar claim about (6.2.75) may be established by reasoning along similar lines, this time making use of the Fatou type result from Proposition 5.12 and the invertibility of the operator S l in the context of (5.2.1). p
6.3 Spaces of Harmonic Fields and Green Type Formulas Following K. Kodaira [69] we shall call the forms of class C 1 that are simultaneously annihilated both by d and δ in an open subset Ω ⊆ M harmonic fields. They make up a distinguished subclass of nullsolutions of the HodgeLaplacian in Ω. For our purposes, given an Ahlfors regular domain Ω ⊂ M, along with l ∈ {0, 1, . . . , n} and p ∈ (0, ∞], it will be useful to consider the space H l,p (Ω) := {u ∈ L1loc (Ω, Λ l T M) : N u ∈ L p (∂Ω),
du = 0 and δu = 0 in D (Ω)}.
(6.3.1)
Then, from (2.2.45) and (2.1.115) we see that γ
2,r
H l,p (Ω) ⊂ ⋂ Hloc (Ω, Λ l T M) ⊂ ⋂ Cloc (Ω, Λ l T M). 1 0 sufficiently small relative to p, as well as the Ahlfors regularity constants and local John constants of Ω. Let ν be the outward unit conormal to Ω.
7.1 Preparatory Results 
287
Then any differential form u satisfying u ∈ C 1 (Ω, Λ l T M), ∆HL u = 0 in Ω, n.t. as well as ν ∨ u∂Ω = 0 on ∂Ω and
N u, N(du) ∈ L p (∂Ω),
n.t. ν ∨ (du)∂Ω = 0 on ∂Ω,
(7.1.68)
necessarily belongs to the space H∨l (Ω). Proof. In the case when p ≥ 2(n − 1)/(n − 2) this is a direct consequence of the last claim in Lemma 7.4. The case when 1 < p < 2(n − 1)/(n − 2) can then be reduced to this one via a bootstrap argument, as follows. Suppose u is as in (7.1.68). Then the integral representation (7.1.50) reduces to just −1
u = − VΠ l u + VδSl+1 [(− 12 I + N l+1 ) −1
+ V Sl [(− 12 I + N l )
n.t. (ν ∧ (Π l u)∂Ω )]
n.t. (ν ∧ (δΠ l u)∂Ω )] −1
− VδSl+1 [(− 12 I + N l+1 )
(7.1.69)
[ν ∧ S l ((− 12 I + N l )
−1
n.t. (ν ∧ (δΠ l u)∂Ω ))]]
in Ω, where all layer potentials appearing above are associated with the domain Ω and the operator ∆HL − V for some strictly positive constant potential V (chosen to be suitably small). From this, (7.1.48), Theorem 3.7, and (3.2.8) we deduce that N u ∈ L p∗ (∂Ω),
where p∗ is as in (7.1.41).
(7.1.70)
To proceed, observe that the operator identities dδSl+1 = −V Sl+1 − δSl+1 d ∂ , −1
d ∂ (− 12 I + N l+1 )
= (− 12 I + N l+2 )
−1
(7.1.71)
d ∂ − Vν ∧ S l+1 (− 12 I + N l+1 )
−1
d ∂ (ν ∧ S l ) = −ν ∧ dS l ,
(7.1.72) (7.1.73)
may be proved making use of (3.2.5) and (3.2.40) (with R l−1 = 0 and Q l = 0 in the present case) for the ﬁrst formula, (3.2.44) (with Q l = 0) for the second formula, and (3.2.18) for the third formula. Applying d to both sides of (7.1.69) and relying on (7.1.71)–(7.1.73) leads to the following integral representation formula for du in Ω: du = − VdΠ l u − V 2 Sl+1 [(− 12 I + N l+1 ) −1
− VδSl+2 [(− 12 I + N l+2 )
−1
−1
−1
(7.1.74)
n.t. d ∂ (ν ∧ (Π l u)∂Ω )]
+ V 2 δSl+2 [ν ∧ S l+1 (− 12 I + N l+1 ) + VdSl [(− 12 I + N l )
n.t. (ν ∧ (Π l u)∂Ω )] n.t. (ν ∧ (Π l u)∂Ω )]
n.t. (ν ∧ (δΠ l u)∂Ω )]
+ V 2 Sl+1 [(− 12 I + N l+1 )
−1 −1
+ VδSl+2 [(− 12 I + N l+2 )
−1
[ν ∧ S l ((− 12 I + N l )
n.t. (ν ∧ (δΠ l u)∂Ω ))]] −1
[(ν ∧ dS l ) ((− 12 I + N l )
− V 2 δSl+2 [ν ∧ S l+1 (− 12 I + N l+1 )
−1
n.t. (ν ∧ (δΠ l u)∂Ω ))]] −1
[ν ∧ S l ((− 12 I + N l )
n.t. (ν ∧ (δΠ l u)∂Ω ))]].
288  7 Solvability of Boundary Problems for the HodgeLaplacian
In concert, (7.1.74), (7.1.48), Theorem 3.7, and (3.2.8) then imply N(du) ∈ L p∗ (∂Ω),
where p∗ is as in (7.1.41).
(7.1.75)
Collectively, the memberships in (7.1.70) and (7.1.75) amount to an improvement over the original assumptions in (7.1.68), and after ﬁnitely many iterations matters may be reduced to the case when p ≥ 2(n − 1)/(n − 2), which has already been treated.
7.2 Solvability Results The following solvability result gets us started on the proof of Theorem 1.1. Proposition 7.6. Let Ω be a regular SKT domain in M with outward unit conormal ν, and ﬁx some l ∈ {0, 1, . . . , n} along with p ∈ (1, ∞). In this context, consider the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { N u, N(du) ∈ L p (∂Ω), { { { p { n.t. { ν ∨ u∂Ω = f ∈ Ltan (∂Ω, Λ l−1 T M), { { { { { p n.t. n.t. ⊥ l l { ν ∨ (du)∂Ω = g ∈ Ltan (∂Ω, Λ T M) ∩ {H∨ (Ω)∂Ω } ,
(7.2.1)
where {. . .}⊥ refers to the annihilator of {. . .} (regarded as a subspace of L p (∂Ω, Λ l T M), ∗ 1/p + 1/p = 1) in L p (∂Ω, Λ l T M) = (L p (∂Ω, Λ l T M)) . Then the following statements are true: (1) The boundary value problem (7.2.1) has always a solution. (2) The space solutions for the homogeneous version of (7.2.1) is precisely H∨l (Ω). p,δ (3) If f ∈ Ltan (∂Ω, Λ l−1 T M) then any solution u of (7.2.1) satisﬁes
(4) If actually
N(δu) ∈ L p (∂Ω).
(7.2.2)
p,δ n.t. ⊥ g ∈ Ltan (∂Ω, Λ l T M) ∩ {H∨l (Ω)∂Ω }
(7.2.3)
then any solution u of (7.2.1) satisﬁes N(δdu), N(dδu) ∈ L p (∂Ω).
(5) Any solution u of (7.2.1) with g = 0 has the property that du = 0 in Ω.
(7.2.4)
7.2 Solvability Results  289
(6) Analogous results to those described in (1)–(5) above hold for the Hodge dual of (7.2.1), i.e., for the boundary value problem u ∈ C 0 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { N u, N(δu) ∈ L p (∂Ω), { { { p { n.t. { ν ∧ u∂Ω = f ∈ Lnor (∂Ω, Λ l+1 T M), { { { { { p n.t. n.t. ⊥ l l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ T M) ∩ {H∧ (Ω)∂Ω } .
(7.2.5)
Finally, all results above continue to be true when Ω is an εSKT domain, provided the parameter ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. For starters observe that there is no loss of generality in assuming that Ω does not contain any connected component of M (since such components may be harmlessly discarded both from Ω and M). Under this assumption it is then possible to ﬁnd a potential V as in the statement of Proposition 7.1. Then Proposition 7.1 ensures that p p n.t. ⊥ whenever f ∈ Ltan (∂Ω, Λ l−1 T M) and g ∈ Ltan (∂Ω, Λ l T M) ∩ {H∨l (Ω)∂Ω } there exist p p l−1 l h ∈ Ltan (∂Ω, Λ T M) along with some k ∈ Ltan (∂Ω, Λ T M) such that (cf. (7.1.4)) h f T l ( ) = ( ). k g
(7.2.6)
For the pair h, k just identiﬁed, if we now deﬁne u as in (7.1.15) (cf. also (7.1.16)) it follows from (7.1.18), (7.1.19), and (7.2.6), that u solves (7.2.1). This proves (1). Next, (2) is a direct consequence of Lemma 7.5. To justify the claim in (3), observe that if p,δ p,δ f ∈ Ltan (∂Ω, Λ l−1 T M) then (7.1.10) implies that h ∈ Ltan (∂Ω, Λ l T M). In concert with (7.1.16) this permits us to compute δu = δSl k + δdSl−1 h − δQ l−1 h = δSl k − dδSl−1 h − δQ l−1 h = δSl k − dSl−1 (δ ∂ h) − δQ l−1 h in Ω,
(7.2.7)
from which (7.2.2) follows for this particular solution. Thanks to (2), the same is true for any other solution. This ﬁnishes the proof of (3). As regards (4), when the boundary data is as in (7.2.3) the regularity results in the p,δ last part of the statement Proposition 7.1 imply that k ∈ Ltan (∂Ω, Λ l T M). Granted this, it follows from (7.1.16) and (3.2.40) that δdu = δdSl k − δdQ l−1 h = −dδSl k − δdQ l−1 h = −dSl (δ ∂ k) − R l−1 k − δdQ l−1 h in Ω.
(7.2.8)
From this, (3.2.8), (3.1.56), and Lemma 9.54 we then deduce that the solution u satisﬁes (7.2.4).
290  7 Solvability of Boundary Problems for the HodgeLaplacian Next, if u solves (7.2.1) with g = 0, then Lemma 7.5 applied to the (l + 1)form du gives that du ∈ H∨l+1 (Ω). In particular, δdu = 0 in Ω,
and
N(du) ∈ ⋂ L q (∂Ω).
(7.2.9)
1 0 sufficiently small relative to p, as well as the Ahlfors regularity constants and local John constants of Ω. Denote the surface measure and outward unit conormal to Ω by σ and ν, respectively. Then any differential form u satisfying u ∈ C 1 (Ω, Λ l T M),
∆HL u = 0 in Ω,
N u, N(du) ∈ L p (∂Ω),
(7.2.11)
n.t. has the property that (du)∂Ω exists at σa.e. point on ∂Ω and n.t. n.t. ⊥ ν ∨ (du)∂Ω ∈ {H∨l (Ω)∂Ω } ,
(7.2.12)
where {. . .}⊥ refers to the annihilator of {. . .} (regarded as a subspace of L p (∂Ω, Λ l T M), ∗ 1/p + 1/p = 1) in L p (∂Ω, Λ l T M) = (L p (∂Ω, Λ l T M)) . Proof. Let u be as in (7.2.11). From Theorem 6.6 (used with V = 0) it follows that both n.t. n.t. u∂Ω and (du)∂Ω exist at σa.e. point on ∂Ω and are pth power integrable on ∂Ω. Invoking Proposition 7.6 with p n.t. f := ν ∨ u∂Ω ∈ Ltan (∂Ω, Λ l−1 T M) and p n.t. ⊥ g := 0 ∈ Ltan (∂Ω, Λ l T M) ∩ {H∨l (Ω)∂Ω }
(7.2.13)
then yields a solution to the boundary value problem w ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL w = 0 in Ω, { { { { N w, N(dw) ∈ L p (∂Ω), { { { { n.t. n.t. { ν ∨ w∂Ω = ν ∨ u∂Ω , { { { { { n.t. { ν ∨ (dw)∂Ω = 0.
(7.2.14)
7.2 Solvability Results  291
Then υ := u − w satisﬁes υ ∈ C 1 (Ω, Λ l T M),
∆HL υ = 0 in Ω, as well as n.t. N u, N(du) ∈ L p (∂Ω), and ν ∨ υ∂Ω = 0 on ∂Ω.
(7.2.15)
Moreover, thanks to item (5) in Proposition 7.6 we also have dυ = 0 in Ω.
(7.2.16)
Granted (7.2.15), Lemma 7.4 then implies that N(δυ) ∈ L p (∂Ω)
and
n.t. (δυ)∂Ω exists at σa.e. point on ∂Ω.
(7.2.17)
Pick an arbitrary ω ∈ H∨l (Ω). Keeping in mind (7.2.16), the last condition in (7.2.15), and n.t. that the membership of ω to H∨l (Ω) entails ν ∨ ω∂Ω = 0 at σa.e. point on ∂Ω, formula (6.3.46) implies n.t. n.t. n.t. n.t. ∫ ⟨ν ∨ (du)∂Ω , ω∂Ω ⟩ dσ = ∫ ⟨ν ∨ (dυ)∂Ω , ω∂Ω ⟩ dσ ∂Ω
∂Ω
n.t. n.t. = ∫ ⟨(δυ)∂Ω , ν ∨ ω∂Ω ⟩ dσ = 0.
(7.2.18)
∂Ω
This proves (7.2.12). We are now in a position to present the proof of Theorem 1.1. Proof of Theorem 1.1. For starters, Lemma 7.7 implies that (1.1.6) is a necessary compatibility condition for the solvability of (BVP1)l . Conversely, that (1.1.6) is also sufficient for the solvability of (BVP1)l follows from item (1) in Proposition 7.6. This ﬁnishes the treatment of item (1). Next, the ﬁrst claim in item (2) is a direct consequence of Lemma 7.5 (incidentally, this also shows that the boundary data f, g determine du and δu uniquely). The claim in (1.1.7) then follows from this and item (v) in Proposition 6.12. The remaining claims in item (2) are clear. ̃ the Before going any further, we make the following convention. Denote by M ̃ ̃ (ﬁnite) union of all connected components of M not contained in Ω, and set Ω := Ω ∩ M ̃ (i.e., Ω is obtained from Ω by discarding all connected components of M contained in Ω). We may then ﬁnd a potential V as in the statement of Proposition 7.1 relative ̃ and M ̃ . Associated with this potential introduce boundary layer potentials as in to Ω ̃ and the ambient manifold M ̃ . The convention we make § 3, relative to the domain Ω for the remainder of the proof is that the action of the said layer potentials is extended ̃ (hence, zero in to the entire original manifold M by taking them to be zero in M \ M every connected component of M contained in Ω). Bearing this in mind, at this point we may conclude (as a byproduct of items (1) and (2) already settled) that any solution of (BVP1)l has the form (compare with (7.1.16))
292  7 Solvability of Boundary Problems for the HodgeLaplacian u = Sl k + dSl−1 h − Q l−1 h + ω in Ω, h∈
p Ltan (∂Ω, Λ l−1 T M),
k∈
for some ω ∈ H∨l (Ω) and
p Ltan (∂Ω, Λ l T M)
with T l (h, k) = (f, g),
(7.2.19)
where the operator T l is as in Proposition 7.1. Moreover, as a consequence of the Open Mapping Theorem, there exists a constant C ∈ (0, ∞) independent of f and g with the property that p p ‖h‖Ltan (∂Ω,Λ l−1 TM) + ‖k‖Ltan (∂Ω,Λ l TM) p p ≤ C‖f‖Ltan (∂Ω,Λ l−1 TM) + C‖g‖Ltan (∂Ω,Λ l TM) .
(7.2.20)
Then (1.1.10) follows from (7.2.19), (7.2.20), and (3.2.8). To ﬁnish the proof of the claims in item (3) there remains to observe that the memberships in (1.1.11) are direct consequences of (2.2.50), while the memberships in (1.1.12) in the case p = 2 are implied by Corollary 6.8. p,δ From item (3) of Proposition 7.6 we see that if f ∈ Ltan (∂Ω, Λ l−1 T M) then any solution u of (BVP1)l satisﬁes N(δu) ∈ L p (∂Ω). The membership in (1.1.15) is then a consequence of (2.2.50), while the pointwise existence of the nontangential boundary n.t. trace (δu)∂Ω is seen from Theorem 6.6 (applied to δu). In addition, the membership in (1.1.16) in the case when p = 2 is seen from Corollary 6.8. Conversely, if (BVP1)l has a solution u which also satisﬁes N(δu) ∈ L p (∂Ω) then p,δ Proposition 2.41 shows that we necessarily have f ∈ Ltan (∂Ω, Λ l−1 T M). From the proof p,δ p,δ l−1 of Proposition 7.1 we also know that if f ∈ Ltan (∂Ω, Λ T M) then h ∈ Ltan (∂Ω, Λ l T M) and there exists a constant C ∈ (0, ∞) independent of f and g such that p ‖h‖L p,δ (∂Ω,Λ l−1 TM) + ‖k‖Ltan (∂Ω,Λ l TM) tan
p ≤ C‖f‖L p,δ (∂Ω,Λ l−1 TM) + C‖g‖Ltan (∂Ω,Λ l TM) .
(7.2.21)
tan
Then (1.1.14) is seen from (7.2.19), (7.2.21), and (3.2.8). This concludes the proof of the claims made in item (4). Turning attention to item (5), if a solution u of (BVP1)l has N(δdu) ∈ L p (∂Ω) then also N(dδu) ∈ L p (∂Ω) since dδu = −δdu in Ω. In turn, the last membership p,δ n.t. and identity (2.4.58) necessarily place g = ν ∨ (du)∂Ω in Ltan (∂Ω, Λ l T M). Also, the memberships in (1.1.20) in the case when p = 2 are implied by Corollary 6.8. p,δ In the opposite direction, if u solves (BVP1)l with g ∈ Ltan (∂Ω, Λ l T M) then from item (4) in Proposition 7.6 we conclude that N(δdu) ∈ L p (∂Ω). Finally, the properties in (1.1.19) may be justiﬁed much as those in (1.1.15). Consider next item (6) in the statement of the theorem. In one direction, it is clear n.t. that du = 0 in Ω forces g = ν ∨ (du)∂Ω = 0. In the converse direction, if g = 0 then by item (5) in Proposition 7.6 and the current item (2) (dealt with earlier in the proof) we deduce that du = 0 in Ω, as wanted. Finally, the last claim in (6) is a consequence of what we have proved so far, in concert with Proposition 6.12 and Proposition 6.14. Let us now treat item (7). In one direction, if δdu = 0 in Ω then, by (2.4.58), g p,0 p,0 necessarily belongs to Ltan (∂Ω, Λ l T M). Conversely, assume that g ∈ Ltan (∂Ω, Λ l T M).
7.2 Solvability Results  293
Then the current item (5) implies that N(dδu) belongs to the space L p (∂Ω). With this in hand, and observing that ∆HL (dδu) = dδ(∆HL u) = 0 in Ω, the Fatoutype result from Theorem 6.6 applies (with V = 0) and gives that n.t. (dδu)∂Ω exists at σa.e. point on ∂Ω.
(7.2.22)
Invoking (2.4.58) we may then write n.t. n.t. ν ∨ (δu)∂Ω = −δ ∂ (ν ∨ u∂Ω ) = −δ ∂ f
(7.2.23)
and, keeping in mind (7.2.22), n.t. n.t. n.t. ν ∨ (dδu)∂Ω = −ν ∨ (δdu)∂Ω = δ ∂ (ν ∨ (du)∂Ω ) = δ ∂ g = 0.
(7.2.24)
Together, these formulas show that the (l − 1)form δu solves (BVP1)l−1 for the pair of boundary data (−δ ∂ f, 0). Consequently, item (6) may be invoked in order to conclude that 0 = dδu = −δdu, and the proof of (7) is completed. To deal with the claims in item (8) suppose ﬁrst that δu = 0 in Ω. Then (2.4.58) p,0 p,0 implies f ∈ Ltan (∂Ω, Λ l−1 T M), and also g ∈ Ltan (∂Ω, Λ l T M) by arguing as in (7.2.24). In addition, integrations by parts based on Theorem 2.36 show that the membership n.t. ⊥ of f to the space {H∨l−1 (Ω)∂Ω } is a necessary condition. Conversely, assuming that f, g are as in (1.1.25), we aim at proving that δu = 0 in Ω. First, from (7) we obtain dδu = −δdu = 0 in Ω which forces δu ∈ H∨l−1 (Ω), on account of (2.4.58). Using this, the fact that du is coclosed, and the annihilation condition satisﬁed by f , we may then invoke Theorem 2.36 in order to write n.t. ∫δu2 dVol = ∫ ⟨f, (δu)∂Ω ⟩ dσ = 0. Ω
(7.2.25)
∂Ω
Hence, δu = 0 in Ω as desired. The part in (8) concerning the problem (BVP5)l is justiﬁed by reasoning as before. Finally, the claim in item (9) follows based on what we have already proved for (BVP1)l and the properties of the Hodge∗ isomorphism. The proof of Theorem 1.1 is therefore ﬁnished. In preparation for proving Theorem 1.2, we bring in the following lemma. Lemma 7.8. Let p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} be arbitrary and assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Then, with p ∈ (1, ∞) such that 1/p + 1/p = 1, one has p p,0 n.t. ⊥ Lnor (∂Ω, Λ l T M) ∩ {H l,p (Ω)∂Ω } ⊆ Lnor (∂Ω, Λ l T M)
(7.2.26)
p p,0 n.t. ⊥ Ltan (∂Ω, Λ l T M) ∩ {H l,p (Ω)∂Ω } ⊆ Ltan (∂Ω, Λ l T M).
(7.2.27)
and
294  7 Solvability of Boundary Problems for the HodgeLaplacian p n.t. ⊥ Proof. Let g be an arbitrary form in Lnor (∂Ω, Λ l T M) ∩ {H l,p (Ω)∂Ω } . We aim at prov2 l+1 ing that d ∂ g = 0. To this end, let ϕ ∈ C (M, Λ T M) be arbitrary. The key idea is to invoke Theorem 1.1 in order to guarantee the existence of a form u satisfying
u ∈ C 1 (Ω, Λ l T M), { { { { { { du = 0 in Ω, { { { δu = 0 in Ω, { { { { { { N u ∈ L p (∂Ω), { { { n.t. { ν ∨ u∂Ω = ν ∨ (δϕ)∂Ω ,
(7.2.28)
where, as usual, ν denotes the outward unit conormal to Ω. Notice that the compatn.t. ⊥ ibility condition ν ∨ (δϕ)∂Ω ∈ {H∨l−1 (Ω)∂Ω } , required for the solvability of (7.2.28), is a consequence of the integration by parts formula from Theorem 2.36. With this in hand, we may write n.t. n.t. ∫ ⟨g, δϕ⟩ dσ = ∫ ⟨ν ∨ g, ν ∨ u∂Ω ⟩ dσ = ∫ ⟨g, u∂Ω ⟩ dσ = 0, ∂Ω H l,p (Ω)
∂Ω
(7.2.29)
∂Ω
n.t. ⊥ {H l,p (Ω)∂Ω } .
and g ∈ In view of (2.4.17), this forces d ∂ g = 0, ﬁnsince u ∈ ishing the proof of (7.2.26). Finally, (7.2.27) follows from (7.2.26) and Hodge duality. Next we take up the task of proving Theorem 1.2. Proof of Theorem 1.2. The necessity of the compatibility condition (1.2.3) is a direct consequence of (6.3.47) in Proposition 6.18. Conversely, assuming that the differenp p tial forms f ∈ Ltan (∂Ω, Λ l T M) and g ∈ Lnor (∂Ω, Λ l T M) are such that f − g satisﬁes the compatibility condition (1.2.3) we shall show that (BVP7)l has a solution. The key observation is that u := υ + w solves (BVP7)l for these data whenever υ solves υ ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL υ = 0 in Ω, { { { { N(υ), N(dυ), N(δυ) ∈ L p (∂Ω), { { { { n.t. { ν ∨ υ∂Ω = 0 on ∂Ω, { { { { { n.t. { ν ∨ (dυ)∂Ω = f on ∂Ω,
(7.2.30)
w ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL w = 0 in Ω, { { { { { dw = 0 in Ω, { { { { N w, N(δw) ∈ L p (∂Ω), { { { { { { ν ∧ wn.t. = 0 on ∂Ω, { ∂Ω { { { { n.t. n.t. { ν ∧ (δw)∂Ω = g − ν ∧ (δυ)∂Ω on ∂Ω.
(7.2.31)
and w solves
7.2 Solvability Results  295
p
Note ﬁrst that (1.2.3) combined with the fact that g ∈ Lnor (∂Ω, Λ l T M) yields n.t. n.t. ∫ ⟨f, ω∂Ω ⟩ dσ = ∫ ⟨f − g, ω∂Ω ⟩ dσ = 0, ∂Ω
∀ ω ∈ H∨l (Ω),
(7.2.32)
∂Ω
since, given any ω ∈ H∨l (Ω), item (10) in Lemma 2.2 together with Proposition 6.12 imp n.t. n.t. ply ω∂Ω = ν ∨ (ν ∧ ω∂Ω ) ∈ Ltan (∂Ω, Λ l T M). Thus, item (1) in Theorem 1.1 applies and ensures the existence of υ as in (7.2.30). Second, problem (7.2.31) above corresponds to the dual under the Hodge starisomorphism of (BVP4)l with boundary data 0 and n.t. ∗(g − ν ∧ (δυ)∂Ω ). As such, based on item (8) in Theorem 1.1 and (2.4.49) we conclude that the problem (7.2.31) is solvable if
However, since
p,0 n.t. g − ν ∧ (δυ)∂Ω ∈ Lnor (∂Ω, Λ l T M).
(7.2.33)
p n.t. g − ν ∧ (δυ)∂Ω ∈ Lnor (∂Ω, Λ l T M),
(7.2.34)
and since by (6.3.47) and the compatibility condition (1.2.3) we have n.t. n.t. n.t. g − ν ∧ (δυ)∂Ω = ν ∨ (dυ)∂Ω − ν ∧ (δυ)∂Ω − (f − g) n.t. ⊥ and belongs to {H l,p (Ω)∂Ω } ,
(7.2.35)
the membership in (7.2.33) is provided by Lemma 7.8. This concludes the treatment of item (1) in Theorem 1.2. To deal with the claims in item (2) of Theorem 1.2 assume ﬁrst that u solves (BVP7)l with f = 0. Then Lemma 7.5 applied to the (l + 1)form du leads to the conclusion that du ∈ H∨l+1 (Ω). Granted this and reasoning as in (7.2.9) and (7.2.10) then proves that du = 0 in Ω. In a similar fashion, we have that if u solves (BVP7)l with g = 0 then necessarily δu = 0 in Ω. As a consequence of what we proved up to this point it follows that each solution u of (BVP7)l is of the form u = υ + w + ω, where υ solves (7.2.30), w solves (7.2.31), and ω ∈ H l,p (Ω).
(7.2.36)
Then representation (7.2.36) immediately gives estimate (1.2.4), while the memberships in (1.2.5) in the case p = 2 are implied by Corollary 6.8. This completes the treatment of item (2) in Theorem 1.2. At this stage, all other claims in the statement of Theorem 1.2 follow from (7.2.36), the regularity properties described in items (4)–(7) of Theorem 1.1, and the observation that u solves (BVP7)l for the boundary data (f, g) if and only if ∗u solves (BVP7)n−l for the boundary data (−∗g, −∗f). Let us now discuss the proof of Theorem 1.3. Proof of Theorem 1.3. The necessity of (1.2.17) for the solvability of (BVP11)l is clear from (2.4.58) and (2.4.59). Conversely, suppose we have f = δ ∂ f and g = d ∂ g for some
296  7 Solvability of Boundary Problems for the HodgeLaplacian differential forms f ∈ Ltan (∂Ω, Λ l−1 T M) and g ∈ Lnor (∂Ω, Λ l+1 T M). Then u := υ + w is a solution to (BVP11)l provided υ and w solve p,δ
p,d
υ ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL υ = 0 in Ω, { { dυ = 0 in Ω, { { { { { { N υ, N(δυ) ∈ L p (∂Ω), { { { n.t. { ν ∨ υ∂Ω = −f on ∂Ω,
(7.2.37)
w ∈ C 1 (Ω, Λ l T M), { { { { { { { ∆HL w = 0 in Ω, { { δw = 0 in Ω, { { { { { { N w, N(dw) ∈ L p (∂Ω), { { { n.t. { ν ∧ w∂Ω = −g on ∂Ω,
(7.2.38)
and
respectively. That (7.2.37) and (7.2.38) are solvable is guaranteed by Theorem 1.1 (compare with (BVP2)l and its Hodge dual). The above reasoning shows that the problem (BVP11)l is solvable if and only if the boundary datum (f, g) belongs to the space p,δ
p,d
δ ∂ [Ltan (∂Ω, Λ l−1 T M)] ⊕ d ∂ [Lnor (∂Ω, Λ l+1 T M)].
(7.2.39)
Since from Theorem 8.6 we know that this is a ﬁnite codimensional subspace of (1.2.16) (whose codimension is bounded independently of p), the ﬁrst claim in the statement of Theorem 1.3 also follows. Finally, the claim pertaining to the inﬁnite dimensionality of the space of solutions of the homogeneous version of (BVP11)l is a consequence of Proposition 6.16. Moving on, we present the proof of Theorem 1.4. Proof of Theorem 1.4. The necessity of (1.2.22) is clear from (2.4.59). For sufficiency, p,d assume that g = d ∂ g for some g ∈ Lnor (∂Ω, Λ l+1 T M), and let υ, w satisfy υ ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL υ = 0 in Ω, { { { δυ = 0 in Ω, { { { { p { { { N υ, N(dυ) ∈ L (∂Ω), { { p,d n.t. l+1 { ν ∧ υ∂Ω = −g ∈ Lnor (∂Ω, Λ T M),
(7.2.40)
7.2 Solvability Results  297
and w ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL w = 0 in Ω, { { { dw = 0 in Ω, { { { { p { { { N w ∈ L (∂Ω), { { p n.t. n.t. l−1 { ν ∨ w∂Ω = f − ν ∨ υ∂Ω ∈ Ltan (∂Ω, Λ T M),
(7.2.41)
respectively. Note that the solvability of (7.2.40) is a consequence of item (6) in Theorem 1.1 and Hodge duality (cf. also (2.4.48)), while the solvability of (7.2.41) follows from item (4) in Proposition 7.6. With the help of (2.4.59) one may then check without difficulty that u := υ + w in Ω (7.2.42) is a solution of (BVP12)l . At this point we may conclude that the problem (BVP12)l p,d is solvable if and only if g belongs to the space d ∂ [Lnor (∂Ω, Λ l+1 T M)]. Given that from p,0 Theorem 8.6 this is known to be a ﬁnite codimensional subspace of Lnor (∂Ω, Λ l+2 T M) whose codimension is bounded independently of p, all claims in item (1) in Theorem 1.4 have been justiﬁed. As regards item (2), assume ﬁrst that u is a solution of the homogeneous version of (BVP12)l . From (8.4.3) in Theorem 8.19 we then conclude that N(δu) ∈ L p (∂Ω). Together with (2.4.58), this implies that δu ∈ H∨l−1 (Ω). Then Proposition 6.12 gives
N(δu) ∈ L p (∂Ω)
(7.2.43)
where p ∈ (1, ∞) is such that 1/p + 1/p = 1. Also, from Theorem 6.6 it follows that n.t. n.t. (δu)∂Ω and u∂Ω exist σa.e. on ∂Ω. Granted these (and keeping in mind (2.2.50)), it follows that the integration by parts formula (2.3.1) is applicable for the forms δu and n.t. u. Since d(δu) = 0 in Ω and ν ∨ u∂Ω = 0, this gives n.t. n.t. ∫δu2 dVol = − ∫ ⟨ν ∨ u∂Ω , (δu)∂Ω ⟩ dσ = 0. Ω
(7.2.44)
∂Ω
Ultimately, this shows that any nullsolution u of (BVP12)l satisﬁes δu = 0 in Ω. As regards the dimensionality of the space of nullsolutions for (1.2.21), consider the mapping assigning du to each nullsolution u of (1.2.21). By design, its image is contained in H∧l+1 (Ω) while what we have just proved above implies that its kernel is precisely H∨l (Ω). Then from the First Isomorphism Theorem of Group Theory and item (iv) in Proposition 6.12 we conclude that the dimension of the space of nullsolutions for (1.2.21) is at most b l (Ω) + b n−l−1 (Ω). All other claims in item (2) follow from this. Turning to item (3) in Theorem 1.4, observe that the equivalence in (1.2.24) is a direct consequence of (8.4.3) in Theorem 8.19. To justify the equivalence in (1.2.25),
298  7 Solvability of Boundary Problems for the HodgeLaplacian p,0 n.t. ⊥ suppose u solves the problem (1.2.26) when f ∈ Ltan (∂Ω, Λ l−1 T M) ∩ {H∨l−1 (Ω)∂Ω } . Then (2.4.58) permits us to write n.t. n.t. (7.2.45) ν ∨ (δu)∂Ω = −δ ∂ (ν ∨ u∂Ω ) = −δ ∂ f = 0,
and since δ(δu) = 0, d(δu) = 0, we conclude that δu ∈ H∨l−1 (Ω). As such, Proposition 6.12 gives N(δu) ∈ ⋂ L q (∂Ω). (7.2.46) 1 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Denote by ν and σ the outward unit conormal and surface measure on ∂Ω. Then, with p ∈ (1, ∞) such that 1/p + 1/p = 1, one has ∫ ⟨δ ∂ f, ν ∨ g⟩ dσ = − ∫ ⟨ν ∧ f, d ∂ g⟩ dσ, ∂Ω
(8.1.1)
∂Ω p ,d
p,δ
for every f ∈ Ltan (∂Ω, Λ l T M) and g ∈ Lnor (∂Ω, Λ l T M). Proof. Associate boundary layer operators with Ω as in § 3, taking the potential V to be a strictly positive constant. By relying on the last part of Theorem 3.7, it is therefore meaningful to consider f := (− 21 I + M l )−1 f ∈ Ltan (∂Ω, Λ l T M), p,δ
p ,d
g := ( 12 I + N l )−1 g ∈ Lnor (∂Ω, Λ l T M).
(8.1.2)
If we now set u := dSl f and υ := δSl g in Ω, then du = 0,
N u, N(δu) ∈ L p (∂Ω),
δυ = 0,
N υ, N(dυ) ∈ L p (∂Ω),
n.t. n.t. n.t. n.t. there exist u∂Ω , (δu)∂Ω , υ∂Ω , (dυ)∂Ω , n.t. n.t. and f = ν ∨ u∂Ω , g = ν ∧ υ∂Ω .
(8.1.3)
Moreover, u, δu ∈ L np/(n−1) (Ω), and υ, dυ ∈ L np /(n−1) (Ω) thanks to (2.2.50), while n.t. n.t. Proposition 2.41 gives δ ∂ f = −ν ∨ (δu)∂Ω and d ∂ g = −ν ∧ (dυ)∂Ω . With these in hand and using integration by parts as in Theorem 2.36 (while also relying on Lemma 2.2 and Lemma 2.8) we may then write n.t. ∫ ⟨δ ∂ f, ν ∨ g⟩ dσ = ∫ ⟨−ν ∨ (δu)∂Ω , ν ∨ g⟩ dσ ∂Ω
∂Ω
n.t. n.t. = − ∫ ⟨(δu)∂Ω , ν ∧ (ν ∨ (ν ∧ υ∂Ω ))⟩ dσ ∂Ω
n.t. n.t. = − ∫ ⟨(δu)∂Ω , ν ∧ υ∂Ω ⟩ dσ = − ∫⟨δu, dυ⟩ dVol ∂Ω
Ω
n.t. n.t. = ∫ ⟨ν ∨ u∂Ω , (dυ)∂Ω ⟩ dσ ∂Ω
n.t. n.t. = − ∫ ⟨ν ∧ (ν ∨ u)∂Ω , −ν ∧ (dυ)∂Ω ⟩ dσ ∂Ω
= − ∫ ⟨ν ∧ f, d ∂ g⟩ dσ, ∂Ω
and the desired conclusion follows.
(8.1.4)
8.1 de Rham Cohomology on Regular SKT Surfaces  317
To continue, we present several useful consequences of Proposition 8.1 in the next four corollaries. The ﬁrst such corollary is a density result, complementing Lemma 2.42. Corollary 8.2. Suppose p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} have been ﬁxed and assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Denote by ν the outward unit conormal to Ω. Then p,δ {ν ∨ ψ∂Ω : ψ ∈ C 1 (M, Λ l+1 T M)} is dense in Ltan (∂Ω, Λ l T M),
(8.1.5)
p,d {ν ∧ ψ∂Ω : ψ ∈ C 1 (M, Λ l−1 T M)} is dense in Lnor (∂Ω, Λ l T M).
(8.1.6)
and
Proof. That
p,δ {ν ∨ ψ∂Ω : ψ ∈ C 1 (M, Λ l+1 T M)} ⊂ Ltan (∂Ω, Λ l T M) p,δ
(8.1.7)
is clear from Proposition 2.41. To justify (8.1.5), assume Λ ∈ (Ltan (∂Ω, Λ l T M)) functional with the property that Λ(ν ∨ ψ∂Ω ) = 0,
∀ ψ ∈ C 1 (M, Λ l+1 T M).
∗
is a
(8.1.8)
From (2.4.25) we know that, with p ∈ (1, ∞) such that 1/p + 1/p = 1, there exist two p p differential forms, g ∈ Ltan (∂Ω, Λ l T M) and h ∈ Ltan (∂Ω, Λ l−1 T M), such that p,δ
Λ(f) = ∫ ⟨f, g⟩ dσ + ∫ ⟨δ ∂ f, h⟩ dσ, ∂Ω
∀ f ∈ Ltan (∂Ω, Λ l T M),
(8.1.9)
∂Ω
where σ denotes the surface measure on ∂Ω. Collectively, (8.1.8), (8.1.9), (2.4.58), and Lemma 2.2 then imply ∫ ⟨ψ, ν ∧ g⟩ dσ = ∫ ⟨δψ, ν ∧ h⟩ dσ, ∂Ω
∀ ψ ∈ C 1 (M, Λ l+1 T M)
(8.1.10)
d ∂ (ν ∧ h) = ν ∧ g.
(8.1.11)
∂Ω
which, in light of (2.4.35) and (2.4.36), forces p,d
ν ∧ h ∈ Lnor (∂Ω, Λ l T M)
and
With this in hand, we may then invoke (8.1.1) from Proposition 8.1 in order to write that Λ(f) = ∫ ⟨f, g⟩ dσ + ∫ ⟨δ ∂ f, h⟩ dσ ∂Ω
∂Ω
= ∫ ⟨ν ∨ (ν ∧ f), g⟩ dσ + ∫ ⟨ν ∨ (ν ∧ δ ∂ f), h⟩ dσ ∂Ω
∂Ω
= ∫ ⟨ν ∧ f, ν ∧ g⟩ dσ + ∫ ⟨ν ∧ δ ∂ f, ν ∧ h⟩ dσ ∂Ω
∂Ω
318  8 Additional Results and Applications
= ∫ ⟨ν ∧ f, d ∂ (ν ∧ h)⟩ dσ + ∫ ⟨ν ∧ δ ∂ f, ν ∧ h⟩ dσ ∂Ω
= 0,
∀f ∈
∂Ω p,δ l Ltan (∂Ω, Λ T M).
(8.1.12)
This shows that the functional Λ actually vanishes identically on the entire Banach p,δ space Ltan (∂Ω, Λ l T M). By a wellknown consequence to the HahnBanach theorem this then proves the density result announced in (8.1.5). Finally, (8.1.6) follows from (8.1.5) and Hodge duality (cf. (2.4.48)). The second corollary alluded to above adds more ﬂexibility in computing δ ∂ f for a p,δ given differential form f ∈ Ltan (∂Ω, Λ l T M) than in the original deﬁnition (2.4.17). Corollary 8.3. Let p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} be arbitrary and assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Denote by σ the surface measure on ∂Ω. Also, suppose u ∈ C 1 (Ω, Λ l−1 T M) is such that
N u, N(du) ∈ L p (∂Ω), where 1/p + 1/p = 1, and
n.t. n.t. there exist u∂Ω , (du)∂Ω at σa.e. point on ∂Ω.
(8.1.13)
p,δ
Then for every f ∈ Ltan (∂Ω, Λ l T M) one has n.t. n.t. ∫ ⟨δ ∂ f, u∂Ω ⟩ dσ = ∫ ⟨f, (du)∂Ω ⟩ dσ. ∂Ω
(8.1.14)
∂Ω
As a consequence, p,δ p,0 n.t. ⊥ δ ∂ [Ltan (∂Ω, Λ l+1 T M)] ⊆ Ltan (∂Ω, Λ l T M) ∩ {H l,p (Ω)∂Ω } p,0 n.t. ⊥ ⊆ Ltan (∂Ω, Λ l T M) ∩ {H∨l (Ω)∂Ω } .
(8.1.15)
Proof. Let ν stand for the outward unit conormal to Ω. Granted the current assumptions, Proposition 2.41 enures that p ,d n.t. g := ν ∧ u∂Ω ∈ Lnor (∂Ω, Λ l T M)
and
n.t. d ∂ g = −ν ∧ (du)∂Ω .
(8.1.16)
p,δ
Pick an arbitrary f ∈ Ltan (∂Ω, Λ l T M). Bearing (8.1.16) in mind and writing n.t. n.t. ∫ ⟨(du)∂Ω , f ⟩ dσ = ∫ ⟨ν ∧ (du)∂Ω , ν ∧ f ⟩ dσ ∂Ω
∂Ω
= − ∫ ⟨ν ∧ f, d ∂ g⟩ dσ,
(8.1.17)
∂Ω
formula (8.1.14) then follows by invoking Proposition 8.1 and Lemma 2.2. Finally, the ﬁrst inclusion in (8.1.15) is implied by (2.4.30) and (8.1.14), while the second inclusion in (8.1.15) is obvious from (6.3.5).
8.1 de Rham Cohomology on Regular SKT Surfaces  319
The third corollary in the list of consequences of Proposition 8.1 advertised above deals with the following wellposedness result. Corollary 8.4. Let p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} be arbitrary and assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. In this context, ﬁx a basis {ω j }1≤j≤b l (Ω) of the vector space H∨l (Ω) . Then for each differential form p,δ
f ∈ δ ∂ [Ltan (∂Ω, Λ l T M)]
(8.1.18)
the boundary value problem u ∈ C 1 (Ω, Λ l T M), { { { { { { δu = 0 in Ω, { { { { { { du = 0 in Ω, { { { N u ∈ L p (∂Ω), { { { { n.t. { ν ∨ u∂Ω = f on ∂Ω, { { { { { ∫Ω ⟨u, ω j ⟩ dVol = 0 for j ∈ {1, . . . , b l (Ω)},
(8.1.19)
has a unique solution. Proof. This is readily seen with the help of the ﬁrst inclusion in (8.1.15) and the last part of item (8) in Theorem 1.1. In the fourth consequence of Proposition 8.1 mentioned earlier we consider the realization of δ ∂ as an unbounded operator and identify its functional analytic adjoint. Corollary 8.5. Let p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} be arbitrary and assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Denote by ν the outward unit conormal to Ω. and assume p ∈ (1, ∞) is such that 1/p + 1/p = 1. In this setting, consider δ ∂ (originally introduced in § 2.4) as an unbounded operator p
p
δ ∂ : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l−1 T M) acting according to p,δ
f → δ ∂ f for every form f ∈ Dom (δ ∂ ) := Ltan (∂Ω, Λ l T M).
(8.1.20)
Then δ ∂ is closed and densely deﬁned and δ∗∂ , its adjoint in the sense of unbounded p
operators between Banach spaces, is the unbounded operator from Ltan (∂Ω, Λ l−1 T M) p
into Ltan (∂Ω, Λ l T M) with domain p ,d
Dom (δ∗∂ ) = ν ∨ Lnor (∂Ω, Λ l T M)
(8.1.21)
and action δ∗∂ (ν ∨ g) = −ν ∨ d ∂ g,
p ,d
for each g ∈ Lnor (∂Ω, Λ l T M).
(8.1.22)
320  8 Additional Results and Applications
In particular, the nullspace and range of (8.1.21) and (8.1.22) are p ,0
Ker δ∗∂ = ν ∨ Lnor (∂Ω, Λ l T M)
and
(8.1.23)
Im δ∗∂ = ν ∨ d ∂ [Lnor (∂Ω, Λ l T M)]. p ,d
Finally, similar results are valid for the realization of d ∂ (originally introduced in p p § 2.4) as an unbounded operator from Lnor (∂Ω, Λ l T M) into Lnor (∂Ω, Λ l+1 T M). Proof. That δ ∂ in (8.1.20) is closed and densely deﬁned follows from (2.4.17) and Lemma 2.42, respectively. Then the identiﬁcations in (8.1.21) and (8.1.22) become consequences of Proposition 8.1. Finally, (8.1.23) is clear from (8.1.22) and (2.4.27). To set the stage for our main result in this section observe from (2.4.38) that the boundary differential operator d ∂ induces the complex d∂
p,d
p,d
⋅ ⋅ ⋅ → Lnor (∂Ω, Λ l−1 T M) → Lnor (∂Ω, Λ l T M) d∂
p,d
→ Lnor (∂Ω, Λ l+1 T M) → ⋅ ⋅ ⋅
(8.1.24)
while (2.4.26) implies that the boundary differential operator δ ∂ induces the complex δ∂
p,δ
p,δ
⋅ ⋅ ⋅ → Ltan (∂Ω, Λ l+1 T M) → Ltan (∂Ω, Λ l T M) δ∂
p,δ
→ Ltan (∂Ω, Λ l−1 T M) → ⋅ ⋅ ⋅
(8.1.25)
The goal is to show that the cohomology groups associated with these complexes are ﬁnite dimensional. This is done in Theorem 8.6 below. For now, we remark that if the Betti numbers b1 (M) and b2 (M) of the manifold M (cf. (3.1.9)) vanish, and if Ω ⊂ M is an arbitrary UR domain, then by combining (2.4.43), (2.4.57), and (2.4.32) we obtain p,0
dim (
Lnor (∂Ω, Λ n−1 T M) p,d
δ ∂ [Lnor (∂Ω, Λ n T M)]
p,0
) = dim (
Lnor (∂Ω, Λ1 T M) p,d
d ∂ [Lnor (∂Ω, Λ0 T M)]
= b0 (∂Ω),
) (8.1.26)
where b0 (∂Ω) denotes the 0th Betti number ∂Ω (i.e., the number of connected components of ∂Ω). Theorem 8.6. Let Ω ⊂ M be a regular SKT domain and ﬁx an arbitrary p ∈ (1, ∞). Then, for every l ∈ {0, 1, . . . , n}, the cohomology group p,d
{f ∈ Lnor (∂Ω, Λ l T M) : d ∂ f = 0} p,d
{d ∂ g : g ∈ Lnor (∂Ω, Λ l−1 T M)}
p,0
=
Lnor (∂Ω, Λ l T M) p,d
d ∂ [Lnor (∂Ω, Λ l−1 T M)]
(8.1.27)
is ﬁnite dimensional, with a bound on the dimension independent of p. As a consequence, the operator p,d
p,0
d ∂ : Lnor (∂Ω, Λ l−1 T M) → Lnor (∂Ω, Λ l T M)
(8.1.28)
8.1 de Rham Cohomology on Regular SKT Surfaces  321
has closed range, and the following operator is Fredholm and injective: p,d
d∂ :
Lnor (∂Ω, Λ l−1 T M) p,0 Lnor (∂Ω, Λ l−1 T M)
p,0
→ Lnor (∂Ω, Λ l T M).
(8.1.29)
Likewise, for every l ∈ {0, 1, . . . , n}, the cohomology group p,δ
{f ∈ Ltan (∂Ω, Λ l T M) : δ ∂ f = 0} p,δ
{δ ∂ g : g ∈ Ltan (∂Ω, Λ l+1 T M)}
p,0
=
Ltan (∂Ω, Λ l T M) p,δ
δ ∂ [Ltan (∂Ω, Λ l+1 T M)]
(8.1.30)
is ﬁnite dimensional, with a bound on the dimension independent of p. As a corollary, the operator p,δ p,0 δ ∂ : Ltan (∂Ω, Λ l+1 T M) → Ltan (∂Ω, Λ l T M) (8.1.31) has closed range, and the following operator is Fredholm and injective: p,δ
δ∂ :
Ltan (∂Ω, Λ l+1 T M) p,0
Ltan (∂Ω, Λ l+1 T M)
p,0
→ Ltan (∂Ω, Λ l T M).
(8.1.32)
In fact, all the above results remain valid when the underlying set Ω is only assumed to be an εSKT domain, provided ε > 0 is sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Proof. Given the nature of the conclusion we seek, there is no loss of generality in assuming that Ω does not contain any connected component of M. Granted this, it is then possible to select a function V as in (3.1.13) which, in addition, satisﬁes supp V ∩ Ω = ⌀.
(8.1.33)
Denote by ν and σ the outward unit conormal and surface measure of Ω, and associate layer potentials as in (3.2) with the potential V as above. p,0 To proceed, ﬁx some l ∈ {0, 1, . . . , n}, pick an arbitrary f ∈ Ltan (∂Ω, Λ l T M) and consider the (l + 1)form u := dSl f ∈ C 1 (Ω, Λ l+1 T M). Then, since (8.1.33) ensures that ∆HL Sl f = 0 in Ω, and since δ ∂ f = 0, formula (3.2.40) permits us to express δu = δdSl f = −dδSl f = −dSl−1 (δ ∂ f) − dR l−1 f = −dR l−1 f in Ω.
(8.1.34)
In particular, N(δu) ∈ L p (∂Ω). In addition to this, the differential form u also satisﬁes ∆HL u = 0 in Ω, du = 0 in Ω, N u ∈ L p (∂Ω), as well as n.t. ν ∨ u∂Ω = (− 12 I + M l )f
and
n.t. ν ∧ u∂Ω = ν ∧ dS l f,
(8.1.35)
by (3.2.25) and (3.2.16). Collectively, these permit us to write integral representation formula (6.3.36) for the current (l + 1)form u which, given the available information, becomes u = − Sl+1 (ν ∧ dR l−1 f ) + δSl+2 (ν ∧ dS l f ) − dSl ((− 21 I + M l )f ) − R l+1 (ν ∧ dS l f ) + Q l ((− 12 I + M l )f ) in Ω.
(8.1.36)
322  8 Additional Results and Applications
If in both sides of (8.1.36) we now go nontangentially to the boundary and then apply ν ∨ ⋅ to the resulting identity, the jumpformulas from § 3.2 yield (− 21 I + M l )f = − ν ∨ S l+1 (ν ∧ dR l−1 f ) + ν ∨ δS l+2 (ν ∧ dS l f) − (− 21 I + M l )(− 21 I + M l )f − ν ∨ R l+1 (ν ∧ dS l f ) + ν ∨ Q l ((− 12 I + M l )f ).
(8.1.37)
Hence, after some algebra, ν ∨ δS l+2 (ν ∧ dS l f) = − 14 f + M 2l f + ν ∨ S l+1 (ν ∧ dR l−1 f ) + ν ∨ R l+1 (ν ∧ dS l f ) − ν ∨ Q l ((− 21 I + M l )f ).
(8.1.38)
Together with (3.2.17), this ultimately proves that p,0
− 14 I + C l = −δ ∂ (ν ∨ S l+2 ∘ (ν ∧ dS l )) on Ltan (∂Ω, Λ l T M),
(8.1.39)
where we have abbreviated C l := M 2l + ν ∨ S l+1 ∘ (ν ∧ dR l−1 ) + ν ∨ R l+1 ∘ (ν ∧ dS l ) − ν ∨ Q l ∘ (− 12 I + M l ).
(8.1.40)
Then from (8.1.40), (3.1.56), and Theorem 3.6 we deduce that p
p
C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M) is a welldeﬁned, linear, and compact operator.
(8.1.41)
In particular, (8.1.41), Lemma 2.42, and Lemma 3.8 imply that p
p
dim Ker (− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)) is a ﬁnite integer, independent of p.
(8.1.42)
p,0
To proceed, we note that Ltan (∂Ω, Λ l T M) embeds continuously into L p (∂Ω, Λ l T M), the operator ν ∧ dS l maps the latter space boundedly into L p (∂Ω, Λ l+2 T M) which, as p,δ indicated in (3.2.17), is mapped by ν ∨ S l+2 boundedly into Ltan (∂Ω, Λ l+1 T M). Altogether, this shows that p,0
p,δ
ν ∨ S l+2 ∘ (ν ∧ dS l ) : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l+1 T M)
(8.1.43)
is a welldeﬁned, linear and bounded operator. Since from (2.4.29) we know that δ ∂ p,δ p,0 further maps Ltan (∂Ω, Λ l+1 T M) continuously into Ltan (∂Ω, Λ l T M), we may conclude that p,0
p,0
δ ∂ (ν ∨ S l+2 ∘ (ν ∧ dS l )) : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M) is a welldeﬁned, linear, and bounded operator.
(8.1.44)
8.1 de Rham Cohomology on Regular SKT Surfaces  323
From this we then deduce that C l = 14 I − δ ∂ (ν ∨ S l+2 ∘ (ν ∧ dS l )) is a bounded mapping p,0 p,0 from Ltan (∂Ω, Λ l T M) into itself. Given that, as noted in (2.4.28), Ltan (∂Ω, Λ l T M) is a p closed subspace of Ltan (∂Ω, Λ l T M), we may conclude from (8.1.41) that p,0
p,0
C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)
(8.1.45)
is a welldeﬁned, linear, and compact operator. As such, the operator p,0
p,0
− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)
(8.1.46)
is Fredholm with index zero. Also, as seen from (8.1.39) and (8.1.43), its image satisﬁes p,0
p,0
Im (− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)) p,δ
⊆ δ ∂ [Ltan (∂Ω, Λ l+1 T M)].
(8.1.47)
Collectively, (8.1.46), (8.1.47), (8.1.42), and the obvious inclusion p,0
p,0
Ker (− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)) p
p
⊆ Ker(− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)),
(8.1.48)
then force p,0
dim (
Ltan (∂Ω, Λ l T M) p,δ
δ ∂ [Ltan (∂Ω, Λ l+1 T M)]
) p,0
p,0
≤ dim CoKer (− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)) p,0
p,0
= dim Ker (− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)) p
p
≤ dim Ker (− 14 I + C l : Ltan (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l T M)) = a ﬁnite integer, independent of p.
(8.1.49)
This proves the ﬁnite dimensionality of the cohomology group in (8.1.30), with a bound independent of p. Having established this, it follows that the linear and bounded operator (8.1.31) has a ﬁnite dimensional cokernel which, in turn, is known to imply that the operator in question has closed range. Of course, what we have just proved shows that the operator (8.1.29) is Fredholm and injective. Finally, the ﬁnite dimensionality of the cohomology group (8.1.27), once again with a bound independent of p, as well as the claims made about the operators (8.1.31) and (8.1.32) are immediate consequences of what we have established in the ﬁrst part of the proof and Hodge duality (cf. (2.4.47)–(2.4.51)). Remark 8.7. In relation to Theorem 8.6 we wish to mention that in the class of Lipschitz domains² a more precise result has been obtained in [86]³ where it has been 2 which is neither contained in, nor contains, the class of regular SKT domains treated here 3 where full advantage is taken of the fact that ∂Ω has an intrinsic Lipschitz manifold structure
324  8 Additional Results and Applications
shown that
if Ω ⊂ M is a Lipschitz domain then p,0
Lnor (∂Ω, Λ l+1 T M) p,d d ∂ [Lnor (∂Ω, Λ l T M)]
l ≅ Hsing (∂Ω; ℝ)
(8.1.50)
for all l ∈ {0, 1, . . . , n} and p ∈ (1, ∞). l (∂Ω; ℝ) is the lth singular homology group of ∂Ω over the reals. In parAbove, Hsing l (∂Ω; ℝ) < ∞, is the lth Betti number of ∂Ω. From this ticular, b l (∂Ω) := dim Hsing perspective, it would be interesting to know if, in the case when Ω ⊂ M is a regular SKT domain, the dimension of the cohomology group (8.1.27) is independent of p.
There are several notable consequences of Theorem 8.6 worth singling out here. First, we note that Theorem 8.6, (2.4.43), and (2.4.57) imply that if the Betti numbers b1 (M) and b2 (M) of M (cf. (3.1.9)) vanish then any εSKT domain Ω ⊂ M, for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω, has the property that ∂Ω has ﬁnitely many connected components⁴. The second consequence of Theorem 8.6 we wish to present is described in Theorem 8.8 below. To set the stage, we start by remarking that identity (8.1.1) suggests the possibility of extending the action of the operator δ ∂ , originally mapping from its p,δ p,δ p,0 “natural” domain Ltan (∂Ω, Λ l T M) into δ ∂ Ltan (∂Ω, Λ l T M) ⊆ Ltan (∂Ω, Λ l−1 T M), to a larger space. Speciﬁcally, we shall consider the weak extension p ,d
p
δ ∂ : Ltan (∂Ω, Λ l T M) → (
ν ∨ Lnor (∂Ω, Λ l T M) ν∨
p ,0 Lnor (∂Ω, Λ l T M)
∗
) ,
1 1 = 1, + p p
(8.1.51)
deﬁned by ⟨⟨δ ∂ f, [ν ∨ g]⟩⟩ := − ∫ ⟨ν ∧ f, d ∂ g⟩ dσ
(8.1.52)
∂Ω p Ltan (∂Ω,
Λ l T M)
p ,d Lnor (∂Ω,
and g ∈ Λ l T M). Here ⟨⟨⋅, ⋅⟩⟩ stands for the natfor any f ∈ ural duality pairing, whereas [ ⋅ ] denotes the projection operator onto the quotient space under discussion. Also, the star superscript indicates the dual space. It is not too difficult to check that, under an appropriate embedding p,δ
δ ∂ [Ltan (∂Ω, Λ l T M)] → (
p ,d
∗
p ,0
) ,
ν ∨ Lnor (∂Ω, Λ l T M) ν ∨ Lnor (∂Ω, Λ l T M)
(8.1.53)
and, as such, the operator in (8.1.52) is a genuine extension of the “old” δ ∂ (introduced in (2.4.17)). In the sequel, we shall frequently use this fact without any special mention. Note that this also justiﬁes our keeping the same notation for both operators. Our main result pertaining to the extension (8.1.51), (8.1.52) of δ ∂ reads as follows.
4 in this regard, it would be interesting to give a direct proof of this topological result
8.1 de Rham Cohomology on Regular SKT Surfaces  325
Theorem 8.8. Let p ∈ (1, ∞) and l ∈ {0, 1, . . . , n} be arbitrary and assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Also, ﬁx some degree l ∈ {0, 1, . . . , n} and assume p ∈ (1, ∞) is such that 1/p + 1/p = 1. Then the operator Ltan (∂Ω, Λ l T M) p,0
Ltan (∂Ω, Λ l T M)
∗
p ,d
p
δ∂ :
→(
ν ∨ Lnor (∂Ω, Λ l T M) p ,0
ν ∨ Lnor (∂Ω, Λ l T M)
)
(8.1.54)
is an isomorphism. Moreover, a similar result is valid for d ∂ . Proof. By the deﬁnition of δ ∂ in (8.1.51) and (8.1.52) and the discussion above, the operator in (8.1.54) is onetoone. The crux of the matter is establishing that this operator is also onto. To this end, we ﬁrst note that its range does not change if the original domain is replaced by p Ltan (∂Ω, Λ l T M) . (8.1.55) p,δ δ ∂ (Ltan (∂Ω, Λ l+1 T M)) Assuming that this is the case, we claim that the following factorization of δ ∂ holds: p
Ltan (∂Ω, Λ l T M) p,δ δ ∂ (Ltan (∂Ω, Λ l+1 T M))
Φ
p ,0
∗
→ (Lnor (∂Ω, Λ l T M)) (d ∂ (ν∧⋅))∗
−−−→ (
p ,d
∗
p ,0
) ,
ν ∨ Lnor (∂Ω, Λ l T M) ν ∨ Lnor (∂Ω, Λ l T M)
(8.1.56)
where Φ is the mapping deﬁned by ⟨⟨Φ([f]), g⟩⟩ := − ∫ ⟨ν ∧ f, g⟩ dσ,
(8.1.57)
∂Ω
for any p
[f] ∈
Ltan (∂Ω, Λ l T M) p,δ δ ∂ (Ltan (∂Ω, Λ l+1 T M))
and
p ,0
g ∈ Lnor (∂Ω, Λ l T M).
(8.1.58)
Note that the mapping Φ is welldeﬁned since, by (8.1.1), the righthand side of (8.1.57) always vanishes when one pairs elements of the form f = δ ∂ h, for arbitrary p,δ p ,0 h ∈ Ltan (∂Ω, Λ l+1 T M), with g ∈ Lnor (∂Ω, Λ l T M). The linearity and continuity of this mapping are immediate. Next, we claim that the mapping Φ is also surjective. To justify this claim, ﬁx p ,0 ∗ some arbitrary functional ℓ ∈ (Lnor (∂Ω, Λ l T M)) . Then by invoking HahnBanach’s extension theorem and Riesz’s representation theorem we may ﬁnd some form f in p p ,0 Lnor (∂Ω, Λ l+1 T M) which satisﬁes ℓ(h) = ∫∂Ω ⟨f, h⟩ dσ for any h ∈ Lnor (∂Ω, Λ l+1 T M). It is then clear that Φ sends [−ν ∨ f] into ℓ, hence Φ is onto.
326  8 Additional Results and Applications
We now examine the second horizontal arrow in sequence (8.1.56). From (8.1.29) in Theorem 8.6 we know that the operator p,d
d∂ :
Lnor (∂Ω, Λ l T M) p,0 Lnor (∂Ω, Λ l T M)
p,0
→ Lnor (∂Ω, Λ l+1 T M)
(8.1.59)
is onetoone and Fredholm. Since the operator p ,d
ν∧⋅:
ν ∨ Lnor (∂Ω, Λ l T M) p ,0
ν ∨ Lnor (∂Ω, Λ l T M)
p ,d
→
Lnor (∂Ω, Λ l T M) p ,0
Lnor (∂Ω, Λ l T M)
(8.1.60)
is obviously an isomorphism, we deduce that (d ∂ (ν ∧ ⋅))∗ in (8.1.56) is onto. In light of all these results, in order to show that δ ∂ in (8.1.54) is onto, we only need to prove that we have the factorization of this operator claimed in (8.1.56). However, p p ,d for f ∈ Ltan (∂Ω, Λ l T M) and g ∈ Lnor (∂Ω, Λ l T M) we write ⟨⟨δ ∂ ([f]), [ν ∨ g]⟩⟩ = − ∫ ⟨ν ∧ f, d ∂ g⟩ dσ = ⟨⟨Φ([f]), d ∂ (ν ∧ (ν ∨ g))⟩⟩ ∂Ω ∗
= ⟨⟨(d ∂ (ν ∧ ⋅)) Φ([f]), [ν ∨ g]⟩⟩
(8.1.61)
thus δ ∂ = (d ∂ (ν ∧ ⋅))∗ ∘ Φ, as desired. Yet another consequence of Theorem 8.6 is the Hodgelike decomposition result on regular SKT surfaces presented in the next theorem. Theorem 8.9. Let Ω ⊂ M be an εSKT domain with ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. Denote by ν the outward unit conormal to Ω and ﬁx a degree l ∈ {0, 1, . . . , n}. Then 2,d l+1 δ ∂ L2,δ T M) and ν ∨ d ∂ Lnor (∂Ω, Λ l T M) tan (∂Ω, Λ (8.1.62) are closed subspaces of L2tan (∂Ω, Λ l T M), the space 2,0
2,0
Ltan (∂Ω, Λ l T M) ∩ (ν ∨ Lnor (∂Ω, Λ l+1 T M)) is ﬁnite dimensional,
(8.1.63)
and 2,δ 2,d L2tan (∂Ω, Λ l T M) = δ ∂ Ltan (∂Ω, Λ l+1 T M) ⊕ [ν ∨ d ∂ Lnor (∂Ω, Λ l T M)] 2,0
2,0
⊕ [Ltan (∂Ω, Λ l T M) ∩ (ν ∨ Lnor (∂Ω, Λ l+1 T M))],
(8.1.64)
where all direct sums are orthogonal. In addition, a similar decomposition (exhibiting analogous features) is valid for the space L 2nor (∂Ω, Λ l T M). As a consequence, one has
8.1 de Rham Cohomology on Regular SKT Surfaces  327
2,δ
2,d
L2 (∂Ω, Λ l T M) = δ ∂ Ltan (∂Ω, Λ l+1 T M) ⊕ [ν ∨ d ∂ Lnor (∂Ω, Λ l T M)] 2,d
2,δ
⊕ d ∂ Lnor (∂Ω, Λ l−1 T M) ⊕ [ν ∧ δ ∂ Ltan (∂Ω, Λ l T M)] 2,0 l l+1 ⊕ [L2,0 T M))] tan (∂Ω, Λ T M) ∩ (ν ∨ L nor (∂Ω, Λ 2,0
2,0
⊕ [Lnor (∂Ω, Λ l T M) ∩ (ν ∧ Ltan (∂Ω, Λ l−1 T M))],
(8.1.65)
where all direct sums are orthogonal. Before presenting the proof of the above theorem, we record and prove a useful abstract functional analytic result. Lemma 8.10. Let H−1 , H0 , H1 be three Hilbert spaces and assume that T : H−1 → H0 and R : H0 → H1 are two linear, closed, densely deﬁned, unbounded operators, with the property that RT = 0. Then H0 = Im T ⊕ Im R∗ ⊕ (Ker T ∗ ∩ Ker R), (8.1.66) where the direct sums are orthogonal in H0 . Proof. This is seen by establishing (via routine arguments) that Im T ⊥ Im R∗ and that ⊥ Ker T ∗ ∩ Ker R = (Im T ⊕ Im R∗ ) . We now turn in earnest to the proof of Theorem 8.9. Proof of Theorem 8.9. To prove (8.1.64), consider δ∂
δ∂
L2tan (∂Ω, Λ l+1 T M) → L2tan (∂Ω, Λ l T M) → L2tan (∂Ω, Λ l−1 T M)
(8.1.67)
where each δ ∂ is viewed as a linear, closed, densely deﬁned, unbounded operator, as described in Corollary 8.5. By relying on the fact that δ ∂ ∘ δ ∂ = 0, in the sense of composition of unbounded operators, Lemma 8.10, and (8.1.23), we readily obtain the version of (8.1.64) in which the ﬁrst two summands are replaced by their respective L2 closures. However, it follows from Theorem 8.6 that the operator δ ∂ in (8.1.31) and the operator d ∂ as in (8.1.28), but with l replaced by l + 1, have closed ranges. As such, the said closures become redundant in the present case, justifying (8.1.62) and ﬁnishing the proof of (8.1.64). Consider next the claim made in (8.1.63). To see that this is true, we shall revisit the proof of Theorem 8.6 and employ notation and results from that setting. Speciﬁcally, observe that thanks to (8.1.39) (used with p = 2) any form 2,0 l l+1 f ∈ L2,0 T M)) tan (∂Ω, Λ T M) ∩ (ν ∨ L nor (∂Ω, Λ
(8.1.68)
may be expressed as f = 4C l f + 4δ ∂ (ν ∨ S l+2 (ν ∧ dS l f))
(8.1.69)
where the operator C l is as in (8.1.40). Given that (8.1.68) implies f = ν ∨ (ν ∧ f)
and
2,0
ν ∧ f ∈ Lnor (∂Ω, Λ l+1 T M),
(8.1.70)
328  8 Additional Results and Applications
by making use of (8.1.69), (3.2.17), formula (8.1.1), and the CauchySchwarz inequality, we may estimate ‖f‖2L2 (∂Ω,Λ l TM) = ∫ f2 dσ = ∫ ⟨f, f ⟩ dσ ∂Ω
∂Ω
= ∫ ⟨4C l f, f ⟩ dσ + 4 ∫ ⟨δ ∂ (ν ∨ S l+2 (ν ∧ dS l f)), ν ∨ (ν ∧ f)⟩ dσ ∂Ω
∂Ω
= ∫ ⟨4C l f, f ⟩ dσ − 4 ∫ ⟨ν ∧ (ν ∨ S l+2 (ν ∧ dS l f)), d ∂ (ν ∧ f)⟩ dσ ∂Ω
∂Ω
= ∫ ⟨4C l f, f ⟩ dσ ≤ 4‖C l f‖L2 (∂Ω,Λ l TM) ‖f‖L2 (∂Ω,Λ l TM) .
(8.1.71)
∂Ω
Hence, ultimately, ‖f‖L2 (∂Ω,Λ l TM) ≤ 4‖C l f‖L2 (∂Ω,Λ l TM) for every form 2,0
2,0
f ∈ Ltan (∂Ω, Λ l T M) ∩ (ν ∨ Lnor (∂Ω, Λ l+1 T M)).
(8.1.72)
Since by (8.1.40) and the present background assumptions, C l may be assumed to be sufficiently close to the space of compact operators on L2tan (∂Ω, Λ l T M), this forces the inclusion 2,0
2,0
Ltan (∂Ω, Λ l T M) ∩ (ν ∨ Lnor (∂Ω, Λ l+1 T M)) → L2tan (∂Ω, Λ l T M)
(8.1.73)
to be compact. With this in hand, Riesz’s theorem applies and gives (8.1.63). Finally, that a decomposition similar to (8.1.64), exhibiting analogous features, is valid for the space L2nor (∂Ω, Λ l T M) follows from what we have just proved and Hodge duality (cf. (2.4.47)–(2.4.51)) keeping in mind (3.1.10). In turn, this decomposition and (8.1.64) then yield (8.1.65) since L2 (∂Ω, Λ l T M) is the direct orthogonal sum of L2tan (∂Ω, Λ l T M) and L2nor (∂Ω, Λ l T M) (cf. (2.4.6)). The circle of ideas and techniques employed in the proof of Theorem 8.6 may be further adapted to prove the remarkable compact embedding result presented in Theorem 8.11 below. In a nutshell, this asserts that even though Ω is only assumed to be an εSKT domain for some small ε > 0 (hence, generally speaking, ∂Ω utterly lacks any type 2,d l l+1 T M)) equipped of manifold structure), the space L2,δ tan (∂Ω, Λ T M) ∩ (ν ∨ L nor (∂Ω, Λ with a natural norm, behaves as typically Sobolev spaces⁵ on compact manifolds do, in the sense that it embeds compactly into L2 (∂Ω, Λ l T M). Theorem 8.11. Let Ω ⊂ M be an εSKT domain with ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. Denote by ν the outward unit conormal to Ω and ﬁx a degree l ∈ {0, 1, . . . , n}.
5 involving a positive amount of smoothness
8.1 de Rham Cohomology on Regular SKT Surfaces  329
Then the space 2,d l l+1 T M)) L2,δ tan (∂Ω, Λ T M) ∩ (ν ∨ L nor (∂Ω, Λ
(8.1.74)
equipped with the norm f  := ‖f‖L2,δ (∂Ω,Λ l TM) + ‖ν ∧ f‖L2,d l+1 TM) nor (∂Ω,Λ
(8.1.75)
tan
embeds compactly into L2 (∂Ω, Λ l T M). Proof. In view of the nature of the claim we are trying to establish, there is no loss of generality in assuming that the domain Ω does not contain any connected component of M. Granted this, it is then possible to select a function V as in (3.1.13) which, in addition, satisﬁes supp V ∩ Ω = ⌀. (8.1.76) In relation to this potential V and the domain Ω, associate layer potentials as in (3.2). l 1 l+1 T M). Pick an arbitrary form f ∈ L2,δ tan (∂Ω, Λ T M) and deﬁne u := d Sl f ∈ C (Ω, Λ Bearing in mind that (8.1.76) entails ∆HL Sl f = 0 in Ω, formula (3.2.40) permits us to express δu = δdSl f = −dδSl f = −dSl−1 (δ ∂ f) − dR l−1 f in Ω. (8.1.77) Consequently, N(δu) ∈ L2 (∂Ω)
and (8.1.78) n.t. ν ∧ (δu)∂Ω = −ν ∧ dS l−1 (δ ∂ f) − ν ∧ dR l−1 f. Moreover, the differential form u satisﬁes ∆HL u = 0 in Ω, du = 0 in Ω, N u ∈ L2 (∂Ω), as well as n.t. n.t. ν ∨ u∂Ω = (− 12 I + M l )f and ν ∧ u∂Ω = ν ∧ dS l f, (8.1.79) by (3.2.25) and (3.2.16). Together, these allow us to write integral representation formula (6.3.36) for the present (l + 1)form u, which takes the form u = − Sl+1 (ν ∧ dS l−1 (δ ∂ f)) − Sl+1 (ν ∧ dR l−1 f ) + δSl+2 (ν ∧ dS l f ) − dSl ((− 12 I + M l )f ) − R l+1 (ν ∧ dS l f ) + Q l ((− 21 I + M l )f )
(8.1.80)
in Ω. By taking the nontangential trace to the boundary of both sides of (8.1.80) and then applying ν ∨ ⋅ yields (by virtue of the jumpformulas from § 3.2) (− 12 I + M l )f = −ν ∨ S l+1 (ν ∧ dS l−1 (δ ∂ f)) − ν ∨ S l+1 (ν ∧ dR l−1 f ) + ν ∨ δS l+2 (ν ∧ dS l f) − (− 12 I + M l )(− 21 I + M l )f − ν ∨ R l+1 (ν ∧ dS l f ) + ν ∨ Q l ((− 12 I + M l )f ).
(8.1.81)
Straightforward algebra and (3.2.17) then permits us to conclude that each differential 2,δ form f ∈ Ltan (∂Ω, Λ l T M) may be written as f = δ ∂ (4ν ∨ S l+2 (ν ∧ dS l f)) + 4M 2l f + 4ν ∨ S l+1 (ν ∧ dS l−1 (δ ∂ f)) + 4ν ∨ S l+1 (ν ∧ dR l−1 f ) + 4ν ∨ R l+1 (ν ∧ dS l f ) − 4ν ∨ Q l ((− 12 I + M l )f ).
(8.1.82)
330  8 Additional Results and Applications
For ease of use, introduce Cl := 4ν ∨ S l+2 ∘ (ν ∧ dS l )
(8.1.83)
and C l := 4ν ∨ S l+1 ∘ (ν ∧ dR l−1 ) + 4ν ∨ S l+1 ∘ (ν ∧ dS l−1 ∘ δ ∂ ) + 4ν ∨ R l+1 ∘ (ν ∧ dS l ) − 4ν ∨ Q l ∘ (− 12 I + M l ),
(8.1.84)
then rewrite (8.1.82) as f = δ ∂ Cl f + 4M 2l f + C l f,
l ∀ f ∈ L2,δ tan (∂Ω, Λ T M).
(8.1.85)
By (3.2.17) we have that 2,δ Cl : L2 (∂Ω, Λ l T M) → Ltan (∂Ω, Λ l+1 T M)
(8.1.86)
is a welldeﬁned, linear and bounded operator, while the ﬁrst conclusion in Proposition 3.5 implies that Cl : L2 (∂Ω, Λ l T M) → L2 (∂Ω, Λ l+1 T M)
(8.1.87)
is a compact operator, Also, Proposition 3.2, (3.2.17), (3.1.56), and Proposition 2.41 show that 2,δ 2,δ l l C l : L tan (∂Ω, Λ T M) → L tan (∂Ω, Λ T M)
(8.1.88)
is a welldeﬁned, linear and bounded operator, and also
2,δ l 2 l C l : L tan (∂Ω, Λ T M) → L (∂Ω, Λ T M)
(8.1.89)
is a compact operator. Finally, assuming that Ω is an εSKT domain with ε > 0 sufficiently small, Theorem 3.6 ensures that there exists 2 l 2 l C l : L tan (∂Ω, Λ T M) → L tan (∂Ω, Λ T M) compact operator 1 2 2 satisfying ‖4M 2l − C l ‖Ltan (∂Ω,Λ l TM)→Ltan (∂Ω,Λ l TM) < 2 .
(8.1.90)
In particular, the estimate in (8.1.90) implies that for each f ∈ L2tan (∂Ω, Λ l T M) we have 1 ‖4M 2l f‖L2 (∂Ω,Λ l TM) ≤ ‖C l f‖L2 (∂Ω,Λ l TM) + 2 ‖f‖L2 (∂Ω,Λ l TM) . 2,δ
2,d
(8.1.91)
To proceed, pick an arbitrary f ∈ Ltan (∂Ω, Λ l T M) ∩ (ν ∨ Lnor (∂Ω, Λ l+1 T M)) and l+1 T M). Making use of (8.1.85), (8.1.86), formula note that this forces ν ∧ f ∈ L2,d nor (∂Ω, Λ (8.1.1), and the CauchySchwarz inequality, we may then estimate
8.1 de Rham Cohomology on Regular SKT Surfaces  331
‖f‖2L2 (∂Ω,Λ l TM) = ∫ f2 dσ = ∫ ⟨f, f ⟩ dσ ∂Ω
∂Ω
∫ ⟨δ ∂ Cl f, f ⟩ dσ
=
+ ∫ ⟨(4M 2l + C l )f, f ⟩ dσ
∂Ω
∂Ω
∫ ⟨δ ∂ Cl f, ν
=
∨ (ν ∧ f)⟩ dσ + ∫ ⟨C l f, f ⟩ dσ
∂Ω
∂Ω
= − ∫ ⟨ν ∧ ≤
Cl f, d ∂ (ν
∧ f)⟩ dσ + ∫ ⟨(4M 2l + C l )f, f ⟩ dσ
∂Ω C l f‖L2 (∂Ω,Λ l+1 TM) ‖ν
+
‖(4M 2l
+
∂Ω
∧ f‖L2,d (∂Ω,Λ l+1 TM) nor
C l )f‖L2 (∂Ω,Λ l TM) ‖f‖L2 (∂Ω,Λ l TM) .
Thanks to (8.1.91) and (8.1.92), for each f ∈ we therefore have
l L2,δ tan (∂Ω, Λ T M)
∩ (ν ∨
(8.1.92)
l+1 T M)) L2,d nor (∂Ω, Λ
‖f‖2L2 (∂Ω,Λ l TM) ≤ (‖Cl f‖L2 (∂Ω,Λ l+1 TM) + ‖C l f‖L2 (∂Ω,Λ l TM) + ‖4M 2l f‖L2 (∂Ω,Λ l TM) ) f  ≤ (‖Cl f‖L2 (∂Ω,Λ l+1 TM) + ‖C l f‖L2 (∂Ω,Λ l TM) 1 + ‖C l f‖L2 (∂Ω,Λ l TM) + 2 ‖f‖L2 (∂Ω,Λ l TM) ) f . At this stage, consider a sequence 2,d l l+1 {f j }j∈ℕ ⊆ L2,δ T M)) tan (∂Ω, Λ T M) ∩ (ν ∨ L nor (∂Ω, Λ
with the property that
sup f j ≤ 1.
(8.1.93)
(8.1.94) (8.1.95)
j∈ℕ
In particular, this implies sup ‖f j ‖L2 (∂Ω,Λ l TM) ≤ sup ‖f j ‖L2,δ (∂Ω,Λ l TM) ≤ 1. j∈ℕ
(8.1.96)
tan
j∈ℕ
Bearing in mind the compactness properties in (8.1.87) and (8.1.89), by restricting ourselves to a suitable subsequence, there is no loss of generality in assuming that {Cl f j }j∈ℕ converges in L2 (∂Ω, Λ l+1 T M), 2 l {C l f j }j∈ℕ converges in L (∂Ω, Λ T M),
{C l f j }j∈ℕ
(8.1.97)
converges in L (∂Ω, Λ T M). 2
l
Writing (8.1.93) with f replaced by f j − f k , where j, k ∈ ℕ are arbitrary yields, in view of (8.1.95) and the linearity of the operators involved, 1 2 ‖f j
− f k ‖2L2 (∂Ω,Λ l TM) ≤ Cl f j − Cl f k ‖L2 (∂Ω,Λ l+1 TM) + ‖C l f j − C l f k ‖L2 (∂Ω,Λ l TM) + ‖C l f j − C l f k ‖L2 (∂Ω,Λ l TM) .
(8.1.98)
332  8 Additional Results and Applications
In concert with the convergence properties recorded in formula (8.1.97), this proves that the sequence {f j }j∈ℕ is Cauchy in L2 (∂Ω, Λ l+1 T M), and the desired conclusion follows. The compact embedding result from Theorem 8.11 is remarkably versatile. For one thing, it provides a direct (and rather satisfactory) proof of the ﬁnite dimensionality claim made in (8.1.63), simply by virtue of the observation that f  = ‖f‖L2 (∂Ω,Λ l TM) for every differential form 2,0
2,0
f ∈ Ltan (∂Ω, Λ l T M) ∩ (ν ∨ Lnor (∂Ω, Λ l+1 T M)).
(8.1.99)
For another thing, in concert with the description of the realization of δ ∂ as an unbounded operator and the identiﬁcation its functional analytic adjoint from Corollary 8.5, the compact embedding result from Theorem 8.11 proves that a great deal of analysis can be carried out on regular SKT surfaces, akin to classical results on smooth manifolds. In this process, the catalyst is Proposition 9.71 containing a collection of abstract results, of functional analytic nature, proved in [46]. In concert with Corollary 8.5 and Theorem 8.11 this yields a wealth of results which we collect in the theorem below. Theorem 8.12. Suppose Ω ⊂ M is an εSKT domain with the parameter ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. Let ν denote the outward unit conormal to Ω and, as in (1.3.2), let G := ⊕nl=0 Λ l T M stand for the Grassmann algebra bundle on the manifold M. In this context, consider the Hilbert space H := L2tan (∂Ω, G), with norm ‖⋅‖H := ‖⋅‖L2 (∂Ω,G) . (8.1.100) Then the following properties hold. (1) If one deﬁnes the unbounded linear operator T : H → H acting according to Tf := δ ∂ f,
2,δ
∀ f ∈ Dom(T) := Ltan (∂Ω, G),
(8.1.101)
then T : H → H is a linear, closed, densely deﬁned unbounded operator,
(8.1.102)
which satisﬁes the cancellation condition⁶ T 2 = 0.
(8.1.103)
The nullspace and range of T are, respectively, 2,0
Ker T = Ltan (∂Ω, G)
and
2,δ
Im T = δ ∂ [Ltan (∂Ω, G)].
Moreover, the operator T has closed range in H. 6 in the sense of composition of unbounded operators
(8.1.104)
8.1 de Rham Cohomology on Regular SKT Surfaces  333
(2) The functional analytic adjoint of T is the operator T ∗ : H → H T ∗ f = −ν ∨ d ∂ (ν ∧ f),
acting according to ∀ f ∈ Dom(T ∗ ) = ν ∨ L2,d nor (∂Ω, G).
(8.1.105)
In particular, the nullspace and range of T ∗ are, respectively, Ker T ∗ = ν ∨ Lnor (∂Ω, G) 2,0
and
Im T ∗ = ν ∨ d ∂ [Lnor (∂Ω, G)]. 2,d
(8.1.106)
Also, the operator T ∗ has closed range in H. 2,d (3) If for every f ∈ L2,δ tan (∂Ω, G) ∩ (ν ∨ L nor (∂Ω, G)) one deﬁnes f  := ‖f‖L2,δ (∂Ω,G) + ‖ν ∧ f‖L2,d nor (∂Ω,G) tan
≈ ‖f‖H + ‖δ ∂ f‖H + ‖d ∂ (ν ∧ f)‖H ,
(8.1.107)
then 2,δ 2,d the space Ltan (∂Ω, G) ∩ (ν ∨ Lnor (∂Ω, G)) equipped with the norm ⋅ embeds compactly into the Hilbert space H.
(8.1.108)
(4) The boundary Dirac operator D ∂ := T + T ∗
(8.1.109)
is an unbounded selfadjoint operator on H, with spectrum Spec(D ∂ ) ⊆ ℝ. Moreover, the space Ker D ∂ is ﬁnite dimensional and may be described as 2,0 2,0 Ker D ∂ = Ltan (∂Ω, G) ∩ (ν ∨ Lnor (∂Ω, G)).
(8.1.110)
(5) For each z ∈ ℂ \ Spec(D ∂ ) there exists a ﬁnite constant C z > 0 with the property that f  ≤ C z ‖(zI − D ∂ )f‖H f ∈
L2,δ tan (∂Ω, G)
for every form (8.1.111)
∩ (ν ∨ L2,d nor (∂Ω, G)).
(6) The boundary Dirac operator D ∂ has compact resolvent and closed range. In fact, 2,0
2,0
⊥
Im D ∂ = (Ltan (∂Ω, G) ∩ (ν ∨ Lnor (∂Ω, G))) .
(8.1.112)
(7) The spectrum Spec(D ∂ ) of the boundary Dirac operator is discrete, and consists of real eigenvalues which can only accumulate at ±∞. (8) The boundary Dirac operator has the property that 2,d D ∂ acting from L2,δ tan (∂Ω, G) ∩ (ν ∨ L nor (∂Ω, G)), equipped with the norm ⋅ from (8.1.107), into the Hilbert space H, is linear, bounded, and Fredholm of index zero.
(8.1.113)
334  8 Additional Results and Applications
(9) One has the boundary Hodge decomposition 2,δ
2,d
H = δ ∂ [Ltan (∂Ω, G)] ⊕ (ν ∨ d ∂ [Lnor (∂Ω, G)]) ⊕ Ker D ∂ ,
(8.1.114)
as well as the boundary Friedrichs decompositions 2,0
ν∨
Ltan (∂Ω, G) = Ker D ∂ ⊕ δ ∂ [Ltan (∂Ω, G)],
2,δ
(8.1.115)
2,0 Lnor (∂Ω, G)
2,d d ∂ [Lnor (∂Ω, G)]),
(8.1.116)
= Ker D ∂ ⊕ (ν ∨
where all direct sums are orthogonal. In particular, one has ﬁnite dimensional cohomology groups 2,0
Ltan (∂Ω, G) δ ∂ [L2,δ tan (∂Ω, G)]
≅
L2,0 nor (∂Ω, G) d ∂ [L2,d nor (∂Ω, G)]
≅ Ker D ∂ .
(8.1.117)
(10) The boundary HodgeLaplacian⁷ ∆ ∂ := −(TT ∗ + T ∗ T) is a nonpositive selfadjoint operator on H.
(8.1.118)
Its domain Dom(∆ ∂ ) may be described as 2,d Dom (∆ ∂ ) = { f ∈ L2,δ tan (∂Ω, G) ∩ (ν ∨ L nor (∂Ω, G)) : 2,d 2,δ δ ∂ f ∈ ν ∨ Lnor (∂Ω, G), ν ∨ d ∂ (ν ∧ f) ∈ Ltan (∂Ω, G)},
(8.1.119)
and the action of the boundary HodgeLaplacian on this domain is explicitly given by ∆ ∂ f = ν ∨ d ∂ (ν ∧ δ ∂ f) + δ ∂ (ν ∨ d ∂ (ν ∧ f)),
∀ f ∈ Dom (∆ ∂ ).
(8.1.120)
Moreover, the image and nullspace of the boundary HodgeLaplacian are, respectively, given by Im ∆ ∂ = (Ker D ∂ )⊥ , Ker ∆ ∂ = Ker D ∂ . (8.1.121) In particular, the operator ∆ ∂ acting from Dom(∆ ∂ ), equipped with the graph norm f → ‖f‖H + ‖∆ ∂ f‖H , into the Hilbert space H, is linear, bounded, and Fredholm of index zero.
(8.1.122)
(11) The squareroot (−∆ ∂ )1/2 of minus the boundary HodgeLaplacian has the property that 2,d Dom ((−∆ ∂ )1/2 ) = L2,δ (8.1.123) tan (∂Ω, G) ∩ (ν ∨ L nor (∂Ω, G)), and Dom (∆ ∂ ) is dense in the latter space when equipped with the norm ⋅ deﬁned in (8.1.107).
7 understood in the sense of composition of unbounded operators
8.1 de Rham Cohomology on Regular SKT Surfaces  335
(12) The boundary HodgeLaplacian ∆ ∂ generates an analytic semigroup on H. (13) The boundary HodgeLaplacian ∆ ∂ has a compact resolvent. Moreover, the spectrum of the operator −∆ ∂ on H consists only of real, nonnegative eigenvalues⁸ 0 ≤ λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λ j ≤ λ j+1 ≤ ⋅ ⋅ ⋅
(8.1.124)
which satisfy λ j → +∞ as j → ∞, and for which the following minmax principle holds for each j ∈ ℕ: λ j = min ( max dim(L j )=j
‖δ ∂ f‖2H + ‖d ∂ (ν ∧ f)‖2H
0=f̸ ∈L j
‖f‖2H
),
(8.1.125)
where the above minimum is taken over all linear subspaces L j of the vector space 2,d L2,δ tan (∂Ω, G) ∩ (ν ∨ L nor (∂Ω, G)) having dimension j. (14) The restriction of the boundary HodgeLaplacian ∆ ∂ : (Ker D ∂ )⊥ → (Ker D ∂ )⊥
(8.1.126)
is selfadjoint, onetoone and onto. (15) If (∆ ∂ )−1 denotes the inverse of the operator (8.1.126) and if π⊥ denotes the orthogonal projection of H onto (Ker D ∂ )⊥ , then the boundary Green operator G := (∆ ∂ )−1 π⊥ : H → H
(8.1.127)
is bounded, selfadjoint, compact, and nonpositive. It satisﬁes G ∘ ∆ ∂ = ∆ ∂ ∘ G = π⊥ .
(8.1.128)
In particular, with π standing for the orthogonal projection of H onto the ﬁnite dimensional space Ker D ∂ , one has the following constructive boundary Hodge decomposition: f = δ ∂ (ν ∨ d ∂ (ν ∧ Gf)) + ν ∨ d ∂ (ν ∧ δ ∂ Gf) + πf,
∀ f ∈ H.
(8.1.129)
(16) One has 2,d 2,0 L2,0 tan (∂Ω, G) ∩ (ν ∨ L nor (∂Ω, G)) → L tan (∂Ω, G)
densely, and
2,0 2,δ 2,0 (ν ∨ Lnor (∂Ω, G)) ∩ Ltan (∂Ω, G) → ν ∨ Lnor (∂Ω, G)
(8.1.130)
densely.
(17) In one deﬁnes 2,0
2,0
ℍ := Ltan (∂Ω, G) ⊕ (ν ∨ Lnor (∂Ω, G)),
(8.1.131)
which is a closed subspace of H ⊕ H (hence Hilbert with the naturally induced inner product from the latter space), and consider the linear unbounded operator T : ℍ → ℍ,
8 listed according to their multiplicity
T := (
0 −ν ∨ d ∂ (ν ∧ ⋅)
δ∂ ) 0
(8.1.132)
336  8 Additional Results and Applications
with domain 2,0
2,d
Dom (T ) := (Ltan (∂Ω, G) ∩ (ν ∨ Lnor (∂Ω, G))) 2,δ ⊕ ((ν ∨ L2,0 nor (∂Ω, G)) ∩ L tan (∂Ω, G))
(8.1.133)
and natural action, then T is welldeﬁned, closed, densely deﬁned and selfadjoint. In particular, T has a compact resolvent so that its spectrum, Spec(T ), is discrete and consists of only real eigenvalues accumulating at ±∞. (18) For any complex number z ∈ ℂ \ Spec (T ) and any forms η, ξ ∈ H, the boundary Maxwell system 2,d 2,δ { f ∈ ν ∨ Lnor (∂Ω, G), g ∈ Ltan (∂Ω, G), { { { −ν ∨ d ∂ (ν ∧ f) − zg = η, { { { { δ g − zf = ξ, { ∂
(8.1.134)
has a unique solution, and there exists a ﬁnite constant C z > 0, independent of η, ξ , such that this solution satisﬁes ‖f‖H + ‖g‖H ≤ C z (‖η‖H + ‖ξ‖H ).
(8.1.135)
8.2 Maxwell’s Equations in Regular SKT Domains In this section we indicate how the classical timeharmonic Maxwell system, considered here on arbitrary regular SKT subdomains of Riemannian manifolds, and with boundary data in L p based spaces with p ∈ (1, ∞) arbitrary, may be subsumed within the theory developed so far. As such, our present work further reﬁnes and augments results in [56, 90, 92] dealing with bounded Lipschitz and C 1 domains in the ﬂat Euclidean setting, as well as results in [86] dealing with Lipschitz subdomains of Riemannian manifolds and with the integrability exponent p near 2. To formulate the boundary value problem for the Maxwell system in a regular SKT domain Ω ⊂ M with outward unit conormal ν, ﬁx an degree l ∈ {0, 1, . . . , n}, an integrability exponent p ∈ (1, ∞), and a wavenumber k ∈ ℂ. This problem then asks for ﬁnding an electric form E along with a magnetic form H subject to E ∈ C 1 (Ω, Λ l Tℂ M), { { { { { { H ∈ C 1 (Ω, Λ l+1 Tℂ M), { { { { { { dE − ikH = 0 in Ω, { { { (MAXk )l { δH + ikE = 0 in Ω, { { { { { { δE = 0 and dH = 0 in Ω, { { { { N(E), N(H) ∈ L p (∂Ω), { { { { p,d n.t. l+1 { ν ∧ E∂Ω = f ∈ Lnor (∂Ω, Λ Tℂ M),
(8.2.1)
8.2 Maxwell’s Equations in Regular SKT Domains 
337
where Tℂ M denotes the complexiﬁed tangent bundle on M. Our main result in this regard reads as follows. Theorem 8.13. Given a regular SKT domain Ω ⊂ M, there exists a sequence of nonnegative real numbers {k j }j with the property that for any l ∈ {0, 1, . . . , n}, p ∈ (1, ∞), and k ∈ ℂ \ {±k j }j , the boundary problem (MAXk )l has a unique solution. Moreover, for some constant C ∈ (0, ∞) independent of f , this solution satisﬁes ‖N(E)‖L p (∂Ω) + ‖N(H)‖L p (∂Ω) ≤ C‖f‖L p,d (∂Ω,Λ l+1 Tℂ M) . nor
(8.2.2)
Finally, analogous results are true when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω, as well as a targeted exponent p. As a preamble to the proof of Theorem 8.13 we shall treat the analogue of (BVP6)l from § 1 written for the HodgeHelmholtz operator ∆HL + k2 , with k ∈ ℂ. Speciﬁcally, we shall prove the following wellposedness result. Theorem 8.14. Let Ω ⊂ M be a regular SKT domain with outward unit conormal ν and for every k ∈ ℂ, l ∈ {0, 1, . . . , n}, and p ∈ (1, ∞) consider the boundary value problem u ∈ C 1 (Ω, Λ l Tℂ M), { { { { { { { (∆HL + k2 )u = 0 in Ω, { { { (BVPk )l { N u, N(du), N(δu) ∈ L p (∂Ω), { { p,d { n.t. { ν ∧ u∂Ω = f ∈ Lnor (∂Ω, Λ l+1 Tℂ M), { { { { { p n.t. l { ν ∧ (δu)∂Ω = g ∈ Lnor (∂Ω, Λ Tℂ M).
(8.2.3)
Then there exists a sequence of nonnegative real numbers {k j }j with the property that for any given k ∈ ℂ \ {±k j }j , l ∈ {0, 1, . . . , n}, and p ∈ (1, ∞) the boundary problem (BVPk )l has a unique solution. Moreover, this solution satisﬁes ‖N u‖L p (∂Ω) + ‖N(δu)‖L p (∂Ω) + ‖N(δu)‖L p (∂Ω) ≤ C (‖f‖L p,d (∂Ω,Λ l+1 Tℂ M) + ‖g‖L p (∂Ω,Λ l Tℂ M) ) nor
(8.2.4)
for some constant C ∈ (0, ∞) independent of f and g. In addition, δu = 0 in Ω ⇐⇒ g = 0 on ∂Ω.
(8.2.5)
In addition, similar results are valid for the Hodgedual problem of (BVPk )l . Once again, analogous results are true when Ω is only assumed to be an εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω, as well as a targeted exponent p. Proof. Consider the HodgeLaplacian on complex coefficient lforms, in the context l ∆HL : H 1,2 (M, Λ l Tℂ M) → H −1,2 (M, Λ l Tℂ M).
(8.2.6)
338  8 Additional Results and Applications Under the assumption that the Riemannian metric has C 2 coefficients, this is a l bounded, nonpositive, formally selfadjoint operator. In particular, (∆HL − λ)−1 ex2 l ists and is a selfadjoint, compact operator on L (M, Λ Tℂ M) for λ ∈ ℝ sufficiently large. This implies the existence of a discrete subset of (−∞, 0], which we denote l l by Spec (∆HL ), which accumulates only at −∞ and so that ∆HL − z is invertible from l 1,2 l −1,2 l H (M, Λ Tℂ M) onto H (M, Λ Tℂ M) whenever z ∈ ℂ \ Spec (∆HL ). Select a count able family of real numbers k j ≥ 0 so that l ). {−(kj )2 }j = ⋃ Spec (∆HL
(8.2.7)
0≤l≤n
Granted this choice, the entire discussion in § 6 can be carried out in connection with the operator L := ∆HL + k2 for any given k ∈ ℂ \ {±kj }j . To stress the dependence on k, we shall write M k , N k , Sk , S k , etc., for the resulting layer potential operators. These operators enjoy all the properties described in § 6 for the case of realvalued potentials V. One should also keep in mind that, since k is constant, the residual terms in (3.1.39) and (3.1.51) vanish. To proceed, note that the assignment k → M k , viewed as a mapping with values in the Banach space of all linear bounded operators on L2tan (∂Ω, Λ l Tℂ M), is holomorphic in ℂ \ {±kj }j . In light of Theorem 3.6 and (the last part of) Theorem 3.7 (both adapted to the case when V is a complex potential) the analytic Fredholm theorem applies to this family and gives that ± 12 I + M k are invertible on L2tan (∂Ω, Λ l Tℂ M) for all k in ℂ \ {±kj }j , except perhaps for a discrete subset {k j }j of ℂ \ {±k j }j . Having established this, the same type of argument as in the proof of Theorem 3.7 gives that ± 12 I + M k are p
p,δ
invertible both on Ltan (∂Ω, Λ l Tℂ M) and Ltan (∂Ω, Λ l Tℂ M) for any p ∈ (1, ∞) provided k ∈ ℂ \ ({±kj }j ∪ {k j }j ). In summary, the argument so far gives the existence of a sequence {k j }j (taken to be {kj }j ∪ {k j }j ) of real, positive numbers, with the property that for any l ∈ {0, 1, . . . , n} we have p ± 12 I + M k is invertible on Ltan (∂Ω, Λ l Tℂ M) p,δ
as well as on the space Ltan (∂Ω, Λ l Tℂ M) for any k ∈ ℂ \ {±k j }j and any p ∈ (1, ∞).
(8.2.8)
Based on (8.2.9), an application of the Hodge ∗isomorphism, this also gives that for any l ∈ {0, 1, . . . , n} we have (with {k j }j as above) p
± 12 I + N k are invertible on Lnor (∂Ω, Λ l Tℂ M) p,d
as well as on the space Lnor (∂Ω, Λ l Tℂ M) for any k ∈ ℂ \ {±k j }j and any p ∈ (1, ∞).
(8.2.9)
Armed with these invertibility results, we can now tackle the ﬁrst part in the conclusion of Theorem 8.14. Regarding existence, if k ∈ ℂ \ {±k j }j then a direct veriﬁcation shows that
8.3 DirichlettoNeumann Operators for the HodgeLaplacian in Regular SKT Domains
 339
−1
u := δSk [( 12 I + N k ) (f − ν ∧ S k ( 12 I + N k )−1 g)] −1
+ Sk [( 12 I + N k ) g] in Ω,
(8.2.10)
solves (BVPk )l and satisﬁes the estimate (8.2.4). In order to prove uniqueness, the ﬁrst observation is that a Greentype integral representation formula holds for any u satisfying the ﬁrst three conditions listed in (8.2.3). Speciﬁcally, arguing as in the proof of (6.3.36) it follows that any such form can be expressed in Ω as n.t. n.t. u = δSk (ν ∧ u∂Ω ) − dSk (ν ∨ u∂Ω )
n.t. n.t. − Sk (ν ∨ (du)∂Ω ) + Sk (ν ∧ (δu)∂Ω ).
(8.2.11)
If u actually solves the homogeneous version of (BVPk )l then the ﬁrst and last terms in the righthand side of (8.2.11) drop off. In the resulting identity, apply d to both sides, go nontangentially to the boundary, then take ν ∨ ⋅ of both sides. In doing so, n.t. we arrive at ( 12 I + M k )(ν ∨ du∂Ω ) = 0 on ∂Ω. Because of (8.2.8), this further implies n.t. n.t. ν ∨ du∂Ω = 0 on ∂Ω. Using this back in (8.2.11) leaves us with u = −dSk (ν ∨ u∂Ω ) in Ω. Once again, going nontangentially to the boundary and taking ν ∨ ⋅ of both sides n.t. n.t. yields ( 12 I + M k )(ν ∨ u∂Ω ) = 0 on ∂Ω so that, by (8.2.8), we also have ν ∨ u∂Ω = 0 on ∂Ω. Having proved this, we deduce from (8.2.11) that u = 0 in Ω, concluding the proof of uniqueness. To justify (8.2.5), note that if u is a solution for (BVPk )l with g = 0 then υ := δu solves the homogeneous version of (BVPk )l−1 . Hence, from what we have proved so far, υ necessarily vanishes, i.e., δu = 0. The proof of Theorem 8.14 is therefore complete. We ﬁnally turn to the task of providing the proof of Theorem 8.13. Proof of Theorem 8.13. Select {k j }j so that the conclusions in Theorem 8.14 are valid, then solve (8.2.3) with f as in (8.2.1) and g = 0. The important observation is that the pair E := u, H := −ik −1 du becomes a solution for (8.2.1). The estimate (8.2.2) is also seen from this. To justify uniqueness, simply note that if (E, H) is a nullsolution for (MAXk )l then u := E solves the homogeneous version of (BVPk )l . Hence, u = 0 which also entails H = 0.
8.3 DirichlettoNeumann Operators for the HodgeLaplacian in Regular SKT Domains We start by making the following deﬁnition. Deﬁnition 8.15. Given some p ∈ (1, ∞), and a degree l ∈ {1, . . . , n}, let Ω ⊂ M be an εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity
340  8 Additional Results and Applications
constants and local John constants of Ω, as well as the given exponent p. Denote by ν and σ the outward unit conormal and surface measure to Ω. In this setting, associate p,δ to an arbitrary f ∈ Ltan (∂Ω, Λ l−1 T M) the differential form n.t. ΛDN (f) := −ν ∧ (δu)∂Ω at σa.e. point on ∂Ω, (8.3.1) where u is any solution of (BVP1)l with boundary datum (f, 0), that is, u ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL u = 0 in Ω, { { { { N u, N(du) ∈ L p (∂Ω), { { { { n.t. { ν ∨ u∂Ω = f on ∂Ω, { { { { { n.t. { ν ∨ (du)∂Ω = 0 on ∂Ω.
(8.3.2)
From items (1) and (2) in Theorem 1.1 we know that the boundary value problem (8.3.2) has a solution, which is unique up to additive forms from H∨l (Ω). As such, f deterp,δ mines δu uniquely. Moreover, since f ∈ Ltan (∂Ω, Λ l−1 T M) to begin with, item (4) of Theorem 1.1 ensures that N(δu) ∈ L p (∂Ω), δu ∈ L np/(n−1) (Ω, Λ l T M)
n.t. and (δu)∂Ω exists at σa.e. point on ∂Ω.
(8.3.3)
In particular, this shows that ΛDN (f) is meaningfully and unambiguously deﬁned in (8.3.1). This is our version of the socalled DirichlettoNeumann operator that has been typically studied in the literature when all structures and objects involved are smooth (cf., e.g., [3, 72, 113, 115], and the references therein) in connection with the problem of electrical impedance tomography⁹, as well as various inverse problems of topological ﬂavor¹⁰. For the endpoint case when l = n, the mapping f → ΛDN (f) deﬁned in (8.3.1) may be canonically identiﬁed (as seen by unraveling deﬁnitions and invoking (2.4.54) and (2.4.55)) with the more familiar mapping taking a scalar function f belonging to p L1 (∂Ω), the L p based Sobolev space of order one on ∂Ω, into the conormal derivative ∂u/∂ν ∈ L p (∂Ω) of the solution u of the Regularity Problem for the LaplaceBeltrami operator in Ω with boundary datum f , i.e., ∆LB u = 0 in Ω, N u, N(∇u) ∈ L p (∂Ω), n.t. u∂Ω = f σa.e. on ∂Ω.
(8.3.4)
The latter problem is a particular case (corresponding to l = 0) of (1.4.25). 9 modeled upon the famous inverse conductivity problem posed by A. P. Calderón in [12] 10 in which one seeks to recover topological information (such as Betti numbers and cohomology structures) pertaining to the underlying domain from the knowledge of its DirichlettoNeumann operator
8.3 DirichlettoNeumann Operators for the HodgeLaplacian in Regular SKT Domains
 341
Our ﬁrst main result in this section, pertaining to the nature and properties of the DirichlettoNeumann map considered in Deﬁnition 8.15, is contained in the following theorem. Theorem 8.16. Fix p ∈ (1, ∞) along with l ∈ {1, . . . , n} and assume that Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given exponent p as well as the Ahlfors regularity constants and local John constants of Ω. Denote by ν and σ the outward unit conormal and surface measure to Ω. Then the DirichlettoNeumann map introduced in Deﬁnition 8.15 induces a welldeﬁned, linear, and bounded operator p,δ
p,0
ΛDN : Ltan (∂Ω, Λ l−1 T M) → Lnor (∂Ω, Λ l T M).
(8.3.5)
This DirichlettoNeumann operator satisﬁes the following properties: p,δ (1) If p, p ∈ (1, ∞) are such that 1/p + 1/p = 1, then for every f ∈ Ltan (∂Ω, Λ l−1 T M) p ,δ
and g ∈ Ltan (∂Ω, Λ l−1 T M) one has ∫ ⟨ΛDN (f), ν ∧ g⟩ dσ = ∫ ⟨ν ∧ f, ΛDN (g)⟩ dσ. ∂Ω
(8.3.6)
∂Ω
(2) Corresponding to p = 2, one has ∫ ⟨ΛDN (f), ν ∧ f ⟩ dσ ≥ 0
l−1 for every f ∈ L2,δ T M). tan (∂Ω, Λ
(8.3.7)
∂Ω
(3) The nullspace of the operator (8.3.5) may be described as p,δ p,0 n.t. Ker (ΛDN : Ltan (∂Ω, Λ l−1 T M) → Lnor (∂Ω, Λ l T M)) = ν ∨ (H l,p (Ω)∂Ω )
(8.3.8)
while the image of the operator (8.3.5) may be described as p,δ p,0 n.t. Im (ΛDN : Ltan (∂Ω, Λ l−1 T M) → Lnor (∂Ω, Λ l T M)) = ν ∧ (H l−1,p (Ω)∂Ω ). (8.3.9)
(4) The DirichlettoNeumann operator (8.3.5) induces a canonical isomorphism p,δ
L (∂Ω, Λ l−1 T M) ∼ n.t. l−1,p ̂ (Ω)∂Ω ). ΛDN : tan n.t. → ν ∧ (H ν ∨ (H l,p (Ω)∂Ω )
(8.3.10)
Moreover, if (̂ ΛDN )−1 denotes the inverse of (8.3.10), then n.t. n.t. δ ∂ ∘ (̂ ΛDN )−1 : ν ∧ (H l−1,p (Ω)∂Ω ) → ν ∨ (H l−1,p (Ω)∂Ω )
(8.3.11)
is a welldeﬁned, linear and bounded operator. (4) One has p,δ
(8.3.12)
p,δ
(8.3.13)
d ∂ ∘ ΛDN = 0 on Ltan (∂Ω, Λ l−1 T M) and ΛDN ∘ δ ∂ = 0 on Ltan (∂Ω, Λ l T M).
342  8 Additional Results and Applications
In order to facilitate the discussion in the proof of Theorem 8.16, in the lemma below we isolate a preparatory result, of independent interest. Lemma 8.17. Pick p ∈ (1, ∞) together with l ∈ {1, . . . , n} and suppose Ω ⊂ M is an εSKT domain with ε > 0 sufficiently small relative to the given p as well as the Ahlfors regularity constants and local John constants of Ω. Then, with ν denoting the outward unit conormal to Ω, one has p,δ n.t. δ ∂ [Ltan (∂Ω, Λ l T M)] ⊆ ν ∨ (H l,p (Ω)∂Ω )
p,0 n.t. ⊥ ⊆ Ltan (∂Ω, Λ l−1 T M) ∩ {H∨l−1 (Ω)∂Ω }
(8.3.14)
and p,d n.t. d ∂ [Lnor (∂Ω, Λ l T M)] ⊆ ν ∧ (H l,p (Ω)∂Ω )
p,0 n.t. ⊥ ⊆ Lnor (∂Ω, Λ l+1 T M) ∩ {H∨l+1 (Ω)∂Ω } .
(8.3.15) p,δ
Proof. To justify the ﬁrst inclusion in (8.3.14), pick an arbitrary f ∈ Ltan (∂Ω, Λ l T M) and invoke item (1) in Theorem 1.1 in order to solve u ∈ C 1 (Ω, Λ l+1 T M), { { { { { { ∆HL u = 0 in Ω, { { { { N u, N(du) ∈ L p (∂Ω), { { { { n.t. { ν ∨ u∂Ω = f on ∂Ω, { { { { { n.t. { ν ∨ (du)∂Ω = 0 on ∂Ω.
(8.3.16)
From item (4) and item (6) in Theorem 1.1 we know that any such solution u satisﬁes N(du) ∈ L p (∂Ω) and du = 0 in Ω. In particular, δu ∈ H l,p (Ω) and since Proposition 2.41 gives n.t. n.t. n.t. δ ∂ f = δ ∂ (ν ∨ u∂Ω ) = −ν ∨ (δu)∂Ω ∈ ν ∨ (H l,p (Ω)∂Ω ), (8.3.17) the ﬁrst inclusion in (8.3.14) follows. To prove the second inclusion in (8.3.14), observe that, thanks to (2.4.30), we have p,0 n.t. ν ∨ (H l,p (Ω)∂Ω ) ⊆ Ltan (∂Ω, Λ l−1 T M).
(8.3.18)
As such, there remains to show that for each ω ∈ H l,p (Ω) and η ∈ H∨l−1 (Ω) we have n.t. n.t. ∫ ⟨ν ∨ ω∂Ω , η∂Ω ⟩ dσ = 0.
(8.3.19)
∂Ω
This, however, is an immediate consequence of Theorem 2.36. At this point, the proof of the double inclusion in (8.3.14) is complete, and (8.3.15) then follows from this and Hodge duality (cf. Lemma 2.2 and the results from § 2.4 in this regard). We are now prepared to present the proof of Theorem 8.16.
8.3 DirichlettoNeumann Operators for the HodgeLaplacian in Regular SKT Domains
 343
p ,δ
p,δ
Proof of Theorem 8.16. Pick some f ∈ Ltan (∂Ω, Λ l−1 T M) and g ∈ Ltan (∂Ω, Λ l−1 T M) where p, p ∈ (1, ∞) are such that 1/p + 1/p = 1. We claim that ∫ ⟨ΛDN (f), ν ∧ g⟩ dσ = ∫⟨δu, δυ⟩ dVol,
(8.3.20)
Ω
∂Ω
where u is a solution of (8.3.2) for the given f , and υ is a solution of υ ∈ C 1 (Ω, Λ l T M), { { { { { { ∆HL υ = 0 in Ω, { { { { N υ, N(dυ) ∈ L p (∂Ω), { { { { n.t. { ν ∨ υ∂Ω = g on ∂Ω, { { { { { n.t. { ν ∨ (dυ)∂Ω = 0 on ∂Ω.
(8.3.21)
Much as before, item (4) of Theorem 1.1 gives that
N(δυ) ∈ L p (∂Ω), δυ ∈ L np /(n−1) (Ω, Λ l T M)
n.t. and (δυ)∂Ω exists at σa.e. point on ∂Ω.
(8.3.22)
Next, observe from item (6) of Theorem 1.1 that any solution u of (8.3.2) necessarily satisﬁes du = 0 in Ω. (8.3.23) In turn, this further forces d(δu) = 0 in Ω.
(8.3.24)
With this in hand, Theorem 2.36 applies to the differential forms δu and υ since, thanks to (8.3.3) and (8.3.22), n.t. n.t. (δu)∂Ω and υ∂Ω exist at σa.e. point on ∂Ω, N(δu) ⋅ N υ ∈ L1 (∂Ω), and δuδυ ∈ L 1 (Ω).
(8.3.25)
In the present setting, the integration by parts formula (2.3.1) yields n.t. n.t. − ∫⟨δu, δυ⟩ dVol = ∫ ⟨ν ∧ (δu)∂Ω , υ∂Ω ⟩ dσ Ω
∂Ω
n.t. n.t. = ∫ ⟨ν ∧ (δu)∂Ω , ν ∧ (ν ∨ υ∂Ω )⟩ dσ ∂Ω
= − ∫ ⟨ΛDN (f), ν ∧ g⟩ dσ,
(8.3.26)
∂Ω
thanks to (8.3.24), (2.4.3)–(2.4.6), (8.3.1), and the ﬁrst boundary condition in (8.3.21). This ﬁnishes the proof of (8.3.20).
344  8 Additional Results and Applications
Since the righthand side of (8.3.20) is symmetric in u and υ, it follows that the lefthand side of (8.3.20) must also be symmetric in f and g. This yields (8.3.6). Let us also note that, when p = 2, identity (8.3.20) with f = g implies ∫ ⟨ΛDN (f), ν ∧ f ⟩ dσ = ∫δu2 dVol ≥ 0
(8.3.27)
Ω
∂Ω 2,δ
for every f ∈ Ltan (∂Ω, Λ l−1 T M). This establishes (8.3.7). p,δ Going further, suppose f ∈ Ltan (∂Ω, Λ l−1 T M) is such that ΛDN (f) = 0. Let u solve (8.3.2). Then by deﬁnition n.t. ν ∧ (δu)∂Ω = 0 at σa.e. point on ∂Ω,
(8.3.28)
which, together with (8.3.24) goes to show that δu ∈ H∧l−1 (Ω). In particular, Proposition 6.12 gives N(δu) ∈ ⋂ L q (∂Ω). (8.3.29) 1 n. Then the DirichlettoNeumann mapping ΨDN , deﬁned in (1.4.43), is a Fredholm operator with index zero in that context. l introduced in (1.4.19) and denote Proof. To justify this claim, recall the space NΩ,V by Ker (SCHNp )l the space of nullsolutions of the Neumann boundary value problem (1.4.31). By unraveling deﬁnitions one may readily verify that l ⊆ Ker (SCHNp )l . NΩ,V
(8.3.39)
In turn, this permits us to consider the mapping p
Φ : Ker (ΨDN : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)) → Φf := u f +
l NΩ,V
for each f ∈
p L1 (∂Ω, Λ l T M)
Ker (SCHNp )l l NΩ,V
(8.3.40)
so that ΨDN f = 0,
where u f is related to f as in (1.4.42). Clearly, Φ is a welldeﬁned and linear mapping. l which then forces Moreover, if the form f belongs to the nullspace of Φ then u f ∈ NΩ,V n.t. f = u f ∂Ω = 0. Hence, Φ is also injective which goes to show that p
dim Ker (ΨDN : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)) ≤ dim (
Ker (SCHNp )l l NΩ,V
)
l < ∞, = dim Ker (SCHNp )l − dim NΩ,V
(8.3.41)
thanks to the fact that the Neumann boundary value problem (1.4.31) is Fredholm solvl ≤ dim NΩl = b l (Ω, M) < ∞. able, and the fact that dim NΩ,V To proceed, let us consider V := {∇ν♯ u : u ∈ C 1 (Ω, Λ l T M), (∆HL − V)u = 0 in Ω,
(8.3.42)
and N u, N(∇u) ∈ L (∂Ω)}. p
The Fatou result proved in Proposition 5.12 ensures that this is a welldeﬁned subspace of L p (∂Ω, Λ l T M). Moreover, since the Neumann boundary value problem (1.4.31) is Fredholm solvable, we ultimately conclude that V is a closed subspace of ﬁnite codimension in L p (∂Ω, Λ l T M).
(8.3.43)
Pick now an arbitrary g ∈ V . By design, there exists a form u ∈ C 1 (Ω, Λ l T M) satisfying (∆HL − V)u = 0 in Ω and N u, N(∇u) ∈ L p (∂Ω), and such that ∇ν♯ u = g. By once again invoking Proposition 5.12, it follows that p n.t. f := u∂Ω exists σa.e. on ∂Ω and belongs to L1 (∂Ω, Λ l T M),
(8.3.44)
8.4 Fatou Type Results with Additional Constraints or Regularity Conditions 
347
where σ := H n−1 ⌊∂Ω denotes the surface measure on ∂Ω. If we now let u f be related l to this f as in (1.4.42), then ω := u − u f ∈ NΩ,V so ΨDN f = ∇ν♯ u f = ∇ν♯ u = g. p L1 (∂Ω,
This proves that V ⊆ Im (ΨDN : (8.3.43), establishes the estimate
Λ l T M)
→
L p (∂Ω,
(8.3.45) Λ l T M))
which, by virtue of
p
dim CoKer (ΨDN : L1 (∂Ω, Λ l T M) → L p (∂Ω, Λ l T M)) ≤ dim (
L p (∂Ω, Λ l T M) ) < ∞. V
(8.3.46)
Collectively, (8.3.41) and (8.3.46) permit us to conclude that the DirichlettoNeumann mapping ΨDN is a Fredholm operator in the context of (1.4.43).
(8.3.47)
A more direct approach to (8.3.47), based on layer potentials, which also shows that the index of ΨDN is zero, is as follows. Given the nature of the conclusion we seek there is no loss of generality in assuming that ∂Ω intersects each connected component of M.
(8.3.48)
̃ as in (3.1.13). AsGranted this, it possible to extend the potential V to a function V sociated with this, consider boundary layer potentials in the domain Ω as before. In p particular, since the operator (5.2.1) is an isomorphism, for any f ∈ L1 (∂Ω, Λ l T M) the differential form u := Sl (S−1 (8.3.49) l f) in Ω solves the Regularity problem with boundary datum f . As such, n.t. ⊤ −1 1 ΨDN f = ΨDN (u∂Ω ) = ∇ν♯ (Sl (S−1 l f)) = (− 2 I + K l )(S l f),
(8.3.50)
which goes to show that −1 ΨDN = (− 12 I + K ⊤ l ) ∘ Sl
(8.3.51)
p as linear and bounded operators from L1 (∂Ω, Λ l T M) into L p (∂Ω, Λ l T M). Given that p 1 ⊤ l p l the operator S−1 l : L 1 (∂Ω, Λ T M) → L (∂Ω, Λ T M) is an isomorphism and − 2 I + K l p l is a Fredholm operator with index zero on L (∂Ω, Λ T M), we conclude that ΨDN is p a Fredholm operator with index zero from L1 (∂Ω, Λ l T M) into L p (∂Ω, Λ l T M). This
ﬁnishes the proof of the proposition.
8.4 Fatou Type Results with Additional Constraints or Regularity Conditions In this section we revisit the topic of Fatou theorems. Compared with Theorem 6.6, the aim here is to study the nature of boundary traces under various additional regularity assumptions. Our ﬁrst result of this nature reads as follows.
348  8 Additional Results and Applications Theorem 8.19. Fix p ∈ (1, ∞) along with l ∈ {0, 1, . . . , n} and assume Ω ⊂ M is an εSKT domain with ε > 0 sufficiently small relative to the given p as well as the Ahlfors regularity constants and local John constants of Ω. Let ν and σ denote, respectively, the outward unit conormal and surface measure of Ω. In this context, consider a differential form satisfying u ∈ C 1 (Ω, Λ l T M),
∆HL u = 0 in Ω,
N u, N(du) ∈ L p (∂Ω).
(8.4.1)
In particular, by Theorem 6.6, the nontangential boundary traces
Then
n.t. n.t. u∂Ω and (du)∂Ω exist at σa.e. point on ∂Ω.
(8.4.2)
p,δ n.t. ν ∨ u∂Ω ∈ Ltan (∂Ω, Λ l−1 T M) ⇐⇒ N(δu) ∈ L p (∂Ω)
(8.4.3)
plus naturally accompanying estimates. In addition, if either condition in (8.4.3) holds n.t. then the nontangential boundary trace (δu)∂Ω also exists at σa.e. point on ∂Ω. As a p,δ
consequence, corresponding to l = 1 (a scenario in which Ltan (∂Ω, Λ0 T M) = L p (∂Ω); cf. (2.4.31)), one has u ∈ C 1 (Ω, Λ1 T M), } } } { N(δu) ∈ L p (∂Ω) and ⇒ { ∆HL u = 0 in Ω, } n.t. } } (δu)∂Ω exists at σa.e. point on ∂Ω. { N u, N(du) ∈ L p (∂Ω)}
(8.4.4)
Moreover, under the assumptions in (8.4.1), the following equivalences are also valid: n.t. (8.4.5) ν ∨ (du)∂Ω = 0 ⇐⇒ du = 0 in Ω, and p,0 n.t. ⊥ n.t. ν ∨ u∂Ω ∈ Ltan (∂Ω, Λ l−1 T M) ∩ {H∨l−1 (Ω)∂Ω } } } ⇐⇒ δu = 0 in Ω, p,0 n.t. ν ∨ (du)∂Ω ∈ Ltan (∂Ω, Λ l T M) }
(8.4.6)
as well as p,δ n.t. ν ∨ (du)∂Ω ∈ Ltan (∂Ω, Λ l T M) ⇐⇒ N(δdu) ∈ L p (∂Ω)
⇐⇒ N(dδu) ∈ L p (∂Ω),
(8.4.7)
plus naturally accompanying estimates. Proof. The leftpointing implication in (8.4.3) is a direct consequence of Proposition 2.41. To prove the rightpointing implication displayed in (8.4.3), let us introduce p,δ n.t. f := ν ∨ u∂Ω ∈ Ltan (∂Ω, Λ l−1 T M). Also, since ∆HL (du) = 0 in Ω, Theorem 6.6 applies n.t. and gives that the nontangential boundary trace (du)∂Ω exists at σa.e. point on ∂Ω. p n.t. Hence, g := ν ∧ (du)∂Ω is welldeﬁned and belongs to Ltan (∂Ω, Λ l T M). In particular, the given form u solves the boundary value problem (1.1.5) with boundary data (f, g).
8.4 Fatou Type Results with Additional Constraints or Regularity Conditions
 349
Granted this, item (4) of Theorem 1.1 may be invoked to conclude that N(δu) ∈ L p (∂Ω). Once this has been established, Theorem 6.6 applies and gives that the nontangential n.t. boundary trace (δu)∂Ω also exists at σa.e. point on ∂Ω. Finally, the equivalences in (8.4.5) and (8.4.6) are seen directly from items (6) and (8) in Theorem 1.1, the ﬁrst the equivalence in (8.4.7) is implied by (8.4.3) (written for du in place of u), while the second equivalence in (8.4.7) is clear from the harmonicity of u and (1.0.3). We wish to remark that, in the geometric setting of Theorem 8.19, we may actually decouple the trace results jointly recorded in (8.4.2) as follows: u ∈ C 0 (Ω, Λ1 T M),} } } n.t. ⇒ u∂Ω exists at σa.e. point on ∂Ω, ∆HL u = 0 in Ω, } } } N u ∈ L p (∂Ω) }
(8.4.8)
u ∈ C 1 (Ω, Λ1 T M),} } } n.t. ⇒ (du)∂Ω exists at σa.e. point on ∂Ω. ∆HL u = 0 in Ω, } } } N(du) ∈ L p (∂Ω) }
(8.4.9)
and
Indeed, (8.4.8) is contained in Theorem 6.6, while (8.4.9) follows by applying (8.4.8) to the harmonic form du. In fact, by the same token we also have u ∈ C 1 (Ω, Λ1 T M),} } } n.t. ⇒ (δu)∂Ω exists at σa.e. point on ∂Ω. ∆HL u = 0 in Ω, } } } N(δu) ∈ L p (∂Ω) }
(8.4.10)
Next, we record a couple of natural consequences of the above theorem. The ﬁrst result, similar in spirit to Theorem 8.19, emphasizes normal boundary traces. Corollary 8.20. Retain the same background geometric setting as in Theorem 8.19. Then for any differential form u satisfying u ∈ C 1 (Ω, Λ l T M), one has and
∆HL u = 0 in Ω,
N u, N(δu) ∈ L p (∂Ω)
(8.4.11)
p,d n.t. ν ∧ u∂Ω ∈ Lnor (∂Ω, Λ l+1 T M) ⇐⇒ N(du) ∈ L p (∂Ω),
(8.4.12)
n.t. ν ∧ (δu)∂Ω = 0 ⇐⇒ δu = 0 in Ω,
(8.4.13)
as well as p,d n.t. ν ∧ (δu)∂Ω ∈ Lnor (∂Ω, Λ l T M) ⇐⇒ N(dδu) ∈ L p (∂Ω)
⇐⇒ N(δdu) ∈ L p (∂Ω),
(8.4.14)
350  8 Additional Results and Applications
and p,0 n.t. n.t. ⊥ ν ∧ u∂Ω ∈ Lnor (∂Ω, Λ l+1 T M) ∩ {H∧l+1 (Ω)∂Ω } } } ⇐⇒ du = 0 in Ω. p,0 n.t. ν ∧ (δu)∂Ω ∈ Lnor (∂Ω, Λ l T M) }
(8.4.15)
Proof. This follows from Theorem 8.19 and Hodge duality. In the second consequence mentioned above we identify natural conditions guaranteeing that the HodgeLaplace partial differential equation ∆HL u = 0 in Ω decouples into δdu = 0 and dδu = 0 in Ω. Corollary 8.21. Assume the same background geometric setting as in Theorem 8.19. Then for any differential form u satisfying u ∈ C 1 (Ω, Λ l T M),
∆HL u = 0 in Ω,
N(du), N(δu) ∈ L p (∂Ω)
(8.4.16)
the nontangential traces n.t. n.t. (du)∂Ω , (δu)∂Ω exist at σa.e. point on ∂Ω,
(8.4.17)
and one has n.t.
N(δdu) ∈ L p (∂Ω) and ν ∨ (du)∂Ω ∈ Ltan (∂Ω, Λ l T M) p,0
⇐⇒ dδu = 0 in Ω ⇐⇒ δdu = 0 in Ω p,0 n.t. ⇐⇒ N(dδu) ∈ L p (∂Ω) and ν ∧ (δu)∂Ω ∈ Lnor (∂Ω, Λ l T M).
(8.4.18)
Proof. Given that ∆HL (du) = 0 and ∆HL (δu) = 0 in Ω, Theorem 6.6 applies and gives that n.t. n.t. the nontangential boundary traces (du)∂Ω and (δu)∂Ω exist at σa.e. point on ∂Ω. p,0 n.t. Next, assuming that N(δdu) ∈ L p (∂Ω) and ν ∨ (du)∂Ω ∈ Ltan (∂Ω, Λ l T M), it follows from Proposition 2.41 and (2.4.27) that n.t. n.t. n.t. 0 = δ ∂ (ν ∨ (du)∂Ω ) = −ν ∨ (δ(du))∂Ω = ν ∨ (d(δu))∂Ω .
(8.4.19)
With this in hand, (8.4.5) allows us to conclude that dδu = 0 in Ω. This proves the rightpointing implication of the ﬁrst equivalence in (8.4.18), and the leftpointing implication of the last equivalence in (8.4.18) is established in a similar fashion (making use of (8.4.13)). All remaining implications in (8.4.18) are clear, so the proof of the corollary is complete. The effect of the change in the integrability exponent of the nontangential trace is studied below. Theorem 8.22. Select p, q ∈ (1, ∞) along with l ∈ {0, 1, . . . , n} and suppose Ω ⊂ M is an εSKT domain with ε > 0 sufficiently small relative to the given p, q, as well as the Ahlfors regularity constants and local John constants of Ω. Deﬁne σ := H n−1 ⌊∂Ω. In this setting, pick a potential V ∈ L r (Ω) for some r > n, which is real and nonnegative
(8.4.20)
8.4 Fatou Type Results with Additional Constraints or Regularity Conditions
 351
and consider a differential form u satisfying u ∈ C 0 (Ω, Λ l T M),
(∆HL − V)u = 0 in Ω,
N u ∈ L p (∂Ω).
(8.4.21)
Then n.t. u∂Ω ∈ L q (∂Ω, Λ l T M) ⇐⇒ N u ∈ L q (∂Ω),
(8.4.22)
q n.t. u∂Ω ∈ L1 (∂Ω, Λ l T M) ⇐⇒ N(∇u) ∈ L q (∂Ω),
(8.4.23)
and
plus naturally accompanying estimates in each case. Moreover, n.t. { (∇u)∂Ω exists σa.e. on ∂Ω, and q ∗ l n.t. { (∇u)∂Ω ∈ L1 (∂Ω, T M ⊗ Λ T M),
N(∇u) ∈ L q (∂Ω) ⇒ {
(8.4.24)
again accompanied by a natural estimate. Proof. Given the nature of the conclusions we seek, there is no loss of generality in assuming that the domain Ω does not contain any connected component of M. Granted this, all implications (as well as all accompanying estimates) may be justiﬁed with the help of (6.2.7)–(6.2.9) in Theorem 6.6, item (3) in Theorem 5.10, Corollary 9.20 (cf. (9.5.56) in this regard), and item (3) in Theorem 4.9. Here is a consequence of the above result, in the spirit of the classical Fractional Integration Theorem. Corollary 8.23. Fix p ∈ (1, ∞) along with l ∈ {0, 1, . . . , n} and let Ω ⊂ M be an εSKT domain with ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. Set σ := H n−1 ⌊∂Ω, and consider a potential V ∈ L r (Ω) for some r > n, which is real and nonnegative. Then any differential form satisfying u ∈ C 1 (Ω, Λ l T M),
(∆HL − V)u = 0 in Ω,
N(∇u) ∈ L p (∂Ω)
(8.4.25)
necessarily has N u ∈ L p∗ (∂Ω),
(8.4.26)
where the exponent p∗ is as in (7.1.41). In addition,
and
n.t. n.t. the nontangential traces u∂Ω , (∇u)∂Ω exist σa.e. on ∂Ω,
(8.4.27)
n.t. (∇u)∂Ω ∈ L p (∂Ω, T ∗ M ⊗ Λ l T M),
(8.4.28)
n.t. u∂Ω ∈ L p∗ (∂Ω, Λ l T M),
plus naturally accompanying estimates.
352  8 Additional Results and Applications
Proof. The membership in (8.4.26) follows from Theorem 8.22 and the fact that, as p seen from (9.5.39), if p∗ is as in (7.1.41) then the embedding L1 (∂Ω) → L p∗ (∂Ω) holds. Having proved (8.4.26), the properties recorded in (8.4.27) and (8.4.28) also follow from Theorem 8.22. p,q
To state our last result in this section, we let B s , where 0 < p, q ≤ ∞ and s ∈ ℝ, denote the classical scale of Besov spaces (cf., e.g., [123] and the references therein). Theorem 8.24. Pick p ∈ (1, ∞), l ∈ {0, 1, . . . , n}, and assume Ω ⊂ M is an εSKT domain with ε > 0 sufficiently small relative to the given p, as well as the Ahlfors regularity constants and local John constants of Ω. In this setting, select a potential V ∈ L∞ (Ω) which is real and nonnegative
(8.4.29)
and consider a differential form satisfying p,1
u ∈ B1/p (Ω, Λ l T M)
and (∆HL − V)u = 0 in Ω.
(8.4.30)
Then one necessarily has N u ∈ L p (∂Ω), plus a naturally accompanying estimate. Proof. To being with, observe that the regularity result recorded in (2.1.115) ensures that u ∈ C γ (Ω, Λ l T M) for every γ < 2. To proceed, let Ω j ↗ Ω as j → ∞ be an approximating sequence of domains, as in Proposition 2.35. Then for each j ∈ ℕ we have uΩ j ∈ C 0 (Ω j , Λ l T M). In particular, if Nj is the nontangential maximal operator rel ative to Ω j , then Nj (uΩ j ) ∈ L p (∂Ω j ) and f j := u∂Ω j ∈ L p (∂Ω j , Λ l T M). Moreover, by invoking [10, (8.18), p. 4408] (applied to each Ω j ) and [123, Theorem 18.6, p. 139] we may conclude that sup ‖f j ‖L p (∂Ω j ,Λ l TM) ≤ C‖u‖B p,1 (Ω,Λ l TM) (8.4.31) 1/p
j∈ℕ
for some constant C = C(Ω, p) ∈ (0, ∞). With this in hand, the same type of argument as in the last part of the proof of Theorem 6.6 applies and shows that u has the integral representation (6.2.7) for some f ∈ L p (∂Ω, Λ l T M) satisfying ‖f‖L p (∂Ω,Λ l TM) ≤ C‖u‖B p,1 (Ω,Λ l TM) .
(8.4.32)
1/p
From this, the desired conclusion follows by virtue of the CalderónZygmund theory from § 9.9.
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property The goal in this section is to introduce weak tangential and normal traces on the boundaries of regular SKT domains satisfying what we call Friedrichs property, and study how the regularity of one inﬂuences the regularity of the other. We begin by formally deﬁning the Friedrichs property.
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property
 353
Deﬁnition 8.25. A nonempty open subset Ω of M is said to have the Friedrichs property provided for any degree l ∈ {0, 1, . . . , n}, any integrability exponent p ∈ (1, ∞), and any differential form u ∈ L p (Ω, Λ l T M) with du ∈ L p (Ω, Λ l+1 T M), one can ﬁnd a sequence u j ∈ C 1 (Ω, Λ l T M), j ∈ ℕ, with the property that u j → u in L p (Ω, Λ l T M) as j → ∞,
(8.5.1)
du j → du in L p (Ω, Λ l+1 T M) as j → ∞.
Classically (cf. K. O. Friedrichs [35] and L. Hörmander [53]), the equality of weak and strong exterior derivatives implies that any Lipschitz domain has the Friedrichs property.
(8.5.2)
By Hodge starduality, it follows from Deﬁnition 8.25 that if the domain Ω ⊂ M has the Friedrichs property then any degree l ∈ {0, 1, . . . , n}, integrability exponent p ∈ (1, ∞), and differential form u ∈ L p (Ω, Λ l T M) with δu ∈ L p (Ω, Λ l−1 T M), it is possible to ﬁnd a sequence u j ∈ C 1 (Ω, Λ l T M), j ∈ ℕ, with the property that u j → u in L p (Ω, Λ l T M) as j → ∞, and
(8.5.3)
δu j → δu in L p (Ω, Λ l+1 T M) as j → ∞.
Consider next an Ahlfors regular domain Ω ⊂ M with surface measure σ and, for each l ∈ {0, 1, . . . , n} and p ∈ (1, ∞), deﬁne W p (Ω, Λ l T M) := {u ∈ C 1 (Ω, Λ l T M) : N u, N(du), N(δu) ∈ L p (∂Ω), n.t. and u∂Ω exists σa.e. on ∂Ω}.
(8.5.4)
Observe that, obviously, C 1 (Ω, Λ l T M) ⊆ W p (Ω, Λ l T M).
(8.5.5)
In this regard, let us also record the following result. Proposition 8.26. Fix l ∈ {0, 1, . . . , n} along with p ∈ (1, ∞) and assume that Ω ⊂ M is an εSKT domain with ε > 0 sufficiently small relative to the given p as well as the Ahlfors regularity constants and local John constants of Ω. Then {u ∈ C 1 (Ω, Λ l T M) : N u, N(∇u) ∈ L p (∂Ω),
(8.5.6)
and ∆HL u = 0 in Ω} ⊆ W (Ω, Λ T M). p
l
Proof. This is a consequence of the deﬁnition in (8.5.4), the pointwise estimate in (9.2.34), and the Fatou type result from Theorem 6.6. Going further, assuming that Ω ⊂ M is an Ahlfors regular domain with surface measure σ, for each degree l ∈ {0, 1, . . . , n} and integrability exponent p ∈ (1, ∞) let us also
354  8 Additional Results and Applications
introduce L p (∂Ω, Λ l T M) := { f ∈ L p (∂Ω, Λ l T M) : ∃ u ∈ W p (Ω, Λ l T M) n.t. such that f = u∂Ω at σa.e. point on ∂Ω}.
(8.5.7)
∗
In the sequel, by (L p (∂Ω, Λ l T M)) we shall denote the algebraic dual of the vector space introduced in (8.5.7). Hence, whenever p ∈ (1, ∞) is such that 1/p + 1/p = 1, the canonical duality between L p (∂Ω, Λ l T M) and L p (∂Ω, Λ l T M) induces a natural inclusion ∗ L p (∂Ω, Λ l T M) ⊆ (L p (∂Ω, Λ l T M)) . (8.5.8) We agree to denote by ((⋅, ⋅)) the duality pairing between functionals in the algebraic ∗ dual (L p (∂Ω, Λ l T M)) and elements in vector space L p (∂Ω, Λ l T M). Let us now make the following deﬁnition. Deﬁnition 8.27. Assume Ω ⊂ M is an Ahlfors regular domain with the Friedrichs property, and denote by ν its outward unit conormal. Also, ﬁx l ∈ {0, 1, . . . , n} along with p, p ∈ (1, ∞) satisfying 1/p + 1/p = 1. In this context, given a differential form u ∈ L p (Ω, Λ l T M)
with δu ∈ L p (Ω, Λ l−1 T M),
one deﬁnes
ν ∨ u ∈ (L p (∂Ω, Λ l−1 T M))
∗
(8.5.9)
(8.5.10)
as the linear functional acting on any f ∈ L p (∂Ω, Λ l−1 T M) according to ((ν ∨ u, f)) := ∫⟨u, dυ⟩ dVol − ∫⟨δu, υ⟩ dVol, Ω
Ω
where υ is any function in W p (Ω, Λ l−1 T M) n.t. such that f = υ∂Ω at σa.e. point on ∂Ω.
(8.5.11)
The functional ν ∨ u is referred to as the weak tangential trace of u on ∂Ω. Likewise, given a differential form u ∈ L p (Ω, Λ l T M)
with du ∈ L p (Ω, Λ l+1 T M),
one deﬁnes
ν ∧ u ∈ (L p (∂Ω, Λ l+1 T M)) as the linear functional acting on any f ∈
∗
L p (∂Ω, Λ l+1 T M)
(8.5.12)
(8.5.13) according to
((ν ∧ u, f)) := ∫⟨du, υ⟩ dVol − ∫⟨u, δυ⟩ dVol, Ω
Ω
where υ is any function in W p (Ω, Λ l+1 T M) n.t. such that f = υ∂Ω at σa.e. point on ∂Ω. The functional ν ∧ u is referred to as the weak normal trace of u on ∂Ω.
(8.5.14)
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property 
355
As regards the above deﬁnition, it should be noted that, for a differential form u as in (8.5.9), the linear functional described in (8.5.10) and (8.5.11) is meaningfully deﬁned, thanks to (2.2.50), and also acts in an unambiguous fashion. The latter claim is a con n.t. sequence of the fact that if υ ∈ W p (Ω, Λ l−1 T M) is such that υ∂Ω = 0 at σa.e. point on ∂Ω then (8.5.15) ∫⟨u, dυ⟩ dVol = ∫⟨δu, υ⟩ dVol. Ω
Ω
In turn, this is justiﬁed by selecting a sequence u j ∈ C 1 (Ω, Λ l T M), j ∈ ℕ, satisfying (8.5.3) and then writing ∫⟨u, dυ⟩ dVol = lim ∫⟨u j , dυ⟩ dVol j→∞
Ω
Ω
= lim ∫⟨δu j , υ⟩ dVol = ∫⟨δu, υ⟩ dVol, j→∞
Ω
(8.5.16)
Ω
where we have relied on (8.5.3) and (2.2.50) in the ﬁrst and third equalities, and have used Theorem 2.36 in the second equality. In a similar manner one can show that, for u as in (8.5.12), the linear functional described in (8.5.13) and (8.5.14) is also meaningfully deﬁned and acts in an unambiguous fashion. It turns out that checking whether the weak tangential trace vanishes requires testing only against functions which are smooth on the closure of the underlying domain. Speciﬁcally, we have the following result. Proposition 8.28. Assume Ω ⊂ M is an Ahlfors regular domain with the Friedrichs property, and ﬁx l ∈ {0, 1, . . . , n} along with p, p ∈ (1, ∞) satisfying 1/p + 1/p = 1. Then, if u is as in (8.5.9), the following statements are equivalent: ∗ (1) ν ∨ u = 0 in (L p (∂Ω, Λ l−1 T M)) ; (2) ∫Ω ⟨u, dυ⟩ dVol = ∫Ω ⟨δu, υ⟩ dVol for every differential form υ ∈ C 1 (Ω, Λ l−1 T M); (3) ∫Ω ⟨u, dυ⟩ dVol = ∫Ω ⟨δu, υ⟩ dVol for every differential form υ ∈ L p (Ω, Λ l−1 T M) with the property that dυ ∈ L p (Ω, Λ l T M). Moreover, if u is as in (8.5.12), the following statements are equivalent: ∗ (1 ) ν ∧ u = 0 in (L p (∂Ω, Λ l+1 T M)) ; (2 ) ∫Ω ⟨du, υ⟩ dVol = ∫Ω ⟨u, δυ⟩ dVol for every differential form υ ∈ C 1 (Ω, Λ l+1 T M);
(3 ) ∫Ω ⟨du, υ⟩ dVol = ∫Ω ⟨u, δυ⟩ dVol for every differential form υ ∈ L p (Ω, Λ l+1 T M) with the property that δυ ∈ L p (Ω, Λ l T M). Proof. For starters, observe that the implication (1) ⇒ (2) is a direct consequence of (8.5.11) and (8.5.5). Consider now the opposite implication. Pick an arbitrary differen tial form υ ∈ W p (Ω, Λ l−1 T M) and note that this entails
υ ∈ L p (Ω, Λ l−1 T M)
and
dυ ∈ L p (Ω, Λ l T M)
(8.5.17)
by (2.2.50), given that Ω is a bounded Ahlfors regular domain. Since Ω also has the Friedrichs property, it is then possible to ﬁnd a sequence υ j ∈ C 1 (Ω, Λ l−1 T M), j ∈ ℕ,
356  8 Additional Results and Applications
such that
υ j → υ in L p (Ω, Λ l−1 T M) and dυ j → dυ in L p (Ω, Λ l T M) as j → ∞.
(8.5.18)
Then, granted the current working hypotheses, we may write ∫⟨u, dυ⟩ dVol = lim ∫⟨u, dυ j ⟩ dVol j→∞
Ω
Ω
= lim ∫⟨δu, υ j ⟩ dVol = ∫⟨δu, υ⟩ dVol, j→∞
Ω
(8.5.19)
Ω
∗
which goes to show that ν ∨ u = 0 in (L p (∂Ω, Λ l−1 T M)) . This proves the implication (2) ⇒ (1). Reasoning as in (8.5.18)(8.5.19) also proves (2) ⇒ (3), while (3) ⇒ (2) is obvious. Finally, the equivalence of (1 ), (2 ), (3 ), is established in a similar fashion. Regarding the compatibility of the weak traces introduced in Deﬁnition 8.27 with the nontangential pointwise traces, we remark that, as seen from Theorem 2.36 and (8.5.11), whenever the differential form u is as in (8.5.9) and, in addition, n.t. N u ∈ L p (∂Ω) and u∂Ω exists at σa.e. point on ∂Ω, it follows that ∗ the weak tangential trace ν ∨ u ∈ (L p (∂Ω, Λ l−1 T M)) coincides n.t. p l−1 with ν ∨ (u∂Ω ) ∈ L (∂Ω, Λ T M) (in the sense of (8.5.8)).
(8.5.20)
In a similar fashion, whenever the differential form u is as in (8.5.12) and, in addition, n.t. N u ∈ L p (∂Ω) and u∂Ω exists at σa.e. point on ∂Ω, it follows that ∗ the weak normal trace ν ∧ u ∈ (L p (∂Ω, Λ l+1 T M)) coincides with n.t. ν ∧ (u∂Ω ) ∈ L p (∂Ω, Λ l+1 T M) (in the sense of (8.5.8)).
(8.5.21)
The stage has been set for stating the main result in this section. Theorem 8.29. Fix l ∈ {0, 1, . . . , n} along with some p ∈ (1, ∞) and assume that Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given exponent p as well as the Ahlfors regularity constants and local John constants of Ω. Also, assume that Ω has the Friedrichs property and denote by ν the outward unit conormal to Ω. In this context, suppose the differential form u satisﬁes u ∈ L p (Ω, Λ l T M),
du ∈ L p (Ω, Λ l+1 T M),
δu ∈ L p (Ω, Λ l−1 T M).
(8.5.22)
Then, with p ∈ (1, ∞) such that 1/p + 1/p = 1, the following statements are equivalent: ∗ (1) the weak normal trace ν ∧ u ∈ (L p (∂Ω, Λ l+1 T M)) , as initially deﬁned in (8.5.14), actually belongs to L p (∂Ω, Λ l+1 T M);
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property  357
∗
(2) the weak tangential trace ν ∨ u ∈ (L p (∂Ω, Λ l−1 T M)) as initially deﬁned in (8.5.11), actually belongs to L p (∂Ω, Λ l−1 T M). Moreover, there exists a constant C = C(Ω, p) ∈ (0, ∞) such that, whenever (1) and (2) above are satisﬁed, max {‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) , ‖ν ∨ u‖L p (∂Ω,Λ l−1 TM) } ≤ C‖u‖L p (Ω,Λ l TM) + C‖du‖L p (Ω,Λ l+1 TM) + C‖δu‖L p (Ω,Λ l−1 TM) + C min {‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) , ‖ν ∨ u‖L p (∂Ω,Λ l−1 TM) }.
(8.5.23)
Finally, pick a realvalued, nonnegative function V ∈ L∞ (M), and denote by Γ l (x, y) −1 the Schwartz kernel of the operator (∆HL − V) (cf. (3.1.5)). Associated with this fundamental solution and the given domain Ω, deﬁne Newtonian potentials as in (7.1.37) and boundary layer potentials as in § 3.2. Then any differential form u satisfying (8.5.22) as well as (1) or (2) above admits the integral representation formula at a.e. point x ∈ Ω: u(x) = − dΠ l−1 (δu)(x) − δΠ l+1 (du)(x) − Π l (Vu)(x) − ∫⟨Q l−1 (x, y), (δu)(y)⟩ dVol(y) Ω
− ∫⟨R l (x, y), (du)(y)⟩ dVol(y) Ω
+ δSl+1 (ν ∧ u)(x) − R l (ν ∧ u)(x) − dSl−1 (ν ∨ u)(x) + Q l−1 (ν ∨ u)(x).
(8.5.24)
As a consequence of (8.5.24) (with V constant) and the mapping properties of the operators involved, it follows that any differential form u satisfying (8.5.22) as well as (1) or (2) above for p = 2, actually belongs to H 1/2,2 (Ω, Λ l T M) and the subelliptic estimate ‖u‖H 1/2,2 (Ω,Λ l TM) ≤ C‖u‖L2 (Ω,Λ l TM) + C‖du‖L2 (Ω,Λ l+1 TM) + C‖δu‖L2 (Ω,Λ l−1 TM)
(8.5.25)
+ C min {‖ν ∧ u‖L2 (∂Ω,Λ l+1 TM) , ‖ν ∨ u‖L2 (∂Ω,Λ l−1 TM) } holds for some ﬁnite constant C = C(Ω) > 0, independent of u. In particular, there exists C ∈ (0, ∞) with the property that if u satisﬁes (8.5.22) with p = 2 and either ν ∨ u = 0 or ν ∧ u = 0 in a weak sense, then the following subelliptic estimate holds: ‖u‖H 1/2,2 (Ω,Λ l TM) ≤ C‖du‖L2 (Ω,Λ l+1 TM) + C‖δu‖L2 (Ω,Λ l−1 TM) + C‖u‖L2 (Ω,Λ l TM) .
(8.5.26)
A few comments are in order here. First, the equivalence (1) ⇔ (2) in the above theorem for arbitrary l ∈ {0, 1, . . . , n} and arbitrary p ∈ (1, ∞) cannot hold in the absence of any type of inﬁnitesimal ﬂatness condition (presently expressed through the proximity of
358  8 Additional Results and Applications
ν to the space of functions of vanishing mean oscillations) for the underlying domain. This is because that even in the class of piecewise smooth domains the veracity of the said equivalence severely restricts the range of p’s. We illustrate this by proving that in conelike domains one can only expect (1) ⇔ (2) to hold in the range 2 ≤ p < ∞. Indeed, for each λ ∈ (0, 1) it is possible to construct a harmonic function υ λ in a cone like domain Ω λ ⊂ ℝ3 with vertex at the origin, satisfying ∇tan υ λ ∈ L∞ (∂Ω) and so that ∇υ λ (x) ≈ xλ−1 as x → 0 (cf. the discussion in [20, p. 167]). Then u λ := dυ λ is closed, coclosed, and satisﬁes ν ∧ u λ ∈ L∞ (∂Ω, Λ2 T M) as well as ν ∨ u λ (x) ≈ xλ−1 as x → 0. Thus, ν ∨ u λ ∈ L p (∂Ω) if and only if p ∉ (2/(1 − λ), 3/(1 − λ)). Now the union of all such intervals when λ ∈ (0, 1) is precisely (2, ∞). In particular, for each p > 2 there exists a conelike (hence, in particular, Lipschitz) domain for which (2) ⇒ (1) fails. A similar construction shows that (1) ⇒ (2) also fails in the range (2, ∞). Second, from the representation formula (8.5.24) and the mapping properties of the integral operators involved (cf. also [82] in this regard) we conclude that space differential forms u ∈ L p (Ω, Λ l T M) satisfying du ∈ L p (Ω, Λ l+1 T M), δu ∈ L p (Ω, Λ l−1 T M) and such that either (1) or (2) in the statement of Theorem 8.29 are satisﬁes, equipped with the norm ‖u‖L p (Ω,Λ l TM) + ‖du‖L p (Ω,Λ l+1 TM) + ‖δu‖L p (Ω,Λ l−1 TM) + min {‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) , ‖ν ∨ u‖L p (∂Ω,Λ l−1 TM) },
(8.5.27)
embeds compactly into L p (Ω, Λ l T M). Results of this nature have found important applications in electromagnetism and hydrodynamics; cf. [76, 106, 122], for related material in more regular settings than the one presently considered. Third, at least in the case when ν ∨ u = 0 or ν ∧ u = 0, and Ω is a C ∞ domain, estimates of the type (8.5.25) go back to Gaffney [37] and Friedrichs [36]. In such a case, due to the smoothness of the underlying domain, H 1/2,2 may be replaced by H 1,2 in the left handside of (8.5.25). More precisely, assuming M is a C ∞ Riemannian manifold and l ∈ {0, 1 . . . , n} is arbitrary, if Ω ⊂ M is a C ∞ domain, ∃ C ∈ (0, ∞) such that ∀ u ∈ L2 (Ω, Λ l T M) satisfying du ∈ L2 (Ω, Λ l+1 T M), δu ∈ L2 (Ω, Λ l−1 T M), as well as either ν ∨ u = 0 or ν ∧ u = 0 in a weak sense on ∂Ω, one has
(8.5.28)
‖u‖H 1,2 (Ω,Λ l TM) ≤ C‖du‖L2 (Ω,Λ l+1 TM) + C‖δu‖L2 (Ω,Λ l−1 TM) + C‖u‖L2 (Ω,Λ l TM) . A more reﬁned version of this result, placing considerably less stringent regularity conditions on the ambient manifold M and the domain Ω has been established in [101, Theorem 4.1] which only requires that M is a C 2 Riemannian manifold and Ω ⊂ M is what the authors call an almost convex domain. Examples of almost convex domains include (see [101] for precise deﬁnitions) domains of class C 1,1 , Lipschitz domains satisfying a local exterior ball condition, and domains with corners.
(8.5.29)
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property 
359
In particular, the estimate in (8.5.28) holds in these classes of domains. This being said, the current exponent 1/2 in (8.5.26) is sharp even in the class of Lipschitz domains and for smooth metrics, as simple counterexamples show. For Lipschitz domains, estimates in the spirit of (8.5.25) have been proved in [55, 86, 101]. Fourth, as a consequence of (8.5.24) (with V supported away from Ω) and the local regularity result recorded in (2.1.116), in the context of Theorem 8.29 we have the following weak characterization of Dirichlet harmonic ﬁelds: {u ∈ C 1 (Ω, Λ l T M) : N u ∈ L p (∂Ω), du = 0 in Ω, δu = 0 in Ω, n.t. and ν ∨ u∂Ω = 0 in L p (∂Ω, Λ l−1 T M)} = {u ∈ L p (Ω, Λ l T M) : du = 0 in Ω, δu = 0 in Ω,
∗
and ν ∨ u = 0 in (L p (∂Ω, Λ l−1 T M)) }.
(8.5.30)
Of course, a similar weak characterization of Neumann harmonic ﬁelds is also valid in the same setting, namely {u ∈ C 1 (Ω, Λ l T M) : N u ∈ L p (∂Ω), du = 0 in Ω, δu = 0 in Ω, n.t. and ν ∧ u∂Ω = 0 in L p (∂Ω, Λ l+1 T M)} = {u ∈ L p (Ω, Λ l T M) : du = 0 in Ω, δu = 0 in Ω,
∗
and ν ∧ u = 0 in (L p (∂Ω, Λ l+1 T M)) }.
(8.5.31)
We are now ready to present the proof of Theorem 8.29. Proof of Theorem 8.29. For starters, observe that it suffices to prove that (1) ⇒ (2) since the opposite implication follows from this and an application of the Hodge starisomorphism. With this goal in mind, ﬁx a strictly positive constant potential V and, associated with this choice of the potential and the given domain Ω, consider boundary layer potentials as in § 3. Assume that u satisﬁes (8.5.22) and that, in addition, (1) holds. To proceed, pick an arbitrary form
f ∈ L p (∂Ω, Λ l−1 T M) ⊆ L p (∂Ω, Λ l−1 T M).
(8.5.32)
n.t. Hence, there exists υ ∈ W p (Ω, Λ l−1 T M) such that f = υ∂Ω at σa.e. point on ∂Ω, p ,d where σ denotes the surface measure on ∂Ω. In particular, ν ∧ f ∈ Lnor (∂Ω, Λ l T M) by (2.4.59). Granted this, Theorem 3.7 ensures the existence of
p ,d
g ∈ Lnor (∂Ω, Λ l T M)
such that ( 12 I + N l )g = ν ∧ f.
(8.5.33)
Thus, if we now set w := δSl g on Ω,
(8.5.34)
it follows that w ∈ C 1 (Ω, Λ l T M),
(∆HL − V)w = 0 in Ω, δw = 0 in Ω, n.t. the nontangential boundary trace w∂Ω exists σa.e. on ∂Ω, n.t. n.t. and ν ∧ w∂Ω = ( 12 I + N l )g = ν ∧ f = ν ∧ υ∂Ω on ∂Ω.
(8.5.35)
360  8 Additional Results and Applications
Moreover, (3.2.40) yields dw = −δSl+1 (d ∂ g) − V Sl g in Ω,
(8.5.36)
so, for some constant C = C(Ω, V, p) ∈ (0, ∞), the CalderónZygmund theory ensures that N w‖L p (∂Ω) ≤ C‖g‖L p (∂Ω,Λ l TM) and (8.5.37) N(dw)‖L p (∂Ω) ≤ C‖g‖ p ,d . l L (∂Ω,Λ TM) nor
Collectively, (8.5.35) and (8.5.37) imply that
w − υ ∈ W p (Ω, Λ l−1 T M)
n.t. ν ∧ (w − υ)∂Ω = 0.
and
(8.5.38)
We claim that ∫⟨u, d(w − υ)⟩ dVol = ∫⟨δu, w − υ⟩ dVol. Ω
(8.5.39)
Ω
To justify this claim, select a sequence u j ∈ C 1 (Ω, Λ l T M), j ∈ ℕ, satisfying (8.5.3) and write ∫⟨u, d(w − υ)⟩ dVol = lim ∫⟨u j , d(w − υ)⟩ dVol
(8.5.40)
j→∞
Ω
Ω
= lim ∫⟨δu j , w − υ⟩ dVol = ∫⟨δu, w − υ⟩ dVol, j→∞
Ω
Ω
where the ﬁrst and third equalities are consequences of (8.5.3) and (2.2.50), while the second equality uses Theorem 2.36 and (8.5.38). Hence, (8.5.39) is proved. We may then write ((ν ∨ u, f)) = − ∫⟨δu, υ⟩ dVol + ∫⟨u, dυ⟩ dVol Ω
Ω
= ∫⟨δu, w − υ⟩ dVol − ∫⟨δu, w⟩ dVol Ω
Ω
− ∫⟨u, d(w − υ)⟩ dVol + ∫⟨u, dw⟩ dVol Ω
Ω
= ∫⟨u, dw⟩ dVol − ∫⟨δu, w⟩ dVol Ω
Ω
= − ∫⟨u, δSl+1 (d ∂ g)⟩ dVol − V ∫⟨u, Sl g⟩ dVol Ω
Ω
− ∫⟨δu, w⟩ dVol Ω
= − ∫⟨du, Sl+1 (d ∂ g)⟩ dVol + ((ν ∧ u, S l+1 (d ∂ g))) Ω
− V ∫⟨u, Sl g⟩ dVol − ∫⟨δu, w⟩ dVol Ω
Ω
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property
 361
= − ∫⟨du, dSl g⟩ dVol + ∫ ⟨ν ∧ u, dS l g⟩ dσ Ω
∂Ω
− V ∫⟨u, Sl g⟩ dVol − ∫⟨δu, w⟩ dVol. Ω
(8.5.41)
Ω
Above, the ﬁrst equality follows from (8.5.11) and the choice of υ. The second equality is plain algebra, while the third equality is a consequence of (8.5.39). Next, the fourth equality uses (8.5.36), whereas the ﬁfth equality is based on (8.5.14) and the observation that n.t. Sl+1 (d ∂ g) ∈ W p (Ω, Λ l+1 T M) and S l+1 (d ∂ g) = (Sl+1 (d ∂ g))∂Ω on ∂Ω. (8.5.42) Finally, the ﬁfth equality in (8.5.41) is seen by reconverting Sl+1 (d ∂ g) into dSl g, and S l+1 (d ∂ g) into dS l g (on account of (3.2.40) and (3.2.42), with R l−1 = 0 and Q l = 0 in the present case) and recalling that, by assumption,
∗
ν ∧ u ∈ L p (∂Ω, Λ l+1 T M) ⊆ (L p (∂Ω, Λ l+1 T M)) .
(8.5.43)
In turn, based on (8.5.41), Hölder’s inequality, (3.2.8), (8.5.37), (2.2.50), and the fact that the principal value singular integral operator dS l is bounded on L p , we are then able to estimate ((ν ∨ u, f)) ≤ C‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) ‖g‖L p (∂Ω,Λ l TM) + C‖du‖L p (Ω,Λ l+1 TM) ‖g‖L p (∂Ω,Λ l TM) + C‖δu‖L p (Ω,Λ l−1 TM) ‖g‖L p (∂Ω,Λ l TM) + C‖u‖L p (Ω,Λ l TM) ‖g‖L p (∂Ω,Λ l TM) .
(8.5.44)
Since, from Theorem 3.9 we also have ‖g‖L p (∂Ω,Λ l TM) ≤ C‖f‖L p (∂Ω,Λ l−1 TM) , we ﬁnally conclude from (8.5.44) that (8.5.45) ((ν ∨ u, f)) ≤ C{‖u‖L p (Ω,Λ l TM) + ‖du‖L p (Ω,Λ l+1 TM) + ‖δu‖L p (Ω,Λ l−1 TM) + ‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) } ‖f‖L p (∂Ω,Λ l−1 TM) ,
for every f ∈ L p (∂Ω, Λ l−1 T M), where the constant C = C(Ω, V, p) ∈ (0, ∞) is independent of the differential forms u and f . The next step in the proof is to show that, in the present setting,
L p (∂Ω, Λ l−1 T M) → L p (∂Ω, Λ l−1 T M) densely.
(8.5.46)
This follows from the density result recorded in (9.4.13), the solvability of the Regularity problem (1.4.25), Proposition 8.26, and the deﬁnitions in (8.5.4)–(8.5.7). Having established (8.5.46), Riesz’s representation theorem applies and gives that ∗ ν ∨ u, originally deﬁned in (8.5.11) as a linear functional in (L p (∂Ω, Λ l−1 T M)) , actually belongs to L p (∂Ω, Λ l−1 T M) and ‖ν ∨ u‖L p (∂Ω,Λ l−1 TM) ≤ C{‖u‖L p (Ω,Λ l TM) + ‖du‖L p (Ω,Λ l+1 TM) + ‖δu‖L p (Ω,Λ l−1 TM) + ‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) },
(8.5.47)
362  8 Additional Results and Applications for some constant C = C(Ω, V, p) ∈ (0, ∞) independent of u. This ﬁnishes the proof of (1) ⇒ (2). As already noted, (2) ⇒ (1) follows from this and Hodge duality which, in view of (8.5.47) also gives ‖ν ∧ u‖L p (∂Ω,Λ l+1 TM) ≤ C{‖u‖L p (Ω,Λ l TM) + ‖du‖L p (Ω,Λ l+1 TM)
(8.5.48)
+ ‖δu‖L p (Ω,Λ l−1 TM) + ‖ν ∨ u‖L p (∂Ω,Λ l−1 TM) }, whenever u is as in (2). Together, (8.5.47) and (8.5.48) imply (8.5.23). To ﬁnish the proof of the theorem, there remains to prove the integral representation formula (8.5.24). With this goal in mind, consider an arbitrary open set O with the property that O ⊂ Ω and pick a scalarvalued function ψ ∈ C 01 (Ω) with the property that ψ ≡ 1 in O. Then for a.e. x ∈ O we may write ∫ ⟨d y Γ l (x, y), ν(y) ∧ u(y)⟩ dσ(y) ∂Ω
= ∫ ⟨(1 − ψ(y))d y Γ l (x, y), ν(y) ∧ u(y)⟩ dσ(y) ∂Ω
= ((ν ∧ u, (1 − ψ)dΓ l (x, ⋅))) = ∫⟨(1 − ψ(y))d y Γ l (x, y), (du)(y)⟩ dVol(y) Ω
− ∫⟨δ y ((1 − ψ(y))d y Γ l (x, y)), u(y)⟩ dVol(y) Ω
= ∫⟨(1 − ψ(y))d y Γ l (x, y), (du)(y)⟩ dVol(y) Ω
− ∫⟨(dψ)(y) ∨ d y Γ l (x, y), u(y)⟩ dVol(y) Ω
− ∫⟨(1 − ψ(y))δ y d y Γ l (x, y), u(y)⟩ dVol(y),
(8.5.49)
Ω
by (8.5.14) and item (4) in Lemma 2.8. In a similar manner, for a.e. x ∈ O we have − ∫ ⟨δ y Γ l (x, y), ν(y) ∨ u(y)⟩ dσ(y) ∂Ω
= − ∫ ⟨(1 − ψ(y))δ y Γ l (x, y), ν(y) ∨ u(y)⟩ dσ(y) ∂Ω
= − ((ν ∨ u, (1 − ψ)δΓ l (x, ⋅))) = ∫⟨(1 − ψ(y))δ y Γ l (x, y), (δu)(y)⟩ dVol(y) Ω
− ∫⟨d y ((1 − ψ(y))δ y Γ l (x, y)), u(y)⟩ dVol(y) Ω
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property 
363
= ∫⟨(1 − ψ(y))δ y Γ l (x, y), (δu)(y)⟩ dVol(y) Ω
+ ∫⟨(dψ)(y) ∧ δ y Γ l (x, y), u(y)⟩ dVol(y) Ω
− ∫⟨(1 − ψ(y))δ y d y Γ l (x, y), u(y)⟩ dVol(y)
(8.5.50)
Ω
By combining (8.5.49) with (8.5.50) we arrive at the conclusion that for a.e. x ∈ O ∫ ⟨d y Γ l (x, y), ν(y) ∧ u(y)⟩ dσ(y) − ∫ ⟨δ y Γ l (x, y), ν(y) ∨ u(y)⟩ dσ(y) ∂Ω
∂Ω
= ∫⟨(1 − ψ(y))d y Γ l (x, y), (du)(y)⟩ dVol(y) Ω
− ∫⟨d y Γ l (x, y), (dψ)(y) ∧ u(y)⟩ dVol(y) Ω
− ∫⟨(1 − ψ(y))Γ l (x, y), V(y)u(y)⟩ dVol(y) Ω
+ ∫⟨(1 − ψ(y))δ y Γ l (x, y), (δu)(y)⟩ dVol(y) Ω
+ ∫⟨δ y Γ l (x, y), (dψ)(y) ∨ u(y)⟩ dVol(y)
(8.5.51)
Ω
Writing, by item (4) in Lemma 2.8, (dψ) ∧ u = d(ψu) − ψdu
and
(dψ) ∨ u = −δ(ψu) + ψδu
(8.5.52)
then permits us to further combine terms naturally in the right handside of (8.5.51) and express ∫ ⟨d y Γ l (x, y), ν(y) ∧ u(y)⟩ dσ(y) − ∫ ⟨δ y Γ l (x, y), ν(y) ∨ u(y)⟩ dσ(y) ∂Ω
∂Ω
= ∫⟨d y Γ l (x, y), (du)(y)⟩ dVol(y) Ω
− ∫⟨d y Γ l (x, y), d(ψu)(y)⟩ dVol(y) Ω
− ∫⟨(1 − ψ(y))Γ l (x, y), V(y)u(y)⟩ dVol(y) Ω
+ ∫⟨δ y Γ l (x, y), (δu)(y)⟩ dVol(y) Ω
− ∫⟨δ y Γ l (x, y), δ(ψu)(y)⟩ dVol(y) Ω
(8.5.53)
364  8 Additional Results and Applications for a.e. x ∈ O. Together, the second term and the last term in the right handside of (8.5.53) then proves that for a.e. x ∈ O we have − ∫⟨d y Γ l (x, y), d(ψu)(y)⟩ dVol(y) − ∫⟨δ y Γ l (x, y), δ(ψu)(y)⟩ dVol(y) Ω
Ω
= − ∫ ⟨d y Γ l (x, y), d(ψu)(y)⟩ dVol(y) M
− ∫ ⟨δ y Γ l (x, y), δ(ψu)(y)⟩ dVol(y) M
= (ψu)(x) + ∫ ⟨Γ l (x, y), V(y)(ψu)(y)⟩ dVol(y) M
= u(x) + ∫⟨Γ l (x, y), V(y)(ψu)(y)⟩ dVol(y).
(8.5.54)
Ω
Given that O was arbitrarily chosen in Ω, we ultimately obtain u(x) = − ∫⟨δ y Γ l (x, y), (δu)(y)⟩ dVol(y) Ω
− ∫⟨d y Γ l (x, y), (du)(y)⟩ dVol(y) Ω
− ∫⟨Γ l (x, y), (Vu)(y)⟩ dVol(y) Ω
+ ∫ ⟨d y Γ l (x, y), ν(y) ∧ u(y)⟩ dσ(y) ∂Ω
− ∫ ⟨δ y Γ l (x, y), ν(y) ∨ u(y)⟩ dσ(y),
for a.e. x ∈ Ω.
(8.5.55)
∂Ω
With this in hand, formula (8.5.24) now readily follows upon recalling (3.1.39) and (3.1.51). We next propose to show that if Ω ⊂ M is an εSKT domain, with ε > 0 sufficiently small relative to the large geometry constants of Ω which, in addition, has the Friedrichs property, and satisﬁes a certain extra topological condition (described in Theorem 8.33), then l (Ω; ℝ), dim H∨l (Ω) ≥ dim Hsing
∀ l ∈ {0, 1, . . . , n},
(8.5.56)
l (Ω; ℝ) is the lth singular homology group of Ω over the reals. This is sigwhere Hsing niﬁcant since, in particular, it implies that in the class of domains just described, there are topological obstructions to the issue of uniqueness for the boundary value problem (1.1.5).
8.5 Weak Tangential and Normal Traces in Regular SKT Domains with Friedrichs Property 
365
The inequality in (8.5.56) is proved in Theorem 8.33. This requires a number of preliminary results. To set the stage, we make the following deﬁnition. Given an open set Ω ⊂ M, along with a degree l ∈ {0, 1, . . . , n} and an exponent p ∈ (1, ∞), denote by d l,p : L p (Ω, Λ l T M) → L p (Ω, Λ l+1 T M) (8.5.57) the L p realization of the exterior derivative operator d acting on lforms, as a linear unbounded operator, with domain Dom (d l,p ) := {u ∈ L p (Ω, Λ l T M) : du ∈ L p (Ω, Λ l+1 T M)},
(8.5.58)
and natural action, d l,p u := du
for every u ∈ Dom (d l,p ).
(8.5.59)
The nature of this operator and its adjoint is discussed in the proposition below. Proposition 8.30. Assume Ω ⊂ M is an Ahlfors regular domain with the Friedrichs property, and ﬁx l ∈ {0, 1, . . . , n} along with p, p ∈ (1, ∞) satisfying 1/p + 1/p = 1. Denote by ν the outward unit conormal to Ω. Then the linear unbounded operator d l,p in (8.5.57)–(8.5.59) is closed, densely deﬁned, and whose adjoint is given by
d∗l,p : L p (Ω, Λ l T M) → L p (Ω, Λ l−1 T M)
(8.5.60)
with domain
Dom (d∗l,p ) := {u ∈ L p (Ω, Λ l T M) : δu ∈ L p (Ω, Λ l−1 T M) p
ν ∨ u = 0 in (L (∂Ω, Λ
(8.5.61) l−1
∗
T M)) },
and action d∗l,p u := δu for every u ∈ Dom (d∗l,p ).
(8.5.62)
Proof. This is seen from deﬁnitions and Proposition 8.28. We continue by recording the following abstract Hodge type decomposition appearing in [46, Proposition 2.9]. Proposition 8.31. Let H be a Hilbert space and suppose T : H → H is a closed, densely deﬁned, unbounded linear operator, satisfying T2 = 0
(8.5.63)
Dom (T) ∩ Dom (T ∗ ) embeds compactly into H,
(8.5.64)
and such that Then T and T ∗ have closed ranges, the space H := Ker T ∩ Ker T ∗
(8.5.65)
366  8 Additional Results and Applications
is ﬁnite dimensional, and H = Im T ⊕ Im T ∗ ⊕ H ,
(8.5.66)
where the direct sums are orthogonal. Furthermore, Ker T = H ⊕ Im T,
Ker T ∗ = H ⊕ Im T ∗
(8.5.67)
where the direct sums are orthogonal, and H≅
Ker T Ker T ∗ . ≅ Im T Im T ∗
(8.5.68)
We shall now specialize the above abstract result to the case of the L2 realization of the exterior derivative operator. Proposition 8.32. Assume Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. In addition, suppose Ω has the Friedrichs property. Then, for every degree l ∈ {0, 1, . . . , n}, the operator d l,2 (deﬁned as in (8.5.57)– (8.5.59) with p = 2) has closed range and Ker d l,2 = Im d l−1,2 ⊕ H∨l (Ω)
(8.5.69)
where the direct sum is orthogonal. Proof. The idea is to apply Proposition 8.31 for the Hilbert space H := ⨁ L2 (Ω, Λ l T M)
(8.5.70)
0≤l≤n
and the linear unbounded operator T : H → H given by T := ⨁ d l,2 .
(8.5.71)
0≤l≤n
As indicated in Proposition 8.30, this is a closed, densely deﬁned operator. From item (1) in Lemma 2.8 it follows that the operator T just described satisﬁes (8.5.63). It also satisﬁes (8.5.64), thanks to Proposition 8.30 and (8.5.25). Granted these, Proposition 8.31 then guarantees that each operator d l,2 has closed range. Moreover, (8.5.69) is seen from (8.5.67) and (8.5.30) (used here with p = 2). We are ﬁnally ready to prove the estimate in (8.5.56). A formal statement reads as follows. Theorem 8.33. Let Ω ⊂ M be an εSKT domain for some ε > 0 sufficiently small relative to the Ahlfors regularity constants and local John constants of Ω. In addition, assume l Ω has the Friedrichs property. For each l ∈ {0, 1, . . . , n}, denote by Hsing (Ω; ℝ) the lth singular homology group of Ω over the reals, and let {γ j }1≤j≤b(l) be smooth lcycles in Ω
8.6 The HodgePoisson Kernel and the HodgeHarmonic Measure

367
l (Ω; ℝ). such that their equivalence classes {[γ j ]}1≤j≤b(l) are linearly independent in Hsing Let ι denote the inclusion of each γ j into M. Finally, assume there exists a smooth domain O ⊂ M such that Ω ⊂ O and the inclusion ȷ : Ω → O has the property that the mapping it induces at the level of singular homology groups l l ȷ∗ : Hsing (Ω; ℝ) → Hsing (O; ℝ) is injective. (8.5.72)
Then, for each l ∈ {0, 1, . . . , n}, the linear mapping assigning to each harmonic ﬁeld u its periods, i.e., Φ l : H∨l (Ω) → ℝb(l) ,
Φ l (u) := ( ∫ ι∗ u) γj
(8.5.73) 1≤j≤b(l)
is surjective. As a consequence, l (Ω; ℝ), b l (Ω) := dim H∨l (Ω) ≥ dim Hsing
∀ l ∈ {0, 1, . . . , n}.
(8.5.74)
Proof. Our hypotheses imply that the set {[γ j ]}1≤j≤b(l) is linearly independent not only l l in Hsing (Ω; ℝ), but also in Hsing (O; ℝ). It follows that these cycles are linearly indepenl (M0 ; ℝ), where M0 is the double of O. dent in Hsing By the classical de Rham theorem (cf. [24], or [48, Chapter II, pp. 92–100]), given any α j ∈ ℝ for 1 ≤ j ≤ b(l), we can ﬁnd a C 1 closed lform w on M0 such that
∫ ι∗ w = α j
for j = 1, . . . , b(l).
(8.5.75)
γj
Next, decompose wΩ ∈ C 1 (Ω, Λ l T M) ⊂ Ker d l,2 as wΩ = υ ⊕ u according to (8.5.69). Then, by Stokes’s theorem, we have Φ l (u) = (α j )1≤j≤b(l) , which goes to prove that Φ l is surjective. In closing, we wish to remark that, in the context of Theorem 8.33, condition (8.5.72) is automatically satisﬁed if, for example, Ω is a Lipschitz domain. Moreover, equality holds in (8.5.74) under suitable additional regularity assumptions on the underlying domain Ω. Indeed, by relying on an abstract de Rham theorem in sheaf theory and arguing as in [86, § 11, p. 85], this can be seen to be the case when every point x ∈ ∂Ω has an open neighborhood O with the property that an L2 Poincaré type lemma holds in the open set O ∩ Ω. In particular, this is the case if Ω is a Lipschitz domain.
8.6 The HodgePoisson Kernel and the HodgeHarmonic Measure In this section we revisit the issue of representing the solution of the Dirichlet problem for the HodgeLaplacian, and shift focus from our earlier layer potentials to what we call the HodgePoisson kernel and the Hodgeharmonic measure associated with a given domain. We begin by making the following deﬁnition.
368  8 Additional Results and Applications Deﬁnition 8.34. Let Ω ⊂ M be Ahlfors regular domain with surface measure σ, and ﬁx a degree l ∈ {0, 1, . . . , n} along with some integrability exponent p ∈ (1, ∞). Call a double form P l (x, y) with x ∈ Ω and y ∈ ∂Ω, of bidegree (l, l), a HodgePoisson kernel for Ω which is amenable to the given exponent p provided
P l (x, ⋅) ∈ Λ l T x M ⊗ L p (∂Ω, Λ l T M) and ‖P l (⋅, y)‖L p (∂Ω,Λ l TM) ∈ y
for each x ∈ Ω,
l L∞ loc (Ω, Λ T M),
(8.6.1)
where 1/p + 1/p = 1, and the integral operator with this kernel, i.e., PIl f(x) := ∫ ⟨P l (x, y), f(y)⟩ dσ(y),
x ∈ Ω,
(8.6.2)
∂Ω
enjoys the following properties for each f ∈ L p (∂Ω, Λ l T M): ∆HL (PIl f) = 0 in Ω,
(8.6.3)
N(PIl f) ∈ L (∂Ω),
(8.6.4)
n.t. (PIl f)∂Ω = f at σa.e. point on ∂Ω.
(8.6.5)
p
In such a scenario, the integral operator PIl from (8.6.2) is going to be called a HodgePoisson integral for Ω, amenable to p. Remarkably, in a regular SKT domain there exists a unique HodgePoisson kernel which is amenable to any exponent p ∈ (1, ∞). A more nuanced result of this nature is stated and proved below. Theorem 8.35. Fix an exponent p ∈ (1, ∞) and assume that Ω ⊂ M is an εSKT domain for some ε > 0 sufficiently small relative to the given exponent p, as well as the Ahlfors regularity constants and local John constants of Ω. Make the assumption that Ω does not contain any connected component of M. Then for each degree l ∈ {0, 1, . . . , n} there exists a unique HodgePoisson kernel P l (x, y) for Ω which is amenable to the given exponent p. Consequently, there exist a unique HodgePoisson integral PIl for Ω which is amenable to p. Moreover, for any given f ∈ L p (∂Ω, Λ l T M), the differential form l u := PIl f ∈ L∞ loc (Ω, Λ T M)
(8.6.6)
is the unique solution of the Dirichlet boundary value problem υ ∈ C 0 (Ω, Λ l T M), N υ ∈ L (∂Ω), p
∆HL υ = 0 in Ω, n.t. υ∂Ω = f on ∂Ω.
(8.6.7)
In particular, there exists a constant C ∈ (0, ∞) such that ‖N(PIl f)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,Λ l TM) ,
∀ f ∈ L p (∂Ω, Λ l T M).
(8.6.8)
8.6 The HodgePoisson Kernel and the HodgeHarmonic Measure

369
Proof. Let V be the potential deﬁned by V := λ 1M\Ω
for some λ > sup ‖Ricx ‖Λ l T x M→Λ l T x M
(8.6.9)
x∈M\Ω
and, for each degree l ∈ {0, 1, . . . , n}, let K l be the principal value double layer on ∂Ω associated as in (4.1.61) with the operator ∆HL − V for this choice of V. Recall that K⊤ l denotes the (real) transposed of K l . Then item (3) in Theorem 5.10 guarantees the −1 on L p (∂Ω, Λ l T M). In particular, for x ∈ Ω and existence of the inverse ( 12 I + K ⊤ l ) y ∈ ∂Ω it makes sense to consider −1
∗ P l (x, y) := (( 12 I + K ⊤ l ) (i Sym (∇ , ν(⋅))∇⋅ Γ l (x, ⋅))) (y).
(8.6.10)
Then there exists a ﬁnite constant C > 0 with the property that ‖P l (⋅, y)‖L p (∂Ω,Λ l TM) ≤ C∇y Γ l (⋅, y)L p (∂Ω,Λ l TM) y y
(8.6.11)
which, on account of (3.1.18), shows that P l (x, y) is a double form of bidegree (l, l) which satisﬁes (8.6.1). If we now consider the integral operator associated with this kernel as in (8.6.2), then for each f ∈ L p (∂Ω, Λ l T M) we may write PIl f(x) = ∫ ⟨P l (x, y), f(y)⟩ dσ(y) ∂Ω −1
∗ = ∫ ⟨(( 12 I + K ⊤ l ) (i Sym (∇ , ν(⋅))∇⋅ Γ l (x, ⋅)))(y) , f(y)⟩ dσ(y) ∂Ω −1
= ∫ ⟨i Sym (∇∗, ν(y))∇y Γ l (x, y), (( 12 I + K l ) f )(y)⟩ dσ(y) ∂Ω −1
= Dl (( 12 I + K l ) f )(x),
for each x ∈ Ω,
(8.6.12)
where the last line makes use of (4.1.6). Hence, for each f ∈ L p (∂Ω, Λ l T M) we have −1
PIl f = Dl (( 12 I + K l ) f ) in Ω.
(8.6.13)
Having established this, we may then conclude that properties (8.6.3)–(8.6.5) hold, thanks to (4.1.9) (bearing in mind that, as seen from (8.6.9), V ≡ 0 in Ω), (4.1.65) (which shows that, in fact, the stronger property (8.6.8) holds), and (4.1.79). At this point we may therefore conclude that, for each l ∈ {0, 1, . . . , n}, the double form P l (x, y) of bidegree (l, l) deﬁned as in (8.6.10) is a HodgePoisson kernel for Ω which is amenable to the given exponent p. Moreover, properties (8.6.3)–(8.6.5) show that, given any form f ∈ L p (∂Ω, Λ l T M), the differential form u deﬁned as in (8.6.6) solves the Dirichlet problem (8.6.7) (which, thanks to Theorem 1.8 and the current assumptions on Ω, is known to be uniquely solvable). Let us now address the issue of uniqueness for the HodgePoisson kernel of bidegree (l, l) for Ω which is amenable to the given exponent p. To this end, suppose
370  8 Additional Results and Applications (1)
(2)
P l (x, y) and P l (x, y) are two such HodgePoisson kernels, and consider their difference (1) (2) Θ l (x, y) := P l (x, y) − P l (x, y). (8.6.14) Pick an arbitrary f ∈ L p (∂Ω, Λ l T M) and deﬁne u(x) := ∫ ⟨Θ l (x, y), f(y)⟩ dσ(y),
x ∈ Ω.
(8.6.15)
∂Ω
Then
u ∈ C 0 (Ω, Λ l T M), N u ∈ L (∂Ω), p
∆HL u = 0 in Ω, n.t. u∂Ω = 0 on ∂Ω,
(8.6.16)
which, by the wellposedness result from Theorem 1.8, forces u = 0 in Ω. In turn, since f ∈ L p (∂Ω, Λ l T M) has been arbitrarily chosen, this implies that for each x ∈ Ω we have (1) (2) Θ l (x, ⋅) = 0 at σa.e. point on ∂Ω. Hence, P l (x, y) = P l (x, y), as wanted. A few comments are in order here. First, it is clear from (8.6.10), and the compatibility of the inverses involved in this formula when considered on various Lebesgue spaces, that the HodgePoisson kernel of bidegree (l, l) for Ω which is amenable to a given exponent p does not actually depend on p. In light of this observation, we agree to simplify the terminology by referring to P l (x, y) just as the HodgePoisson kernel of bidegree (l, l) for Ω. Second, it is remarkable that even though both Γ l (x, y) from (3.1.16) and K l from (4.1.61) depend on the choice of the potential V, the HodgePoisson kernel of bidegree (l, l) for Ω deﬁned as in (8.6.10) using these ingredients ultimately is unique, hence, independent of the choice of V as well. Third, in the context of Theorem 8.35, given a degree l ∈ {0, 1, . . . , n}, for each x ∈ Ω deﬁne the Hodgeharmonic measure with pole at x in a domain Ω as the double formvalued measure ω xl := P l (x, ⋅)σ (8.6.17) where P l (x, y) is the HodgePoisson kernel of bidegree (l, l) for Ω. This permits us to express the HodgePoisson integral PIl for Ω as PIl f(x) = ∫ ⟨f(y), dω xl (y)⟩,
x ∈ Ω,
(8.6.18)
∂Ω
and also write the unique solution of the Dirichlet problem (1.4.2) in the form u(x) = ∫ ⟨f, dω xl ⟩
for each x ∈ Ω.
(8.6.19)
∂Ω
This is formally analogous to the classical scalar case, and opens the door to studying the nature of the solution of the Dirichlet problem (1.4.2) through the prism of the Hodgeharmonic measure associated with the domain Ω.
9 Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Here we discuss some further useful material from differential geometry, geometric measure theory, harmonic analysis, functional analysis, partial differential equations, and Clifford analysis. Sections 9.1 and 9.2 treat connections on vector bundles over a Riemannian manifold M, ﬁrst for general vector bundles, then speciﬁcally for the bundles TM and Λ l TM. In Section 9.3 we introduce the Riemann curvature tensor on such a Riemannian manifold, and an associated Ricci operator, which is seen to arise to compare the HodgeLaplacian and the Bochner Laplacian. Sections 9.4 and 9.5 present results on L p Sobolev spaces of functions on ∂Ω, ﬁrst when ∂Ω is an Ahlfors regular domain in ℝn , then when ∂Ω is an Ahlfors regular domain in a Riemannian manifold M. Section 9.6 is devoted to various integration by parts formulas involving functions on ∂Ω. Section 9.7 discusses an elliptic regularity result yielding solutions in H 1/2,2 (Ω). Section 9.9 presents basic estimates for single and double layer potentials acting on functions on ∂Ω when Ω is a UR domain, ﬁrst for model operators, then for potentials arising from second order elliptic operators with coefficients of limited smoothness. The applicability of such results depends on the invertibility of members of an appropriate class of second order elliptic operators, discussed in Section 9.10. Our analysis depends not only on boundedness of various operators on L p (∂Ω) from Section 9.9, for UR domains, but also on compactness results for certain important special cases, described in Section 9.11. Here the specialization from UR domains to regular SKT domains comes to the fore. More generally, operators analyzed here are close to compact if Ω is an εSKT domain, for small ε. This chapter ends with three short sections, one on a sharp divergence theorem, one outlining the structure of Clifford algebras, and one on closed, densely deﬁned linear operators T on a Hilbert space, satisfying T 2 = 0, discussing the selfadjoint operator T + T ∗ and the connection with the selfadjoint operator TT ∗ + T ∗ T + I, of use in the analysis of the HodgeLaplacian.
9.1 Connections and Covariant Derivatives on Vector Bundles We aim to describe connections that arise on various vector bundles over a manifold M. We develop this notion for general vector bundles here, in preparation for special
372  9 Further Tools from Geometry and Analysis connections on the bundles TM and Λ l TM, when M is a Riemannian manifold, which will be treated in the following section. Further material on this topic can be found in [118, Appendix C]. To start, let M be a C 1 smooth manifold of dimension n, equipped with a continuous Riemannian metric (throughout, the summation convention is used) g jk dx j ⊗ dx k ,
g jk ∈ C 0 .
(9.1.1)
As is customary, for each pair of indices j, k ∈ {1, . . . , n} denote by g jk the (j, k)entry −1 in the matrix [(g jk )1≤j,k≤n ] and set g := det[(g jk )1≤j,k≤n ]. In particular, dVol = √g dx is the volume element on M in the local coordinates used to describe the Riemannian metric as in (9.1.1). Recall the musical isomorphisms (aka metric identiﬁcations) between tangent and cotangent vectors given by TM ∋ X = X j ∂ j → X ♭ := g jk X k dx j ∈ T ∗ M, T ∗ M ∋ ξ = ξ j dx j → ξ ♯ := g jk ξ j ∂ k ∈ TM.
(9.1.2)
These satisfy, for each X, Y ∈ TM, ξ, η ∈ T ∗ M, and i, j ∈ {1, . . . , n}, ⟨ξ ♯ , X⟩ = ξ(X) = ⟨ξ, X ♭ ⟩, (ξ ♯ )♭ = ξ,
X ♭ (Y) = ⟨X, Y⟩ = ⟨X ♭ , Y ♭ ⟩,
⟨ξ ♯ , η♯ ⟩ = ⟨ξ, η⟩ = ξ(η♯ ),
(∂ j )♭ = g jk dx k , dx j = g jk (∂ k )♭ ,
(dx j )♯ = g jk ∂ k , ⟨dx j , (∂ i )♭ ⟩ = δ ij ,
(X ♭ )♯ = X,
∂ j = g jk (dx k )♯ ,
(9.1.3)
⟨(dx j )♯ , ∂ i ⟩ = δ ij .
In relation to these, let us also note that for any scalar function f ∈ C 1 (M) and any vector ﬁeld X ∈ C 0 (M, TM), we have grad f := ∂ j fg jk ∂ k = ∂ j f(dx j )♯ = (df)♯ ,
df = (grad f)♭ ,
X(f) = ⟨grad f, X⟩ = df(X) = ⟨(df)♯ , X⟩ = ⟨df, X ♭ ⟩.
(9.1.4)
Moreover, for any scalar functions f, g and any X ∈ C 0 (M, TM), we have the product formula X(fg) = X(f)g + fX(g). (9.1.5) If the metric tensor on the manifold M is of class C 1 then the Christoffel symbols are locally given for 1 ≤ r, j, k ≤ n, r := Γ jk
1 n rs ∂g sj ∂g sk ∂g jk + − ∑g ( ). 2 s=1 ∂x k ∂x j ∂x s
(9.1.6)
It is wellknown that for each k ∈ {1, . . . , n} we have n
j
∑ Γ jk = j=1
∂ k (√g ) . √g
(9.1.7)
9.1 Connections and Covariant Derivatives on Vector Bundles 
373
The ﬁrstorder partial derivatives of the metric tensor may be recovered from the metric together with the Christoffel symbols as follows: n
n
i i ∂ j g kℓ = ∑ g iℓ Γ jk + ∑ g ik Γ jℓ , i=1
∀ j, k, ℓ ∈ {1, . . . , n},
(9.1.8)
i=1 n
n
∂ j g kℓ = − ∑ g ki Γ ijℓ − ∑ gℓi Γ ijk , i=1
∀ j, k, ℓ ∈ {1, . . . , n},
(9.1.9)
i=1
as it may be veriﬁed based on (9.1.6) and a straightforward computation. For later purposes it is useful to note here that (9.1.9) implies n
n
j
n
k , ∑ g ik Γ ik = − ∑ ∂ k g jk − ∑ g ij Γ ik i,k=1
k=1
∀ j ∈ {1, . . . , n}.
(9.1.10)
i,k=1
Also, the Christoffel symbols are symmetric in the lower indices, i.e., Γ ijk = Γ jik
for every i, j, k ∈ {1, . . . , n}.
(9.1.11)
Moving on, recall that the classical Divergence Theorem asserts that for any scalar function f ∈ C 01 (M) and any C 1 vector ﬁeld X we have ∫ X(f) dVol = − ∫ f div X dVol, M
(9.1.12)
M
where div X := g −1/2 ∂ j (g 1/2 X j ) = ∂ j X j + Γ jk X k j
if locally X = X j ∂ j .
(9.1.13)
The latter also implies the product formula div(fX) = X(f) + f div X, ∀ X ∈ C 1 (M, TM),
∀ f scalarvalued.
(9.1.14)
For further reference, let us observe that k div ∂ j = Γ jk
for every j ∈ {1, . . . , n},
(9.1.15)
From (9.1.5), (9.1.12), and (9.1.15) we also deduce that k ∂⊤j = −∂ j − Γ jk
for every j ∈ {1, . . . , n}.
(9.1.16)
Finally, a simple integration by parts argument based on formula (9.1.12) shows that div(u♯ ) = −δu,
∀ u ∈ C 1 (M, T ∗ M) ≡ C 1 (M, Λ1 TM).
(9.1.17)
Moving on, consider E → M, a Hermitian vector bundle over M, ﬁx ξ ∈ T ∗ M, and introduce the operator of tensor multiplication by the covector ξ , denoted by m ξ , given by m ξ : E → T ∗ M ⊗ E, m ξ (u) := ξ ⊗ u ∀ u ∈ E. (9.1.18)
374  9 Further Tools from Geometry and Analysis
Then its adjoint map m∗ξ : T ∗ M ⊗ E → E
(9.1.19)
satisﬁes m∗ξ (η ⊗ u) = ⟨ξ, η⟩u
∀ η ∈ T ∗ M and ∀ u ∈ E.
(9.1.20)
Hence, m∗ξ ∘ m η = ⟨ξ, η⟩I
∀ ξ, η ∈ T ∗ M,
(9.1.21)
where I denotes the identity operator on E. Fix next ξ, η, ω ∈ T ∗ M along with u ∈ E, then based on (9.1.20) and (9.1.18) write (m η ∘ m∗ξ )(ω ⊗ u) = m η (⟨ξ, ω⟩u) = ⟨ξ, ω⟩η ⊗ u.
(9.1.22)
This shows that m η ∘ m∗ξ = ⟨ξ, ⋅⟩m η
∀ ξ, η ∈ T ∗ M.
(9.1.23)
̂ j }1≤j≤n dual bases in TM and T ∗ M and recall the Consider next {X j }1≤j≤n and {X ̂ k ∈ T ∗ M and each u ∈ E, musical isomorphism ♭ introduced in (9.1.2). For each ω = ω k X we may write ̂ j ⊗ u = ω k ⟨X ♭ , X ̂ k ⟩X ̂j ⊗ u (m X̂ j ∘ m∗X♭ )(ω ⊗ u) = ⟨X ♭j , ω⟩X j j
̂ j ⊗ u = ω k δ jk X ̂ k (X j )X ̂j ⊗ u = ωk X ̂ j ⊗ u = ω ⊗ u, = ωj X
(9.1.24)
where the ﬁrst equality uses (9.1.23), the third identity follows from the ﬁrst formula in ̂ k (X j ) = δ jk . Consequently, (9.1.3), and the fourth identity uses the dual bases property X m X̂ j ∘ m∗X♭ is the identity operator on T ∗ M ⊗ E.
(9.1.25)
j
Hence, in particular, m dx j ∘ m∗∂♭ is the identity operator on T ∗ M ⊗ E.
(9.1.26)
j
We continue by introducing the notion of connection, which turns out to be a ﬁrstorder differential operator¹. Deﬁnition 9.1. A connection on a vector bundle E → M is a ℂlinear map ∇ : C 1 (M, E) → C 0 (M, T ∗ M ⊗ E) ≡ C 0 (M, Hom (TM, E))
(9.1.27)
which satisﬁes Leibniz’s product rule ∇(fu) = f ∇u + df ⊗ u for each u ∈ C 1 (M, E) and for each f ∈ C 1 (M) scalar function.
(9.1.28)
1 when working with differential operators acting between vector bundles over the manifold M, it is useful to keep in mind that the sections of each vector bundle form a sheaf
9.1 Connections and Covariant Derivatives on Vector Bundles 
375
Taking the contraction of a connection ∇ by a vector ﬁeld X yields the covariant derivative along X. This is made precise below. Deﬁnition 9.2. Given a connection ∇ on a Hermitian vector bundle E → M, along with a C 1 vector ﬁeld X ∈ TM, deﬁne the covariant derivative along X (for sections of E) as the mapping ∇X : C 1 (M, E) → C 0 (M, E) given by ∇X := m∗X♭ ∘ ∇.
(9.1.29)
Hence, given a connection ∇ : C 1 (M, E) → C 0 (M, T ∗ M ⊗ E), we have ∇X u = (∇u)(X),
∀ u ∈ C 1 (M, E), ∀ X ∈ C 0 (M, TM).
(9.1.30)
Also, taking adjoints of the two sides in the last formula appearing in (9.1.29) yields (∇X )∗ = ∇∗ ∘ m X♭ ,
∀ X ∈ C 1 (M, TM).
(9.1.31)
As a consequence of this and (9.1.18) we see that (∇X )∗ u = ∇∗ (X ♭ ⊗ u),
∀ u ∈ C 1 (M, E), ∀ X ∈ C 0 (M, TM).
(9.1.32)
Other basic properties of connections and covariant derivatives are collected in the proposition below. Proposition 9.3. Given a Hermitian vector bundle E → M, the following properties hold: (1) There exists a connection ∇ on E. ̃ on E (2) If ∇ denotes the connection on E from item (1) then any other connection ∇ 0 ∗ ̃ satisﬁes ∇ − ∇ = A where A ∈ C (M, Hom (E, T M ⊗ E)). Conversely, if ∇ is any connection on E and A ∈ C 0 (M, Hom(E, T ∗ M ⊗ E)), then ∇ + A continues to be a connection on E. (3) Any connection ∇ on E is a ﬁrstorder differential operator. In particular, supp (∇u) ⊆ supp u
for each u ∈ C 1 (M, E),
(9.1.33)
and, for each O ⊆ M open set, one may deﬁne the restriction of ∇ to C 1 (O, EO ) via setting for each u ∈ C 1 (O, EO ) ∇u := (∇υ)O ,
∀ υ ∈ C 1 (M, E) such that υO = u.
(9.1.34)
(4) For any connection ∇ on E there holds Sym (∇, ξ) = i m ξ
for each ξ ∈ T ∗ M,
(9.1.35)
where m ξ is as in (9.1.18). Thus, the principal symbol of ∇ is given by Sym (∇, ξ)u = iξ ⊗ u,
∀ ξ ∈ T ∗ M, ∀ u ∈ E.
(9.1.36)
In particular, the principal symbol of ∇ is injective, although in general it fails to be surjective.
376  9 Further Tools from Geometry and Analysis (5) Let ∇ be a connection on E. Then, for each {X j }1≤j≤n local basis in TM and with ̂ j }1≤j≤n denoting its dual coframe in T ∗ M, there holds {X ̂ j ⊗ ∇X . ∇=X j
(9.1.37)
In particular, ∇ may be expressed in local coordinates as ∇ = dx j ⊗ ∇∂ j ,
(9.1.38)
and for any local orthonormal basis {X j }1≤j≤n in TM one has ∇ = X ♭j ⊗ ∇X j .
(9.1.39)
(6) Recall (9.1.18). Then for any connection ∇ on E there holds [∇, ψ] = m dψ
for each scalar function ψ ∈ C 1 (M).
(9.1.40)
(7) For any C 1 vector ﬁelds X, Y on M, any u, υ ∈ E, and any scalar function f ∈ C 1 (M), the covariant derivative associated with a connection ∇ on E as in Deﬁnition 9.2 satisﬁes ∇X+Y u = ∇X u + ∇Y u, ∇fX u = f ∇X u, ∇X (u + υ) = ∇X u + ∇X υ,
(9.1.41)
∇X (fu) = f ∇X u + X(f)u. (8) If ∇∗ denotes the adjoint of a connection ∇ on E, then Sym (∇∗, ξ) = −i m∗ξ
for each ξ ∈ T ∗ M.
(9.1.42)
Hence, the principal symbol of the mapping ∇∗ : C 1 (M, T ∗ M ⊗ E) ≡ C 1 (M, Hom (TM, E)) → C 0 (M, E)
(9.1.43)
is given by Sym (∇∗, ξ)(η ⊗ u) = −i⟨ξ, η⟩u,
∀ ξ, η ∈ T ∗ M, ∀ u ∈ E.
(9.1.44)
(9) For each vector ﬁeld X = X j ∂ j ∈ TM, the covariant derivative ∇X associated with a connection ∇ on E as in Deﬁnition 9.2 is a ﬁrstorder differential operator whose principal symbol is given by Sym (∇X , ξ) = iξ(X) I = iX j ξ j I = i⟨X ♭ , ξ⟩ I = i⟨X, ξ ♯ ⟩ I,
∀ ξ = ξ j dx j ∈ T ∗ M,
(9.1.45)
where ⟨⋅, ⋅⟩ denotes the pointwise inner product in T ∗ M, and I is the identity operator. Hence, as a consequence of (9.1.45) and (2.1.65), one also has Sym ((∇X )∗, ξ ) = −iξ(X),
∀ ξ ∈ T ∗ M.
(9.1.46)
9.1 Connections and Covariant Derivatives on Vector Bundles 
377
(10) Let ∇ be a connection on E and ﬁx a local frame {e α }α for E in an open set O ⊂ M. In this context, introduce the connection coefficients of ∇ (with respect to the αβ frame {e α }α in O) as the functions γ j deﬁned by the requirement that αβ
∇∂ j e β = γ j e α
for all β, j.
(9.1.47)
Then for any C 1 section u = u α e α of E and vector ﬁeld X = X j ∂ j in O there holds αβ
∇X u = (X j ∂ j u α + X j γ j u β )e α .
(9.1.48)
In particular, in local coordinates, for each j ∈ {1, . . . , n} one has αβ
∇∂ j u = (∂ j u α + γ j u β )e α
for each u = u α e α ∈ E.
(9.1.49)
(11) For each C 1 vector ﬁeld X on M, the covariant derivative ∇X associated with a connection ∇ on E as in Deﬁnition 9.2 is ‘almost’ metric in the sense that X(⟨u, υ⟩E ) = ⟨∇X u, υ⟩E + ⟨u, ∇X υ⟩E + C X (u, υ),
(9.1.50)
for every u, υ ∈ C 1 (M, E), where C X is a bilinear form on E. Call the connection ∇ metric provided C X = 0 for every X. In particular, if ∇ is metric then X(⟨u, υ⟩E ) = ⟨∇X u, υ⟩E + ⟨u, ∇X υ⟩E for every vector ﬁeld X ∈ C 0 (M, TM) and sections u, υ ∈ C 1 (M, E).
(9.1.51)
(12) With C X deﬁned for each vector ﬁeld X ∈ TM as in item (11), for each section u ∈ E ̃ X (u) ∈ Hom(E, E) deﬁned by the requirement consider C ̃ X (u)⟩ C X (υ, u) = ⟨υ, C
for every section υ ∈ C 0 (M, E).
(9.1.52)
Then, given a connection ∇ on E, for each C 1 covector ξ ∈ T ∗ M and every C 1 section u ∈ E, the adjoint of ∇ satisﬁes ∗ ̃ ξ ♯ (u). ∇∗ (ξ ⊗ u) = (∇ξ ♯ ) u = −∇ξ ♯ u − (div ξ ♯ )u − C
(9.1.53)
In particular, if ∇ is metric then ∇∗ (ξ ⊗ u) = −∇ξ ♯ u − (div ξ ♯ )u for every ξ ∈ C 1 (M, T ∗ M) and u ∈ C 1 (M, E), and
if ∇ is metric then (∇X )∗ u = −∇X u − (div X)u for every X ∈ C 1 (M, TM) and u ∈ C 1 (M, E).
(9.1.54)
(9.1.55)
Proof. To prove (1), we start by considering a locally ﬁnite atlas A on M and {ψO }O∈A , a partition of unity subordinate to A. Let also {eO α }α be a local frame in O for E. Then, O in O for some coefficients u O ∈ C 1 (O), deﬁne for each u ∈ C 1 (M, E) such that u = uO e α α α O ∇u := ∑ ψO duO α ⊗ eα . O∈A
(9.1.56)
378  9 Further Tools from Geometry and Analysis It follows that the assignment u → ∇u considered in (9.1.56) is an unambiguously deﬁned ℂlinear map from C 1 (M, E) into C 0 (M, T ∗ M ⊗ E). To prove that this is a connection, there remains to show that Leibniz’s product rule (9.1.28) is satisﬁed. To this O O O end ﬁx f ∈ C 1 (M) and u ∈ C 1 (M, E) such that u = uO α e α in O. Then, since fu = (fu α )e α O 1 in O with fu α ∈ C (O), we may write using (9.1.56) O O O O ∇(fu) = ∑ ψO d(fuO α ) ⊗ e α = ∑ ψ O (u α df + fdu α ) ⊗ e α O∈A
O∈A
O O O = df ⊗ ( ∑ ψO uO α e α ) + f ( ∑ ψ O du α ⊗ e α ) O∈A
O∈A
= df ⊗ ( ∑ ψO uO ) + f ∇u O∈A
= df ⊗ u + f ∇u,
(9.1.57)
O O where the second equality uses the fact that d(fuO α ) = u α df + fdu α , the fourth equality is based again on (9.1.56), and the ﬁfth equality follows bearing in mind that {ψO }O∈A is a partition of unity. This shows that ∇ satisﬁes (9.1.28) and completes the proof of (1). ̃ are two connections on the bunTurning our attention to (2) assume that ∇ and ∇ ̃ Clearly A is a ℂlinear map from C 1 (M, E) into C 0 (M, T ∗ M ⊗ E) dle E and set A := ∇ − ∇. and our goal is to show that A ∈ C 0 (M, Hom(E, T ∗ M ⊗ E)). To see this, observe that
̃ ̃ ̃ A(fu) = (∇ − ∇)(fu) = ∇(fu) − ∇(fu) = f(∇ − ∇)(u) = fAu, ∀ f ∈ C 1 (M)
and ∀ u ∈ C 1 (M, E),
(9.1.58)
where the second equality above follows from Leibniz’s rule (9.1.28) written for both ∇ ̃ To proceed, consider a local frame {e α }α for E, and let a αβ ’s be the coefficients and ∇. j
αβ
of Ae α with respect to the basis {dx j ⊗ e β }j,β of T ∗ M ⊗ E, i.e., Ae α = a j dx j ⊗ e β . Then by (9.1.58) and linearity we have αβ
Au = A(u α e α ) = u α Ae α = a j u α dx j ⊗ e β ∀ u ∈ C 1 (M, E) such that u = u α e α in M,
(9.1.59)
which goes to show that A ∈ C 0 (M, Hom (E, T ∗ M ⊗ E)), as desired. Next, consider ∇ a connection on E and A ∈ C 0 (M, Hom (E, T ∗ M ⊗ E)). Then clearly ∇ + A is a ℂlinear map from C 1 (M, E) into C 0 (M, T ∗ M ⊗ E). In addition ∇ satisﬁes Leibniz’s product rule (9.1.28) and A satisﬁes (9.1.58). This immediately shows that Leibniz’s product rule for ∇ + A is valid, completing the proof of (2). The main claim in item (3), namely that any connection ∇ on E is a ﬁrstorder differential operator, is a direct consequence of the second part of item (2) and (9.1.56). As regards item (4), ﬁx an arbitrary point x ∈ M and pick some arbitrary ξ ∈ T x∗ M and u ∈ Ex . Also, for some neighborhood O of x, select ̃ ∈ C 1 (O, E) such that u ̃ (x) = u, and u ψ ∈ C 1 (O) scalar function so that ψ(x) = 0 and dψ(x) = ξ.
(9.1.60)
9.1 Connections and Covariant Derivatives on Vector Bundles 
379
Then, based on (2.1.61), Leibniz’s product rule (9.1.28), and (9.1.60), we may write ̃ )x = iξ ⊗ u Sym (∇, ξ)u = i∇(ψ u
(9.1.61)
proving formula (9.1.36). Consider next the main claim made in item (5). Given a vector ̂ j (X) for every j hence, ﬁeld X = a j X j ∈ TM, its coefficients may be expressed as a j = X for every section u ∈ E, ̂ j (X)∇X u = (X ̂ j ⊗ ∇X u)(X). ∇X u = a j ∇X j u = X j j
(9.1.62)
Thanks to (9.1.30), this implies (9.1.37). Next, in light of (9.1.18), Leibniz’s product rule (9.1.28) may be reformulated as the commutator identity (9.1.40) stated in item (6). Turning to item (7), note that for any C 1 vector ﬁeld X on TM, any C 1 section u of E, and any scalar function f ∈ C 1 (M), ∇fX u = m∗(fX)♭ (∇u) = f m∗X♭ (∇u) = f ∇X u,
(9.1.63)
proving the second formula in (9.1.41). Also, from (9.1.29), Leibniz’s product rule (9.1.28), (9.1.20), and (9.1.4), we obtain ∇X (fu) = m∗X♭ (∇(fu)) = m∗X♭ (f ∇u) + m∗X♭ (df ⊗ u) = f m∗X♭ (∇u) + ⟨X ♭ , df⟩u = f ∇X u + X(f)u,
(9.1.64)
proving the fourth formula in (9.1.41). Since the remaining formulas in (9.1.41) are direct consequences of deﬁnitions, this ﬁnishes the treatment of item (7). Next, the ﬁrst claim in (8) is seen from (9.1.35), bearing in mind that Sym (∇∗, ξ) is the Hermitian adjoint of Sym (∇, ξ). With this in hand, formula (9.1.44) follows with the help of (9.1.20). All claims in item (9) are clear from (9.1.29), (9.1.20), (9.1.21), the fact that ∇ is a ﬁrstorder differential operator, and (9.1.36). For example, from (9.1.35) and (9.1.21) we obtain Sym (∇X , ξ) = Sym (m∗X♭ ∘ ∇, ξ ) = m∗X♭ ∘ Sym (∇, ξ) = i m∗X♭ ∘ m ξ = i⟨X ♭ , ξ⟩ = iξ(X),
(9.1.65)
for every vector ﬁeld X ∈ TM and every covector ξ ∈ T ∗ M. Going further, we note that (9.1.49) is readily implied by (9.1.47) and the last formula in (9.1.41). In turn, (9.1.49) is seen to imply (9.1.48), ﬁnishing the proof of the claims in item (10). αβ To prove the claim in item (11), let γ j be the connection coefficients of ∇ with respect to some ﬁxed local orthonormal frame {e α }α for E. Then, given any two sections locally expressed as u = u α e α ∈ E and υ = υ α e α ∈ E, making use of (9.1.48) for any vector ﬁeld X = X j ∂ j ∈ TM we may compute αβ
⟨∇X u, υ⟩E + ⟨u, ∇X υ⟩E = (X j ∂ j u α + X j γ j u β )υ α αβ
+ (X j ∂ j υ α + X j γ j υ β )u α .
(9.1.66)
380  9 Further Tools from Geometry and Analysis
On the other hand, from (9.1.5) we deduce that X(⟨u, υ⟩E ) = X(u α υ α ) = X(u α )υ α + u α X(υ α ) = X j (∂ j u α )υ α + X j u α (∂ j υ α ).
(9.1.67)
From (9.1.66) and (9.1.67) it is then visible that (9.1.50) holds if we take C X to be the bilinear form on E given by αβ
αβ
αβ
βα
C X (u, υ) := X j γ j u β υ α + X j γ j υ β u α = X j (γ j + γ j )u α υ β .
(9.1.68)
To deal with the claim in item (12), let u, υ be two C 1 sections of E, and pick a C 1 covector ξ , a section of T ∗ M. Without loss of generality assume they are all compactly supported in M. Then making use of (9.1.29) and (9.1.50) we may compute ⟨∇υ, ξ ⊗ u⟩ = ⟨∇υ, m ξ (u)⟩ = ⟨m∗ξ (∇υ), u⟩ = ⟨∇ξ ♯ υ, u⟩ = ξ ♯ (⟨υ, u⟩) − ⟨υ, ∇ξ ♯ u⟩ − C ξ ♯ (υ, u).
(9.1.69)
Integrating the most extreme sides of (9.1.69) over M and employing the Divergence Theorem (9.1.12) then yields ∫⟨∇υ, ξ ⊗ u⟩ dVol = ∫ ξ ♯ (⟨υ, u⟩) dVol − ∫⟨υ, ∇ξ ♯ u⟩ dVol − ∫ C ξ ♯ (υ, u) dVol M
M
M
M
̃ ξ ♯ (u)⟩ dVol. = ∫⟨υ, −∇ξ ♯ u − (div ξ )u − C ♯
(9.1.70)
M
In turn, the second equality in (9.1.53) now follows from (9.1.70) given that υ is arbitrary. Finally, the ﬁrst equality in (9.1.53) is a consequence of (9.1.32) and (9.1.3). We continue by proving the following useful unique continuation property for connections on vector bundles. Proposition 9.4. Let E → M be a C 1 vector bundle equipped with a connection ∇. Suppose O ⊆ M is an open connected set and assume that u ∈ C 1 (O, E) is a section satisfying ∇u = 0 everywhere in O
(9.1.71)
there exists x0 ∈ O such that u(x0 ) = 0 ∈ Ex0 .
(9.1.72)
and
Then necessarily u vanishes identically in O. Proof. Work in a local coordinate patch U ⊆ O near x0 and assume c : (−1, 1) → U of the form c(t) = (x1 (t), . . . , x n (t)) for t ∈ (−1, 1) is a continuously differentiable curve such that c(0) = x0 . Write u = u α e α , where {e α }α is a local frame for E and ﬁx α. Then, in light of (9.1.49), condition (9.1.71) implies ∂u α αβ = −γ j u β in U, ∂x j
(9.1.73)
9.2 The Extension of the LeviCivita Connection to Differential Forms
for each j ∈ {1, . . . , n}. Multiply (9.1.73) by of the chain rule)
dx j dt

381
and add over j to obtain (with the help
d α αβ [u (c(t))] = −γ j (c(t))xj (t)u β (c(t)) dt
for t ∈ (−1, 1).
(9.1.74)
Now set y α (t) := u α (c(t)) and a αβ (t) := −γ j (c(t))xj (t) for t ∈ (−1, 1). Then (9.1.74) combined with (9.1.72) becomes the ﬁrstorder system of equations: αβ
y (t) = F(y(t), t)
for t ∈ (−1, 1), y(0) = 0,
(9.1.75)
where y = (y α )α and F : ℝrankE × (−1, 1) → ℝrankE is deﬁned by F(y, t) := (a αβ (t)y β )α for (y, t) ∈ ℝrankE × (−1, 1). Note that F is a continuous function and linear in the y variable. As such, standard uniqueness results for ordinary differential equations apply and ultimately yield the desired conclusion.
9.2 The Extension of the LeviCivita Connection to Differential Forms Here we discuss the LeviCivita connection on TM, when M is a Riemannian manifold, and show how this gives rise to natural connections on the bundles Λ l TM. For scalarvalued functions f , we set ∇X f := X(f) for X ∈ TM. In particular, if we regard ∇f as a tangent vector via the requirement that ⟨∇f, X⟩ = ∇X f for every X ∈ TM, it follows from (9.1.4) that ∇f = grad f = (df)♯ . (9.2.1) In particular, we have the Leibniz (product) formula ∇(fg) = g∇f + f ∇g. Next, recall that the LeviCivita connection on vector ﬁelds is the map ∇X (deﬁned for each X ∈ C 0 (M, TM)) sending vector ﬁelds to vector ﬁelds which satisﬁes: (i) [linearity] ∇X Y is linear both in X ∈ C 0 (M, TM) and in Y ∈ C 1 (M, TM), over the reals; (ii) [homogeneity] ∇fX Y = f ∇X Y for f scalarvalued function and X, Y ∈ C 1 (M, TM); (iii) [Leibniz formula] ∇X (fY) = f ∇X Y + (Xf)Y for f scalar and X, Y ∈ C 1 (M, TM); (iv) [torsionfree property] ∇X Y − ∇Y X = [X, Y] for all X, Y ∈ C 1 (M, TM); (v) [metric property] X(⟨Y, Z⟩) = ⟨∇X Y, Z⟩ + ⟨Y, ∇X Z⟩ for all X, Y, Z ∈ C 1 (M, TM). The fundamental theorem of Riemannian geometry asserts that there exists a unique mapping ∇ : C 1 (M, TM) → C 0 (M, T ∗ M ⊗ TM) enjoying these properties. In local coordinates we have ∇∂ i ∂ j = ∇∂ j ∂ i ∀ i, j ∈ {1, . . . , n}, (9.2.2) thanks to the fact that the LeviCivita connection is torsionfree and Schwarz’s theorem (9.2.3) [∂ i , ∂ j ] = 0, ∀ i, j ∈ {1, . . . , n}.
382  9 Further Tools from Geometry and Analysis
This also may be seen from (9.1.11) and the formula (compare with (9.1.47)) ∇∂ i ∂ j = Γ ijk ∂ k
∀ i, j ∈ {1, . . . , n}.
(9.2.4)
The canonical way of extending the action of the LeviCivita connection ∇ to act on 1forms is to assign to each u ∈ C 1 (M, Λ1 TM) ≡ C 1 (M, T ∗ M) and each X ∈ C 0 (M, TM) the 1form ∇X u deﬁned via the requirement that (∇X u)(Y) := X(⟨u♯ , Y⟩) − ⟨u♯ , (∇X Y)⟩ = X(u(Y)) − u(∇X Y),
∀ Y ∈ TM.
(9.2.5)
One can check that the righthand side is tensorial in Y (i.e., at each point x ∈ M it only depends on Y at x), so this deﬁnition is meaningful and, that in local coordinates, k ∇∂ j dx k = −Γ jℓ dxℓ ,
∀ j, k ∈ {1, . . . , n},
(9.2.6)
k are the Christoffel symbols introduced in (9.1.6). It is immediate from deﬁniwhere Γ jℓ tions that for each 1form u ∈ Λ1 TM ≡ T ∗ M one has ♭
♯
∇X u = (∇X (u♯ ))
or, equivalently, (∇X u) = ∇X (u♯ ).
(9.2.7)
That is, the LeviCivita connection is lifted at the level of 1forms via musical isomorphisms. Let us also note here that, as seen from (9.2.7), ∇X (Y ♭ ) = (∇X Y)
♭
for every X, Y ∈ C 1 (M, TM).
(9.2.8)
Moving on, the action of ∇ from (9.2.5) may be canonically extended to forms of arbitrary degrees, yielding an operator ∇ : C 1 (M, Λ l TM) → C 0 (M, T ∗ M ⊗ Λ l TM),
ℓ ∈ {0, 1, . . . , n},
(9.2.9)
in an inductive manner that ensures the validity of the product formula ∇X (u ∧ υ) = (∇X u) ∧ υ + u ∧ (∇X υ)
(9.2.10)
for any two differential forms u, υ, and any vector ﬁeld X. Alternatively, for each lform u and vector ﬁeld X, one can directly deﬁne the lform ∇X u via the formula (∇X u)(Y1 , . . . , Y l ) = X(u(Y1 , . . . , Y l )) l
− ∑ u(Y1 , . . . , Y j−1 , ∇X Y j , Y j+1 , . . . , Y l ),
(9.2.11)
j=1
for every vector ﬁelds Y1 , . . . , Y l . From formulas (9.2.4), (9.2.11), and (2.1.22) we deduce that, if u = ∑I=l u I dx I locally, then for each j ∈ {1, . . . , n} we may locally express
∇∂ j u = ∑ (∇∂ j u)I dx I I=l
(9.2.12)
9.2 The Extension of the LeviCivita Connection to Differential Forms 
383
where, for each ordered array I = (i1 , . . . , i l ), l
n
(i ,...,i r−1 ,k,i r+1 ,...,i l ) k Γ ir j
(∇∂ j u)I = ∂ j u I − ∑ ∑ ∑ ε J 1 J=l r=1 k=1
uJ .
(9.2.13)
The extension of the LeviCivita connection ∇ just described turns out to be metric, i.e., for any two lforms u, υ with C 1 coefficients there holds X⟨u, υ⟩ = ⟨∇X u, υ⟩ + ⟨u, ∇X υ⟩,
∀ X ∈ C 0 (M, TM),
(9.2.14)
where ⟨⋅, ⋅⟩ stands here for the pairing in Λ l TM. With the help of (9.2.13), this equality may be easily checked at some arbitrary point x∗ ∈ M by working in a normal coordinate system near x∗ , i.e., a chart (x1 , . . . , x n ) near x∗ satisfying the following properties: x j (x∗ ) = 0, g ij (x∗ ) = δ ij , Γ ijk (x∗ ) = 0, (9.2.15) (∂ i g jk )(x∗ ) = 0, (∂ i g jk )(x∗ ) = 0, g = 1 near x∗ . The fact that the (extension of the) LeviCivita connection ∇ is metric implies (cf. (9.1.55)) (∇X )∗ = −∇X − div X, ∀ X ∈ C 1 (M, TM), (9.2.16) which, in light of (9.2.16) and (9.1.13), entails k (∇∂ j )∗ = −∇∂ j − Γ jk
∀ j ∈ {1, . . . , n}.
(9.2.17)
It is also useful to observe that (9.2.14) implies X(u2 ) = 2⟨∇X u, u⟩ for every form u and every X ∈ C 0 (M, TM).
(9.2.18)
In turn, this goes to show that if u is a differential form in a connected open subset O of M then ∇u = 0 in O ⇒ u is constant in O. (9.2.19) Going further, from (9.2.13), (2.1.37), plus some algebra we readily deduce that, given any differential form u locally expressed as u = ∑I=l u I dx I , then for every index j ∈ {1, . . . , n} we locally have
n
k ∇∂ j u = ∑ {∂ j u I − ( ∑ Γ jk )u I + ∑ I=l
n I ∑ Γ ijk ε kI iJ u J }dx
L=l i,k=1
k=1 n
k = ∑ ∂ j u I dx I − ∑ Γ js g ik dx s ∧ (dx i ∨ u). I=l
(9.2.20)
k,s,i=1
As a consequence, taking into account the symmetry of the Christoffel symbols in the lower indices and the antisymmetry of the exterior product, we may easily see from (2.1.57) and (9.2.20) that n
du = ∑ dx j ∧ ∇∂ j u. j=1
(9.2.21)
384  9 Further Tools from Geometry and Analysis Having established this, it is worth noting that for any ℓform u and any vector ﬁelds X1 , . . . , Xℓ+1 from (2.1.8) we deduce that n
(du)(X1 , . . . , Xℓ+1 ) = (−1)ℓ ∑ ((∇∂ j u) ∧ dx j )(X1 , . . . , Xℓ+1 ) j=1 n
= (−1)ℓ ∑
∑ sign (π) (∇∂ j u)(X π(1) , . . . , X π(ℓ) )dx j (X π(ℓ+1) )
j=1 π∈SHℓ,1
= (−1)ℓ ∑ sign (π) (∇X π(ℓ+1) u)(X π(1) , . . . , X π(ℓ) ) π∈SHℓ,1 ℓ+1
= (−1)ℓ ∑
∑ sign (π) (∇X i u)(X1 , . . . , X i−1 , X i+1 , . . . , Xℓ+1 )
i=1 π∈SHℓ,1 π(ℓ+1)=i ℓ+1
= ∑ (−1)i+1 (∇X i u)(X1 , . . . , X i−1 , X i+1 , . . . , Xℓ+1 ),
(9.2.22)
i=1
since, for each i ∈ {1, . . . , n}, there is only one (ℓ, 1)shuffle π satisfying π(ℓ + 1) = i, and this turns out to have sign (π) = (−1)ℓ+j+1 . We seek similar characterizations for the action of the codifferential operator δ. This requires a short detour and we begin by observing that ∇ dVol = 0.
(9.2.23)
Indeed, given any j ∈ {1, . . . , n}, it follows that ∇∂ j dVol is an nform which, thanks to (9.2.11), (2.1.4), (9.2.4), and (9.1.7) satisﬁes (∇∂ j dVol)(∂1 , . . . , ∂ n ) = ∂ j (dVol(∂1 , . . . , ∂ n )) n
− ∑ dVol(∂1 , . . . , ∂ k−1 , ∇∂ j ∂ k , ∂ k+1 , . . . , ∂ n ) k=1 n l = ∂ j (√g ) − ∑ Γ jk dVol(∂1 , . . . , ∂ k−1 , ∂ l , ∂ k+1 , . . . , ∂ n ) k,l=1 n k √g = 0. = ∂ j (√g ) − ∑ Γ jk
(9.2.24)
k=1
This proves (9.2.23). To proceed, consider two arbitrary lforms u, υ and, given any vector ﬁeld X, make use of the last line in (2.1.27) in order to write ∇X (u ∧ (∗υ)) = ∇X (⟨u, υ⟩ dVol).
(9.2.25)
Now, on the one hand we have ∇X (u ∧ (∗υ)) = (∇X u) ∧ (∗υ) + u ∧ (∇X (∗υ))
(9.2.26)
9.2 The Extension of the LeviCivita Connection to Differential Forms 
385
by the product formula (9.2.10), while on the other hand we have ∇X (⟨u, υ⟩ dVol) = X(⟨u, υ⟩) dVol + ⟨u, υ⟩ ∇X (dVol) = ⟨∇X u, υ⟩ dVol + ⟨u, ∇X υ⟩ dVol = (∇X u) ∧ (∗υ) + u ∧ ∗(∇X υ)
(9.2.27)
by the product formula (9.2.10), the metric property (9.2.14), the vanishing property (9.2.23), and the last line in (2.1.27). In concert, formulas (9.2.25)–(9.2.27) ultimately prove that ∗ (∇X u) = ∇X (∗u) for every form u and vector ﬁeld X. (9.2.28) In turn, from (9.2.10), (9.2.28), and item (9) in Lemma 2.2, we conclude that ∇X (u ∨ υ) = (∇X u) ∨ υ + u ∨ (∇X υ)
(9.2.29)
for any two differential forms u, υ. At this stage, we deduce from (9.2.21), item (2) in Lemma 2.8, item (9) in Lemma 2.2, and (9.2.28) that n
δu = − ∑ dx j ∨ ∇∂ j u.
(9.2.30)
j=1
This entails (upon recalling (2.1.54) and (9.1.3)) that for every ℓform u and every collection of vector ﬁelds X1 , . . . , Xℓ−1 , n
(δu)(X1 , . . . , Xℓ−1 ) = − ∑ (dx j ∨ ∇∂ j u)(X1 , . . . , Xℓ−1 ) j=1 n
= − ∑ (∇∂ j u)((dx j )♯ , X1 , . . . , Xℓ−1 ) j=1 n
= − ∑ (∇∂ j u)(g jk ∂ k , X1 , . . . , Xℓ−1 ) j,k=1 n
= − ∑ g jk (∇∂ j u)(∂ k , X1 , . . . , Xℓ−1 ) j,k=1 n
= − ∑ (∇k u)(∂ k , X1 , . . . , Xℓ−1 ),
(9.2.31)
k=1
where we have abbreviated ∇k := ∑nj=1 g jk ∇∂ j . Via duality, (9.2.30), (9.2.17) yield the following companion of (9.2.21): n
n
j=1
j=1
k du = − ∑ (∇∂ j )∗ (dx j ∧ u) = ∑ (∇∂ j + Γ jk )(dx j ∧ u),
(9.2.32)
while (9.2.21), (9.2.17) yield the following companion of (9.2.30): n
n
j=1
j=1
k δu = ∑ (∇∂ j )∗ (dx j ∨ u) = − ∑ (∇∂ j + Γ jk )(dx j ∨ u).
(9.2.33)
386  9 Further Tools from Geometry and Analysis Formulas (9.2.21), (9.2.30) also show that if u is a differential form of class C 1 in an open subset O of M then du + δu ≤ C∇u pointwise in any compact subset of O.
(9.2.34)
In particular, if u is a differential form of class C 1 in an open subset O of M then ∇u = 0 in O ⇒ du = 0 and
δu = 0 in O.
(9.2.35)
9.3 The BochnerLaplacian and Weintzenböck’s Formula Having indicated in § 9.2 how the LeviCivita connection may be canonically extended to differential forms, we now consider the BochnerLaplacian ∆BL := −∇∗ ∇,
(9.3.1)
where ∇ is the said extension. In a ﬁrst stage, the goal is to ﬁnd various local expressions for this elliptic secondorder operator. To get started, by relying on (9.1.38), (9.1.54), (9.1.3), (9.1.15), and (9.1.10), for each given differential form u we may write (using the familiar summation convention over repeated indices) ∆BL u = −∇∗ ∇u = −∇∗ (dx j ⊗ ∇∂ j u) = ∇(dx j )♯ ∇∂ j u + (div(dx j )♯ )∇∂ j u = g jk ∇∂ k ∇∂ j u + (∂ r g jr + g jl Γ lrr )∇∂ j u j
= g jk ∇∂ j ∇∂ k u + g ab Γ ab ∇∂ j u i ∇∂ i )u. = g jk (∇∂ j ∇∂ k − Γ jk
(9.3.2)
On the other hand, given any two lforms u, υ supported in a coordinate patch on M we have ∫⟨(∇∂ j )∗ g jk ∇∂ k u, υ⟩ dVol = ∫ g jk ⟨∇∂ k u, ∇∂ j υ⟩ dVol M
M
= ∫ g jk ⟨(∂ k ⊗ ∇u) , (∂ j ⊗ ∇υ)⟩ dVol M
= ∫ g jk ⟨∂ k , ∂ j ⟩⟨∇u, ∇υ⟩ dVol M
= ∫ g jk g jk ⟨∇u, ∇υ⟩ dVol M
= ∫⟨∇∗ ∇u, υ⟩ dVol, M
(9.3.3)
9.3 The BochnerLaplacian and Weintzenböck’s Formula  387
proving that ∆BL u = −(∇∂ j )∗ g jk ∇∂ k u.
(9.3.4)
From (9.3.2)–(9.3.4) we may therefore conclude that² i ∇∂ i ). ∆BL = −∇∗ ∇ = −(∇∂ j )∗ g jk ∇∂ k = g jk (∇∂ j ∇∂ k − Γ jk
(9.3.5)
In passing, we note that the last equality above may be veriﬁed directly making use of (9.2.17) and (9.1.10). Moving on, recall the curvature of a Riemannian manifold M with a C 2 metric tensor, deﬁned as R(X, Y) := ∇X ∇Y − ∇Y ∇X − ∇[X,Y] = [∇X , ∇Y ] − ∇[X,Y] ,
X, Y ∈ C 1 (M, TM).
(9.3.6)
At ﬁrst sight, R(X, Y) is a differential operator of order ≤ 1. However, a closer inspection reveals that actually R(X, Y) is zeroth order. (9.3.7) Speciﬁcally, this is tensorial in the sense that for every X, Y, Z, W ∈ C 1 (M, TM) the expression K(X, Y, Z, W)(x) := ⟨R(X, Y)Z, W⟩T x M , x ∈ M, (9.3.8) only depends on the values of X, Y, Z, W at the point x (without involving the values of the partial derivatives of the components of these vector ﬁelds at x). The mapping (X, Y, Z, W) → K(X, Y, Z, W) is called the Riemann curvature tensor of M. One has the ﬁrst Bianchi identity R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0,
∀ X, Y, Z ∈ TM
(9.3.9)
or, equivalently, K(X, Y, Z, W) + K(Y, Z, X, W) + K(Z, X, Y, W) = 0,
(9.3.10)
for every X, Y, Z, W ∈ TM. In addition, for every X, Y, Z, W ∈ TM one has K(X, Y, Z, W) = −K(Y, X, Z, W), K(X, Y, Z, W) = −K(X, Y, W, Z),
(9.3.11)
K(X, Y, Z, W) = K(Z, W, X, Y).
2 A word of caution is in order here. Speciﬁcally, a number of authors write the BochnerLaplacian as g jk ∇∂ j ∇∂ k (or just g jk ∇j ∇k ), though ∇∂ j ∇∂ k does not denote the composition of ∇∂ j with ∇∂ k but rather ℓ ∇(∂ j ,∂ k ) := ∇∂ j ∇∂ k − Γ jk ∇∂ℓ .
388  9 Further Tools from Geometry and Analysis As a consequence, if in local coordinates (x1 , . . . , x n ) we deﬁne the coefficients of the Riemann curvature tensor by setting, for each i, j, k, ℓ ∈ {1, . . . , n}, R ijkℓ := K(∂ i , ∂ j , ∂ k , ∂ℓ ) = K(∂ k , ∂ℓ , ∂ i , ∂ j ) = ⟨R(∂ k , ∂ℓ )∂ i , ∂ j ⟩ = ⟨[∇∂ k , ∇∂ℓ ]∂ i , ∂ j ⟩,
(9.3.12)
then R ijkℓ = −R jikℓ ,
R ijkℓ = −R ijℓk ,
R ijkℓ = R kℓij ,
∀ i, j, k, ℓ ∈ {1, . . . , n},
(9.3.13)
and R ijkℓ + R jkiℓ + R kijℓ = 0,
∀ i, j, k, ℓ ∈ {1, . . . , n}.
(9.3.14)
Furthermore, for every i, j, k, ℓ ∈ {1, . . . , n} one has n s s r s r s − ∂ j Γ ik + ∑ (Γ jk Γ ir − Γ ik Γ jr ))g sℓ . R ijkℓ = (∂ i Γ jk
(9.3.15)
r=1
For further reference let us also remark here that the last equality in (9.3.12) yields n
[∇∂ k , ∇∂ℓ ]∂ i = ∑ R ijkℓ g js ∂ s ,
∀ i, k, ℓ ∈ {1, . . . , n}.
(9.3.16)
j,s=1
On a related note, recall that the components S jk of the classical Ricci tensor S of the Riemannian manifold M are deﬁned by contracting the components of the Riemann curvature tensor according to n
S jk := ∑ R ijkℓ gℓ i ,
∀ j, k ∈ {1, . . . , n}.
(9.3.17)
i,ℓ=1
Then S : TM × TM → ℝ is the unique bilinear mapping with the property that, in local coordinates, S(∂ j , ∂ k ) = S jk for each j, k ∈ {1, . . . , n}. Thus, n
S(X, Y) = ∑ a j b k S jk j,k=1 n
for all
n
X = ∑ a j ∂ j ∈ TM, j=1
(9.3.18)
Y = ∑ b k ∂ k ∈ TM. k=1
It is worth noting that the classical Ricci tensor S is symmetric, i.e., S(X, Y) = S(Y, X) for every X, Y ∈ TM. Indeed, this comes down to the symmetry property S jk = S kj for all j, k ∈ {1, . . . , n} which, in turn, may be justiﬁed by writing for any given j, k ∈ {1, . . . , n} n
n
i,ℓ=1
i,ℓ=1
n
n
i,ℓ=1
i,ℓ=1
S jk = ∑ R ijkℓ gℓ i = ∑ Rℓjki g iℓ = ∑ R kiℓj gℓ i = ∑ R ikjℓ gℓ i = S kj , thanks to (9.3.13).
(9.3.19)
9.3 The BochnerLaplacian and Weintzenböck’s Formula  389
The speciﬁc manner in which the commutators of covariant derivatives of differential forms may be expressed in terms of the curvature is presented below. Proposition 9.5. Given any l ∈ {0, 1, . . . , n}, Ricci’s identity n
[∇∂ i , ∇∂ j ]u = − ∑ R ijkℓ dx k ∧ (dxℓ ∨ u),
(9.3.20)
k,ℓ=1
holds for each u ∈ C 1 (M, Λ l TM) and each i, j ∈ {1, . . . , n}. Proof. Formula (9.3.20) is trivially true when l = 0 (as both sides vanish), so consider the case when l = 1. In this scenario, it suffices to take u = (∂ r )♭ for some r ∈ {1, . . . , n}, for which we may write n
♭
[∇∂ i , ∇∂ j ](∂ r )♭ = ([∇∂ i , ∇∂ j ]∂ r ) = ( ∑ R rkij g ks ∂ s )
♭
k,s=1 n
n
= − ∑ R ijkr g ks (∂ s )♭ = − ∑ R ijkr g ks g sℓ dxℓ j,s=1
j,s,ℓ=1
n
n
= − ∑ R ijkr dx k = − ∑ R ijkℓ dx k δℓr k=1
k,ℓ=1
n
= − ∑ R ijkℓ dx k ∧ (dxℓ ∨ (∂ r )♭ ),
(9.3.21)
k,ℓ=1
thanks to (9.2.8), (9.3.16), and (9.1.3). This shows that (9.3.20) holds when l = 1. With this in hand, the general case when l ∈ {1, . . . , n} may then be easily handled by induction on l as soon as the following identities are proved for every i, j ∈ {1, . . . , n}: [∇∂ i , ∇∂ j ](u ∧ υ) = ([∇∂ i , ∇∂ j ]u) ∧ υ + u ∧ ([∇∂ i , ∇∂ j ]υ), n
(9.3.22)
n
∑ R ijkℓ dx k ∧ (dxℓ ∨ (u ∧ υ)) = ( ∑ R ijkℓ dx k ∧ (dxℓ ∨ u)) ∧ υ k,ℓ=1
k,ℓ=1 n
+ u ∧ ( ∑ R ijkℓ dx k ∧ (dxℓ ∨ υ)),
(9.3.23)
k,ℓ=1
for any forms u, υ, of arbitrary degrees. In this regard, we note that (9.3.23) is an immediate consequence of Lemma 2.3, while (9.3.22) readily follows from the product formula (9.2.10). This completes the proof of Ricci’s identity (9.3.20). Moving on, we can now introduce the Weitzenböck operator on the given Riemannian manifold. Deﬁnition 9.6. In local coordinates, the Weitzenböck operator Ric is deﬁned by n
Ric := ∑ R ijkℓ dx i ∧ (dx j ∨ (dx k ∧ (dxℓ ∨ ⋅ )))
(9.3.24)
i,j,k,ℓ=1
where the R ijkℓ ’s are the components of the Riemann curvature tensor from (9.3.12).
390  9 Further Tools from Geometry and Analysis
Several basic properties of the Weitzenböck operator are listed below. Proposition 9.7. For each l ∈ {0, 1, . . . , n}, formula (9.3.24) induces a linear operator Ric : Λ l TM → Λ l TM
(9.3.25)
whose writing in local coordinates involve coefficients that are real and depend linearly on the Riemann curvature. In addition, the operator in question is symmetric, in the natural sense that (9.3.26) ⟨Ric u, υ⟩ = ⟨u, Ric υ⟩ for all u, υ ∈ Λ l TM. Moreover, ∗ Ric = Ric ∗ .
(9.3.27)
Also, Ric vanishes identically on scalar functions, as well as on nforms,
(9.3.28)
while on 1forms it may be identiﬁed with the classical Ricci tensor S in the sense that (Ric u)(X) = S(X, u♯ ) = S(u♯ , X)
for all u ∈ Λ1 TM, X ∈ TM.
(9.3.29)
In particular, ⟨Ric u, υ⟩ = S(u♯ , υ♯ ) for all u, υ ∈ Λ1 TM.
(9.3.30)
Proof. The symmetry property (9.3.26) is readily justiﬁed with the help of (9.3.24), (9.3.13), and item (8) in Lemma 2.2. To establish (9.3.27), for an arbitrary lform u, items (9) and (7) in Lemma 2.2 along with (2.1.23) permit us to write n
∗(Ric u) = ∑ R ijkℓ dx i ∨ (dx j ∧ (dx k ∨ (dxℓ ∧ (∗u)))) i,j,k,ℓ=1 n
= ∑ R ijkℓ dx i ∨ (dx j ∧ (g kℓ (∗u) − dxℓ ∧ (dx k ∨ (∗u)))) i,j,k,ℓ=1 n
= − ∑ R ijkℓ dx i ∨ (dx j ∧ (dxℓ ∧ (dx k ∨ (∗u)))) i,j,k,ℓ=1 n
= − ∑ R ijkℓ (g ij (dxℓ ∧ (dx k ∨ (∗u))) − dx j ∧ (dx i ∨ (dxℓ ∧ (dx k ∨ (∗u))))) i,j,k,ℓ=1 n
= ∑ R ijkℓ dx j ∧ (dx i ∨ (dxℓ ∧ (dx k ∨ (∗u)))) i,j,k,ℓ=1 n
= ∑ R jiℓk dx j ∧ (dx i ∨ (dxℓ ∧ (dx k ∨ (∗u)))) i,j,k,ℓ=1
= Ric (∗u),
(9.3.31)
9.3 The BochnerLaplacian and Weintzenböck’s Formula  391
where, in the third and ﬁfth equalities we have used the symmetry of the matrix (2.1.5) together with the antisymmetry properties of the components of the Riemann curvature tensor recorded in (9.3.13), with the latter antisymmetry properties also used in the nexttolast equality above. That Ric vanishes identically on scalar functions is clear from (9.3.24). In concert with (9.3.27), this also shows that Ric vanishes identically on nforms. As regards formula (9.3.32), we ﬁrst note that as a consequence of (9.3.24), (2.1.23), and (9.3.17), whenever u = ∑ni=1 u i dx i ∈ Λ1 TM we have n
Ric u = ∑ S jk g ki u i dx j .
(9.3.32)
i,j,k=1 n Hence, given any X = ∑ℓ=1 aℓ ∂ℓ ∈ TM, we may invoke (9.3.18) and (9.1.3) in order to compute n
n
(Ric u)(X) = ∑ S jk g ki u i aℓ dx j (∂ℓ ) = ∑ S jk g ki u i a j i,j,k,ℓ=1
i,j,k=1
n
= ∑ S jk (u♯ )k a j = S(X, u♯ ),
(9.3.33)
j,k=1
as wanted. Finally, (9.3.30) is a consequence of (9.3.29) and the properties of the musical isomorphisms (from (9.1.2) and (9.1.3)). This brings us to the main result in this section, presented in the theorem below. Theorem 9.8. When acting on differential forms of arbitrary degree, the HodgeLaplacian and the BochnerLaplacian on the manifold M are related via Weitzenböck’s formula (9.3.34) ∆HL = ∆BL − Ric where Ric is the Weitzenböck operator on M. In the particular case of 1forms, this becomes Bochner’s formula ∆HL u = ∆BL u − S(u♯ , ⋅)
for all u ∈ C 2 (M, Λ1 TM),
(9.3.35)
where S(u♯ , ⋅) is regarded as the 1form acting according to TM ∋ X → S(u♯ , X) ∈ ℝ. Proof. Throughout, we agree to denote by [A, B] := AB − BA and {A, B} := AB + BA, respectively, the commutator and anticommutator of two generic operators A, B. In particular, the identities {AB, C} = {A, C}B + A[B, C],
(9.3.36)
{A, BC} = B{A, C} + [A, B]C,
(9.3.37)
[A, BC] = [A, B]C + B[A, C],
(9.3.38)
∗
∗
∗
[A, B] = −[A , B ],
(9.3.39)
392  9 Further Tools from Geometry and Analysis
(where the superscipt star denotes the adjoint) are readily veriﬁed. Then, using the summation convention over repeated indices, with the help of (2.1.94), (9.2.21), and (9.3.36) we may write ∆HL = −{d, δ} = −{dx i ∧ ∇∂ i , δ} = I + II
(9.3.40)
where I := −{dx i ∧ , δ}∇∂ i
and
II := −dx i ∧ [∇∂ i , δ].
(9.3.41)
To transform I, use (9.2.33) and (9.3.37) to write I = −{dx i ∧ , (∇∂ j )∗ dx j ∨}∇∂ i = −((∇∂ j )∗ {dx i ∧ , dx j ∨} + [dx i ∧ , (∇∂ j )∗ ]dx j ∨)∇∂ i .
(9.3.42)
To proceed, note that for every i, j ∈ {1, . . . , n} item (7) in Lemma 2.2 and (2.1.23) give {dx i ∧ , dx j ∨} = g ij ,
(9.3.43)
[dx i ∧ , (∇∂ j )∗ ] = −[dx i ∧ , ∇∂ j ] = [∇∂ j , dx i ∧].
(9.3.44)
while from (9.2.17) we have
Moreover, for every differential form u, from the product formula (9.2.10) and (9.2.6) we deduce that [∇∂ j , dx i ∧]u = ∇∂ j (dx i ∧ u) − dx i ∧ (∇∂ j u) i = (∇∂ j (dx i )) ∧ u = −Γ jℓ dxℓ ∧ u.
(9.3.45)
Hence i dxℓ ∧ [∇∂ j , dx i ∧] = −Γ jℓ
(9.3.46)
which, in concert with (9.3.43) and (9.3.4), permits us to rewrite I in (9.3.42) as i I = −(∇∂ j )∗ g ij ∇∂ i + Γ jℓ dxℓ ∧ (dx j ∨ ∇∂ i ) i = ∆BL + Γ jℓ dxℓ ∧ (dx j ∨ ∇∂ i ).
(9.3.47)
Returning to II in (9.3.41), recall (9.2.30) and make use of (9.3.38), as well as (9.2.17), (9.3.39) and item (8) in Lemma 2.2, to further transform this term as follows: II = dx i ∧ [∇∂ i , dx j ∨ ∇∂ j ] = dx i ∧ ([∇∂ i , dx j ∨]∇∂ j + dx j ∨ [∇∂ i , ∇∂ j ]) = dx i ∧ ([−(∇∂ i )∗, (dx j ∧ )∗ ]∇∂ j + dx j ∨ [∇∂ i , ∇∂ j ]) ∗
= dx i ∧ ([∇∂ i , dx j ∧] ∇∂ j + dx j ∨ [∇∂ i , ∇∂ j ]) j
∗
= dx i ∧ ((−Γ iℓ dxℓ ∧) ∇∂ j + dx j ∨ [∇∂ i , ∇∂ j ]) j
= dx i ∧ (−Γ iℓ dxℓ ∨ ∇∂ j + dx j ∨ [∇∂ i , ∇∂ j ]).
(9.3.48)
9.4 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting
 393
In view of Ricci’s identity from Proposition 9.5 and the deﬁnition of Ric in (9.3.24) we obtain j
II = −Γ iℓ dx i ∧ (dxℓ ∨ ∇∂ j ) − R ijkℓ dx i ∧ (dx j ∨ (dx k ∧ (dxℓ ∨ ))) j
= −Γ iℓ dx i ∧ (dxℓ ∨ ∇∂ j ) − Ric.
(9.3.49)
After readjusting notation (and invoking the symmetry of the Christoffel symbols in the lower indices) we therefore arrive at the conclusion that i II = −Γ jℓ dxℓ ∧ (dx j ∨ ∇∂ i ) − Ric.
(9.3.50)
At this stage, formula (9.3.34) follows from (9.3.40), (9.3.47), and (9.3.50). Finally, (9.3.35) is a particular case of (9.3.34), thanks to (9.3.29). In closing, we note that as a consequence of (9.3.34), item (5) in Lemma 2.8, and (9.3.27), it follows that the BochnerLaplacian commutes with the Hodge staroperator, i.e., ∗ ∆BL = ∆BL∗ . (9.3.51)
9.4 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting Here the goal is to review Sobolev spaces on the boundaries of Ahlfors regular domains in the Euclidean setting. To set the stage, assume that Ω ⊂ ℝn is an open set of locally ﬁnite perimeter satisfying H n−1 (∂Ω \ ∂∗ Ω) = 0. With σ := H n−1 ⌊∂Ω, this implies that the geometric measure theoretic outward unit normal ν = (ν1 , . . . , ν n ) to Ω is deﬁned σa.e. on ∂Ω. To proceed, for each j, k ∈ {1, . . . , n} consider the ﬁrstorder tangential derivative operator ∂ τ jk acting on an arbitrary function φ ∈ C01 (ℝn ) by ∂ τ jk φ := ν j [∂ k φ]∂Ω − ν k [∂ j φ]∂Ω
on ∂Ω.
(9.4.1)
Note that if also ψ ∈ C01 (ℝn ), then (with L n denoting the ndimensional Lebesgue measure in ℝn ) we have: ∫ (∂ τ jk φ)ψ dσ = ∫ {ν j (∂ k φ)ψ − ν k (∂ j φ)ψ} dσ ∂Ω
∂Ω
= ∫{(∂ k φ)(∂ j ψ) − (∂ j φ)(∂ k ψ)} dL n Ω
(9.4.2)
= ∫ {ν k φ(∂ j ψ) − ν j φ(∂ k ψ)} dσ ∂Ω
= − ∫ φ(∂ τ jk ψ) dσ. ∂Ω
394  9 Further Tools from Geometry and Analysis
Above, the second and third identities are implied by the GaussGreen formula for smooth, compactly supported vector ﬁelds, in domains of locally ﬁnite perimeter (cf. [31]; see also Theorem 9.66), ∫ ν j F k dσ = ∫ ∂ j F k dL n ,
(9.4.3)
Ω
∂Ω
applied ﬁrst to F k := (∂ k φ)ψ, and its counterpart with k and j switched, so the resulting integral is ∫{(∂ k φ)(∂ j ψ) + (∂ j ∂ k φ)ψ − (∂ j φ)(∂ k ψ) − (∂ k ∂ j φ)ψ} dL n ,
(9.4.4)
Ω
and the resulting cancellation yields the second line in (9.4.2), provided φ, ψ ∈ C02 (ℝn ). This gives (9.4.2) for such φ, ψ, and a limiting argument gives (9.4.2) for φ, ψ ∈ C01 (ℝn ). For later use, we recast this argument. We set up the vector ﬁelds G jk := (∂ k φ)ψej − (∂ j φ)ψek , H jk := φ(∂ j ψ)ek − φ(∂ k ψ)ej ,
(9.4.5)
where {e1 , . . . , en } is the standard orthonormal basis of ℝn . Then div G jk = div H jk = (∂ k φ)(∂ j ψ) − (∂ j φ)(∂ k ψ), ν ⋅ G jk = (∂ τ jk φ)ψ,
ν ⋅ H jk = −φ(∂ τ jk ψ),
(9.4.6)
Then (9.4.2) can be rewritten ∫ (∂ τ jk φ)ψ dσ = ∫ ν ⋅ G jk dσ ∂Ω
∂Ω
= ∫ div G jk dL n Ω
(9.4.7)
= ∫ ν ⋅ H jk dσ ∂Ω
= − ∫ φ(∂ τ jk ψ) dσ, ∂Ω
a sequence of identities that applies directly to all φ, ψ ∈ C01 (ℝn ), using the Divergence Theorem from [31] (cf. also the version recorded in Theorem 9.66). The above considerations suggest making the following deﬁnition. Suppose Ω ⊂ ℝn is an open set of locally ﬁnite perimeter satisfying H n−1 (∂Ω \ ∂∗ Ω) = 0. Deﬁne σ := H n−1 ⌊∂Ω and denote by ν = (ν1 , . . . , ν n ) its geometric measure theoretic outward p unit normal. In this setting, given f ∈ L p (∂Ω), p ∈ [1, ∞], we say that f ∈ L1 (∂Ω) provided that for each j, k ∈ {1, . . . , n} there exists f jk ∈ L p (∂Ω) such that ∫ (∂ τ jk φ)f dσ = − ∫ φf jk dσ, ∂Ω
∂Ω
∀ φ ∈ C01 (ℝn ).
(9.4.8)
9.4 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting  395
In such a case, we say ∂ τ jk f := f jk
(9.4.9)
a piece of notation which allows us to recast (9.4.8) as ∫ f(∂ τ jk φ) dσ = − ∫ (∂ τ jk f)φ dσ, ∂Ω
∀ φ ∈ C01 (ℝn ).
(9.4.10)
∂Ω
In particular, ∂ τ jk f = −∂ τ kj f . Since whenever g ∈ L p (∂Ω), p ∈ [1, ∞], is such that ∀ φ ∈ C0∞ (ℝn ),
∫ φg dσ = 0,
(9.4.11)
∂Ω
we necessarily have g = 0 at σa.e. point on ∂Ω (as seen from Lemma 2.11), it follows that ∂ τ jk f is unambiguously deﬁned in (9.4.9). Moreover, by (9.4.2), or (9.4.7), if we set f := ψ∂Ω , with ψ ∈ C01 (ℝn ), then p
f ∈ L1 (∂Ω)
∂ τ jk f = ν j [∂ k ψ]∂Ω − ν k [∂ j ψ]∂Ω ,
and
∀ j, k ∈ {1, . . . , n}.
(9.4.12)
Based on this observation and Lemma 2.11 we may then conclude that p
L1 (∂Ω) → L p (∂Ω) continuously and densely, for every p ∈ [1, ∞).
(9.4.13)
Next, if for p ∈ (1, ∞) we deﬁne ∗
p
p
L−1 (∂Ω) := (L1 (∂Ω)) ,
1 p
+
1 p
= 1,
(9.4.14)
p
it turns out that L1 (∂Ω) is a reﬂexive space whenever 1 < p < ∞, and p
L p (∂Ω) → L−1 (∂Ω) continuously and densely,
∀ p ∈ (1, ∞).
(9.4.15)
The space (9.4.14) has been characterized in [50, (3.6.39), p. 2682]. Speciﬁcally, it has been shown there that, whenever 1 < p, p < ∞ satisfy 1/p + 1/p = 1, there exists a ﬁnite constant C = C(Ω, p) > 0 such that p
∀ f ∈ L−1 (∂Ω) there exist f0 , f jk ∈ L p (∂Ω),
1 ≤ j < k ≤ n,
p such that ‖f0 ‖L p (∂Ω) + ∑ ‖f jk ‖L p (∂Ω) ≤ C‖f‖L−1 (∂Ω)
and (9.4.16)
1≤j 0 with the property that ± n.t. (u )(x) − (u± n.t. )(y) ≤ C dist (x, y) ⋅ [N(∇u± )(x) + N(∇u± )(y)], ∂Ω ∂Ω for each z ∈ ∂Ω, each r ∈ (0, R), and σa.e. points x, y ∈ B r (z) ∩ ∂Ω.
(9.4.78)
If we now deﬁne g := max {N(∇u− ), N(∇u+ )}, from (9.4.77) and (9.4.78) we conclude that 0 ≤ g ∈ L p (∂Ω), ‖g‖L p (∂Ω) ≤ C‖f‖L1p (∂Ω) , and we have f(x) − f(y) ≤ C dist (x, y) ⋅ [g(x) + g(y)], for all z ∈ ∂Ω, r ∈ (0, R), and σa.e. x, y ∈ B r (z) ∩ ∂Ω.
(9.4.79)
Next, pick some r ∈ (0, R) arbitrary, and abbreviate ∆ r := B r (z) ∩ ∂Ω, f ∆ r := ∫−∆ f dσ, r where z ∈ ∂Ω is arbitrary. Then p − f − f ∆ r dσ) (∫
1/p
1/p
− ∫ − f(x) − f(y)p dσ(y) dσ(x)) ≤ (∫
∆r
∆r ∆r 1/p
− gp dσ) ≤ Cr( ∫
≤ Cr1−(n−1)/p ‖g‖L p (∆ r ) .
(9.4.80)
∆r
In the particular case when p = n − 1, the exponent of r vanishes, which permits us to conclude that, for f ∈ L1n−1 (∂Ω), 1/(n−1)
lim+ {
R→0
sup r∈(0,R), z∈∂Ω
n−1 − f − f ∆ r dσ) (∫
}
∆r
≤ C lim+ { R→0
sup r∈(0,R), z∈∂Ω
‖g‖L n−1 (∆ r ) } = 0.
(9.4.81)
The last equality above is based on the observation that since the ﬁnite measure gn−1 dσ is absolutely continuous with respect to σ, for every ε > 0 there exists ρ > 0
9.4 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting
 405
with the property that ∫ gn−1 dσ < ε
whenever r ∈ (0, ρ).
(9.4.82)
∆r
From (9.4.81) it follows that every function f ∈ L1n−1 (∂Ω) has vanishing mean oscillations. This proves that the inclusion in (9.4.76) is a welldeﬁned linear operator. To show that this inclusion is also bounded (recall that VMO(∂Ω) is a closed subspace of BMO(∂Ω), hence a Banach space when endowed with the norm in the latter space), observe that (9.4.80) with p = n − 1 allow us to write 1/(n−1)
sup r∈(0,R), z∈∂Ω
n−1 − f − f ∆ r dσ) (∫ ∆r
≤ C‖g‖L n−1 (∂Ω) ≤ C‖f‖L1n−1 (∂Ω) .
(9.4.83)
The continuity of the inclusion in (9.4.76) now follows from (9.4.83) and the JohnNirenberg inequality. Under suitable geometric assumptions on the underlying domain, it becomes possible to show that control of the nontangential maximal operator of the gradient of a given function implies that the function in question has nontangential pointwise traces a.e. on the boundary. This topic is considered in our next proposition. Proposition 9.19. Let Ω ⊂ ℝn be a bounded interior NTA domain whose boundary is Ahlfors regular, satisfying H n−1 (∂Ω \ ∂∗ Ω) = 0. Set σ := H n−1 ⌊∂Ω, and ﬁx κ ∈ (0, ∞) along with p ∈ (0, ∞]. In this setting, suppose u ∈ C 1 (Ω) has Nκ (∇u) ∈ L p (∂Ω). Then n.t. Nκ u ∈ L p (∂Ω), u∂Ω exists at σa.e. point on ∂Ω, (9.4.84) and
n.t. u∂Ω L p (∂Ω) ≤ Nκ uL p (∂Ω) .
(9.4.85)
Proof. Let κ̃ , R ∈ (0, ∞) be associate to the given setting as in Lemma 2.22. Consider a measurable set A ⊆ ∂Ω, of full measure (relative to σ), with the property that Ñκ (∇u)(x) < ∞
and
x ∈ Γ κ (x)
for every x ∈ A.
(9.4.86)
That such a set exists is guaranteed by assumptions, Proposition 2.13, (2.2.34), and (2.2.35). Pick an arbitrary x ∈ A. Then for every y0 , y1 ∈ Γ κ (x) ∩ B R/2 (x) consider γ as in (2.2.60). In concert with the Fundamental Theorem of Calculus, this permits us to estimate u(y1 ) − u(y0 ) ≤ ∫ ∇u ds ≤ Ñκ (∇u)(x) ⋅ length(γ) γ
≤ C Ñκ (∇u)(x) ⋅ max {dist (x, y0 ), dist (x, y1 )},
(9.4.87)
406  9 Further Tools from Geometry and Analysis
where ds denotes the arclength measure on γ. Having established this, elementary real analysis then shows (reasoning by contradiction) that the numerical sequence {u(y j )}j∈ℕ is Cauchy whenever {y j }j∈ℕ ⊆ Γ κ (x) is a sequence of points with the property that y j → x as j → ∞. This ultimately shows that lim
Γ κ (x)∋y→x
u(y) exists for σa.e. x ∈ ∂Ω.
(9.4.88)
To proceed, deﬁne r := 12 min {R, diam Ω} and ﬁx an arbitrary point x ∈ ∂Ω. The interior corkscrew condition guarantees the existence of some constant C o ∈ (1, ∞), independent of x, with the property that there exists z ∈ B r (x) such that B r/C o (z) ⊆ Ω. Note that it is possible to choose κ o ∈ (0, ∞) independent of x, z such that actually B r/C o (z) ⊆ Γ κ o (x). Let κ̃ o ∈ (0, ∞) be associate to this κ o as in Lemma 2.22. Then, arguing much as in (9.4.87), for every y ∈ Γ κ o (x) ∩ B r (x) we may estimate u(y) ≤ u(z) + C Ñκ o (∇u)(x) ⋅ max {dist (x, y) , dist (x, z)} ≤ sup u + C Ñκ o (∇u)(x),
(9.4.89)
O
where C = C(Ω) ∈ (0, ∞) is independent of x, y, z and O := {w ∈ Ω : dist (w, ∂Ω) > r/C o }.
(9.4.90)
In turn, (9.4.89) implies sup
y∈Γ κ o (x)∩B r (x)
u(y) ≤ sup u + C Ñκ o (∇u)(x), O
∀ x ∈ ∂Ω.
(9.4.91)
From (9.4.91) we deduce that there exists some compact subset K of Ω with the property that Ñκ o u ≤ sup u + C Ñκ o (∇u) everywhere on ∂Ω. (9.4.92) K
Bearing in mind that ∂Ω has ﬁnite measure, from assumption, (2.2.38), and Proposition 2.13, we then conclude that Nκ u ∈ L p (∂Ω). In concert with (9.4.88), this ﬁnishes the justiﬁcation of the properties listed in (9.4.84). Having established these, estimate (9.4.85) becomes a consequence of (2.2.39) and (2.2.40). As in the past, stronger assumptions on the domain allow us to place the nontangential poitwise trace in a Sobolev space. Speciﬁcally, we have the following result. Corollary 9.20. Suppose Ω ⊂ ℝn is a bounded open set satisfying a twosided local John condition and whose boundary is Ahlfors regular. Then any function u ∈ C 1 (Ω) satisfying N(∇u) ∈ L p (∂Ω) for some p ∈ (1, ∞) and has the property that p n.t. N u ∈ L p (∂Ω), u∂Ω exists and belongs to L1 (∂Ω) (9.4.93) and
n.t. p u∂Ω L (∂Ω) ≤ C (‖N u‖L p (∂Ω) + ‖N(∇u)‖L p (∂Ω) ), 1 for some ﬁnite constant C > 0, independent of u.
(9.4.94)
9.4 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Euclidean Setting  407
Proof. This follows by combining Proposition 9.19 with Proposition 9.17. We conclude this section by considering the issue of integration by parts inside a principal value singular integral on the boundary of a UR domain. Proposition 9.21. Suppose Ω ⊂ ℝn is a bounded UR domain and set σ := H n−1 ⌊∂Ω. Consider a complexvalued function b ∈ C ∞ (ℝn \ {0}) which is even and positive homogeneous of degree 2 − n. Also, ﬁx an integrability exponent p ∈ (1, ∞) and pick some arbitrary j, k ∈ {1, . . . , n}. p Then for every function f ∈ L1 (∂Ω) one has P.V. ∫ ∂ τ jk (y) [b(x − y)]f(y) dσ(y) ∂Ω
= ∫ b(x − y)(∂ τ kj f)(y) dσ(y)
for σa.e. x ∈ ∂Ω.
(9.4.95)
∂Ω
In particular, corresponding to f = 1, there holds P.V. ∫ ∂ τ jk (y) [b(x − y)] dσ(y) = 0 for σa.e. x ∈ ∂Ω.
(9.4.96)
∂Ω
Proof. For each g ∈ L p (∂Ω) deﬁne Bg(x) := ∫ b(x − y)g(y) dσ(y),
x ∈ Ω.
(9.4.97)
∂Ω
Denote by ν = (ν 1 , . . . , ν n ) the geometric measure theoretic outward unit normal to p Ω. Then, if f ∈ L1 (∂Ω), the fact that for any ﬁxed x ∈ Ω the function b(x − ⋅) is of class C 1 near ∂Ω allows us to write −∂ k B(ν j f)(x) + ∂ j B(ν k f)(x) = ∫ ∂ τ jk (y) [b(x − y)]f(y) dσ(y) ∂Ω
= ∫ b(x − y)(∂ τ kj f)(y) dσ(y) ∂Ω
= B(∂ τ kj f)(x),
∀ x ∈ Ω.
(9.4.98)
On the other hand, at σa.e. x ∈ ∂Ω the jumprelations established in [50] yield 1 n.t. ∂ k b(ν(x))ν j (x)f(x) ((∂ k B(ν j f))∂Ω )(x) = ̂ 2i + P.V. ∫ ∂ x k [b(x − y)](ν j f)(y) dσ(y) ∂Ω
1̂ = b(ν(x))ν j (x)ν k (x)f(x) 2 − P.V. ∫ ν j (y)∂ y k [b(x − y)]f(y) dσ(y), ∂Ω
(9.4.99)
408  9 Further Tools from Geometry and Analysis
and
1̂ n.t. ((∂ j B(ν k f))∂Ω )(x) = b(ν(x))ν j (x)ν k (x)f(x) 2 − P.V. ∫ ν k (y)∂ y j [b(x − y)]f(y) dσ(y).
(9.4.100)
∂Ω
Consequently, for σa.e. x ∈ ∂Ω we have n.t. ∫ b(x − y)(∂ τ kj f)(y) dσ(y) = (B(∂ τ kj f))∂Ω (x) ∂Ω
n.t. = (−∂ k B(ν j f) + ∂ j B(ν k f))∂Ω (x)
n.t. n.t. = − ((∂ k B(ν j f))∂Ω )(x) + ((∂ j B(ν k f))∂Ω )(x) = P.V. ∫ ν j (y)∂ y k [b(x − y)]f(y) dσ(y) ∂Ω
− P.V. ∫ ν k (y)∂ y j [b(x − y)]f(y) dσ(y) ∂Ω
= P.V. ∫ ∂ τ jk (y) [b(x − y)] dσ(y),
(9.4.101)
∂Ω
as wanted.
9.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting This section continues the discussion pertaining to Sobolev spaces on boundaries of Ahlfors regular domains, initiated in § 9.4, by considering the manifold setting. Let M be a C 2 manifold equipped with a C 1 Riemannian metric, and assume that Ω ⊂ M is an open set which is of locally ﬁnite perimeter. This means that, if d is the exterior derivative operator on M and 1Ω denotes the characteristic function of the set Ω, we have μ := d1Ω (9.5.1) is a locally ﬁnite T ∗ Mvalued measure. It follows from the RadonNikodym theorem that μ = −ν σ, where σ is a locally ﬁnite positive measure, supported on ∂Ω, and where ν ∈ L∞ (∂Ω, σ) is an T ∗ Mvalued function, satisfying ν(x)T x∗ M = 1 for σa.e. x ∈ ∂Ω.
(9.5.2)
We ﬁnd it convenient to locally identify Ω ⊂ M with its Euclidean image under coordinate charts, which continues to be a domain of locally ﬁnite perimeter (cf. [51]), still denoted Ω. In such local coordinates, we claim that³ if 3 compare with [50, § 5.1 p. 2763, and § 5.3 p. 2773]
9.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting  409
(νEj )1≤j≤n is the outward unit normal to ∂Ω with respect to the Euclidean metric in ℝn ,
(9.5.3)
then (with the summation convention over repeated indices understood) we have ν = ν j dx j where ν j = G−1/2 νEj
G := g rs νEr νEs = (ν j ν j )−1 ,
and
(9.5.4)
and the unit outward normal to ∂Ω with respect to the Riemannian metric (9.1.1) is given by (9.5.5) ν♯ = ν j ∂ j ∈ TM where ν j := g jk ν k = g jk G−1/2 νEk . To justify these claims, consider a point x ∈ ∂∗ Ω. At this location, ∂Ω has an approximate tangent plane T x ∂Ω, and for every vector ﬁeld t = t k ∂ k in it we have 0 = ν(t) = (ν j dx j )(t k ∂ k ) = ν j t j = ⟨(t j )1≤j≤n , (ν j )1≤j≤n ⟩ℝn .
(9.5.6)
Hence, if we regard T x ∂Ω as a linear subspace in ℝn , by identifying t = t k ∂ k ∈ T x ∂Ω with the vector (t j )1≤j≤n ∈ ℝn , then the Euclidean unit normal νE = (νEj )1≤j≤n to the latter space is parallel to (ν j )1≤j≤n . Hence, for some positive scalar G, we have ν j = G−1/2 νEj ,
∀ j ∈ {1, . . . , n}.
(9.5.7)
To ﬁnd concrete formulas for G, observe that 1 = g(ν, ν) = ν j ν k g jk = G−1 g jk νEj νEk ⇒ G = g jk νEj νEk ,
(9.5.8)
1 = ⟨νE , νE ⟩ = Gν j ν j ⇒ G = (ν j ν j )−1 .
(9.5.9)
and
This completes the proof of the claims in (9.5.4). In turn, (9.5.5) readily follows from (9.5.4), (9.1.2), and (9.1.3). Moving on, it has been shown in [50, § 5.3, Proposition 5.7, p. 2774] that σ = HMn−1 ⌊∂Ω,
(9.5.10)
where HMn−1 is the (n − 1)dimensional Hausdorff measure associated with the geodesic distance induced by the metric (9.1.1) on M, and that in local coordinates σ = √g G1/2 σE ,
(9.5.11)
σE := Hℝn−1 ⌊∂Ω n
(9.5.12)
where
is the surface measure induced by the ﬂatspace Euclidean metric δ jk on ∂Ω. Let ∂∗ Ω denote the reduced boundary of Ω, in the Euclidean sense. Then it follows from (9.5.11) and (9.5.4) that σ is supported on ∂∗ Ω. and ν(x)T x∗ M = 1 for each x ∈ ∂∗ Ω. p
Moving on, in order to consider the spaces L1 (∂Ω), 1 ≤ p ≤ ∞, in the case when Ω is a relatively compact subdomain of the C 1 manifold M, we need to establish the
410  9 Further Tools from Geometry and Analysis invariance of the Euclidean scale of Sobolev spaces introduced earlier under C 1 diffeomorphisms of the Euclidean ambient. In the process, we need to know how surface integrals transform under C 1 diffeomorphisms. This aspect has been addressed in [51, Theorem 3.2], recalled below. Proposition 9.22. Let Ω ⊂ ℝn be a domain of locally ﬁnite perimeter, O an open neighborhood of Ω, and let F : O → ℝn be an orientation preserving C 1 diffeomorphism onto its image. ̃ := F(Ω) is a set of locally ﬁnite perimeter and if ν, ̃ν and σ, σ ̃ are, respectively, Then Ω ̃ then the outward unit normals and surface measures on ∂Ω and ∂ Ω, ⊤
(DF −1 ) (ν ∘ F −1 ) ̃ν = (DF −1 )⊤ (ν ∘ F −1 )
(9.5.13)
(with the convention that the right side of (9.5.13) is zero whenever ν ∘ F −1 = 0), and ̃ = (DF −1 )⊤ (ν ∘ F −1 ) (det DF) ∘ F −1  F∗ σ, σ
(9.5.14)
where F∗ σ, the pushforward of σ via F, is deﬁned by the requirement that (F∗ σ)(E) := σ(F −1 (E)),
∀ E ⊆ ∂Ω Borel set.
(9.5.15)
̃ σ ̃ ) with compact support formula (9.5.14) entails Observe that for every⁴ g ∈ L1 (∂ Ω, ̃ = ∫ (g ∘ F)(DF −1 ∘ F)⊤ ν det(DF) dσ. ∫ g dσ ̃ ∂Ω
(9.5.16)
∂Ω
We now turn to the result that permits us to consider Sobolev spaces on boundaries of locally ﬁnite perimeter subsets of C 1 manifolds. This actually provides an explicit formula expressing the way in which tangential derivatives transform under a C 1 diffeomorphic change of variables in the Euclidean ambient. To state it, let us agree that for any two vectors a, b ∈ ℝn with a = (a1 , . . . , a n ) and b = (b1 , . . . , b n ) the symbol a ⊗ b stands for the n × n matrix whose (r, s) entry is given by (a ⊗ b)rs := a r b s ,
r, s ∈ {1, . . . , n}.
(9.5.17)
be a bounded open set satisfying H \ ∂∗ Ω) = 0 Proposition 9.23. Let Ω ⊂ n−1 and, as usual, set σ := H ⌊∂Ω. Also, let O an open neighborhood of Ω and suppose that F : O → ℝn is an orientation preserving C 1 diffeomorphism onto its image ̃ := F(Ω). Ω Then ̃ is an open set of locally ﬁnite perimeter Ω (9.5.18) ̃ = 0. ̃ \ ∂∗ Ω) and H n−1 (∂ Ω ℝn
n−1 (∂Ω
̃ Moreover, if ∂Ω is an Ahlfors regular set then so is ∂ Ω. 4 using a piece of selfexplanatory notation, in order to emphasize the dependence of the scale of Lebesgue spaces considered on the background measure
9.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting  411
̃ and f is a measurable function deﬁned on ∂Ω, then ̃ := H n−1 ⌊∂ Ω Furthermore, if σ for each p ∈ [1, ∞] one has⁵ p p ̃ ̃) f ∈ L1 (∂Ω, σ) ⇐⇒ f ∘ F −1 ∈ L1 (∂ Ω, σ
(9.5.19)
and ‖f‖L1p (∂Ω,σ) ≈ ‖f ∘ F −1 ‖L p (∂ Ω, ̃ σ ̃) . 1
p
Moreover, if Ω is actually a UR domain then for every function f ∈ L1 (∂Ω, σ) with 1 ≤ p ≤ ∞ and every pair of indices j, k ∈ {1, . . . , n}, one has ∂ ̃τ jk (f ∘ F −1 ) =
((DF −1 )⊤ [(ν ⊗ ∇Etan f − ∇Etan f ⊗ ν) ∘ F −1 ](DF −1 ))jk (DF −1 )⊤ (ν ∘ F −1 )
.
(9.5.20)
̃ given by ̃ν j ∂ k − ̃ν k ∂ j if ̃ν = (̃ν j )j is the Above, ∂ ̃τ jk is the tangential derivative on ∂ Ω E ̃ outward unit normal to Ω, and ∇tan f is the (Euclidean) tangential gradient of f , deﬁned as the vector n
∇Etan f := (− ∑ ν k ∂ τ jk f ) k=1
.
(9.5.21)
1≤j≤n
̃ is an Ahlfors regProof. For starters, the claims in (9.5.18) (as well as the fact that ∂ Ω ular set if ∂Ω is so) have been proved in [51]. Also, (9.5.16) implies ‖f‖L p (∂Ω,σ) ≈ ‖f ∘ F −1 ‖L p (∂ Ω, ̃ σ ̃) .
(9.5.22)
Handling tangential derivatives is more delicate. To set the stage, abbreviate A := (A rs )1≤r, s≤m := (DF −1 ∘ F)⊤ = ((DF)−1 )
⊤
and J := det(DF).
(9.5.23)
p
Also, ﬁx p ∈ [1, ∞] and f ∈ L1 (∂Ω, σ). Then for every φ ∈ C01 (ℝn ) and j, k ∈ {1, . . . , n} we may write based on (9.5.16) and (9.5.13) that ̃ = ∫ (f ∘ F −1 ){̃ν j ∂ k φ − ̃ν k ∂ j φ} dσ ̃ ∫ (f ∘ F −1 )∂ ̃τ jk φ dσ ̃ ∂Ω
(9.5.24)
̃ ∂Ω
= ∫ f {(Aν)j (∂ k φ) ∘ F − (Aν)k (∂ j φ) ∘ F} J dσ. ∂Ω
Since chain rule gives that for every r ∈ {1, . . . , n}, (∂ r φ) ∘ F = ∂ s (φ ∘ F)(DF)−1 sr = ∂ s (φ ∘ F)A rs ,
(9.5.25)
the expression in the last set of curly brackets in (9.5.24) may be written as (Aν)j (∂ k φ) ∘ F − (Aν)k (∂ j φ) ∘ F = A jℓ νℓ ∂ s (φ ∘ F)A ks − A ks ν s ∂ℓ (φ ∘ F)A jℓ = A jℓ A ks ∂ τℓs (φ ∘ F).
(9.5.26)
5 again, using a piece of selfexplanatory notation, in order to emphasize the dependence of the scale of Sobolev spaces on the background measure
412  9 Further Tools from Geometry and Analysis We record our progress: for every φ ∈ C01 (ℝn ) and j, k ∈ {1, . . . , n} there holds ̃ = ∫ fA jℓ A ks ∂ τℓs (φ ∘ F) J dσ. ∫ (f ∘ F −1 )∂ ̃τ jk φ dσ ̃ ∂Ω
(9.5.27)
∂Ω
In relation to (9.5.27) we wish to observe that specializing this identity to the case when f = 1 yields ∫ A jℓ A ks ∂ τℓs (φ ∘ F) J dσ = 0.
(9.5.28)
∂Ω
This proves that if F is actually a C 2 diffeomorphism then ∂ τℓs (A jℓ A ks J) = 0 on ∂Ω.
(9.5.29)
In order to be able to take advantage of (9.5.29) we need to regularize F. As a preamble, we recall the elementary fact that any C 1 function is locally Lipschitz (i.e., Lipschitz when restricted to any compact subset of its original open domain of deﬁnition). Applying this to F and F −1 it follows that for each compact set K ⊂ O the function F : K → F(K) is biLipschitz, i.e., there exists λ K ∈ (1, ∞) such that λ−1 K x − y ≤ F(x) − F(y) ≤ λ K x − y,
∀ x, y ∈ K.
(9.5.30)
To proceed, pick θ ∈ C0∞ (ℝn ) such that supp θ ⊂ B(0, 1), θ ≥ 0, and ∫ θ = 1, and for each ε > 0 set θ ε (x) := ε−n θ(x/ε), x ∈ ℝn . Also, introduce Ω ε := {x ∈ ℝn : dist (x, Ω) < ε}, Then for each ε ∈ (0,
1 2
ε > 0.
(9.5.31)
dist (Ω, ∂O)) and each x ∈ Ω ε , set
F ε (x) := ∫ θ ε (x − z)F(z) dz = ∫ θ ε (z)F(x − z) dz. O
(9.5.32)
B(0, ε)
Hence F ε ∈ C ∞ (Ω ε ) and the upperinequality in (9.5.30) ensures that for each compact set K ⊂ O there exists C K ∈ (0, ∞) such that sup F ε − F ≤ εC K .
(9.5.33)
K
Together, (9.5.30) and (9.5.33) then readily imply that for each compact set K ⊂ O there exists ε K ∈ (0, ∞) such that F ε : K → F ε (K) is biLipschitz provided 0 < ε < ε K . Ultimately, this shows that there exists ε o > 0 with the property that ε ∈ (0, ε o ) ⇒ F ε is a C ∞ diffeomorphism from Ω ε onto F ε (Ω ε ). Aε
(9.5.34)
Next, denote by A ε , J ε the objects associated to F ε as in (9.5.23) and note that → A, J ε → J uniformly on ∂Ω as ε → 0+ . Also, A ε , J ε are C ∞ in a neighborhood
9.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting
 413
of ∂Ω. Based on these observations, as well as (9.5.34), (9.5.29), and (9.5.16), we may then compute ∫ fA jℓ A ks ∂ τℓs (φ ∘ F) J dσ = lim+ ∫ fA εjℓ A εks ∂ τℓs (φ ∘ F ε ) J ε dσ ε→0
∂Ω
∂Ω
= − lim+ ∫ (φ ∘ F ε )∂ τℓs (fA εjℓ A εks J ε ) dσ ε→0
∂Ω
= − lim+ ∫ (φ ∘ F ε )(∂ τℓs f)A εjℓ A εks J ε dσ ε→0
∂Ω
− lim+ ∫ (φ ∘ F ε ) f ∂ τℓs (A εjℓ A εks J ε ) dσ ε→0
∂Ω
= − lim+ ∫ (φ ∘ F ε )(∂ τℓs f)A εjℓ A εks J ε dσ ε→0
∂Ω
= − ∫ (φ ∘ F)(∂ τℓs f)A jℓ A ks J dσ ∂Ω
= − ∫( ̃ ∂Ω
A jℓ A ks ∂ τℓs f ̃. ) ∘ F −1 φ dσ Aν
(9.5.35)
This proves that A jℓ A ks ∂ τℓs f ) ∘ F −1 Aν [(DF −1 )⊤ ]jℓ [(DF −1 )⊤ ]ks (∂ τℓs f) ∘ F −1
∂ ̃τ jk (f ∘ F −1 ) = ( =
(DF −1 )⊤ (ν ∘ F −1 )
.
(9.5.36)
̃ σ ̃ ), give Since, as seen from (9.5.22), the last expression in (9.5.36) belongs to L p (∂ Ω, p that ∂ τℓs f ∈ L (∂Ω, σ), we conclude from this analysis and (9.5.22) that lefttoright implication in (9.5.19) holds. Of course, the opposite implication follows from what we have just proved used for the C 1 diffeomorphism F −1 . The equivalence in (9.5.19) is also implicit in the above argument. Hence, (9.5.19) is established. Next, if Ω is a UR domain, it has been shown in [50, Lemma 3.40, p. 2883] that ∂ τ rs f = ν r (∇Etan f)s − ν s (∇Etan f)r ,
∀ r, s ∈ {1, . . . , n}.
(9.5.37)
Then the identity in (9.5.20) follows from (9.5.37) and (9.5.36), after some simple algebra. Based on the module property mentioned earlier in (9.4.38) and the diffeomorphism invariance just presented in Propositions 9.22 and 9.23 one can then deﬁne L p based p Sobolev space of order one L1 (∂Ω) on the boundary of an open set Ω ⊂ M of locally ﬁnite perimeter via localization involving a smooth partition of unity and pullback to
414  9 Further Tools from Geometry and Analysis
the Euclidean model using coordinate charts. This scale of spaces continue to enjoy similar properties as in the Euclidean setting, discussed in § 9.4. More specialized properties hold when stronger conditions are placed on the underlying domain. Speciﬁcally, assume that⁶ Ω ⊂ M is a relatively compact Ahlfors regular domain satisfying a twosided local John condition.
(9.5.38)
In such a case, we have from (9.4.46) that, in a quantitative sense, if p ∈ (1, n − 1), L p (∂Ω) for p∗ := (n−1)p { n−1−p { q p L1 (∂Ω) ⊂ { L (∂Ω) for all q ∈ (1, ∞) if p = n − 1, { r n−1 if p ∈ (n − 1, ∞), { C (∂Ω) for r := 1 − p ∗
(9.5.39)
where n is the dimension of M. In the same setting, Proposition 9.18 gives that, corresponding to the case when p = n − 1, the inclusion L1n−1 (∂Ω) → VMO(∂Ω) is welldeﬁned, linear and bounded.
(9.5.40)
In addition, in the aforementioned setting it follows from (9.4.47) and (9.4.48) the following inclusion operators are welldeﬁned, linear and compact (where p∗ is as in (9.5.39)): p L1 (∂Ω) → L r (∂Ω) if 1 < p < n − 1 and 0 < r < p∗ , (9.5.41) L1 (∂Ω) → L∞ (∂Ω) p
if n − 1 < p < ∞.
(9.5.42)
We also note here that, by [50, Proposition 4.29, p. 2721], if Ω is an open set satisfying a twosided local John condition and whose boundary is compact and Ahlfors regular then p {ψ∂Ω : ψ ∈ C 1 (Ω)} is dense in L1 (∂Ω), for each p ∈ (1, ∞).
(9.5.43)
Going further, we remark that the scale of L p based Sobolev space of order one on boundary of an open set Ω of locally ﬁnite perimeter in a manifold M adapts naturally to the case when the functions involved take values in a given Hermitian vector bundle p E → M. In such a scenario, we denote this brand of Sobolev space by L1 (∂Ω, E). Also, much as (9.4.14), for p ∈ (1, ∞) we deﬁne p
p
∗
L−1 (∂Ω, E) := (L1 (∂Ω, E)) ,
1 p
+
1 p
= 1.
(9.5.44)
These scales of spaces then naturally inherit basic properties of their Euclidean counterparts (discussed in § 9.4). In particular, p
L p (∂Ω, E) → L−1 (∂Ω, E) continuously and densely, 6 these hypotheses imply Ω is a UR domain
∀ p ∈ (1, ∞).
(9.5.45)
9.5 Sobolev Spaces on Boundaries of Ahlfors Regular Domains: The Manifold Setting  415
In fact, it is possible to provide an intrinsic description of the Sobolev space p L1 (∂Ω, E). To state such a result, we need the following piece of notation. Given two continuous vector ﬁelds X, Y ∈ TM, for every φ ∈ C01 (M, E) deﬁne ∂ τ XY φ := ν(X)(∇Y φ)∂Ω − ν(Y)(∇X φ)∂Ω on ∂Ω,
(9.5.46)
where ∇ is a connection (or covariant derivative ) on E, with continuous connection coefficients. In particular, if in local coordinates the outward unit conormal ν is exn pressed as ν = ∑ℓ=1 νℓ dxℓ , then we agree to redenote ∂ τ XY deﬁned as in (9.5.46) corresponding to the choice X := ∂ j and Y := ∂ k simply as ∂ τ jk . That is, for every φ ∈ C 01 (M, E) deﬁne ∂ τ jk φ := ν j (∇∂ k φ)∂Ω − ν k (∇∂ j φ)∂Ω locally on ∂Ω, (9.5.47) which is in agreement with the Euclidean case considered in (9.4.1). Proposition 9.24. Let M be a C 2 manifold of dimension n, equipped with a C 1 Riemannian metric. Assume that E → M is a C 1 Hermitian vector bundle over M and denote by ⟨⋅, ⋅⟩E the pointwise (real) pairing in the ﬁbers of E. Also, suppose that Ω ⊂ M is an open set of locally ﬁnite perimeter with the property that HMn−1 (∂Ω \ ∂∗ Ω) = 0,
(9.5.48)
and ﬁx an integrability exponent p ∈ [1, ∞]. p Then a function f ∈ L p (∂Ω, E) belongs to the Sobolev space L1 (∂Ω, E) if and only if for any two C 1 vector ﬁelds X, Y ∈ TM there exists a function h XY ∈ L p (∂Ω, E) with the property that ∫ ⟨f, ∂ τ XY φ⟩E dσ = ∫ ⟨h XY , φ∂Ω ⟩E dσ, ∂Ω
∀ φ ∈ C01 (M, E),
(9.5.49)
∂Ω
where, as before, σ := HMn−1 ⌊∂Ω is the surface measure on ∂Ω. Moreover, the function h XY ∈ L p (∂Ω, E) doing the job in (9.5.49) is unique, and naturally accompanying estimates are valid for each implication. Proof. Write ν = ν j dx j for the unit outward conormal to ∂Ω and assume that X = X j ∂ j and Y = Y k ∂ k are two arbitrary C 1 vector ﬁelds on M. Select a coordinate patch U ⊂ M over which E may be trivialized using a smooth local orthonormal frame {e α }α . Finally, ﬁx a relatively compact subset O of U, and pick a scalarvalued cutoff function ψ in C01 (U) which is identically one in O. Then, for every C 1 section φ = φ α e α of E with compact support in O, from (9.5.46), (9.1.49), and (9.5.4) we deduce that on U ∩ ∂Ω we have αβ αβ ∂ τ XY φ = ν j X j (Y k ∂ k φ α + Y k γ k φ β )e α − ν k Y k (X j ∂ j φ α + X j γ j φ β )e α = G−1/2 νEj X j (Y k ∂ k φ α + Y k γ k φ β )e α αβ
− G−1/2 νEk Y k (X j ∂ j φ α + X j γ j φ β )e α αβ
416  9 Further Tools from Geometry and Analysis = G−1/2 X j Y k (νEj ∂ k − νEk ∂ j )φ α e α + G−1/2 X j Y k (νEj γ k − νEk γ j )φ β e α αβ
αβ
= G−1/2 X j Y k (∂Eτ jk φ α )e α + G−1/2 X j Y k (νEj γ k − νEk γ j )φ β e α , αβ
αβ
(9.5.50)
with ∂Eτ jk := νEj ∂ k − νEk ∂ j , tangential ﬁrstorder operator to ∂Ω in the Euclidean setting. p Assume now that some function f ∈ L1 (∂Ω, E), 1 ≤ p ≤ ∞, has been given. Hence, p α α f = f e α with each component f satisfying ψf α ∈ L1 (∂Ω, σ). Then ∫ ⟨f, ∂ τ XY φ⟩E dσ = ∫ X j Y k (∂Eτ jk φ α )ψf α √g dσE ∂Ω
∂Ω αβ
αβ
+ ∫ X j Y k (νEj γ k − νEk γ j )φ β f α √g dσE
(9.5.51)
∂Ω
and, granted the current assumptions on X, Y and the metric (9.1.1), there exist functions h α ∈ L p (∂Ω, σE ) with the property that ∫ X j Y k (∂Eτ jk φ α )ψf α √g dσE = ∫ h α φ α dσE . ∂Ω
(9.5.52)
∂Ω
Thus, ∫ ⟨f, ∂ τ XY φ⟩E dσ = ∫ ⟨hO XY , φ⟩E dσ ∂Ω
for all φ ∈ C01 (O, E),
(9.5.53)
∂Ω
if we take hO XY :=
1 βα βα h α e α + ψX j Y k (ν j γ k − ν k γ j )f β e α ∈ L p (∂Ω, E). √gG1/2
(9.5.54)
1 Appropriately patching the functions hO XY using a C partition of unity then yields a function h XY ∈ L p (∂Ω, E) for which (9.5.49) holds. Conversely, assume that f ∈ L p (∂Ω, E) has the property that for any two C 1 vector ﬁelds X, Y ∈ TM there exists a function h XY ∈ L p (∂Ω, E) such that (9.5.49) holds. Much as before, work in local coordinates and, given j, k ∈ {1, . . . , n}, pick X := ∂ j , Y := ∂ k . Then (9.5.51) gives
∫ ⟨h XY , φ⟩E dσ = ∫ (∂Eτ jk φ α )ψf α √g dσE ∂Ω
∂Ω αβ
αβ
+ ∫ (ν j γ k − ν k γ j )φ β f α √g dσ
(9.5.55)
∂Ω
C01 (O, E).
p
for all functions φ ∈ This proves that ψf α √g ∈ L1 (∂Ω, σE ) hence, further, p α E that ψf ∈ L1 (∂Ω, σ ), granted the regularity assumption on the metric. This suffices p to conclude that f belongs to the Sobolev space L1 (∂Ω, E). Finally, (a slight version of) Lemma 2.11 gives that the function h XY ∈ L p (∂Ω, E) doing the job in (9.5.49) is unique, while the fact that there are naturally accompanying estimates for each implication is implicit in the above calculations.
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  417
We conclude this section by mentioning that the Euclidean results formulated in Proposition 9.17, Proposition 9.19, and Corollary 9.20, have all natural counterparts for subdomains of Riemannian manifolds.
(9.5.56)
A concrete generalization of Proposition 9.16 to the manifold setting is discussed later, in Proposition 9.36.
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains Compared with Proposition 9.24, we desire a more constructive boundary integration by parts formula involving boundary Sobolev spaces of order one, and which permits the consideration of more general ﬁrst order (tangential) differential operators. This is accomplished in later, in (9.6.79). We begin building in that direction by ﬁrst proving the following key result. Theorem 9.25. Let M be a C 2 manifold equipped with a Riemannian metric, and consider three Hermitian vector bundles, E, F, H, over M, of class C 2 . Assume all metrics involved are of class C 1 . Suppose P : C 1 (M, E) → C 0 (M, H)
and
Q : C 1 (M, H) → C 0 (M, F)
(9.6.1)
are two ﬁrstorder differential operators with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part, having the property that their principal symbols satisfy the cancellation condition Sym (Q, ξ) Sym (P, ξ) = 0,
∀ ξ ∈ T ∗ M.
(9.6.2)
Denote by P ⊤ : C 1 (M, H) → C 0 (M, E) and Q⊤ : C 1 (M, F) → C 0 (M, H) the (real ) transposed of P, Q (considered in the usual sense on the manifold M). Next, let Ω ⊂ M be a relatively compact Ahlfors regular domain, set σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Finally, pick a Lebesgue measurable section u : Ω → E satisfying N u ∈ L1 (∂Ω),
Pu ∈ L1loc (Ω, H), N(Pu) ∈ L1 (∂Ω), n.t. n.t. and the traces u∂Ω , (Pu)∂Ω exist at σa.e. point on ∂Ω. Then
(9.6.3)
QP : C 1 (M, E) → C 0 (M, F) and P⊤ Q⊤ : C 1 (M, F) → C 0 (M, E) are ﬁrstorder differential operators,
(9.6.4)
418  9 Further Tools from Geometry and Analysis and for every function φ ∈ C 1 (Ω, F) one has n.t. ∫ ⟨i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ
(9.6.5)
F
∂Ω
n.t. = ∫ ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω ⟩ dσ E
∂Ω
n.t. + ∫ ⟨i Sym (QP, ν)u∂Ω , φ∂Ω ⟩ dσ F
∂Ω
n.t. = ∫ ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω − i Sym (P⊤ Q⊤, ν)φ∂Ω ⟩ dσ, E
∂Ω
where all principal symbols are taken in the sense of ﬁrstorder differential operators. Moreover, in the case when φ is actually compactly supported in a local coordinate patch O on M, with local coordinates (x j )1≤j≤n ∈ O, formula (9.6.5) may be written as n.t. ∫ ⟨i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ F
n.t. = − ∫ ⟨u∂Ω , ν j P⊤ (i Sym (Q⊤, dx j )φ)∂Ω ⟩ dσ,
∂Ω
E
(9.6.6)
∂Ω
where ν = ν j dx j in O.
Proof. We debut by making several observations. First, (9.6.4) is a direct consequence of (9.6.2), (2.1.68), and (2.1.64). Second, from Proposition 2.14 and the ﬁrst line in (9.6.3) we have (9.6.7) u ∈ L n/(n−1) (Ω, E) and Pu ∈ L n/(n−1) (Ω, H). Third, since C02 (M, F) is dense in C 1 (Ω, F), it suffices to establish formula (9.6.5) when φ ∈ C02 (M, F). Assume that this is the case and consider the vector ﬁeld F⃗ : Ω → TM deﬁned by asking that ⃗
T x∗ M (ξ, F(x))T x M
= ⟨i SymQ (x, ξ)(Pu)(x), φ(x)⟩Fx − ⟨u(x), i SymP⊤ (x, ξ)(Q⊤ φ)(x)⟩Ex − ⟨i SymQP (x, ξ)u(x), φ(x)⟩Fx
(9.6.8)
for a.e. x ∈ Ω and each ξ ∈ T x∗ M. The linearity of the righthand side in ξ ensures that this is a welldeﬁned object. Also, (9.6.7) and the regularity of φ permit us to conclude that F⃗ ∈ L1loc (Ω, TM). In fact, we may locally express F⃗ = F j ∂ j with F j (x) = ⟨i SymQ (x, dx j )(Pu)(x), φ(x)⟩Fx − ⟨u(x), i SymP⊤ (x, dx j )(Q⊤ φ)(x)⟩Ex − ⟨i SymQP (x, dx j )u(x), φ(x)⟩Fx .
(9.6.9)
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  419
Next, ﬁx a scalar function ψ ∈ C 01 (Ω) and compute the distributional pairing ⃗
D (Ω) (div F, ψ)D(Ω)
= − D (Ω) (F,⃗ grad ψ)D(Ω) ⃗ = − ∫ T x M ⟨(grad ψ)(x), F(x)⟩ T x M dVol(x) Ω
⃗ = − ∫ T x∗ M ((grad ψ)♭ (x), F(x)) T x M dVol(x) Ω
⃗ = − ∫ T x∗ M (dψ(x), F(x)) T x M dVol(x) Ω
= − ∫⟨i Sym(Q, dψ)Pu, φ⟩F dVol Ω
+ ∫⟨u, i Sym(P⊤, dψ)Q⊤ φ⟩E dVol Ω
+ ∫⟨i Sym(QP, dψ)u, φ⟩F dVol Ω
= I + II + III,
(9.6.10)
where, in view of (9.6.4) and (2.1.69), we may take I := ∫⟨[Q, ψ]Pu, φ⟩F dVol, Ω
II := − ∫⟨u, [P⊤, ψ]Q⊤ φ⟩E dVol, Ω
III := − ∫⟨[QP, ψ]u, φ⟩F dVol.
(9.6.11)
Ω
Thanks to Lemma 2.7 and the local regularity properties of u recorded in (9.6.7) we may integrate by parts, without boundary terms to obtain (bearing in mind that ψ is scalarvalued) ⊤
I = ∫⟨u, P⊤ [Q, ψ] φ⟩E dVol Ω
= − ∫⟨u, P⊤ [Q⊤, ψ]φ⟩E dVol,
(9.6.12)
Ω
where the last equality is a consequence of the fact that, in general, [A, B]⊤ = −[A⊤, B⊤ ].
(9.6.13)
Another application of (9.6.13) gives III = ∫⟨u, [P⊤ Q⊤, ψ]φ⟩E dVol. Ω
(9.6.14)
420  9 Further Tools from Geometry and Analysis
We also have − P⊤ [Q⊤, ψ]φ − [P⊤, ψ]Q⊤ φ + [P⊤ Q⊤, ψ]φ = 0.
(9.6.15)
This is seen by writing out all commutators (which is allowed since we are presently assuming φ ∈ C02 (M, F)) and canceling liketerms. From (9.6.10)(9.6.15) we may then conclude that div F⃗ = 0 in D (Ω). (9.6.16) In addition, (9.6.8) and the original assumptions on u imply that the nontangential n.t. trace F⃗ ∂Ω exists σa.e. on ∂Ω and ⃗ n.t. T ∗ M (ν, F ∂Ω )TM
n.t. = ⟨i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩
F
n.t. − ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω ⟩ E n.t. − ⟨i Sym (QP, ν)u∂Ω , φ∂Ω ⟩ . F
(9.6.17)
Since φ is continuously differentiable with compact support, (9.6.8) also gives that N(F)⃗ ≤ C(N u + N(Pu)) on ∂Ω.
(9.6.18)
Granted the original assumptions on u, this shows that N(F)⃗ belongs to L1 (∂Ω). With this in hand, the ﬁrst equality in formula (9.6.5) now follows from Theorem 9.67, bearing in mind (9.6.17) and (9.6.16). The second equality in formula (9.6.5) is a consequence of what we have just proved and (2.1.65). As regards (9.6.6), work locally in O. Write ν = ν j dx j then use this to express i Sym (P⊤, ν)Q⊤ − i Sym (P⊤ Q⊤, ν) = ν j {i Sym (P⊤, dx j )Q⊤ − i Sym (P⊤ Q⊤, dx j )}.
(9.6.19)
We now claim that for every scalarvalued function ψ ∈ C 1 (O) we have i Sym (P⊤, dψ)Q⊤ − i Sym (P⊤ Q⊤, dψ) = −P⊤ i Sym (Q⊤, dψ).
(9.6.20)
Indeed, repeated applications of (2.1.69) permit us to transform the lefthand side of (9.6.20) as −[P⊤, ψ]Q⊤ + [P⊤ Q⊤, ψ] = −P⊤ ψQ⊤ + ψP⊤ Q⊤ + P⊤ Q⊤ ψ − ψP⊤ Q⊤ = P⊤ [Q⊤, ψ] = −P⊤ i Sym (Q⊤, dψ),
(9.6.21)
as wanted. At this stage, (9.6.6) becomes a direct consequence of (9.6.5) and (9.6.19)– (9.6.20). The proof of the theorem is therefore complete. Purely on algebraic grounds, Theorem 9.25 selfimproves (as to apply to families of operators) in the manner described in our next corollary.
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  421
Corollary 9.26. Let M be a C 2 manifold equipped with a Riemannian metric, and let E, F, and Hj for j ∈ {1, . . . , N}, be Hermitian vector bundles over M, of class C 2 . Assume all metrics involved are of class C 1 . Suppose that for j ∈ {1, . . . , N} P j : C 1 (M, E) → C 0 (M, Hj )
and
Q j : C 1 (M, Hj ) → C 0 (M, F)
(9.6.22)
are ﬁrstorder differential operators with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part, having the property that their principal symbols satisfy the cancellation condition N
∑ Sym (Q j , ξ) Sym (P j , ξ) = 0,
∀ ξ ∈ T ∗ M.
(9.6.23)
j=1
For j ∈ {1, . . . , N}, let P⊤j : C 1 (M, Hj ) → C 0 (M, E) and Q⊤j : C 1 (M, F) → C 0 (M, Hj ) be the (real) transposes of P j , Q j (considered in the usual sense on the manifold M). Also, let Ω ⊂ M be a relatively compact Ahlfors regular domain, set σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Finally, pick a Lebesgue measurable section u : Ω → E satisfying n.t. N u ∈ L1 (∂Ω), the trace u ∂Ω exists at σa.e. point on ∂Ω (9.6.24) and, for each j ∈ {1, . . . , N}, P j u ∈ L1loc (Ω, Hj ), N(P j u) ∈ L1 (∂Ω), and n.t. the trace (P j u)∂Ω exists at σa.e. point on ∂Ω.
(9.6.25)
Then N
∑ Q j P j : C 1 (M, E) → C 0 (M, F) j=1 N
and
(9.6.26)
∑ P⊤j Q⊤j : C 1 (M, F) → C 0 (M, E) j=1
are ﬁrstorder differential operators, and for every function φ ∈ C 1 (Ω, F) one has N
n.t. ∑ ∫ ⟨i Sym (Q j , ν)(P j u)∂Ω , φ∂Ω ⟩ dσ
j=1
F
∂Ω
N
n.t. = ∑ ∫ ⟨u∂Ω , i Sym (P⊤j , ν)(Q⊤j φ)∂Ω ⟩ dσ j=1
E
∂Ω N
n.t. + ∫ ⟨i Sym ( ∑ Q j P j , ν)u∂Ω , φ∂Ω ⟩ dσ j=1
∂Ω
(9.6.27)
F
N
N
j=1
j=1
n.t. = ∫ ⟨u∂Ω , ∑ i Sym (P⊤j , ν)(Q⊤j φ)∂Ω − i Sym ( ∑ P⊤j Q⊤j , ν)φ∂Ω ⟩ dσ, ∂Ω
E
where all principal symbols are taken in the sense of ﬁrstorder differential operators.
422  9 Further Tools from Geometry and Analysis
Moreover, in the case when φ is actually compactly supported in a local coordinate patch O on M, with local coordinates (x k )1≤k≤n ∈ O, formula (9.6.5) may be written as N
n.t. ∑ ∫ ⟨i Sym (Q j , ν)(P j u)∂Ω , φ∂Ω ⟩ dσ
j=1
(9.6.28)
F
∂Ω
n
N
n.t. = − ∫ ⟨u∂Ω , ∑ ∑ ν k P⊤j (i Sym (Q⊤j , dx k )φ)∂Ω ⟩ dσ, ∂Ω
E
j=1 k=1
where ν = ν k dx k in O. Proof. Consider the Hermitian vector bundle H := H1 ⊕ ⋅ ⋅ ⋅ ⊕ HN and deﬁne the ﬁrstorder differential operators ̃ : C 1 (M, E) → C 0 (M, H), P ̃ := (P j u)1≤j≤N , Pu and
∀ u ∈ C 1 (M, E),
(9.6.29)
̃ : C 1 (M, H) → C 0 (M, F), Q N
̃ j )1≤j≤N := ∑ Q j υ j , Q(υ
∀ (υ j )1≤j≤N ∈ C 1 (M, H).
(9.6.30)
j=1
Their transposes are given, respectively, by ̃ ⊤ : C 1 (M, H) → C 0 (M, E), P N
̃ ⊤ (υ j )1≤j≤N = ∑ P⊤ υ j , P j
∀ (υ j )1≤j≤N ∈ C 1 (M, H),
(9.6.31)
j=1
and
̃⊤ : C 1 (M, F) → C 0 (M, H), Q ̃⊤ φ = (Q⊤ φ)1≤j≤N , Q j
∀ φ ∈ C 1 (M, F).
(9.6.32)
Note that, by design and (9.6.23), for every u ∈ C 1 (M, E) we have ̃ ξ) Sym (P, ̃ ξ) (Sym (P j , ξ)u) ̃ ξ)u = Sym (Q, Sym (Q, 1≤j≤N N
= ∑ Sym (Q j , ξ) Sym (P j , ξ)u j=1
=0
for all ξ ∈ T ∗ M,
(9.6.33)
and N
̃ Pu ̃ j u)1≤j≤N = ∑ Q j P j u. ̃ = Q(P Q
(9.6.34)
j=1
Bearing all these identiﬁcations in mind, all desired conclusions follow directly from ̃ ̃ Q). Theorem 9.25 (applied to the operators P,
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  423
One signiﬁcant instance when the cancellation condition (9.6.23) is automatically satisﬁed is singled out next. Corollary 9.27. Let M be a C 2 manifold equipped with a Riemannian metric, and let E, F be Hermitian vector bundles over M, of class C 2 . Assume all metrics involved are of class C 1 . Suppose that P : C 1 (M, E) → C 0 (M, F) (9.6.35) is a ﬁrstorder differential operator with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part. Denote its transpose by P ⊤ : C 1 (M, F) → C 0 (M, E). Also, let Ω ⊂ M be a relatively compact Ahlfors regular domain, set σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Pick a Lebesgue measurable section u : Ω → E satisfying n.t.
N u ∈ L1 (∂Ω), the trace u ∂Ω exists at σa.e. point on ∂Ω,
∇u ∈ L1loc (Ω, T ∗ M ⊗ E), N(∇u) ∈ L1 (∂Ω), and n.t. the trace (∇u)∂Ω exists at σa.e. point on ∂Ω.
(9.6.36)
F Finally, ﬁx a C 1 vector ﬁeld X and denote by ∇E X , ∇X , the covariant derivatives along X for sections of E and F, respectively (cf. Deﬁnition 9.2). Then F 2 0 P∇E (9.6.37) X − ∇X P : C (M, E) → C (M, F )
is a ﬁrstorder differential operator, and for every function φ ∈ C 1 (Ω, F) one has n.t. n.t. ∫ ⟨−ν(X)(Pu)∂Ω − i Sym (P, ν)(∇E X u)∂Ω , φ ∂Ω ⟩ dσ F
∂Ω
n.t. ∗ ⊤ = ∫ ⟨u∂Ω , i Sym (P⊤, ν)((∇F X ) φ)∂Ω − ν(X)(P φ)∂Ω ∂Ω
⊤ F + i Sym (P∇E X − ∇X P, ν) φ ∂Ω ⟩ dσ, E
(9.6.38)
where all principal symbols are taken in the sense of ﬁrstorder differential operators. Proof. This is seen directly from Corollary 9.26, specialized to the case when N := 2, P1 := P,
Q1 :=
H1 := F ,
∇F X,
H2 := E,
P2 := −∇E X,
Q2 := P,
(9.6.39)
bearing in mind that, thanks to (9.1.45), the cancellation condition (9.6.23) is automatically satisﬁed in the present setting. It is useful to note that Corollary 9.26 implies the following local integration by parts formula on the boundary. Corollary 9.28. Let M be a C 2 manifold equipped with a Riemannian metric, and let E, F be Hermitian vector bundles over M, of class C 2 . Assume all metrics involved are of class C 1 . Also, let Ω ⊂ M be a relatively compact Ahlfors regular domain, deﬁne
424  9 Further Tools from Geometry and Analysis σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Next, ﬁx an open neighborhood O of a boundary point x0 ∈ ∂Ω, which is contained in a local coordinate patch on M. Denote by (x j )1≤j≤n the local coordinates in O. Suppose that P j : C 1 (O, E) → C 0 (O, F),
1 ≤ j ≤ n,
(9.6.40)
are ﬁrstorder differential operators on O with top coefficients from C 1 (O) and lower order coefficients from C 0 (O), having the property that their principal symbols satisfy the cancellation condition n
∑ ξ j SymP j (x, ξ) = 0,
∀ x ∈ O, ∀ ξ = ξ j dx j ∈ T x∗ M.
(9.6.41)
j=1
For each j ∈ {1, . . . , n}, denote by P⊤j : C 1 (O, F) → C 0 (O, E) the (real ) transpose of P j . Finally, pick a Lebesgue measurable section u : Ω → E satisfying N u ∈ L1 (O ∩ ∂Ω),
n.t. the trace u ∂Ω exists at σa.e. point on O ∩ ∂Ω
(9.6.42)
and, for each j ∈ {1, . . . , n}, P j u ∈ L1loc (O ∩ Ω, F), N(P j u) ∈ L1 (O ∩ ∂Ω), and n.t. the trace (P j u)∂Ω exists at σa.e. point on O ∩ ∂Ω.
(9.6.43)
Then for every function φ ∈ C01 (O, F) one has n
n
n.t. n.t. ∑ ∫ ⟨ν j (P j u)∂Ω , φ∂Ω ⟩ dσ = ∑ ∫ ⟨u∂Ω , ν j (P⊤j φ)∂Ω ⟩ dσ,
j=1
F
∂Ω
j=1
E
(9.6.44)
∂Ω
where ν = ν j dx j in O. Proof. The idea is to apply Corollary 9.26 with N := n, Hj := F, and Q j := ∇∂ j for each j ∈ {1, . . . , n}, where ∇ is a connection in F. Note that for each ξ = ξ j dx j ∈ T ∗ M we may compute n
n
∑ Sym (Q j , ξ) Sym (P j , ξ) = ∑ Sym (∇∂ j , ξ) Sym (P j , ξ) j=1
j=1 n
= i ∑ ξ(∂ j ) Sym (P j , ξ) j=1 n
= i ∑ ξ j Sym (P j , ξ) j=1
= 0,
(9.6.45)
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  425
thanks to (9.1.45), (2.1.19), and (9.6.41). Hence, the cancellation condition (9.6.23) holds for this choice of operators. Given that by (9.1.45) and (2.1.19) we also have n
n
∑ i Sym (Q j , ν)P j = ∑ i Sym(∇∂ j , ν)P j j=1
j=1 n
= − ∑ ν(∂ j )P j j=1 n
= − ∑ νj Pj ,
(9.6.46)
j=1
and n
n
n
n
∑ ∑ ν k P⊤j i Sym (∇⊤∂ j, dx k ) = − ∑ ∑ ν k P⊤j i Sym (∇∂ j , dx k ) j=1 k=1
⊤
j=1 k=1 n
n
= ∑ ∑ ν k P⊤j dx k (∂ j ) j=1 k=1 n
n
= ∑ ∑ δ jk ν k P⊤j j=1 k=1 n
= ∑ ν j P⊤j ,
(9.6.47)
j=1
formula (9.6.44) becomes an immediate consequence of (9.6.28) for the present choices of operators. Before stating our next result, the reader is reminded that m ξ denotes the operator of tensor multiplication with a covector ξ ∈ T ∗ M (applied to sections in a given Hermitian vector bundle over M), and that m∗ξ stands for the adjoint of m ξ . See the discussion pertaining to (9.1.18)–(9.1.26) in this regard. Corollary 9.29. Let M be a C 2 manifold equipped with a Riemannian metric, and let E, F be Hermitian vector bundles over M, of class C 2 . Assume all metrics involved are of class C 1 . Also, let Ω ⊂ M be a relatively compact Ahlfors regular domain, deﬁne σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. In this setting, consider a ﬁrstorder differential operator P : C 1 (M, E) → C 0 (M, T ∗ M ⊗ F)
(9.6.48)
whose top order coefficients are C 1 and the zero order coefficients are continuous, having the property that⁷ m∗ξ Sym (P, ξ) = 0, ∀ ξ ∈ T ∗ M. (9.6.49) 7 in light of (9.1.42), condition (9.6.49) may be interpreted as Sym (∇∗, ξ) Sym (P, ξ) = 0 for all ξ ∈ T ∗ M, which is in agreement with (9.6.2) when Q = ∇∗ .
426  9 Further Tools from Geometry and Analysis Next, ﬁx an open neighborhood O of a boundary point x0 ∈ ∂Ω, which is contained in a local coordinate patch on M, where the local coordinates are denoted (x j )1≤j≤n . With P as above, associate the ﬁrstorder differential operator ̃ : C 1 (O, F) → C 0 (O, T ∗ M ⊗ E) P
(9.6.50)
C 1 (O , F )
whose action on an arbitrary section υ ∈ is described in the local coordinates in O as ̃ := ∂♭ ⊗ (ℙ⊤ (dx j ⊗ υ)), Pυ (9.6.51) j where P⊤ : T ∗ M ⊗ F → E denotes the (real ) transpose of (9.6.48). In other words, ̃ = m ♭ P⊤ m dx in O. P j ∂
(9.6.52)
j
Finally, pick a Lebesgue measurable section u : Ω → E satisfying n.t. N u ∈ L1 (O ∩ ∂Ω), the trace u ∂Ω exists σa.e. on O ∩ ∂Ω Pu ∈ L1loc (O ∩ Ω, T ∗ M ⊗ F), N(Pu) ∈ L1 (O ∩ ∂Ω), and n.t. the nontangential trace (Pu)∂Ω exists σa.e. on O ∩ ∂Ω.
(9.6.53)
Then for every function φ ∈ C01 (O, F) one has n.t. n.t. ̃ ⟩ dσ. ∫ ⟨m∗ν (Pu)∂Ω , φ∂Ω ⟩ dσ = ∫ ⟨u∂Ω , m∗ν (Pφ) ∂Ω F
E
∂Ω
(9.6.54)
∂Ω
Proof. For each j ∈ {1, . . . , n} consider the ﬁrstorder differential operator P j := m∗dx j P.
P j : C 1 (O, E) → C 0 (O, F),
Then (9.6.49) implies that for each covector ξ = ξ j dx j ∈ n
n
j=1
j=1
T∗ M
(9.6.55)
we have
∑ ξ j Sym (P j , ξ) = ∑ ξ j m∗dx j Sym (P, ξ) = m∗ξ Sym (P, ξ) = 0,
(9.6.56)
which shows that the cancellation condition (9.6.41) holds for this choice of operators. Write ν = ν j dx j in O. Upon noting that n
n
j=1
j=1
∑ ν j P j = ∑ ν j m∗dx j P = m∗ν P,
(9.6.57)
and that, thanks to (9.6.52), (9.1.21), (9.1.3), and (9.6.55), n
n
̃ = ∑ m∗ν m ♭ P⊤ m dx = ∑ ⟨ν, ∂♭ ⟩ P⊤ m dx m∗ν P j j ∂ j j=1
j
j=1
n
n
= ∑ ν k ⟨dx k , ∂♭j ⟩ P⊤ m dx j = ∑ ν k δ jk P⊤ m dx j j,k=1
j,k=1
n
n
j=1
j=1
= ∑ ν j P⊤ m dx j = ∑ ν j P⊤j , formula (9.6.54) now follows directly from (9.6.44).
(9.6.58)
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  427
A version of the boundary integration by parts formula established in Theorem 9.25 in Ahlfors regular domains continues to hold in the larger class of domains of locally ﬁnite perimeter, albeit for a more restrictive class of functions. Speciﬁcally, we have the following result. Proposition 9.30. Let the Riemannian manifold M, the Hermitian vector bundles E, F, H, and the ﬁrstorder differential operators P, Q be as in Theorem 9.25. Consider an open set Ω ⊂ M of locally ﬁnite perimeter. Introduce σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Then for every functions u ∈ C 1 (M, E) and φ ∈ C01 (M, F) one has ∫ ⟨i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ F
∂∗ Ω
= ∫ ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω ⟩ dσ + ∫ ⟨i Sym (QP, ν)u∂Ω , φ∂Ω ⟩ dσ E F ∂∗ Ω
∂∗ Ω
= ∫ ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω − i Sym(P⊤ Q⊤, ν)φ∂Ω ⟩ dσ, E
(9.6.59)
∂∗ Ω
where all principal symbols appearing above are taken in the sense of ﬁrstorder differential operators. Moreover, in the case when φ is actually compactly supported in a local coordinate patch O on M, with local coordinates (x j )1≤j≤n ∈ O, formula (9.6.59) may be written as ∫ ⟨i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ F
∂∗ Ω
= − ∫ ⟨u∂Ω , ν j P⊤ (i Sym (Q⊤, dx j )φ)∂Ω ⟩ dσ, E
(9.6.60)
∂∗ Ω
where ν = ν j dx j in O. As a corollary, both (9.6.59) and (9.6.60) hold with ∂∗ Ω replaced by ∂Ω provided HMn−1 (∂Ω \ ∂∗ Ω) = 0. Proof. This is proved by reasoning much as we did in the proof of Theorem 9.25, with Theorem 9.67 replaced by Theorem 9.66. Appropriately specializing Theorem 9.25 yields the following useful result. Corollary 9.31. Let M be a C 2 manifold equipped with a Riemannian metric, and let E be a Hermitian vector bundle over M, of class C 2 . Assume all metrics involved are of class C 1 . Suppose P, Q : C 1 (M, E) → C 0 (M, E) are ﬁrstorder differential operators with C 1 coefficients having the property that their principal symbols commute, in the sense that Sym (P, ξ) Sym (Q, ξ) = Sym (Q, ξ) Sym (P, ξ), ∀ ξ ∈ T ∗ M. (9.6.61) Denote by P ⊤, Q⊤ : C 1 (M, E) → C 0 (M, E) the transposes of P, Q (considered in the usual sense on the manifold M).
428  9 Further Tools from Geometry and Analysis Next, let Ω ⊂ M be a relatively compact Ahlfors regular domain and denote by ν and σ the (geometric measure theoretic) outward unit conormal and surface measure on ∂Ω. Finally, pick a Lebesgue measurable section u : Ω → E satisfying N u ∈ L1 (∂Ω),
Pu, Qu ∈ L1loc (Ω, E), N(Pu), N(Qu) ∈ L1 (∂Ω), n.t. n.t. n.t. and the traces u∂Ω , (Pu)∂Ω , (Qu)∂Ω exist at σa.e. point on ∂Ω.
(9.6.62)
Then the commutator [P, Q] is a ﬁrstorder differential operator on sections of E, and for every function φ ∈ C 1 (Ω, E) one has n.t. n.t. ∫ ⟨i Sym (P, ν)(Qu)∂Ω − i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ E
∂Ω
n.t. = − ∫ ⟨u∂Ω , i Sym(P⊤, ν)(Q⊤ φ)∂Ω − i Sym (Q⊤, ν)(P⊤ φ)∂Ω ⟩ dσ E
∂Ω
n.t. + ∫ ⟨i Sym ([P, Q], ν)u∂Ω , φ∂Ω ⟩ dσ, E
(9.6.63)
∂Ω
where all principal symbols are taken in the sense of ﬁrstorder differential operators. Proof. With E, P, Q as in the statement, consider the Hermitian vector bundles F := E and H := E ⊕ E. Also, deﬁne the ﬁrstorder differential operators ̃ : C 1 (M, E) → C 0 (M, H), P ̃ := (Qu, −Pu), Pu and
∀ u ∈ C 1 (M, E),
̃ : C 1 (M, H) → C 0 (M, F), Q ̃ Q(υ, w) := Pυ + Qw,
∀ (υ, w) ∈ C 1 (M, H).
(9.6.64)
(9.6.65)
Their transposes are given, respectively, by ̃ ⊤ : C 1 (M, H) → C 0 (M, E), P ̃ ⊤ (υ, w) = Q⊤ υ − P⊤ w, P and ̃⊤ φ Q
∀ (υ, w) ∈ C 1 (M, H),
̃⊤ : C 1 (M, F) → C 0 (M, H), Q = (P⊤ φ, Q⊤ φ), ∀ φ ∈ C 1 (M, F).
(9.6.66)
(9.6.67)
Note that, by design and (9.6.61), for every u ∈ E we have ̃ ξ) Sym (P, ̃ ξ)(Sym (Q, ξ)u, − Sym (P, ξ)u) ̃ ξ)u = Sym (Q, Sym (Q, = Sym (P, ξ) Sym (Q, ξ)u − Sym (Q, ξ) Sym (P, ξ)u =0 and
for all ξ ∈ T ∗ M,
̃ Pu ̃ ̃ = Q(Qu, Q −Pu) = PQu − QPu = [P, Q]u.
(9.6.68)
(9.6.69)
In light of these identiﬁcations, the desired conclusions are now seen directly from Theorem 9.25.
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  429
For a more restrictive class of functions, there is also a version of the boundary integration by parts formula, originally proved in Corollary 9.31 in Ahlfors regular domains, in the larger class of domains of locally ﬁnite perimeter. Corollary 9.32. Let the Riemannian manifold M, the Hermitian vector bundle E, and the ﬁrstorder differential operators P, Q : C 1 (M, E) → C 0 (M, E) be as in Corollary 9.31. Let Ω ⊂ M be an open set of locally ﬁnite perimeter. Deﬁne σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Then for every functions u ∈ C 1 (M, E) and φ ∈ C01 (M, E) one has ∫ ⟨i Sym (P, ν)(Qu)∂Ω − i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ E ∂∗ Ω
= − ∫ ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω − i Sym (Q⊤, ν)(P⊤ φ)∂Ω ⟩ dσ E ∂∗ Ω
+ ∫ ⟨i Sym ([P, Q], ν)u∂Ω , φ∂Ω ⟩ dσ, E
(9.6.70)
∂∗ Ω
where all principal symbols are taken in the sense of ﬁrstorder differential operators. In particular, (9.6.70) holds with ∂∗ Ω replaced by ∂Ω provided HMn−1 (∂Ω \ ∂∗ Ω) = 0. Proof. For the class of functions presently considered, the boundary integration by parts formula (9.6.70) is established by reasoning much as in the proof of (9.6.63), with Theorem 9.66 now replacing Theorem 9.67. A useful particular case of Corollary 9.32 is singled out below. Corollary 9.33. Let M be a C 2 manifold equipped with a Riemannian metric, and let E be a Hermitian vector bundle over M of class C 2 . Assume all metrics are of class C 1 . Also, let Ω ⊂ M be an open set of locally ﬁnite perimeter satisfying HMn−1 (∂Ω \ ∂∗ Ω) = 0. Deﬁne σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Then for every pair of functions φ ∈ C 1 (M, E), ψ ∈ C01 (M, E), and every pair of C 1 vector ﬁelds X, Y there holds ∫ ⟨∂ τ XY φ, ψ⟩E dσ = − ∫ ⟨φ, ∂ τ XY ψ⟩E dσ ∂Ω
(9.6.71)
∂Ω
− ∫ ⟨φ, ψ⟩E (ν(X) div Y − ν(Y) div X + ν([X, Y])) dσ. ∂Ω
In particular, locally one has ∫ ⟨∂ τ jk φ, ψ⟩E dσ = − ∫ ⟨φ, ∂ τ jk ψ⟩E dσ ∂Ω
∂Ω n
ℓ ℓ − ν k Γ jℓ − ∑ ∫ ⟨φ, ψ⟩E (ν j Γ kℓ ) dσ. ℓ=1
for every j, k ∈ {1, . . . , n}.
∂Ω
(9.6.72)
430  9 Further Tools from Geometry and Analysis
Proof. Observe that, via a routine density argument, it suffices to prove (9.6.71) in the case when φ, ψ ∈ C 02 (M, E). In such a scenario, the desired formula is implied by (9.6.70) (used with φ, ψ in place of u, φ) specialized to P := ∇X , Q := ∇Y upon observing that such a choice satisﬁes (9.6.61) (cf. (9.1.45)) and entails i Sym (P, ν)(Qφ)∂Ω − i Sym (Q, ν)(Pφ)∂Ω = i Sym (∇X , ν)(∇Y φ)∂Ω − i Sym (∇Y , ν)(∇X φ)∂Ω = −ν(X)(∇Y φ)∂Ω + ν(Y)(∇X φ)∂Ω = −∂ τ XY φ
(9.6.73)
by virtue of (9.1.45) and (9.5.46), as well as i Sym (P⊤, ν)(Q⊤ ψ)∂Ω − i Sym (Q⊤, ν)(P⊤ ψ)∂Ω = i Sym (∇X , ν)((∇Y ψ)∂Ω + ((div Y)ψ)∂Ω ) − i Sym (∇Y , ν)((∇X ψ)∂Ω + ((div X)ψ)∂Ω ) = − ν(X)((∇Y ψ)∂Ω + ((div Y)ψ)∂Ω ) + ν(Y)((∇X ψ)∂Ω + ((div X)ψ)∂Ω ) = − ∂ τ XY ψ + (ν(Y)(div X)∂Ω − ν(X)(div Y)∂Ω )(ψ∂Ω )
(9.6.74)
by virtue of (9.2.16), (9.1.45), and (9.5.46), and also i Sym ([P, Q], ν) = i Sym ([∇X , ∇Y ], ν) = i Sym (∇[X,Y] + R(X, Y), ν) = i Sym (∇[X,Y] , ν) = −ν([X, Y]),
(9.6.75)
thanks to (9.3.6)–(9.3.7) and (9.1.45). Finally, (9.6.72) is obtained by particularizing (9.6.71) to the case when X = ∂ j , Y = ∂ k , keeping in mind (9.5.47) and (9.1.15). Comparing (9.5.49) with (9.6.71) then suggests making the following deﬁnition. Deﬁnition 9.34. Suppose M is a C 2 manifold equipped with a Riemannian metric, and consider a Hermitian vector bundle E over M of class C 2 . Assume all metrics involved are of class C 1 . Also, let Ω ⊂ M be an open set of locally ﬁnite perimeter satisfying HMn−1 (∂Ω \ ∂∗ Ω) = 0. Introduce σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. p Then for any function f ∈ L1 (∂Ω, E) with 1 < p < ∞, and any two compactly supported C 1 vector ﬁelds X, Y, deﬁne ∂ τ XY f := −h XY − (ν(X) div Y − ν(Y) div X + ν([X, Y]))f,
(9.6.76)
where the function h XY ∈ L p (∂Ω, E) is uniquely associated with f and X, Y as in (9.5.49).
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  431
Thanks to Corollary 9.33, this deﬁnition is then consistent with that made in (9.5.47) for functions in C 1 (M, E). Other basic properties of the operators ∂ τ XY considered in (9.6.76) are contained in the proposition below. Proposition 9.35. In the context of Deﬁnition 9.34, p
∂ τ XY : L1 (∂Ω, E) → L p (∂Ω, E),
1 < p < ∞,
(9.6.77)
is a welldeﬁned, linear and bounded operator Also, p
∂ τ YX f = −∂ τ XY f,
∀ f ∈ L1 (∂Ω, E).
(9.6.78)
p
Moreover, given an arbitrary function f ∈ L1 (∂Ω, E) with 1 < p < ∞, for every section φ ∈ C01 (M, E) and every C 1 vector ﬁelds X, Y ∈ TM one has ∫ ⟨∂ τ XY f, φ⟩E dσ = − ∫ ⟨f, ∂ τ XY φ⟩E dσ ∂Ω
(9.6.79)
∂Ω
− ∫ ⟨f, φ⟩E (ν(X) div Y − ν(Y) div X + ν([X, Y])) dσ, ∂Ω
where σ stands for the surface measure on ∂Ω. Hence, locally for every j, k ∈ {1, . . . , n}, ∫ ⟨∂ τ jk f, φ⟩E dσ = − ∫ ⟨f, ∂ τ jk φ⟩E dσ ∂Ω
∂Ω n
ℓ ℓ − ν k Γ jℓ − ∑ ∫ ⟨f, φ⟩E (ν j Γ kℓ ) dσ. ℓ=1
(9.6.80)
∂Ω
Finally, for each p ∈ (1, ∞), p {φ∂Ω : φ ∈ C01 (M, E)} ⊂ L1 (∂Ω, E),
(9.6.81)
and, given any φ ∈ C01 (M, E), ∂ τ XY (φ∂Ω ), considered in the sense of (9.6.76) by viewing φ∂Ω p in the space L1 (∂Ω, E), agrees on ∂Ω with the righthand side of (9.5.46), i.e., with the function ν(X)(∇Y φ)∂Ω − ν(Y)(∇X φ)∂Ω .
(9.6.82)
Proof. The claims in (9.6.77) and (9.6.78) are clear from deﬁnitions, while (9.6.79) follows from (9.6.76) and (9.6.71). Next, the inclusion in (9.6.81) is seen from (9.6.71) and Proposition 9.24. Finally, the claim in (9.6.82) is a consequence of (9.6.71) and (9.6.76). We are now in a position to generalize the Euclidean result formulated in Proposition 9.16 to the manifold setting.
432  9 Further Tools from Geometry and Analysis Proposition 9.36. Let M be a C 2 manifold equipped with a Riemannian metric, and let E be a Hermitian vector bundle over M of class C 2 . Assume all metrics are of class C 1 . Also, let Ω ⊂ M be a relatively compact Ahlfors regular domain and denote by ν and σ the (geometric measure theoretic) outward unit conormal and surface measure on ∂Ω. Pick some p ∈ [1, ∞] and suppose u : Ω → E is Lebesgue measurable, N u ∈ L p (∂Ω),
N(∇u) ∈ L p (∂Ω),
(9.6.83)
n.t. n.t. and u∂Ω , (∇u)∂Ω exist σa.e. on ∂Ω. Then
p n.t. the function u∂Ω belongs to L1 (∂Ω, E) and n.t. p u∂Ω L (∂Ω,E) ≤ C(N uL p (∂Ω) + N(∇u)L p (∂Ω) ) 1
(9.6.84)
for some ﬁnite constant C > 0 independent of u. Moreover, for every two C 1 vector ﬁelds X, Y one has n.t. n.t. n.t. ∂ τ XY (u∂Ω ) = ν(X)(∇Y u)∂Ω − ν(Y)(∇X u)∂Ω σa.e. on ∂Ω.
(9.6.85)
In particular, for every j, k ∈ {1, . . . , n}, one locally has n.t. n.t. n.t. ∂ τ jk (u∂Ω ) = ν j (∇∂ k u)∂Ω − ν k (∇∂ j u)∂Ω σa.e. on ∂Ω.
(9.6.86)
Proof. Clearly, n.t. n.t. u∂Ω ∈ L p (∂Ω, E), (∇u)∂Ω ∈ L p (∂Ω, T ∗ M ⊗ E), and n.t. n.t. u∂Ω L p (∂Ω,E) + (∇u)∂Ω L p (∂Ω,T ∗ M⊗E) ≤ C(N uL p (∂Ω) + N(∇u)L p (∂Ω) ),
(9.6.87)
for some ﬁnite constant C > 0 independent of u. To proceed, ﬁx an arbitrary function φ ∈ C01 (M, E) along with two C 1 vector ﬁelds X, Y. Then, granted the assumptions on u, from formula (9.6.63) used with P = ∇X , Q = ∇Y , and (9.6.73), (9.6.74) we obtain n.t. ∫ ⟨u∂Ω , ∂ τ XY φ⟩ dσ E
∂Ω
n.t. n.t. = − ∫ ⟨ν(X)(∇Y u)∂Ω − ν(Y)(∇X u)∂Ω , φ∂Ω ⟩ dσ E
∂Ω
n.t. − ∫ ⟨u∂Ω , φ∂Ω ⟩ (ν(X) div Y − ν(Y) div X + ν([X, Y])) dσ. E
(9.6.88)
∂Ω
From this, it follows that the function n.t. n.t. h XY := − ν(X)(∇Y u)∂Ω + ν(Y)(∇X u)∂Ω
n.t. − (ν(X) div Y − ν(Y) div X + ν([X, Y]))u∂Ω ∈ L p (∂Ω, E)
(9.6.89)
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  433
n.t. does the job in (9.5.49). Together with Proposition 9.24, this shows that u∂Ω belongs p to the Sobolev space L1 (∂Ω, E). Having established this, formula (9.6.76) from Deﬁnition 9.34 then gives that n.t. n.t. ∂ τ XY (u∂Ω ) = −h XY − (ν(X) div Y − ν(Y) div X + ν([X, Y]))u∂Ω n.t. n.t. = ν(X)(∇Y u)∂Ω − ν(Y)(∇X u)∂Ω .
(9.6.90)
This justiﬁes (9.6.85). Finally, the estimate in the second line of (9.6.84) is implicit in what we have proved so far. There is a more general phenomenon here at play, which we would like to describe. To set things up, let the Riemannian manifold M, the Hermitian vector bundles E, F, H, and the ﬁrstorder differential operators P, Q be as in Theorem 9.25. That is, P : C 1 (M, E) → C 0 (M, H) and Q : C 1 (M, H) → C 0 (M, F)
(9.6.91)
are two ﬁrstorder differential operators with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part, having the property that their principal symbols satisfy the cancellation condition Sym (Q, ξ) Sym (P, ξ) = 0,
∀ ξ ∈ T ∗ M.
(9.6.92)
Also, consider an open set Ω ⊂ M of locally ﬁnite perimeter with the property that HMn−1 (∂Ω \ ∂∗ Ω) = 0. Introduce σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Hence, if we locally express P = A j ∇∂ j + A 0
and
Q = B k ∇∂ k + B 0
(9.6.93)
for some matrices A j , B k , the cancellation property (9.6.92) reads ξ j ξ k B k A j = 0 for every cotangent vector ξ = ξℓ dxℓ .
(9.6.94)
In turn, this is equivalent to having B k A j = −B j A k
for every j, k ∈ {1, . . . , n}.
(9.6.95)
Granted this, we may then write i Sym (Q, ν)P = −B k A j ν k ∇∂ j − νℓ Bℓ A0 = 12 (B k A j ν j ∇∂ k − B k A j ν k ∇∂ j ) − νℓ Bℓ A0 = 12 B k A j ∂ τ jk − νℓ Bℓ A0 .
(9.6.96)
Bearing (9.6.77) in mind, for each p ∈ (1, ∞) we then proceed to deﬁne the bounded linear mapping p ∂ τQ,P : L1 (∂Ω, E) → L p (∂Ω, F), (9.6.97) ∂ τQ,P f := 12 B k A j ∂ τ jk f − νℓ Bℓ A0 f.
434  9 Further Tools from Geometry and Analysis
From (9.6.97), (9.6.96), (9.6.81), and (9.6.82) we may then conclude that Q,P ∂ τ (ψ∂Ω ) = i Sym (Q, ν)(Pψ)∂Ω ,
∀ ψ ∈ C01 (M, E).
(9.6.98)
Let us also note that since the (ordered pair of) differential operators Q⊤, P⊤ satisfy analogous properties as the original P, Q, we may deﬁne P⊤, Q⊤
∂τ
p
: L1 (∂Ω, F) → L p (∂Ω, E),
p ∈ (1, ∞).
(9.6.99)
∀ φ ∈ C01 (M, F).
(9.6.100)
in a similar fashion to (9.6.97) and conclude that (φ∂Ω ) = i Sym (P⊤, ν)(Q⊤ φ)∂Ω ,
P⊤, Q⊤
∂τ
We are now ready to state and prove the following versatile boundary integration by parts formula. Theorem 9.37. Let M be a C 2 manifold equipped with a Riemannian metric, and consider three Hermitian vector bundles, E, F, H, over M, of class C 2 . Suppose all metrics involved are of class C 1 . Assume P : C 1 (M, E) → C 0 (M, H) and Q : C 1 (M, H) → C 0 (M, F)
(9.6.101)
are two ﬁrstorder differential operators with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part, having the property that their principal symbols satisfy Sym (Q, ξ) Sym (P, ξ) = 0, ∀ ξ ∈ T ∗ M. (9.6.102) As in the past, denote by P⊤ : C 1 (M, H) → C 0 (M, E) and Q⊤ : C 1 (M, F) → C 0 (M, H) the (real) transposes of P, Q (considered in the usual sense on the manifold M). Next, let Ω ⊂ M be an open set of locally ﬁnite perimeter with the property that n−1 HM (∂Ω \ ∂∗ Ω) = 0. Introduce σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. p Then for any f ∈ L1 (∂Ω, E) with 1 < p < ∞, and any φ ∈ C01 (M, F) one has Q,P ∫ ⟨∂ τ f, φ∂Ω ⟩F dσ ∂Ω
(φ∂Ω )⟩E dσ + ∫ ⟨i Sym (QP, ν)f, φ∂Ω ⟩E dσ
P⊤, Q⊤
= ∫ ⟨f, ∂ τ ∂Ω
∂Ω ⊤
P ,Q
= ∫ ⟨f, ∂ τ
(φ∂Ω ) − i Sym (P⊤ Q⊤, ν)φ∂Ω ⟩E dσ
⊤
∂Ω
= ∫ ⟨f, i Sym (P⊤, ν)(Q⊤ φ)∂Ω − i Sym (P⊤ Q⊤, ν)φ∂Ω ⟩E dσ,
(9.6.103)
∂Ω
where all principal symbols are taken in the sense of ﬁrstorder differential operators.
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  435
p
Proof. Fix φ ∈ C01 (M, F) along with f ∈ L1 (∂Ω, E) for some 1 < p < ∞. There is no loss of generality in assuming that φ is supported in a coordinate patch. Working locally, by making use of (9.6.97) and (9.6.80) we may write Q,P ∫ ⟨∂ τ f, φ∂Ω ⟩F dσ = ∫ ⟨ 12 B k A j ∂ τ jk f − νℓ Bℓ A0 f, φ∂Ω ⟩F dσ ∂Ω
∂Ω
= − ∫ ⟨f, 12 ∂ τ jk (A⊤j B⊤k φ∂Ω )⟩E dσ ∂Ω n
ℓ ℓ − ν k Γ jℓ − ∑ ∫ ⟨f, 12 A⊤j B⊤k φ∂Ω ⟩E (ν j Γ kℓ ) dσ ℓ=1
∂Ω
⊤ − ∫ ⟨f, νℓ A⊤ 0 B ℓ φ ∂Ω ⟩E dσ.
(9.6.104)
∂Ω
Given the goal we have in mind, in view of (9.6.100) it remains to show that i Sym (P⊤, ν)(Q⊤ φ)∂Ω − i Sym (P⊤ Q⊤, ν)φ∂Ω = − 12 ∂ τ jk (A⊤j B⊤k φ∂Ω ) n
ℓ ℓ − ∑ 12 A⊤j B⊤k φ∂Ω (ν j Γ kℓ − ν k Γ jℓ ) ℓ=1
⊤ − νℓ A⊤ 0 B ℓ φ ∂Ω .
(9.6.105)
Thanks to Lemma 2.11, it suffices to check that the two sides of (9.6.105) match when paired with the restriction to ∂Ω of an arbitrary section ψ ∈ C01 (M, E). To this end, ﬁx such a section ψ and write − ∫ ⟨ψ∂Ω , 12 ∂ τ jk (A⊤j B⊤k φ∂Ω )⟩E dσ ∂Ω n
ℓ ℓ − ν k Γ jℓ − ∑ ∫ ⟨ψ∂Ω , 12 A⊤j B⊤k φ∂Ω ⟩E (ν j Γ kℓ ) dσ ℓ=1
∂Ω
⊤ − ∫ ⟨ψ∂Ω , νℓ A⊤ 0 B ℓ φ ∂Ω ⟩E dσ ∂Ω
= ∫ ⟨ 12 B k A j ∂ τ jk (ψ∂Ω ) − νℓ Bℓ A0 ψ∂Ω , φ∂Ω ⟩F dσ ∂Ω
Q,P = ∫ ⟨∂ τ (ψ∂Ω ), φ∂Ω ⟩F dσ ∂Ω
= ∫ ⟨i Sym (Q, ν)(Pψ)∂Ω , φ∂Ω ⟩F dσ ∂Ω
= ∫ ⟨ψ∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω ⟩F dσ ∂Ω
− ∫ ⟨ψ∂Ω , i Sym (P⊤ Q⊤, ν)φ∂Ω ⟩F dσ. ∂Ω
(9.6.106)
436  9 Further Tools from Geometry and Analysis
Above, the ﬁrst equality is a consequence of (9.6.72), the second equality follows from (9.6.97), the third equality is implied by (9.6.98), while the last equality is seen from (9.6.59). Having proved (9.6.106), we conclude that (9.6.105) holds, and this ﬁnishes the proof of the theorem. Here is a special case of Theorem 9.37 when the cancellation condition (9.6.102) is automatically satisﬁed. Corollary 9.38. Let M be a C 2 manifold equipped with a Riemannian metric, and let E, F be Hermitian vector bundles, over M, of class C 2 . Suppose all metrics involved are of class C 1 . Assume P : C 1 (M, F) → C 0 (M, E) (9.6.107) is a ﬁrstorder differential operator with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part, and denote by P ⊤ : C 1 (M, E) → C 0 (M, F) its transpose (considered in the usual sense on the manifold M). F Next, ﬁx a C 1 vector ﬁeld X and denote by ∇E X , ∇X , the covariant derivatives along X for sections of E and F, respectively (cf. Deﬁnition 9.2). Also, introduce the ﬁrstorder differential operators Pu := (P⊤ u, −∇E X u),
P : C 1 (M, E) → C 0 (M, F ⊕ E), Q : C 1 (M, F ⊕ E) → C 0 (M, F),
⊤ Q(υ, w) := ∇F X υ + P w.
(9.6.108)
Finally, assume that Ω ⊂ M is an open set of locally ﬁnite perimeter with the property that HMn−1 (∂Ω \ ∂∗ Ω) = 0. Set σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. Then the operators P, Q satisfy Sym (Q, ξ) Sym (P, ξ) = 0,
∀ ξ ∈ T ∗ M.
(9.6.109)
Consequently, for each p ∈ (1, ∞), the mapping Q,P
∂τ
p
: L1 (∂Ω, E) → L p (∂Ω, F),
(9.6.110)
considered as in (9.6.97) is welldeﬁned, linear and continuous. Moreover, for any two p functions, f ∈ L1 (∂Ω, E) with 1 < p < ∞, and φ ∈ C01 (M, F), one has Q,P ∗ ∫ ⟨∂ τ f, φ∂Ω ⟩F dσ = ∫ ⟨f, i Sym (P, ν)((∇F X ) φ)∂Ω − ν(X)(Pφ)∂Ω ∂Ω
∂Ω
⊤ ⊤ ⊤ E + i Sym (∇F X P − P ∇X , ν) φ ∂Ω ⟩E dσ,
(9.6.111)
where all principal symbols are taken in the sense of ﬁrstorder differential operators. Proof. This is a direct consequence of Theorem 9.37 and (9.1.45). The conclusion in Theorem 9.37 may be strengthened to a genuine integration by parts formula on the boundary, involving functions belonging to Sobolev spaces for Hölder conjugate exponents, whenever more is assumed on the underlying domain. Speciﬁcally, we have the following corollary.
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  437
Corollary 9.39. Retain the assumptions on the Riemannian manifold M, the Hermitian vector bundles E, F, H, and the ﬁrstorder differential operators P, Q from Theorem 9.37. This time, strengthen the assumptions on the underlying domain by asking that Ω is an open subset of M satisfying a twosided local John condition and whose boundary is compact and Ahlfors regular. p q Then for any functions f ∈ L1 (∂Ω, E) and g ∈ L1 (∂Ω, F) where 1 1 n + ≤ p q n−1
p, q ∈ (1, ∞) and
if 1 < p, q < n − 1,
(9.6.112)
one has Q,P P ∫ ⟨∂ τ f, g⟩F dσ = ∫ ⟨f, ∂ τ ∂Ω
⊤
, Q⊤
g⟩E dσ − ∫ ⟨f, i Sym (P⊤ Q⊤, ν)g⟩E dσ,
∂Ω
(9.6.113)
∂Ω
where the principal symbol in the last integral above is taken in the sense of ﬁrstorder differential operators. Proof. Under the current assumptions on Ω, the embeddings in (9.5.39) imply that
p L1 (∂Ω, E)
n−1−p { if 1 < p < n − 1, { (n−1)p { { ∗ for p := { any r ∈ (1, ∞) if p = n − 1, (9.6.114) { { { ∞ if n − 1 < p < ∞, {
∗
→ L p (∂Ω, E)
and q L 1 (∂Ω, F)
n−1−q { if 1 < q < n − 1, { (n−1)q { { ∗ for q := { any r ∈ (1, ∞) if q = n − 1, { { { ∞ if n − 1 < q < ∞, {
∗
→ L q (∂Ω, F)
(9.6.115)
with all inclusions continuous. On the other hand, the conditions imposed on p, q in (9.6.112) ensure that the assignments ∗
L p (∂Ω, F) ⊗ L q (∂Ω, F) ∋ (h, k) → ∫ ⟨h, k⟩F dσ ∈ ℂ,
(9.6.116)
∂Ω ∗
L p (∂Ω, F) ⊗ L q (∂Ω, F) ∋ (h, k) → ∫ ⟨h, k⟩F dσ ∈ ℂ,
(9.6.117)
∂Ω ∗
∗
L p (∂Ω, F) ⊗ L q (∂Ω, F) ∋ (h, k) → ∫ ⟨h, k⟩F dσ ∈ ℂ,
(9.6.118)
∂Ω p
q
are all continuous. Given any f ∈ L1 (∂Ω, E) and g ∈ L1 (∂Ω, F), the boundary integration by parts formula (9.6.113) then follows by relying on the density result recorded in (9.5.43), the boundedness of the operator in (9.6.97) together with the continuity of the embedding in (9.6.115) and the assignment in (9.6.116), Theorem 9.37, the boundedness of the operator in (9.6.99) together with the continuity of the imbeddings in (9.6.114), (9.6.115) and the assignments in (9.6.117), (9.6.118).
438  9 Further Tools from Geometry and Analysis
It is also possible to prove a generalization of Proposition 9.36 of the sort discussed below. Theorem 9.40. Let M be a C 2 manifold equipped with a Riemannian metric, and consider three Hermitian vector bundles, E, F, H, over M, of class C 2 . Suppose all metrics involved are of class C 1 . Also, let Ω ⊂ M be a relatively compact Ahlfors regular domain and denote by ν and σ the (geometric measure theoretic) outward unit conormal and surface measure on ∂Ω. Pick some p ∈ [1, ∞] and suppose u : Ω → E is Lebesgue measurable, N u ∈ L p (∂Ω),
N(∇u) ∈ L p (∂Ω),
(9.6.119)
n.t. n.t. and u∂Ω , (∇u)∂Ω exist σa.e. on ∂Ω. Also, consider two ﬁrstorder differential operators P : C 1 (M, E) → C 0 (M, H) and
(9.6.120)
Q : C 1 (M, H) → C 0 (M, F)
with C 1 coefficients for the top part and C 0 coefficients for the zeroth order part, having the property that their principal symbols satisfy the cancellation condition Sym (Q, ξ) Sym (P, ξ) = 0,
∀ ξ ∈ T ∗ M.
(9.6.121)
p n.t. Then the function u∂Ω belongs to the Sobolev space L1 (∂Ω, E) and, at σa.e. point on ∂Ω, one has n.t. Q,P n.t. ∂ τ (u∂Ω ) = i Sym (Q, ν)(Pu)∂Ω . (9.6.122)
p n.t. Proof. That f := u∂Ω belongs to L1 (∂Ω, E) has been already established in Proposition 9.36. To prove the identity in (9.6.122), pick an arbitrary function φ ∈ C01 (M, F). Also, as before, denote by P⊤ : C 1 (M, H) → C 0 (M, E) and Q⊤ : C 1 (M, F) → C 0 (M, H) the (real ) transposes of P, Q (considered in the usual sense on the manifold M). Then on account of (9.6.103), (9.6.100), and (9.6.5) we may write
n.t. P⊤, Q⊤ Q,P n.t. ∫ ⟨∂ τ (u∂Ω ), φ∂Ω ⟩F dσ = ∫ ⟨u∂Ω , ∂ τ (φ∂Ω )⟩E dσ ∂Ω
∂Ω
n.t. − ∫ ⟨u∂Ω , i Sym (P⊤ Q⊤, ν)φ∂Ω ⟩E dσ ∂Ω
n.t. = ∫ ⟨u∂Ω , i Sym (P⊤, ν)(Q⊤ φ)∂Ω ⟩E dσ ∂Ω
n.t. − ∫ ⟨u∂Ω , i Sym (P⊤ Q⊤, ν)φ∂Ω ⟩E dσ ∂Ω
n.t. = ∫ ⟨i Sym (Q, ν)(Pu)∂Ω , φ∂Ω ⟩ dσ. F
(9.6.123)
∂Ω
With this in hand, (a slight version of) Lemma 2.11 applies and yields (9.6.122).
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  439
Let us now exemplify the manner in which the considerations pertaining to (9.6.93)(9.6.100) work in the case when the ﬁrstorder differential operators P, Q are, respectively, given by d : Λ l TM → Λ l+1 TM,
d : Λ l+1 TM → Λ l+2 TM,
(9.6.124)
where l ∈ {0, 1, . . . , n} is some ﬁxed arbitrary number. As usual, consider an open set Ω ⊂ M of locally ﬁnite perimeter satisfying HMn−1 (∂Ω \ ∂∗ Ω) = 0. Also, introduce σ := HMn−1 ⌊∂Ω, and denote by ν the (geometric measure theoretic) outward unit conormal to Ω. In the above context, condition (9.6.102) is satisﬁed (thanks to item (3) in Lemma 2.8 and item (4) in Lemma 2.2). In light of (9.2.21), the analogue of (9.6.96) in the current setting becomes 1 n i Sym (Q, ν)P = −ν ∧ d = − ∑ dx j ∧ dx k ∧ (ν j ∇∂ k − ν k ∇∂ j ) 2 j,k=1 =−
1 n ∑ dx j ∧ dx k ∧ ∂ τ jk . 2 j,k=1
(9.6.125)
Consider the tangential operator ∂ τd,d deﬁned as in (9.6.97) for P, Q as in (9.6.124). Given p any function f ∈ L1 (∂Ω, Λ l TM) with p ∈ (1, ∞), we may then deﬁne (ν ∧ d)f := −∂ τd,d f =
1 n ∑ dx j ∧ dx k ∧ ∂ τ jk f ∈ L p (∂Ω, Λ l+2 TM). 2 j,k=1
(9.6.126)
Then (9.6.98) ensures that (ν ∧ d)(ϕ∂Ω ) = ν ∧ (dϕ)∂Ω ,
∀ ϕ ∈ C01 (M, Λ l TM).
(9.6.127)
Granted these, the general theory developed earlier applies and yields the integration by parts on the boundary formulas discussed below. Proposition 9.41. Let M be a C 2 manifold of dimension n, equipped with a C 1 Riemannian metric. Assume Ω ⊂ M is an open set of ﬁnite perimeter satisfying (9.5.48). Denote by σ := HMn−1 ⌊∂Ω the surface measure on ∂Ω, and by ν the outward unit conormal to Ω. Also, ﬁx an arbitrary degree l ∈ {0, 1, . . . , n} along with some integrability exponent p ∈ (1, ∞). p Then for every f ∈ L1 (∂Ω, Λ l TM) and every ϕ ∈ C01 (M, Λ l+2 TM) one has ∫ ⟨(ν ∧ d)f, ϕ⟩ dσ = − ∫ ⟨f, ν ∨ δϕ⟩ dσ. ∂Ω
(9.6.128)
∂Ω
Consequently, p
p,d
ν ∧ L1 (∂Ω, Λ l TM) → Lnor (∂Ω, Λ l+1 TM) continuously p
and d ∂ (ν ∧ f) = −(ν ∧ d)f for every f ∈ L1 (∂Ω, Λ l TM).
(9.6.129)
440  9 Further Tools from Geometry and Analysis In a similar manner, ν ∨ δ induces a welldeﬁned linear and bounded operator in the context p p ν ∨ δ : L1 (∂Ω, Λ l TM) → L1 (∂Ω, Λ l−2 TM), (9.6.130) and
p
p,δ
ν ∨ L1 (∂Ω, Λ l TM) → Ltan (∂Ω, Λ l−1 TM) continuously p
and δ ∂ (ν ∨ f) = −(ν ∨ δ)f for every f ∈ L1 (∂Ω, Λ l TM).
(9.6.131)
Proof. Formula (9.6.128) is a direct consequence of Theorem 9.37 (applied with P, Q as in (9.6.124)), and all other claims follow from this and deﬁnitions. We continue the discussion in this section by presenting the following intrinsic characterization of scalar Sobolev spaces. Proposition 9.42. Let M be a C 2 manifold of dimension n, equipped with a C 1 Riemannian metric. Assume Ω ⊂ M is an open set of locally ﬁnite perimeter satisfying (9.5.48). Denote by σ := HMn−1 ⌊∂Ω the surface measure on ∂Ω, and by ν the outward unit conormal to Ω. Finally, ﬁx an integrability exponent p ∈ (1, ∞) and let p ∈ (1, ∞) be such that 1/p + 1/p = 1. Then p
L1 (∂Ω) = {f ∈ L p (∂Ω) : there exists C ∈ (0, ∞) such that ∫ f⟨ν, δϕ⟩ dσ ≤ Cϕ p ∂Ω L (∂Ω,Λ2 TM) ,
∀ ϕ ∈ C01 (M, Λ2 TM)}.
(9.6.132)
∂Ω
Proof. Given an arbitrary differential form ϕ ∈ C01 (M, Λ2 TM), work in local coordinates and write ϕ = ∑nj,k=1 ϕ jk dx j ∧ dx k . Then n
n
δϕ = ∑ g rj ∂ r ϕ jk dx k − ∑ g sk ∂ s ϕ jk dx j j,k,r=1
(9.6.133)
j,k,s=1
hence n
n
i,j,k,r=1
ℓ,j,k,s=1
ν ∨ δϕ = ∑ g rj g ik ν i ∂ r ϕ jk − ∑ g sk gℓj νℓ ∂ s ϕ jk n
= ∑ g rj g ik (ν i ∂ r − ν r ∂ i )ϕ jk i,j,k,r=1 n
n
n
= ∑ (ν i ∂ r − ν r ∂ i )( ∑ g rj g ik ϕ jk ) + ∑ ϕ jk (ν i ∂ r − ν r ∂ i )(g rj g ik ). i,r=1
j,k=1
(9.6.134)
i,j,k,r=1
Note that n
n
n
n
∑ (ν i ∂ r − ν r ∂ i )( ∑ g rj g ik ϕ jk ) = ∑ (ν i ∇∂ r − ν r ∇∂ i )( ∑ g rj g ik ϕ jk ) i,r=1
j,k=1
i,r=1 n
j,k=1 n
= ∑ ∂ τ ir ( ∑ g rj g ik ϕ jk ). i,r=1
j,k=1
(9.6.135)
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  441
Consider next the linear mapping n
(ϕ jk )1≤j,k≤n → ( ∑ g rj g ik ϕ jk )
,
(9.6.136)
1≤i,r≤n
j,k=1
acting between vector spaces of equal ﬁnite dimension. Note that n
n
∑ g ik ( ∑ g rj ϕ jk ) = 0 for every i, r k=1
(9.6.137)
j=1
forces ∑nj=1 g rj ϕ jk = 0 for every k, r, hence further, ϕ jk = 0 for every j, k. Thus, the mapping in question is injective which then implies that the assignment (9.6.136) is, in fact, a linear isomorphism. Having established this, the desired conclusion may now be seen from (9.6.134) with the help of Proposition 9.24 (bearing in mind the convention made in (9.5.47)). Moving on, we discuss the notion of tangential gradient for functions belonging to the brand of Sobolev spaces introduced earlier. Speciﬁcally, given open set Ω ⊂ M of locally ﬁnite perimeter satisfying (9.5.48), with outward unit conormal ν, we deﬁne the p tangential gradient of a function f ∈ L1 (∂Ω), with p ∈ [1, ∞], according to⁸ ♯
♯
∇tan f := −(ν ∨ d ∂ (fν)) = (ν ∨ (ν ∧ d)f ) ,
(9.6.138)
where the musical isomorphisms are as in (9.1.2). In particular, this implies that ♭
ν ∧ (∇tan f ) = (ν ∧ d)f.
(9.6.139)
It is desirable to have a more transparent way of describing the vector ﬁeld ∇tan f . To accomplish this, we ﬁrst note that from (9.6.126) we have ν ∨ (ν ∧ d)f =
1 n ∑ ν ∨ (dx j ∧ dx k )∂ τ jk f 2 j,k=1
(9.6.140)
and from item (7) of Lemma 2.2 we deduce that, for each j, k ∈ {1, . . . , n}, ν ∨ (dx j ∧ dx k ) = ⟨ν, dx j ⟩ dx k − ⟨ν, dx k ⟩ dx j
(9.6.141)
Bearing in mind that ∂ τ jk f is antisymmetric in j, k (cf. (9.6.78)), from (9.6.140), (9.6.141), and (2.1.23) we see that n
n
j,k=1
j,k,ℓ=1
ν ∨ (ν ∧ d)f = ∑ ⟨ν, dx j ⟩∂ τ jk f dx k = ∑ νℓ gℓj ∂ τ jk f dx k .
8 compare with (9.5.21) in the Euclidean setting
(9.6.142)
442  9 Further Tools from Geometry and Analysis
From this, (9.6.138), and (9.1.3) we then conclude that n
♯
∇tan f = (ν ∨ (ν ∧ d)f ) = ∑ νℓ gℓj ∂ τ jk f (dx k )♯ j,k,ℓ=1 n
= ∑ νℓ g j,k,ℓ,s=1
ℓj ks
g ∂ τ jk f ∂ s .
(9.6.143)
Hence, ∇tan f = ∑ns=1 (∇tan f)s ∂ s is a vector ﬁeld with components n
(∇tan f)s = ∑ νℓ gℓj g ks ∂ τ jk f j,k,ℓ=1 n
= ∑ (ν♯ )j g ks ∂ τ jk f,
∀ s ∈ {1, . . . , n}.
(9.6.144)
j,k=1
Let us also note here that, as a consequence of the ﬁrst line in (9.6.143) and (9.1.3), we have n
((∇tan f)♭ )k = ∑ (ν♯ )j ∂ τ jk f,
∀ k ∈ {1, . . . , n}.
(9.6.145)
j=1
Proposition 9.43. Let M be a C 2 manifold of dimension n, equipped with a C 1 Riemannian metric. Assume that Ω ⊂ M is an open subset of M satisfying a twosided local John condition and whose boundary is compact and Ahlfors regular. Denote by ν the outward p unit conormal to Ω. In this context, ﬁx an arbitrary function f ∈ L1 (∂Ω) with p ∈ (1, ∞). Then, locally, for any a, c ∈ {1, . . . , n} one has ∂ τ ac f = ν a g bc (∇tan f)b − ν c g da (∇tan f)d = ν a ((∇tan f)♭ )c − ν c ((∇tan f)♭ )a .
(9.6.146)
Proof. Thanks to the density result recorded in (9.5.43) it suffices to prove the ﬁrst equality in (9.6.146) in the case when f = φ∂Ω for some φ ∈ C 1 (M). In this scenario, ﬁx some arbitrary a, c ∈ {1, . . . , n}. Making use of (9.6.144) and (9.5.47), we may compute ν a g bc (∇tan f)b = ν a νℓ g bc gℓj g kb ∂ τ jk (φ∂Ω ) = ν a νℓ gℓj ∂ τ jc (φ∂Ω ) = ν a νℓ gℓj ν j (∂ c φ)∂Ω − ν a νℓ gℓj ν c (∂ j φ)∂Ω = ν a (∂ c φ)∂Ω − ν a ν c νℓ gℓj (∂ j φ)∂Ω ,
(9.6.147)
since νℓ gℓj ν j = ⟨ν, ν⟩ = 1. Likewise,
hence
as desired.
ν c g da (∇tan f)d = ν c (∂ a φ)∂Ω − ν a ν c νℓ gℓj (∂ j φ)∂Ω ,
(9.6.148)
ν a g bc (∇tan f)b − ν c g da (∇tan f)d = ν a (∂ c φ)∂Ω − ν c (∂ a φ)∂Ω = ∂ τ ac (φ∂Ω ) = ∂ τ ac f,
(9.6.149)
9.6 Integrating by Parts on the Boundaries of Ahlfors Regular Domains  443
To set the stage for our last result in this section, assume that the ambient manifold M is compact and recall that b l (M) stands for the lth Betti number of M (cf. (3.1.9)). Proposition 9.44. Assume that M is a compact C 2 manifold equipped with a C 1 Riemannian metric, with the property that b1 (M) = b2 (M) = 0.
(9.6.150)
Let Ω ⊂ M be a UR domain and ﬁx some integrability exponent p ∈ [1, ∞]. p Then for every f ∈ L1 (∂Ω) one has ∇tan f = 0 on ∂Ω ⇐⇒ f is locally constant on ∂Ω.
(9.6.151)
Proof. The leftpointing implication in (9.6.151) is always true (i.e., makes no use of the topological conditions in (9.6.150)), as may be seen from (9.6.138) and (9.6.127). The signiﬁcance of the topological condition in (9.6.150) is that a portion of the discussion in § 3.2, pertaining to layer potential operators for the HodgeLaplacian (associated with the de RhamHodge formalism), may be carried out in the case when V ≡ 0. Speciﬁcally, as seen from (3.1.11), in the present context the HodgeLaplacian has the property that ∆HL : H 1,2 (M, Λ l TM) → H −1,2 (M, Λ l TM) is an invertible operator for l ∈ {1, 2}.
(9.6.152)
Granted this, the Schwartz kernel Γ l (x, y) of the inverse ∆−1 HL of the unperturbed HodgeLaplacian in (9.6.152) is meaningful for l ∈ {1, 2}. In particular, we may associate single layer potentials Sl with integral kernels Γ l (x, y) for each l ∈ {1, 2}. Since in the present context the right handside of (3.1.45) (written for l = 1 and V = 0) vanishes, we conclude that R1 (⋅, ⋅) = 0. Together with (3.1.55), this implies that Q1 (⋅, ⋅) = 0. In turn, in the context of the second formula in (3.2.40) (written for l = 1), this implies a genuine commutation identity, of the following sort: dS1 g = S2 (d ∂ g)
p,d
for every g ∈ Lnor (∂Ω, Λ1 TM).
(9.6.153)
To start in earnest the proof of the rightpointing implication in (9.6.151), assume p f ∈ L1 (∂Ω) is such that ∇tan f = 0 on ∂Ω. Introduce Ω+ := Ω and Ω− := M \ Ω, and deﬁne the scalarvalued functions u± := δS1 (νf) in Ω± .
(9.6.154)
Then, thanks to (2.1.117), we have u± ∈ C 1 (Ω± ). Also, based on (9.6.129), (9.6.153), (9.6.139), and assumptions we may write du± = dδS1 (νf) = −δdS1 (νf) = −δS2 (d ∂ (νf)) = δS2 ((ν ∧ d)f ) = δS2 (ν ∧ (∇tan f)♯ ) = 0 in Ω± .
(9.6.155)
444  9 Further Tools from Geometry and Analysis
This implies that u± are locally constant in Ω± . Bearing in mind that, as seen from the jumpformula (3.2.15), n.t. n.t. u+ ∂Ω − u− ∂Ω = 12 ν ∨ (νf) + 12 ν ∨ (νf) = f on ∂Ω,
(9.6.156)
we then conclude that f is locally constant on ∂Ω.
9.7 A Global Sobolev Regularity Result The purpose of this section is to establish the global Sobolev regularity result stated in Theorem 9.45 below. Let M be a C 2 manifold of dimension n. Also, given a vector bundle E → M and an open set Ω ⊂ M, we let H 1/2,2 (Ω, E) be the L2 based Sobolev space of smoothness 1/2 of sections of E over Ω, deﬁned via restriction, namely as {uΩ : u ∈ H 1/2,2 (M, E)}. Theorem 9.45. Let E, F → M be two Hermitian vector bundles. Assume that the metrics on E, F and M are Lipschitz. Consider an elliptic, secondorder, differential operator L mapping sections of E into sections of F such that, when written in local coordinates as αβ
αβ
Lu = (∑ ∑ ∂ j a jk ∂ k u β + ∑ ∑ b j ∂ j u β + ∑ d αβ u β ) , j
j,k β
β
β
(9.7.1)
α
where u = (u β )β , its coefficients exhibit the following type of regularity: αβ
a jk ∈ Lip,
αβ
b j ∈ H 1,r ,
d αβ ∈ L r
for some r > max {n, 2}.
(9.7.2)
Finally, ﬁx an arbitrary interior NTA domain Ω ⊂ M. Then there exists some constant C = C(Ω, L) ∈ (0, ∞) such that for every section u ∈ L 2 (Ω, E) satisfying Lu = 0 in Ω one has 1/2
‖u‖H 1/2,2 (Ω,E) ≤ C‖u‖L2 (Ω,E) + C (∫ (∇u)(x)2 dist (x, ∂Ω) dVol(x))
.
(9.7.3)
Ω
In the proof of this theorem we shall make used of an interior elliptic estimate that appears in [97, Proposition 3.4]. For the reader’s convenience we record this below. As in the past, a barred integral indicates an average. Proposition 9.46. Retain the assumptions on M, E, F, and L from Theorem 9.45. Pick an arbitrary nonempty open set Ω ⊂ M along with some p ∈ (r/(r − 1), r), and let the p section u ∈ Lloc (Ω, E) satisfy Lu = 0 in Ω. Then whenever x ∈ Ω and 0 < t < 2 dist (x, ∂Ω)/3 one has 1/p −1
‖∇u‖L∞ (B(x,t)) ≤ Ct (
− u(y) − u(x) dVol(y)) ∫ p
B(x,3t/2)
where the constant C ∈ (0, ∞) is independent of u, x and t.
+ Ct1−n/r u(x),
(9.7.4)
445
9.7 A Global Sobolev Regularity Result 
Having recorded this, we now turn to the task of proving Theorem 9.45. Proof of Theorem 9.45. Let us abbreviate ρ(x) := dist (x, ∂Ω),
∀ x ∈ M,
(9.7.5)
and, whenever x ∈ Ω and 0 < t < ρ(x)/2, deﬁne the oscillation − u(y) − u(x) dVol(y). osc (u, x, t) := ∫
(9.7.6)
B(x,t)
Granted the current assumptions on the domain Ω, from [110, Corollary 1, p. 398] we know that for any measurable function u : Ω → E one has ‖u‖H 1/2,2 (Ω,E)
1/2 ρ(⋅)/2 dt 2 ≈ ‖u‖L2 (Ω,E) + ( ∫ [osc (u, ⋅, t)] 2 ) . t L2 (Ω) 0
(9.7.7)
To proceed, ﬁx u ∈ L2 (Ω, E) satisfying Lu = 0 in Ω. Then for every x ∈ Ω and every t ∈ (0, ρ(x)/2) we may estimate − (∇u)(y) dVol(y) − u(y) − u(x) dVol(y) ≤ Ct ∫ osc (u, x, t) = ∫ B(x,t)
B(x,t)
≤ Ct‖∇u‖L∞ (B(x,t)) ≤ Ct‖∇u‖L∞ (B(x,ρ(x)/2)) 1/2 −1
≤ Ctρ(x) (
− ∫
u(y) − u(x) dVol(y)) 2
+ Ctρ(x)1−n/r u(x)
B(x,3ρ(x)/4) 1/2
− ∫
≤ Ct(
(∇u)(y) dVol(y)) 2
+ Ctρ(x)1−n/r u(x),
(9.7.8)
B(x,3ρ(x)/4)
by Poincaré’s inequality (used twice) and Proposition 9.46. This implies that whenever x ∈ Ω and t ∈ (0, ρ(x)/2) we have [osc (u, x, t)]2
1 − (∇u)(y)2 dVol(y) + Cρ(x)2−(2n)/r u(x)2 . ≤C ∫ t2
(9.7.9)
B(x,3ρ(x)/4)
Consequently,
1/2 ρ(⋅)/2 2 dt ≤ I + II, ( ∫ [osc (u, ⋅, t)] 2 ) t L2 (Ω) 0
(9.7.10)
where 1/2
ρ(x)/2
I := C ( ∫ ( ∫ ( Ω
0
− ∫ B(x,3ρ(x)/4)
(∇u)(y)2 dVol(y)) dt) dVol(x))
(9.7.11)
446  9 Further Tools from Geometry and Analysis
and 1/2
ρ(x)/2
II := C ( ∫ ( ∫ (ρ(x)2−(2n)/r u(x)2 ) dt) dVol(x))
.
(9.7.12)
0
Ω
Note that 1/2
I ≤ C ( ∫ ρ(x)(
− ∫
Ω
B(x,3ρ(x)/4)
(∇u)(y)2 dVol(y)) dVol(x)) 1/2
≤ C ( ∫ ∫ ρ(x)
1−n
(∇u)(y) 1dist (x,y)≤3ρ(x)/4 dVol(y) dVol(x)) 2
.
(9.7.13)
Ω Ω
Since whenever dist (x, y) ≤ 3ρ(x)/4 we necessarily have ρ(x) ≈ ρ(y), we may further estimate 1/2
I ≤ C ( ∫ ∫ ρ(y)
1−n
(∇u)(y) 1dist (x,y)≤cρ(y) dVol(y) dVol(x)) 2
Ω Ω 1/2
≤ C ( ∫ ρ(y)1−n (∇u)(y)2 ( ∫ 1dist (x,y)≤cρ(y) dVol(x)) dVol(y)) Ω
Ω 1/2
≤ C ( ∫ ρ(y)(∇u)(y) dVol(y)) 2
,
(9.7.14)
Ω
which suits our purposes. There remains to observe that, since r ≥ (2n)/3, 1/2
II ≤ C ( ∫ ρ(x)3−(2n)/r u(x)2 dVol(x))
≤ C‖u‖L2 (Ω,E) .
(9.7.15)
Ω
At this stage, (9.7.3) follows from (9.7.7), (9.7.10), (9.7.14), and (9.7.15). This ﬁnishes the proof of the theorem.
9.8 The PV Harmonic Double Layer on a UR Domain Here we will deal with a bounded UR domain Ω ⊂ ℝn , with surface measure given by σ := H n−1 ⌊∂Ω and outward unit normal ν, and discuss some results on the existence of the principal value harmonic double layer potential Kf(x) := P.V.
1 ⟨ν(y), y − x⟩ f(y) dσ(y), ∫ ω n−1 x − yn ∂Ω
x ∈ ∂Ω,
(9.8.1)
9.8 The PV Harmonic Double Layer on a UR Domain

447
where ω n−1 denotes the area of the unit sphere S n−1 ⊂ ℝn , as the limit as ε → 0+ of the truncated operators K ε f(x) :=
1 ω n−1
⟨ν(y), y − x⟩ f(y) dσ(y), x − yn
∫
x ∈ ∂Ω.
(9.8.2)
∂Ω\B ε (x)
In the process, we make use G. David’s estimate ‖K∗ f‖L p (∂Ω) ≤ C p ‖f‖L p (∂Ω) ,
1 < p < ∞,
(9.8.3)
where the maximal operator K∗ is deﬁned for each f ∈ L p (∂Ω) as (K∗ f)(x) := sup (K ε f)(x), ε>0
∀ x ∈ ∂Ω.
(9.8.4)
Given any f ∈ L p (∂Ω), with 1 < p < ∞, one desires to show that the principal value (9.8.1) exists σa.e. on ∂Ω as the limit (Kf)(x) := lim (K ε f)(x) ε↘0
at σa.e. x ∈ ∂Ω.
(9.8.5)
As in [50, § 3.3], via (9.8.4) one is reduced to showing this for f ∈ Lip (∂Ω), and from this to showing it for f ≡ 1. Then one examines (K ε 1)(x) for x ∈ ∂∗ Ω. Formula⁹ (3.3.15) on p. 2655 of [50] (itself a consequence of the divergence theorem) implies (K ε 1)(x) = ε−(n−1) ⋅
surface measure of ∂B ε (x) ∩ Ω c , ω n−1
(9.8.6)
where Ω c := ℝn \ Ω. As noted in [50], it follows from (9.8.6) and [50, Proposition 3.3, p. 2628] that, whenever x ∈ ∂∗ Ω, there exists a Lebesgue measurable set Ox ⊆ (0, 1], of density 1 at 0,
(9.8.7)
such that lim
ε↘0, ε∈Ox
(K ε 1)(x) =
1 . 2
(9.8.8)
In [50] the authors assert that they can remove the restriction ε ∈ Ox in (9.8.8), but details were not given there. We will give such details here. This will involve sharpening [50, Proposition 3.3, p. 2628]. Before doing this, we make some comments on what one can accomplish with the weaker result (9.8.8). Namely, for each ε ∈ (0, 1] we can deﬁne an operator ̃ K ε acting on arbitrary functions f ∈ L p (∂Ω), with 1 < p < ∞, according to ε
2 (̃ K ε f)(x) := ∫ (K τ f)(x) dτ, ε ε/2
9 written with n in place of n + 1 and used here with R = ∞
∀ x ∈ ∂Ω.
(9.8.9)
448  9 Further Tools from Geometry and Analysis
It then follows from (9.8.8) and the bound on K∗ 1 that lim (̃ K ε 1)(x) = ε↘0
1 , 2
∀ x ∈ ∂∗ Ω.
(9.8.10)
This then leads to the following slightly weaker replacement for (9.8.5). Proposition 9.47. Let Ω ⊂ ℝn be a bounded UR domain. Given any function f ∈ L p (∂Ω), with 1 < p < ∞, the limit (̃ Kf)(x) := lim (̃ K ε f)(x) exists at σa.e. point x ∈ ∂Ω. ε↘0
(9.8.11)
Then, with ̃ Kf(x) so deﬁned, ̃ Kf in L p (∂Ω) as ε ↘ 0. K ε f → ̃
(9.8.12)
With this result in hand, the treatment of nontangential limits in [50, Proposition 3.28, p. 2654] proceeds with only minor changes. Similar considerations apply to [50, Theorem 3.32, p.2662] and [50, Theorem 3.33, p. 2669]. However, our current goal is to provide the details on the following improvement of (9.8.8): 1 lim (K ε 1)(x) = , ∀ x ∈ ∂∗ Ω, (9.8.13) ε↘0 2 which then yields the following improvement of Proposition 9.47. Proposition 9.48. In the setting of Proposition 9.47, the convergence results (9.8.11) and K replaced by K). (9.8.12) hold with ̃ K ε replaced by K ε (and ̃ As mentioned above, we do this by improving [50, Proposition 3.3, p. 2628]. We recall the setup, involving a class of domains more general than UR domains. Here, Ω ⊂ ℝn is a bounded open set with ﬁnite perimeter, satisfying H n−1 (∂Ω \ ∂∗ Ω) = 0. Let denote ∂ T Ω denote the set of points in ∂Ω where an approximate tangent plane exists (cf. the discussion in [50, pp. 2626, 2627]). We take x ∈ ∂∗ Ω ⊆ ∂ T Ω,
(9.8.14)
which, as in [50, (3.1.3), p. 2627], implies that, for each δ > 0, H
n−1
({y ∈ ∂Ω ∩ B r (x) : dist (y, π x ) > δx − y}) = o(r n−1 ) as r → 0+ ,
(9.8.15)
where π x is the (n − 1)dimensional hyperplane through x, orthogonal to the outwardpointing unit normal ν(x). (Cf. [50, Proposition 3.2, p. 2628].) We set π −x := {y ∈ ℝn : ⟨ν(x), y − x⟩ ≤ 0}, π+x := {y ∈ ℝn : ⟨ν(x), y − x⟩ ≥ 0}.
(9.8.16)
Note that π−x would locally coincide with Ω if ∂Ω were ﬂat in a neighborhood of x. Then we take W(x, r) := ∂B r (x) ∩ [Ω △ π−x ], (9.8.17)
9.8 The PV Harmonic Double Layer on a UR Domain

449
where U △ V denotes the symmetric difference (U \ V) ∪ (V \ U) of two arbitrary sets U, V. The following improves on [50, Proposition 3.3, p. 2628] and, consequently, implies the improvement (9.8.13) of (9.8.8). Proposition 9.49. If Ω ⊂ ℝn is a UR domain then, in the notation described above, lim r−(n−1) H
n−1
r→0+
(W(x, r)) = 0,
∀ x ∈ ∂∗ Ω.
(9.8.18)
By comparison, [50, Proposition 3.3, p. 2628] has the conclusion that there exists a Lebesgue measurable set Ox ⊆ (0, 1], with density 1 at 0, such that lim
r→0, r∈Ox
r−(n−1) H
n−1
(W(x, r)) = 0.
(9.8.19)
This result will play a role in the proof of Proposition 9.49. Proof of Proposition 9.49. For δ > 0, let Aδ (x) := {y ∈ ℝn : dist (y, π x ) ≤ δx − y}.
(9.8.20)
Then there exists C ∈ (0, ∞) such that whenever 0 < r ≤ 1 H
n−1
(∂B r (x) ∩ Aδ (x)) ≤ Cδr n−1 .
(9.8.21)
Thus H
n−1
(W(x, r)) ≤ Cδr n−1 + H
n−1
(W(x, r) \ Aδ (x)).
(9.8.22)
as r → 0+ .
(9.8.23)
We can rewrite (9.8.15) as H
n−1
([B2r (x) ∩ ∂Ω] \ Aδ (x)) = o(r n−1 )
By (9.8.19), there exists r δ > 0 such that, for r ∈ (0, r δ ), there exists ̃r = ̃r(r) such that r−(n−1) H
n−1
(W(x, r)) ≤ δ,
and
r < ̃r < 2r.
(9.8.24)
Returning to (9.8.22), we have the task of estimating I1 (r) := H
n−1
(∂B r (x) ∩ [Ω \ π−x ] \ Aδ (x)),
I2 (r) := H
n−1
(∂B r (x) ∩ [π−x \ Ω] \ Aδ (x)),
(9.8.25)
for r ∈ (0, r δ ]. To estimate I1 (r), we write ∂B r (x) ∩ [Ω \ π−x ] \ Aδ (x) = S1 ∪ S2 ,
(9.8.26)
a disjoint union of sets deﬁned as follows. For each point z in the left side of (9.8.26), take the ray from x to z and extend outward, until the ray hits ∂B̃r (x), say at ̃z. (Such a ray is contained in π +x \ Aδ (x).) If ̃z ∈ Ω, we say z ∈ S1 . Otherwise, we say z ∈ S2 . Note that z ∈ S1 ⇒ ̃z ∈ W(x, ̃r ). (9.8.27)
450  9 Further Tools from Geometry and Analysis
We deduce from (9.8.24) that H
n−1
(S1 ) ≤ δr n−1 .
(9.8.28)
On the other hand, if z ∈ S2 , the ray from z to ̃z must cross from Ω to Ω c , so it must cross ∂Ω. We deduce from (9.8.23) that H
n−1
(S2 ) = o(r n−1 )
as r → 0+ .
(9.8.29)
Hence, for r ∈ (0, r δ ], I1 (r) ≤ δr n−1 + o(r n−1 ).
(9.8.30)
∂B r (x) ∩ [π −x \ Ω] \ Aδ (x) = T1 ∪ T2 ,
(9.8.31)
To estimate I2 (r), we write
a disjoint union of sets deﬁned as follows. For each point z in the left side of (9.8.31), take the ray from x to z, and extend outward, until the ray hits ∂B̃r (x), say at ̃z. (This time, such a ray is contained in π −x \ Aδ (x).) If ̃z ∉ Ω, we say z ∈ T1 . Otherwise we say z ∈ T2 . Note that (9.8.32) z ∈ T1 ⇒ ̃z ∈ W(x, ̃r). We deduce from (9.8.24) that H
n−1
(T1 ) ≤ δr n−1 .
(9.8.33)
On the other hand, if z ∈ T2 , the ray from z to ̃z must cross from Ω c to Ω, so it must cross ∂Ω. We deduce from (9.8.23) that H
n−1
(T2 ) = o(r n−1 ) as r → 0+ .
(9.8.34)
Hence, for r ∈ (0, r δ ], I2 (r) ≤ δr n−1 + o(r n−1 ).
(9.8.35)
Putting together (9.8.22), (9.8.25), (9.8.30), and (9.8.35), we get H
n−1
(W(x, r)) ≤ (C + 2)δr n−1 + o(r n−1 ) for r ∈ (0, r δ ],
(9.8.36)
hence lim sup r−(n−1) H r→0+
n−1
(W(x, r)) ≤ (C + 2)δ.
(9.8.37)
Since (9.8.37) holds for each δ > 0, we have the desired conclusion (9.8.18). In closing, we wish to remark that the improvement recorded in Proposition 9.49 over [50, Proposition 3.3, p. 2628] permits the treatment of principalvalue and nontangential limits from [50, Proposition 3.28, p. 2654], [50, Theorem 3.32, p.2662] and [50, Theorem 3.33, p. 2669], as originally intended there. In turn, the results just mentioned play a key role in the analysis undertaken next, in § 9.9. Moreover, the basic formula (9.8.13) is also directly relevant for the proof of Theorem 6.1 (see (6.1.70)).
9.9 CalderónZygmund Theory on UR Domains on Manifolds
 451
9.9 CalderónZygmund Theory on UR Domains on Manifolds Recall that a pseudodifferential operator Q(x, D) with symbol q(x, ξ) in Hörmander’s m is given by the oscillatory integral¹⁰ class S 1,0 ̂ (ξ)e i⟨x, ξ⟩ dξ Q(x, D)u = (2π)−n/2 ∫ q(x, ξ)u = (2π)−n ∬ q(x, ξ)e i⟨x−y, ξ⟩ u(y) dy dξ.
(9.9.1)
m , deﬁned by requiring that Here, we are concerned with a smaller class of symbols, Scl the (matrixvalued) function q(x, ξ) has an asymptotic expansion of the form
q(x, ξ) ∼ q m (x, ξ) + q m−1 (x, ξ) + ⋅ ⋅ ⋅ ,
(9.9.2)
with q j smooth in x and ξ and homogeneous of degree j in ξ (for ξ ≥ 1). Call q m (x, ξ), i.e., the leading term in (9.9.2), the principal symbol of q(x, D). In fact, we shall ﬁnd it convenient to work with classes of symbols that only exhibit a limited amount of regularity in the spatial variable (while still C ∞ in the Fourier variable). Speciﬁcally, for each r ≥ 0 we deﬁne m C r S1,0 := {q(X, ξ) : ‖D αξ q(⋅, ξ)‖C r ≤ C α (1 + ξ)m−α , ∀ α}.
(9.9.3)
m Denote by OP C r S1,0 the class of pseudodifferential operators associated with such m symbols. As before, we write OP C r Scl for the subclass of classical pseudodifferm r ential operators in OP C S1,0 whose symbols can be expanded as in (9.9.2), where m−j
q j (x, ξ) ∈ C r S1,0 is homogeneous of degree j in ξ for ξ ≥ 1, j = m, m − 1, . . . . Finally, m m for the space of all formal adjoints of operators in OP C r Scl . we set ØP C r Scl Next, given two Hermitian vector bundles E, F → M, over an ndimensional manm ifold M, denote by OP C r S1,0 (E, F) the collection of pseudodifferential operators mapping sections of the vector bundle E into sections of the vector bundle F which, in local coordinates of M and over local trivializations of E, F, can be represented as matrices m m with entries from OP C r S1,0 . Finally, ØP C r Scl (E, F) will consist of all formal adjoints m r of operators in OP C Scl (F , E). Going further, for a classical pseudodifferential operator Q(x, D) ∈ OP C0 S−1 cl (E , F ), let k Q ∈ D (M × M, F ⊗ E) ∩ C 0 (M × M \ diag, F ⊗ E) (9.9.4) stand for its Schwartz kernel¹¹, and let Sym (Q, ⋅) : T ∗ M \ 0 → Hom (E, F),
Sym (Q, ⋅) ∈ C0 S−1 cl (E , F ),
denote its principal symbol.
10 where “hat” stands for the Fourier transform in ℝn 11 hence, D (M) ⟨Qu, υ⟩D(M) = D (M×M) ⟨k Q , υ ⊗ u⟩D(M×M) for all test functions u, υ on M
(9.9.5)
452  9 Further Tools from Geometry and Analysis Fix an Ahlfors regular domain Ω ⊆ M and denote by σ := H n−1 ⌊∂Ω its boundary surface measure. The scale of Lebesgue spaces like L p (∂Ω, E), for p ∈ (0, ∞], are deﬁned naturally. In this context, we introduce the integral operator acting on f ∈ L1 (∂Ω, E) according to KQ f(x) := ∫ ⟨k Q (x, y), f(y)⟩Ey dσ(y),
x ∈ M \ ∂Ω.
(9.9.6)
∂Ω
The principal value version of (9.9.6), formally written as K Q f(x) = P.V. ∫ ⟨k Q (x, y), f(y)⟩Ey dσ(y),
x ∈ ∂Ω,
(9.9.7)
∂Ω
is deﬁned via K Q f(x) := lim+ K Q,ε f(x), ε→0
x ∈ ∂Ω,
(9.9.8)
where, for each ε > 0, the truncated singular integral operator K Q,ε is given by K Q,ε f(x) := ∫ ⟨k Q (x, y), f(y)⟩Ey dσ(y),
x ∈ ∂Ω,
(9.9.9)
y∈∂Ω, d(x,y)>ε
with d(x, y) standing for the geodesic distance between x and y in M, with respect to some ﬁxed background metric. In this connection, let us also introduce the maximal operator K Q,∗ f(x) := sup K Q,ε f(x), x ∈ ∂Ω. (9.9.10) ε>0
Similar considerations apply to pseudodifferential operators in ØP C0 S−1 cl (E , F ). To state the ﬁrst result of this section we need some more notation. We set Ω+ := Ω, n.t. Ω− := M \ Ω, and let ⋅ ∂Ω± be the nontangential boundary trace operators on ∂Ω± , that is n.t. u∂Ω± (x) := lim u(y), x ∈ ∂Ω, (9.9.11) Γ± (x)∋y→x
where Γ± (x) ⊆ Ω± are appropriate nontangential approach regions with vertex at x. Finally, N stands for the nontangential maximal operator deﬁned for continuous sections u : Ω± → E by N u(x) := sup u(y), x ∈ ∂Ω. (9.9.12) y∈Γ± (x)
Theorem 9.50. Let Ω ⊂ M be a UR domain satisfying ∂(Ω) = ∂Ω, with outward unit conormal ν ∈ T ∗ M and surface measure σ := H n−1 ⌊∂Ω. Also, let E, F → M be two Hermitian vector bundles, and let Q(x, D) ∈ OP C0 S−1 cl (E , F ) be a classical pseudodifferential operator with the property that its principal symbol Sym (Q, ξ) is odd in the cotangent variable ξ ∈ T ∗ M. Then for each function f ∈ L1 (∂Ω, E) the principal value integral deﬁning K Q f(x) exists at σalmost every boundary point x ∈ ∂Ω and K Q : L p (∂Ω, E) → L p (∂Ω, F)
(9.9.13)
9.9 CalderónZygmund Theory on UR Domains on Manifolds
 453
is a welldeﬁned, linear, and bounded operator whenever p ∈ (1, ∞). Corresponding to p = 1, K Q : L1 (∂Ω, E) → L1,∞ (∂Ω, F) is bounded. (9.9.14) In fact, the sublinear maximal operators K Q,∗ : L p (∂Ω, E) → L p (∂Ω, F) K Q,∗ : L (∂Ω, E) → L 1
1,∞
if 1 < p < ∞,
(∂Ω, F),
(9.9.15)
are also welldeﬁned and bounded. As a consequence of this and Lebesgue’s Dominated Convergence Theorem, for every f ∈ L p (∂Ω, E) with p ∈ (1, ∞) one has lim K Q,ε f = K Q f
ε→0+
in L p (∂Ω, F).
(9.9.16)
Also, with K Q⊤ denoting the singular integral operator associated with Q⊤ in the same manner K Q has been associated with Q in (9.9.8), for each p ∈ (1, ∞) one has ⊤
(K Q ) = K Q⊤ , as operators from L p (∂Ω, F) into L p (∂Ω, E).
(9.9.17)
Furthermore, for every p ∈ (1, ∞) there exists a constant C ∈ (0, ∞) such that ‖N(KQ f)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω, E) ,
∀ f ∈ L p (∂Ω, E),
(9.9.18)
∀ f ∈ L1 (∂Ω, E).
(9.9.19)
and, corresponding to p = 1, ‖N(KQ f)‖L1,∞ (∂Ω) ≤ C‖f‖L1 (∂Ω, E) ,
Moreover, given any f ∈ L1 (∂Ω, E), the function KQ f has a nontangential boundary trace at almost every boundary point, in the precise sense that n.t. = ∓ 12 i Sym (Q, ν)f + K Q f, KQ f ∂Ω± σa.e. on ∂Ω,
(9.9.20)
∀ f ∈ L1 (∂Ω, E).
In addition, if actually Q(x, D) ∈ OP C1 S−1 cl (E , F ) then there exists C ∈ (0, ∞) such that for every f ∈ L2 (∂Ω, E) one has 2 ∫ ∇(KQ f)(x) dist (x, ∂Ω) dVol(x) ≤ C ∫ f2 dσ. M\∂Ω
(9.9.21)
∂Ω
Finally, similar results are valid for pseudodifferential operators in ØP C0 S−1 cl (E , F ). Proof. The problem localizes and, given the invariance of the class of pseudodifferential operators and symbols under discussion, it can be transported in to the Euclidean setting via coordinate mappings. By ﬁxing local frames in E and F, the operator Q(x, D) can be locally identiﬁed, in a canonical manner, with a matrix of classical pseudodifferential operators (Q jk (x, D))j,k , while k Q (x, y) can be identiﬁed with a matrixvalued
454  9 Further Tools from Geometry and Analysis distribution (b jk (x, y))j,k . The latter is such that the Schwartz kernel of Q jk (x, D) with respect to the Euclidean volume form is b jk (x, y)√ g(y). Going further, Ω may be locally identiﬁed with a UR domain in ℝn . While we retain the same notation, Ω, for the latter domain, there are now two measures operating on ∂Ω, namely the original σ and the Euclidean surface measure, which we denote by σE . They are related to one another as in (9.5.11) and (9.5.12). Moreover, the original differential geometric outward unit normal ν and the Euclidean outward unit normal, denoted by νE , are related to one another as in (9.5.3)–(9.5.5). Consequently, KQ can be identiﬁed with the operatorvalued matrix (Bjk )j,k where, for x ∈ Ω± , 1/2
Bjk f(x) := ∫ b jk (x, y)√ g(y)(∑ g jk (y)νEj (y)νEk (y))
f(y) dσE (y).
(9.9.22)
j,k
∂Ω
Let us consider the jump formula (9.9.20). With the above notation, the jumpformulas proved in the class of variable coefficient CalderónZygmund operators in UR domain in the Euclidean setting from [50] gives that, at σE a.e. point x ∈ ∂Ω, n.t. (Bjk f Ω± )(x) = ∓ 12 i Sym (Q jk , νE (x))(∑ g jk (x)νEj (x)νEk (x))
1/2
f(x)
j,k
1/2
+ lim+ ∫ b jk (x, y)√ g(y)(∑ g jk (y)νEj (y)νEk (y)) ε→0
y∈∂Ω d(x,y)>ε
f(y) dσE (y)
j,k
= ∓ 12 i Sym (Q jk , νE (x))f(x) + K jk f(x),
(9.9.23)
where we have used the fact that Sym (Q jk , ξ) is homogeneous of degree −1 in ξ , and we have denoted by K jk the operator given by the principalvalue integral in (9.9.23). Under the identiﬁcation K Q ≡ (K jk )j,k , the analysis above amounts to n.t. (KQ f ∂Ω± )(x) = ∓ 12 i Sym (Q, ν(x))f(x) + lim+ ∫ ⟨k Q (x, y), f(y)⟩y dσ(y), ε→0
(9.9.24) at σa.e. x ∈ ∂Ω,
y∈∂Ω d(x,y)>ε
which is precisely (9.9.20). The claims pertaining to (9.9.13)–(9.9.19) follow in a similar fashion from the corresponding theory for singular integrals with variable kernel in UR domains in the Euclidean setting from [50]. Also, the L2 square function estimate (9.9.21) may be established in an analogous manner by relying on the Euclidean result recorded in Proposition 9.51 below. Finally, consider (9.9.17). In this regard, we shall ﬁrst show that for every ε > 0 ﬁxed we have (K Q,ε )⊤ = K Q⊤,ε . (9.9.25)
9.9 CalderónZygmund Theory on UR Domains on Manifolds  455
With this in mind, observe that (k Q )⊤ (y, x) = k Q⊤ (x, y)
in D (M × M, E ⊗ F).
(9.9.26)
Indeed, for two arbitrary test functions u ∈ C 0 (M, F) and υ ∈ C 0 (M, E) we have ⟨(k Q )⊤ (y, x), u(x) ⊗ υ(y)⟩E⊗F = ⟨k Q (x, y), υ(x) ⊗ u(y)⟩F⊗E = ⟨Qu, υ⟩F = ⟨Q⊤ υ, u⟩E
= ⟨k Q⊤ (x, y), u(x) ⊗ υ(y)⟩E⊗F
(9.9.27)
from which (9.9.26) follows. Next, ﬁx some arbitrary f ∈ L p (∂Ω, E), g ∈ L p (∂Ω, F), with 1 < p, p < ∞ satisfying 1/p + 1/p = 1, and use Fubini’s Theorem along with (9.9.26) in order to write ∫ ⟨(K Q,ε )⊤ g, f ⟩E dσ = ∫ ⟨g, K Q,ε f ⟩F dσ ∂Ω
∂Ω
= ∫ ⟨g(x), ∫ ⟨k Q (x, y), f(y)⟩Ey dσ(y)⟩ ∂Ω
Fx
dσ(x)
y∈∂Ω d(x,y)>ε
=
⟨k Q (x, y), g(x) ⊗ f(y)⟩Fx ⊗Ey d(σ ⊗ σ)(x, y)
∬
(x,y)∈∂Ω×∂Ω d(x,y)>ε
=
⟨(k Q )⊤ (y, x), f(x) ⊗ g(y)⟩Ex ⊗Fy d(σ ⊗ σ)(x, y)
∬
(x,y)∈∂Ω×∂Ω d(x,y)>ε
=
⟨k Q⊤ (x, y), f(x) ⊗ g(y)⟩Ex ⊗Fy d(σ ⊗ σ)(x, y)
∬
(x,y)∈∂Ω×∂Ω d(x,y)>ε
= ∫ ⟨ ∫⟨k Q⊤ (x, y), g(y)⟩Fy dσ(y) , f(x)⟩ ∂Ω
Ex
dσ(x)
y∈∂Ω d(x,y)>ε
= ∫ ⟨K Q⊤,ε g, f ⟩E dσ.
(9.9.28)
∂Ω
This proves (9.9.26). Having established this, we may now rely on formula (9.9.16) as well as its counterpart for the operators associated with Q⊤ in order to write that, for every section g ∈ L p (∂Ω, F) with p ∈ (1, ∞), we have ⊤
(K Q ) g = lim+ (K Q,ε )⊤ g = lim+ K Q⊤,ε g ε→0
ε→0
= K Q⊤ g weakly in L p (∂Ω, E). ⊤
Thus, (K Q ) g = K Q⊤ g as functions in L p (∂Ω, E), ﬁnishing the proof of (9.9.17).
(9.9.29)
456  9 Further Tools from Geometry and Analysis Here is the Euclidean version of the L2 square function estimate for integral operators with variable coefficient kernels in uniformly rectiﬁable sets which has been invoked in the proof of Theorem 9.50. For a proof of Proposition 9.51 the reader is referred to [49]. Proposition 9.51. There exists a positive integer M = M(n) with the following signiﬁcance. Let Σ be a compact, uniformly rectiﬁable set in ℝn and assume that U ⊆ ℝn is a bounded, open neighborhood of Σ. Consider a function (ℝn \ {0}) × U ∋ (z, y) → b(z, y) ∈ ℂ
(9.9.30)
which is odd and (positively) homogeneous of degree 1 − n in the variable z ∈ ℝn \ {0}, and which has the property that ∂ αz b(z, y) is continuous and bounded on S n−1 × U for α ≤ M.
(9.9.31)
Finally, deﬁne the variable kernel integral operator Bf(x) := ∫ b(x − y, y)f(y) dσ(y),
x ∈ U \ Σ,
(9.9.32)
Σ n−1 ⌊Σ
is the measure induced by the (n − 1)dimensional Hausdorff meawhere σ := H sure on Σ. Then there exists a constant C ∈ (0, ∞) depending only on n, the UR character of Σ, the diameter of U, and ‖∂ αz b‖L∞ (S n−1 ×U) for α ≤ M, such that ∫ ∇Bf(x)2 dist (x, Σ) dx ≤ C ∫f2 dσ, U\Σ
∀ f ∈ L2 (Σ).
(9.9.33)
Σ
Going further, we pass from Theorem 9.50 to estimates for single and double layer potentials associated to an elliptic operator with coefficients of limited smoothness. Before stating this result we make the following convention. Suppose E, F → M are two vector bundles. Given a double distribution E ∈ D (M × M, E ⊗ F) along with two ̃ : E → E, writing differential operators P : F → F and P ̃ x ⊗ P y )E(x, y) (P
(9.9.34)
̃ acts in the variable x on the columns of E(x, y) while P acts in the indicates that P variable y on the rows of E(x, y) (where, in local coordinates over which E, F trivialize, the distribution E(x, y) has been identiﬁed with a matrix E αβ (x, y)e α ⊗ f β using local frames {e α }α , {f β }β in E and F). Theorem 9.52. Let E, F → M be two Hermitian vector bundles over the manifold¹² M. It is assumed that the metric structures on E, F and M are Lipschitz continuous. In this 12 as usual, assumed to have real dimension n
9.9 CalderónZygmund Theory on UR Domains on Manifolds 
457
context, consider an elliptic, secondorder, differential operator L mapping sections of E into sections of F such that, when written in local coordinates as αβ
αβ
Lu = (∑ ∑ ∂ j a jk ∂ k u β + ∑ ∑ b j ∂ j u β + ∑ d αβ u β ) , j
j,k β
β
(9.9.35)
α
β
its coefficients satisfy αβ
a jk ∈ C 1 ,
b j ∈ L∞ , αβ
d αβ ∈ L r for somer > n.
(9.9.36)
In addition, assume that in local coordinates the coefficients of L ∗ , the Hermitian adjoint of L, also satisfy (9.9.36). Also, suppose that L : H 1,2 (M, E) → H −1,2 (M, F) is invertible
(9.9.37)
and let E ∈ D (M × M, E ⊗ F) denote the Schwartz kernel of L−1 .
(9.9.38)
Next, let Ω be an arbitrary UR domain in M satisfying ∂(Ω) = ∂Ω, with outward unit conormal ν ∈ T ∗ M and surface measure σ := H n−1 ⌊∂Ω. Deﬁne action of the single layer operator S and of its boundary version S on an arbitrary function f ∈ L1 (∂Ω, F) as Sf(x) := ∫ ⟨E(x, y), f(y)⟩Fy dσ(y),
x ∈ M \ ∂Ω.
(9.9.39)
x ∈ ∂Ω.
(9.9.40)
∂Ω
and, respectively, Sf(x) := ∫ ⟨E(x, y), f(y)⟩Fy dσ(y), ∂Ω
Moreover, for a ﬁrstorder differential operator P ∈ Diff 1 (F , F) with bounded measurable coefficients, consider the integral operator with kernel (Ix ⊗ P y )E(x, y), i.e., for each function f ∈ L1 (∂Ω, F) set Af(x) := ∫ ⟨(Ix ⊗ P y )E(x, y), f(y)⟩Fy dσ(y),
x ∈ M \ ∂Ω.
(9.9.41)
∂Ω
For each ε > 0 also consider the truncated singular integral operator A ε acting on each f ∈ L1 (∂Ω, F) according to A ε f(x) := ∫ ⟨(Ix ⊗ P y )E(x, y), f(y)⟩Fy dσ(y),
x ∈ ∂Ω,
(9.9.42)
y∈∂Ω, d(x,y)>ε
with d(x, y) standing for the geodesic distance between x and y in M, with respect to some background metric. Finally, introduce the maximal operator A∗ f(x) := sup A ε f(x), ε>0
and recall that, as usual, Ω+ := Ω, Ω− := M \ Ω.
x ∈ ∂Ω,
(9.9.43)
458  9 Further Tools from Geometry and Analysis
Then the following claims are true. (1) There exists a constant C ∈ (0, ∞) with the property that ‖N(Sf)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,F) , ‖N(Sf)‖L p (∂Ω) ≤ C‖f‖
p L−1 (∂Ω,F)
∀ f ∈ L p (∂Ω, F), ,
∀f ∈
p L−1 (∂Ω, F),
‖N(∇Sf)‖L p (∂Ω) ≤ C‖f‖L p (∂Ω,F) ,
∀ f ∈ L (∂Ω, F),
‖N(∇Sf)‖L1,∞ (∂Ω) ≤ C‖f‖L1 (∂Ω,F) ,
∀ f ∈ L (∂Ω, F).
p
if p ∈ [1, ∞],
(9.9.44)
if p ∈ (1, ∞),
(9.9.45)
if p ∈ (1, ∞),
(9.9.46)
1
(9.9.47)
In fact, a more nuanced version of (9.9.44) holds. Namely, if n = dim M ≥ 3 one has: if p ∈ (1, n − 1) and 1/q = 1/p − 1/(n − 1) there exists C ∈ (0, ∞) so that ‖N(Sf)‖L q (∂Ω) ≤ C‖f‖L p (∂Ω,F) , ∀ f ∈ L p (∂Ω, F), while for each p ∈ (n − 1, ∞) there exists C ∈ (0, ∞) such that N(Sf)L∞ (∂Ω) ≤ C‖f‖L p (∂Ω,F) , ∀ f ∈ L p (∂Ω, F), and, corresponding to p = n − 1, for every q ∈ (1, ∞) there holds N(Sf)L q (∂Ω) ≤ C q ‖f‖L n−1 (∂Ω,F) , ∀ f ∈ L n−1 (∂Ω, F),
(9.9.48)
while corresponding to the critical value p = 1 one has 1 N(Sf)L(n−1)/(n−2),∞ (∂Ω) ≤ C‖f‖L1 (∂Ω,F) , ∀ f ∈ L (∂Ω, F). In addition, in the case when n = 2 one has N(Sf)L∞ (∂Ω) ≤ C‖f‖L p (∂Ω,F) , ∀ f ∈ L p (∂Ω, F) N(Sf)L q (∂Ω) ≤ C q ‖f‖L1 (∂Ω,F) , ∀ f ∈ L1 (∂Ω, F)
if p ∈ (1, ∞), for all q ∈ (1, ∞).