Geometry, Analysis and Dynamics on Sub-riemannian Manifolds 3037191627, 978-3-03719-162-0, 9783037191637, 3037191635

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Geometry, Analysis and Dynamics on Sub-riemannian Manifolds
 3037191627, 978-3-03719-162-0, 9783037191637, 3037191635

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Series of Lectures in Mathematics

Volume I Davide Barilari, Ugo Boscain and Mario Sigalotti Editors Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds.

The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.

ISBN 978-3-03719-162-0

www.ems-ph.org

Barilari et al. Vol. I | Rotis Sans | Pantone 287, Pantone 116 | 170 x 240 mm | RB: 16 ?? mm

Davide Barilari, Ugo Boscain and Mario Sigalotti, Editors

In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology.

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds, Volume I

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds Volume I Davide Barilari Ugo Boscain Mario Sigalotti Editors

EMS Series of Lectures in Mathematics Edited by Ari Laptev (Imperial College, London, UK) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form Armen Sergeev, Lectures on Universal Teichmüller Space Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, 3-Manifold Groups Hans Triebel, Tempered Homogeneous Function Spaces Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink and Jürgen Sander, Four Faces of Number Theory Alberto Cavicchioli, Friedrich Hegenbarth and Dušan Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds Volume I Davide Barilari Ugo Boscain Mario Sigalotti Editors

Editors: Prof. Davide Barilari Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris 7, Denis Diderot 5 rue Thomas Mann 75205 Paris 13 Cedex France E-mail: [email protected]

Prof. Mario Sigalotti INRIA Saclay Centre de Mathématiques Appliquées École Polytechnique Route de Saclay 91128 Palaiseau Cedex France E-mail: [email protected]

Prof. Ugo Boscain CNRS Centre de Mathématiques Appliquées, Ecole Polytechnique Route de Saclay 91128 Palaiseau Cedex France E-mail: [email protected]

2010 Mathematics Subject Classification: Primary: 53C17; Secondary: 35H10, 60H30, 49J15 Key words: sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths

ISBN 978-3-03719-162-0 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2016 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland

Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org

Typeset using the authors’ TEX files: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface This book, divided into two volumes, collects different cycles of lectures given at the IHP Trimester “Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds”, held at Institut Henri Poincaré in Paris, and the CIRM Summer School “SubRiemannian Manifolds: From Geodesics to Hypoelliptic Diffusion”, held at Centre Internationale de Rencontres Mathématiques, in Luminy, during fall 2014. Sub-Riemannian geometry is a generalization of Riemannian geometry, whose birth dates back to Carathéodory’s 1909 seminal paper on the foundations of Carnot thermodynamics, followed by E. Cartan’s 1928 address at the International Congress of Mathematicians in Bologna. Sub-Riemannian geometry is characterized by non-holonomic constraints: distances are computed by minimizing the length of curves whose velocities belong to a given subspace of the tangent space. From the theoretical point of view, subRiemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics. Let us mention geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics and optimal control (motion planning, robotics, nonholonomic mechanics, quantum control) and another to image processing, biology and vision. Even if, nowadays, sub-Riemannian geometry is recognized as a transverse subject, researchers working in different communities are still using quite different language. The aim of these lectures is to collect reference material on sub-Riemannian structures for the use of both researchers and graduate students. Starting from basic definitions and extending up to the frontiers of research, this material reflects the point of view of authors with different backgrounds. The exchanges among the participants of the IHP Trimester and of the CIRM school are reflected here by several connections and interplays between the different chapters. This will hopefully reduce the existing gap in language between the different communities and favour the future development of the field. The notes of Francesco Serra Cassano give an extensive presentation of geometric measure theory in Carnot groups. The first part of the notes discusses differential calculus for maps between Carnot groups in relation with the underlying metric structure. The text then focuses on differential calculus within Carnot groups and uses it to investigate intrinsic regular and Lipschitz surfaces in Carnot groups and their relation with rectifiability. The final section deals with sets of finite perimeter and with the related notions of reduced and minimal boundary. An application to minimal graphs in Heisenberg groups is developed.

vi

Preface

The lecture notes by Nicola Garofalo are a quite comprehensive compendium of results in geometric analysis. In the first part, starting from basic examples and definitions of sub-Riemannian manifolds, length-spaces and Carnot groups, he discusses, in the sub-Riemannian context, Sobolev spaces, BV functions and Sobolev embedding theorems, passing through isoperimetric inequalities. In the second part he discusses classical results in geometric analysis in Riemannian manifolds and the now classical contributions by Folland–Stein, Rothschild–Stein and Nagel–Stein– Wainger in the sub-Riemannian context. Besides giving estimates for the fundamental solution of the heat equation, the goal is to discuss Li–Yau inequalities and curvature dimensional inequalities in the sub-Riemannian case. The lecture notes by Fabrice Baudoin study hypoelliptic diffusion operators from the viewpoint of geometric analysis. The main focus is on sub-Riemannian Laplacians that arise as horizontal Laplacians of a Riemannian foliation. For this kind of operator an extensive theory is developed, with special attention to subelliptic Weitzenböck identities and different applications, from Li–Yau inequalities to spectral gap inequalities and the Bonnet–Myers theorem. The last section is devoted to the analysis of some Kolmogorov-type hypoelliptic diffusion operators and hypocoercive estimates. Davide Barilari Ugo Boscain Mario Sigalotti

Contents 1

Some topics of geometric measure theory in Carnot groups . . . . . . . . . Francesco Serra Cassano 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 An introduction to Carnot groups . . . . . . . . . . . . . . . . . . . 3 Differential calculus on Carnot groups . . . . . . . . . . . . . . . . 4 Differential calculus within Carnot groups . . . . . . . . . . . . . . 5 Sets of finite perimeter and minimal surfaces in Carnot groups . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 20 34 83 108

2

Hypoelliptic operators and some aspects of analysis and geometry of subRiemannian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Nicola Garofalo 1 Sub-Riemannian geometry and hypoelliptic operators . . . . . . . . 123 2 Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3 Fundamental solutions and the Yamabe equation . . . . . . . . . . . 147 4 Carnot–Carathéodory distance . . . . . . . . . . . . . . . . . . . . 160 5 Sobolev and BV spaces . . . . . . . . . . . . . . . . . . . . . . . . 175 6 Fractional integration in spaces of homogeneous type . . . . . . . . 189 7 Fundamental solutions of hypoelliptic operators . . . . . . . . . . . 202 8 The geometric Sobolev embedding and the isoperimetric inequality . 212 9 The Li–Yau inequality for complete manifolds with Ricci ≥ 0 . . . . 216 10 Heat semigroup approach to the Li–Yau inequality . . . . . . . . . 224 11 A heat equation approach to the volume doubling property . . . . . 233 12 A sub-Riemannian curvature-dimension inequality . . . . . . . . . 239 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

3

Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Fabrice Baudoin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 2 Riemannian foliations and their Laplacians . . . . . . . . . . . . . 261 3 Horizontal Laplacians and heat kernels on model spaces . . . . . . 268 4 Transverse Weitzenböck formulas . . . . . . . . . . . . . . . . . . 282 5 The horizontal heat semigroup . . . . . . . . . . . . . . . . . . . . 292 6 The horizontal Bonnet–Myers theorem . . . . . . . . . . . . . . . . 302 7 Riemannian foliations and hypocoercivity . . . . . . . . . . . . . . 308 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Chapter 1

Some topics of geometric measure theory in Carnot groups

Francesco Serra Cassano1 To my parents

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 An introduction to Carnot groups . . . . . . . . . . . . . . . . . 3 Differential calculus on Carnot groups . . . . . . . . . . . . . . 4 Differential calculus within Carnot groups . . . . . . . . . . . . 5 Sets of finite perimeter and minimal surfaces in Carnot groups . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 3 20 34 83 108

1 Introduction These notes aim at illustrating some results achieved in geometric measure theory in Carnot groups. They are an extended version of part of the course Geometric Measure Theory given during the “Geometry, Analysis and Dynamics on subRiemannian Manifolds” trimester, held in Paris in September 2014 at the Institut Henri Poincaré. First of all, I would like to thank the organizers of the trimester Andrei Agrachev, Davide Barilari, Ugo Boscain, Yacine Chitour, Frederic Jean, Ludovic Rifford, and Mario Sigalotti, for their kind invitation, as well to IHP for its backing. It is also a great pleasure for me to acknowledge the help and support of several friends of mine who have made this work possible: first of all, most of the results [email protected] Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123, Trento, Italy. F.S.C. is supported by MIUR, Italy, GNAMPA of the INdAM, University of Trento, Italy and by MAnET Marie Curie Initial Training Network Grant 607643–FP7-PEOPLE-2013-ITN. Part of the work was done while F.S.C. was visiting at the Institut Henri Poincaré, Paris. He wishes to thank the IHP for its hospitality.

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Francesco Serra Cassano

presented here have been achieved jointly with Bruno Franchi and Raul Serapioni. Our long joint work has been always an invaluable source of scientific as well as human enrichment. Without their contribution and friendship, I would never have been able to “attack” this hard subject. I have to thank them also for allowing me to widely quote both our joint papers and their latest ones. I have appreciated the support of other friends and collaborators with whom I have shared fruitful discussions during the last 15 years and whose work is mentioned here: Andrei Agrachev, Giovanni Alberti, Luigi Ambrosio, Zoltan Balogh, Stefano Bianchini, Francesco Bigolin, Ugo Boscain, Luca Capogna, Laura Caravenna, Jih-Hsin Cheng, Giovanna Citti, Bernd Kirchheim, Enrico Le Donne, Nicola Garofalo, Piotr Hajłasz, Pekka Koskela, Bruce Kleiner, Gian Paolo Leonardi, Valentino Magnani, Andrea Malchiodi, Pertti Mattila, Francescopaolo Montefalcone, Michele Miranda jr., Roberto Monti, Pierre Pansu, Fabio Paronetto, Scott Pauls, Andrea Pinamonti, Severine Rigot, Manuel Ritoré, Jeremy Tyson, Davide Vittone, Paul Yang. . . and many others whom I am likely to have left out, for which I apologize. These notes are not intended to provide a thorough—not even a partial—survey of geometric measure theory in Carnot groups, since they deal with only some topics of this field. Anyone interested in an exhaustive overview of this subject, as well as in a full bibliography, sharp statements, and detailed proofs, may refer to the PhD theses of Magnani [153], Monti [181], and Montefalcone [175], as well as to the recent ones of Bigolin [38], Vittone [221], and Pinamonti [196]. For more specific facets we can merely recommend to the reader the monographs [73, 128, 129, 125, 124, 216, 220, 180, 50, 43, 2], and the papers [10, 11, 13, 48, 68, 99, 105, 106, 118, 130, 140, 191, 190, 192, 193, 223] and the references therein. Since these notes especially focus on some issues of geometric measure theory in Carnot groups, they of course leave out many other important topics, for which reference will be made to special in-depth studies. By way of example, coarea formulas (see, for instance, [153, 155]), fractal sets (see, for instance, [217, 23, 27, 30, 209, 81]), variational formulas for minimal surfaces (see, for instance, [69, 178, 133, 134, 56, 213, 187, 206, 54, 114, 115, 179]), isoperimetric sets (see, for instance, [148, 71, 206, 182, 185, 205, 116, 184, 87]), Bernstein’s problem for minimal surfaces (see, for instance, [120, 56, 33, 187, 72, 134, 117]), currents and Rumin’s complex (see, for instance, [11, 106, 207, 208, 21, 110]), curvatures (see, for instance, [50, 2, 136, 199, 34, 3]), mass transportation and optimal transport (see, for instance, [14, 201, 84, 79]), applications to theoretical computer science, geometry of Banach spaces, mathematical models in neuroscience (see, for instance, [52, 64]). Finally we warmly thank Simonetta Rivelli for invaluable support in English.

1 Some topics of geometric measure theory in Carnot groups

3

2 An introduction to Carnot groups 2.1 Definition and first properties. The present subsection is largely taken from [105] and [102] (see also [104]). A Carnot (or stratified) group G of step k (see [85, 135, 186, 129, 190, 219, 220, 43]) is a connected, simply connected Lie group of dimension n whose Lie algebra g admits a step-k stratification, i.e., there exist linear subspaces V1 , . . . ,Vk such that g = V1 ⊕ · · · ⊕ Vk ,

[V1 ,Vi ] = Vi+1 ,

Vk , {0},

Vi = {0} if i > k,

(2.1)

where [V1 ,Vi ] is the subspace of g generated by the commutators [X,Y ] with X ∈ V1 and Y ∈ Vi . Let mi = dim Vi , for i = 1, . . . , k and hi = m1 + · · · + mi with h0 = 0 and, clearly, hk = n. Choose a basis e1 , . . . , en of g adapted to the stratification, that is, such that eh j−1 +1 , . . . , eh j is a base of Vj for each j = 1, . . . , k. Let X = {X1 , . . . , X n } be the family of left-invariant vector fields such that X i = ei . Given (2.1), the subset X1 , . . . , X m1 generates by commutations all the other vector fields; we will refer to X1 , . . . , X m1 as generating vector fields of the group. The exponential map is a one-to-one map from g onto G, i.e., any p ∈ G can be written in a unique way as p = exp(p1 X1 + · · · + pn X n ). Using these exponential coordinates, we identify p with the n-tuple (p1 , . . . , pn ) ∈ Rn and we identify G with (Rn , ·) where the explicit expression of the group operation · is determined by the Campbell–Hausdorff formula (see [43, Section 2.2.2]) and some of its features are described in Proposition 2.3 below. If p ∈ G and i = 1, . . . , k, we put pi = (ph i−1 +1 , . . . , ph i ) ∈ Rm i , so that we can also identify p with [p1 , . . . , pk ] ∈ Rm1 × · · · × Rm k = Rn . The subbundle of the tangent bundle TG that is spanned by the vector fields X1 , . . . , X m1 plays a particularly important role in the theory; it is called the horizontal bundle HG. The fibers of HG are HG x = span {X1 (x), . . . , X m1 (x)},

x ∈ G.

A sub-Riemannian structure is defined on G, endowing each fiber of HG with a scalar product h·, ·i x and with a norm | · | x that make the basis X1 (x), . . . , X m1 (x) Pm1 an orthonormal basis. That is, if v = i=1 vi X i (x) = (v1 , . . . , vm1 ) and w = P 1 Pm1 2 w X (x) = (w , . . . , w ) are in HG , then hv, wi x := m 1 m1 x i=1 i i j=1 v j w j and |v| x := hv, vi x . The sections of HG are called horizontal sections, a vector of HG x is a horizontal vector, while any vector in TG x that is not horizontal is a vertical vector. Each horizontal section is identified by its canonical coordinates with respect to this moving

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Francesco Serra Cassano

frame X1 (x), . . . , X m1 (x). This way, a horizontal section φ is identified with a function φ = (φ1 , . . . , φ m1 ) : Rn → Rm1 . When dealing with two such sections φ and ψ whose argument is not explicitly written, we drop the index x in the scalar product writing hψ, φi for hψ(x), φ(x)i x . The same convention is adopted for the norm. Two important families of diffeomorphisms of G are the so-called intrinsic translations and the intrinsic dilations of G. For any x ∈ G, the (left) translation τx : G → G is defined as z 7→ τx z := x · z. For any λ > 0, the dilation δ λ : G → G, is defined as δ λ (x 1 , . . . , x n ) = (λ α1 x 1 , . . . , λ α n x n ),

(2.2)

where α i ∈ N is called the homogeneity of the variable x i in G (see [86, Chap. 1] ) and is defined as α j = i whenever hi−1 + 1 < j ≤ hi , (2.3) hence 1 = α1 = · · · = α m1 < α m1 +1 = 2 ≤ · · · ≤ α n = k. Example 2.1 (Euclidean spaces). The Euclidean space Rn can be thought of as a (trivial) abelian step-1 Carnot group with respect to its structure of a vector space. Namely, the group law can be defined by the standard sum p + q, if p, q ∈ Rn , and the dilations defined as λ p, if p ∈ Rn and λ > 0. The standard base of the Lie algebra E of Rn is given by the left-invariant vector fields X j = ∂ j , j = 1, . . . , n. All commutator relations in E are trivial. Example 2.2 (Heisenberg group). The simplest example of a Carnot group is provided by the Heisenberg group Hn = R2n+1 . The reader can find an exhaustive introduction to this structure as well as applications in [216]. A point p ∈ Hn is denoted by p = (p1 , . . . , p2n , p2n+1 ) ∈ R2n+1 . For p, q ∈ Hn , we define the group operation given by 2n

1X p · q = * p1 + q1 , . . . , p2n + q2n , p2n+1 + q2n+1 + (pi qi+n − pi+n qi ) + 2 , i=1 and the family of (nonisotropic) dilations δ λ (p) := (λ p1 , . . . , λ p2n , λ 2 p2n+1 )

∀p ∈ Hn , λ > 0.

The standard base of the Lie algebra of Hn , denoted h n , is given by the left-invariant vector fields pn+ j X j = ∂j − ∂2n+1 , j = 1, . . . , n, 2 pj Yj = ∂n+ j + ∂2n+1 , j = 1, . . . , n, 2 T = ∂2n+1 ,

5

1 Some topics of geometric measure theory in Carnot groups

the only nontrivial commutator relations being [X j ,Yj ] = T,

j = 1, . . . , n.

Thus the vector fields X1 , . . . , X n ,Y1 , . . . ,Yn satisfy Hörmander’s rank condition, and Hn is a step-2 Carnot group, the stratification of the Lie algebra of left-invariant vector fields being given by h n = V1 ⊕ V2 , V1 = span {X1 , . . . , X n ,Y1 , . . . ,Yn }

and V2 = span {T }.

We collect in the following proposition some more-or-less elementary properties of the group operation and of the canonical vector fields. Proposition 2.3. The group product has the form x · y = x + y + Q (x, y)

∀x, y ∈ Rn

(2.4)

where Q = (Q1 , . . . , Qn ) : Rn × Rn → Rn and each Qi is a homogeneous polynomial of degree α i with respect to the intrinsic dilations of G defined in (2.2), that is, Qi (δ λ x, δ λ y) = λ α i Qi (x, y) ∀x, y ∈ G. Moreover, (i) Q is anti-symmetric, that is,

Qi (p, q) = −Qi (−q, −p) ∀ p, q ∈ G. (ii) ∀x, y ∈ G,

Q1 (x, y) = · · · = Qm1 (x, y) = 0, Q j (x, 0) = Q j (0, y) = 0 and Q j (x, x) = Q j (x, −x) = 0, Q j (x, y) = Q j (x 1 , . . . , x h i−1 , y1 , . . . , yh i−1 ), Qi (p, q) =

X

if

1h l i where qi, j (x) = ∂Q ∂y j (x, y)| y=0 , so that if h`−1 < j ≤ h` , then qi, j (x) = qi, j (x 1 , . . . , x h l−1 ) and qi, j (0) = 0.

By (2.1), the rank of the Lie algebra generated by X1 , . . . , X m1 is n; hence X = (X1 , . . . , X m1 ) is a system of smooth vector fields satisfying Hörmander’s condition. Finally, it is useful to write G = G1 ⊕ G2 ⊕ · · · ⊕ G k ,

(2.9)

where Gi = exp(Vi ) = Rm i is the ith layer of G and to write p ∈ G as [p1 , . . . , pk ], with pi ∈ Gi . According to this,   p · q = p1 + q1 , p2 + q2 + Q2 (p, q), . . . , pk + q k + Qk (p, q)

∀ p, q ∈ G, (2.10)

where Qi : Rn × Rn → Rm i , i = 2, . . . , k is defined as  Qi (p, q) := Qh i−1 +1 , . . . , Qh i , with Q j : Rn × Rn → R, j = 1, . . . , n being the functions defined in Proposition 2.3. Remark 2.5. An interesting metric characterization of a Carnot group has been obtained in [144]. In the following, by a Carnot group G we will mean a couple (Rn , ·), in exponential coordinates, equipped with a basis of its Lie algebra G , X1 , . . . , X m1 , X m1 +1 , . . . , X n satisfying the previous properties.

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1 Some topics of geometric measure theory in Carnot groups

2.2 Metrics on Carnot groups Definition 2.6. Given a Carnot group G = (Rn , ·), a distance d on G is said to be invariant if (i) d : Rn × Rn → [0, +∞) is continuous with respect to the Euclidean topology; (ii) d is left-invariant with respect the family of left translations and 1-homogeneous with respect to the one of dilations, that is, d(g · p, g · q) = d(p, q),

d(δ λ (p), δ λ (q)) = λd(p, q)

(2.11)

for all p, q, g ∈ G and all λ > 0. A Carnot group G can be equivalently endowed by a homogeneous norm k·k, that is, a continuous function k·k : G ≡ Rn → [0, +∞) (with respect to the Euclidean topology), such that kpk = 0

⇐⇒

p = 0;

kp k = kpk, kδ λ (p)k = λ kpk, ∀p ∈ G, ∀λ > 0; kp · qk ≤ kpk + kqk ∀p, q ∈ G. −1

(2.12) (2.13) (2.14)

From a homogeneous norm k·k, it is possible to induce a distance in G as follows: d(p, q) = d(q−1 · p, 0) = kq−1 · pk

∀p, q ∈ G.

(2.15)

Proposition 2.7 ([43, Corollary 5.1.5]). Let G be a Carnot group. Then two invariant distances d and d˜ on G are equivalent each other, that is, there exists C > 1 such that for all x, y ∈ G, ˜ y) ≤ C d(x, y). C −1 d(x, y) ≤ d(x, Given a Carnot group G and an invariant metric d on it, we will denote by Ud (p,r) and Bd (p,r), respectively, the open and closed ball centered at p and with radius r > 0, that is, Ud (p,r) = {q ∈ G : d(p, q) < r } ,

Bd (p,r) = {q ∈ G : d(p, q) ≤ r } .

(2.16)

Proposition 2.8 ([105, Proposition 2.4]). Let G be a Carnot group endowed with an invariant distance d. Then diam(Bd (p,r)) = 2r

∀p ∈ G, r > 0.

Moreover, if µ is a Radon measure on G, s-homogeneous with respect to the family of dilations for some s > 0, then µ(∂Bd (0,r)) = 0 ∀r > 0.

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Francesco Serra Cassano

Lets us now introduce a couple of relevant invariant metrics on a Carnot group. Definition 2.9. Let G be a Carnot group and let X1 , . . . , X m1 be a generating system of vector fields. An absolutely continuous curve γ : [0,T] → G is a subunit (or admissible) curve with respect to X1 , . . . , X m1 if there exist measurable real functions P c1 (s), . . . , cm1 (s), s ∈ [0,T] such that j c2j ≤ 1 and γ(s) ˙ =

m1 X

c j (s)X j (γ(s)),

for a.e. s ∈ [0,T].

j=1

Definition 2.10. If p, q ∈ G, we define their Carnot–Carathéodory (CC) distance as d c (p, q) := inf {T > 0 : there exists a subunit curve γ with γ(0) = p, γ(T ) = q} . By Chow’s theorem (see [43, Chap. 17]), the set of subunit curves joining p and q is not empty, furthermore d c is a distance on G that induces the Euclidean topology (see [43, Theorem 19.1.3] and Proposition 2.15 below). We stress explicitly that, in general, Carnot–Carathéodory distances are not Euclidean at any scale, and hence not Riemannian. A beautiful proof can be found in [211] (for a more general statement see also [159]). Definition 2.11. Given a metric space (X, d) and a curve γ : [a, b] → X, its total variation is, by definition, Var(γ) =

sup

k−1 X

a ≤t 1 < ··· 0 then Ln (δ λ (E)) = λ Q Ln (E). Finally it holds that

Ln (Bd (p,r)) = Ln (Ud (p,r)) = r Q Ln (Bd (p, 1)) = r Q Ln (Bd (0, 1)).

(2.20)

Proof. Let us simply define the balls Bd ≡ B and Ud ≡ U. Notice that, by Proposition 2.15, Ln (Bd (p,r)) < ∞ for each p ∈ G, r > 0. Let us consider the diffeomorphisms τx : Rn → Rn and δ λ : Rn → Rn . Because of the group structure, it is easy to see that det J (τx ) = 1, det J (δ λ ) = λ Q ,

1 Some topics of geometric measure theory in Carnot groups

11

where J (ψ) denotes the Jacobian matrix associated to a regular map ψ : Rn → Rn . Thus the first two statements easily follow. Moreover,

Ln (B(p,r)) = Ln (τp (B(0,r))) = Ln (δr (B(0, 1))) = r Q Ln (B(0, 1)). Finally, by Proposition 2.8, it follows that Ln (B(p,r)) = Ln (U (p,r)).



2.3 Hausdorff measures in a metric space: Applications to Carnot groups 2.3.1 Hausdorff measures in metric spaces. Let us recall below three different notions of Hausdorff measures in a metric space. The notions of Hausdorff and spherical Hausdorff measures are classical and they are obtained by Carathéodory’s construction as in [83, Section 2.10.2]; the one of centered Hausdorff measure is more recent (see [209]). Let us introduce some notation and notions. Throughout this paper (X, d) will be a separable metric space, and B(x,r) := {y ∈ X : d(x, y) ≤ r } is the closed ball with center a and radius r > 0. The diameter of a set E ⊂ X is denoted as diam(E) := sup {d(x, y) : x, y ∈ E} . If µ is an outer measure in X and A ⊂ X, then the restriction of µ to A is denoted as µ

A(E) = µ( A ∩ E)

if E ⊂ X.

We will assume the following condition on the diameter of closed balls: there exist constants 0 < %0 ≤ 2 and δ0 > 0 such that, for all r ∈ (0, δ0 ) and x ∈ X, diam(B(x,r)) = %0 r. For m > 0, we define α m :=

(2.21)

π m/2 , Γ( m2 + 1)

with Γ being the Euler function and β m := α m %−m 0 .

(2.22)

Remark 2.20. In the case where m is an integer, the constant α m turns out to be equal to the m-dimensional Lebesgue measure of a unit closed ball of Rm . The reason for this normalization is to get equality (2.47) in the Euclidean case.

12

Francesco Serra Cassano

We repeat the well-known definitions of Hausdorff measures mainly to stress their differences from the less-known notion of centered spherical Hausdorff measure. Definition 2.21. Let A ⊂ X, m ∈ [0, ∞), δ ∈ (0, ∞), and let β m be the constant (2.22). m ( A), where (i) Hdm ( A) := limδ→0 Hd,δ [ (X m Hd,δ ( A) = inf β m diam(Ei ) m : A ⊂ Ei , i

) diam(Ei ) ≤ δ .

i

We will call Hdm m-dimensional Hausdorff measure. m ( A), where (ii) Sdm ( A) := limδ→0 Sd,δ [ (X m Sd,δ ( A) = inf β m diam(B(x i ,r i )) m : A ⊂ B(x i ,r i ), i

i

)

diam(B(x i ,r i )) ≤ δ . We will call S m the m-dimensional spherical Hausdorff measure. (iii) m Cdm ( A) := sup Cd,0 (F), F ⊆A

m m m (F) := lim where Cd,0 δ→0+ C d,δ (F), and, in turn, C d,δ (F) = 0 if F = ∅ and for F , ∅, [ (X m Cd,δ (F) = inf β m diam(B(x i ,r i )) m : F ⊂ B(x i ,r i ), i

i

x i ∈ F,

) diam(B(x i ,r i )) ≤ δ .

We will call Cdm the m-dimensional centered Hausdorff measure. The centered Hausdorff measures Cdm were introduced in [209], when X = Rn and d is the Euclidean measure, to estimate more efficiently the Hausdorff dimension of self-similar fractal sets. A detailed study of different types of centered Hausdorff measures and of their properties has been carried out in [81], in the setting of a general metric space. Notice that the set function C0m is not necessarily monotone (see [209, Sect. 4]). It is well known that Hdm and Sdm are metric (outer) measures, being obtained by the classical Carathéodory construction (see [83, 2.10.1] or [168, Theorem 4. 2]). Recall that an outer measure µ on X is said to be metric if µ( A ∪ B) = µ( A) + µ(B)

if dist( A, B) > 0.

1 Some topics of geometric measure theory in Carnot groups

13

Also the measures Cdm are metric measures in any metric space, but this fact, proved in [81], is not as immediate as for H m and S m . It holds that (see [81])

Hdm ≤ Sdm ≤ Cdm ≤ 2m Hdm .

(2.23)

In particular, the three measures Hdm , Sdm , and Cdm are equivalent. Remark 2.22. Notice that if d 1 and d 2 are two equivalent metrics on X, then Hd1 and Hd2 are equivalent, as well as Sd1 and Sd2 , Cd1 and Cd2 . 2.3.2 Hausdorff (or metric) dimension in metric spaces Definition 2.23 (Hausdorff dimension). Let (X, d) be a (separable) metric space and let A ⊆ X. We define the Hausdorff (or metric) dimension of A as the value ) ( Hdim( A) := inf m ∈ [0, +∞) : Hdm ( A) = 0 . Remark 2.24. Since Hdm , Sdm , and Cdm are equivalent, the notion of Hausdorff dimension can be equivalently stated by means of measure Sdm or Cdm . Also, the notion of metric dimension is stable with respect to equivalent metrics on X. Clearly the Hausdorff dimension has the natural properties of monotonicity and stability with respect the countable union: Hdim( A) ≤ Hdim(B) = sup Hdim( Ai )

Hdim(∪∞ i=1 Ai )

if A ⊂ B ⊂ X; if Ai ⊂ X i = 1, 2, . . . .

(2.24) (2.25)

i

To state the definition in other words, Hdim( A) is the unique number (it may be ∞ in some metric space) for which s < Hdim( A) s > Hdim( A)

⇒ ⇒

H s ( A) = ∞, H s ( A) = 0.

(2.26) (2.27)

At the borderline case s = Hdim( A) we cannot have any general nontrivial information about the value H s ( A): all three cases H s ( A) = 0, 0 < H s ( A) < ∞, and H s ( A) = ∞ are admissible. But if for some given A we can find s such that 0 < H s ( A) < ∞, then s = Hdim( A). Let us now propose a simple criterion in order to calculate the Hausdorff dimension in a metric space. Definition 2.25 (Ahlfors regularity). Let (X, d) be a metric space and let µ be a Radon measure on X. We say that the metric measure space (X, d, µ) is Ahlfors regular of dimension Q if a1 r Q ≤ µ(B(x,r)) ≤ a2 r Q for suitable positive constants ai (i = 1, 2).

∀x ∈ X,r > 0

(2.28)

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Francesco Serra Cassano

Theorem 2.26. Let (X, d, µ) be an Ahlfors-regular metric measure space of dimension Q, that is, (2.28) holds. Then Hdim(X ) = Q. For the proof of this result we will need a basic Vitali covering result in metric space, the proof of which can be found in [168]. Theorem 2.27. Let (X, d) be a separable metric space and let F be an arbitrary family of balls (closed or open) in X such that sup {diam(B) : B ∈ F } < ∞. Then there exists a countable, disjoint subfamily F 0 ⊆ F , ∪ B ∈F B ⊆ ∪ B ∈F 0 5B, where 5B denotes the ball with same center as B and radius five times the radius of B. Proof of Theorem 2.26. It suffices to prove that, for a given x 0 ∈ X, 0 < SdQ (B(x 0 ,r)) < ∞ ∀r > 0.

(2.29)

From (2.29), it will follow that Hdim(B(x 0 ,r)) = Q for each r > 0. Therefore, since X = ∪∞ h=1 B(x 0 , h), by (2.25), we can get the desired conclusion. Fix r > 0 and let 0 < δ < r. Consider a covering of closed balls F = {Bi : i ∈ N} of B(x 0 ,r) with Bi = B(yi ,r i ), diam(Bi ) = %0 r i ≤ δ for each i ∈ N. Without loss of generality, we can assume that B(x 0 ,r) ∩ Bi , ∅ for each i ∈ N. From this assumption, we can infer that Bi ⊂ B(x 0 , 2r)

∀i ∈ N.

(2.30)

Applying Theorem 2.27 to the family of balls F , we get that there exists a countable, disjoint subfamily of F , {Bi h :, h = 1, . . . , N }, where N can be a finite integer or infinity, such that N B(x 0 ,r) ⊂ ∪∞ (2.31) i=1 Bi ⊂ ∪h=1 5Bi h . From the definition of the spherical Hausdorff measure, (2.31), (2.28), and (2.30), it follows that

SdQ (B(x 0 ,r)) ≤ βQ (5%0 ) Q

N X i=1



N C1 µ(∪i=1 Bi h )

r iQh ≤ C1

N X

µ(Bi h )

(2.32)

i=1

≤ C1 µ(B(x 0 , 2r)) ≤ C1 a2 2 r

Q Q

< ∞,

1 Some topics of geometric measure theory in Carnot groups

15

where C1 = a1−1 β m (5%0 ) Q . Thus the right-hand inequality of (2.29) follows. To get the left-hand inequality of (2.29), observe that, for a given covering of balls {Bi : i ∈ N} of B(x 0 ,r), with Bi = B(yi ,r i ), diam(Bi ) = %0 r i ≤ δ for each i ∈ N, it follows that 0 < a1 r Q ≤ µ(B(x 0 ,r)) ≤

∞ X

µ(B(yi ,r i )) ≤ C2

i=1

∞ X

βQ diam(Bi ) Q ,

(2.33)

i=1

−1 %−Q . From (2.33) it follows that where C2 = a2 βQ 0 Q 0 < a1 r Q ≤ µ(B(x 0 ,r)) ≤ C2 Sd,δ (B(x 0 ,r))

∀δ > 0.

Taking the limit as δ → 0 in the previous inequality we get 0 < a1 r Q ≤ µ(B(x 0 ,r)) ≤ C2 SdQ (B(x 0 ,r))

∀r > 0.

(2.34) 

Remark 2.28. Notice that, from (2.32) and (2.34), it follows that if (X, d, µ) is an Ahlfors metric measure space of dimension Q, so is the metric measure space (X, d, SdQ ). Finally let us estimate the metric dimension of a Lipschitz curve in a metric space, which is the simplest example of a 1-dimensional regular submanifold in a metric space. More precisely, the following theorem holds. Theorem 2.29. Suppose that γ : [a, b] → (X, d) is a Lipschitz curve and let Γ = γ([a, b]). Then

H1d (Γ) ≤ Sd1 (Γ) ≤ Cd1 (Γ) ≤ Var(γ) < +∞, and equality holds if γ is injective. Proof. We have only to show that

Cd1 (Γ) ≤ Var(γ).

(2.35)

Indeed, from [17, Theorem 4.4.2], it follows that

H1d (Γ) = Var(γ)

if γ is injective.

Thus, from (2.23) and (2.35), we get the desired conclusion. Let us prove (2.35). By a reparametrization of γ (see [17, Theorem 4.2.1]), we can assume that a = 0, b = Var(γ) with metric derivative of γ, | γ| ˙ = 1 a.e. on [a, b]. In particular, observe that γ is 1-Lipschitz and Var(γ, [t, s]) = s − t

∀ a ≤ t ≤ s ≤ b.

(2.36)

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Francesco Serra Cassano

Given δ > 0, choose N ∈ N such that Var(γ)/N < δ and let h := Var(γ)/N and Ji := [i h, (i+1) h], i = 0, . . . , N −1. Let Bi := B(pi , h/%0 ) with pi = γ((2i+1) h/2), i = 0, . . . , N − 1, where %0 is the constant in (2.21). Then, by (2.21), diam(Bi ) = h,

i =, 0, . . . , N − 1

(2.37)

and N −1 Γ ⊆ ∪i=1 Bi .

(2.38)

Indeed, notice that, by (2.36), for each i = 0, . . . , N − 1, !! 2i + 1 2i + 1 h h d γ(t), γ h ≤ t − h ≤ ≤ 2 2 2 % 0

∀ t ∈ Ji .

Thus γ(Ji ) ⊂ Bi for each i = 0, . . . , N − 1, and (2.38) follows. Therefore, from (2.38) and (2.37), we get that, for each E ⊆ Γ, 1 1 Cd,δ (E) ≤ Cd,δ (Γ) ≤

N −1 X

diam(Bi ) = Var(γ)

∀ δ > 0.

(2.39)

i=0

Passing to the limit as δ → 0 in (2.39), we get that 1 Cd,0 (E) ≤ Var(γ) ∀ E ⊆ Γ.

Finally, taking the supremum over all subsets E ⊆ Γ in the previous inequality, (2.35) follows.  2.3.3 Applications to Carnot groups. Now we are going to apply Hausdorff measures and metric dimension issues to the case when X = G is a Carnot group equipped with an invariant distance d. Let G be a Carnot group equipped with an invariant distance d. Let us first observe that (2.21) is satisfied with %0 = 2 by Proposition 2.8. Moreover, because of the invariance of distance d, all three Hausdorff measures are left-invariant with respect to the family of left translations τx (x ∈ G), and m-homogeneous with respect to the family of 1-parameter dilations δ λ (λ > 0), that is, if µ = Hdm , or µ = Sdm , or µ = Cdm , then, for each A ⊂ G, µ(τx ( A)) = µ( A),

µ(δ λ ( A)) = λ m µ( A)

∀x ∈ G, λ > 0.

(2.40)

We denote by H m the m-dimensional Hausdorff measure obtained from the Euclidean distance in Rn ' G, by Hcm the m-dimensional Hausdorff measure obtained m the m-dimensional Hausdorff measure obfrom the distance d c in G, and by H∞ m , and C m tained from the distance d ∞ in G. Analogously, S m , C m , Scm , Ccm , S∞ ∞ denote the corresponding spherical and centered Hausdorff measures. As an immediate consequence of Theorem 2.26 and (2.20), we get the calculation of the metric dimension of a Carnot group, proved for the first time in [174].

1 Some topics of geometric measure theory in Carnot groups

17

Theorem 2.30. Let G = (Rn , ·) be a Carnot group endowed with an invariant metric d. Then Hdim(G) = Q, where Q denotes the homogeneous dimension of G; see (2.19). Remark 2.31. Observe that n ≤ Q and the equality holds iff G is a step-1 Carnot group, that is, G ≡ Rn is isomorphic to the Euclidean space Rn . In the other case, namely when G is a Carnot group of step k ≥ 2, the topological dimension n is strictly less than the metric dimension Q. For instance, in the case of the Heisenberg group Hn , Hdim(Hn ) = Q = dim V1 + 2 dim V2 = 2n + 2. This gap is more evidence that CC geometry differs from the Euclidean one and a new feeling for the CC geometry requirements. In particular, we should change the Euclidean feeling that a regular submanifold of a Carnot group still keeps both the topological and metric dimensions equal. We will discuss this issue later in Section 4. Proof of Theorem 2.30. From (2.20) it follows that the metric measure space (G, d, Ln ) is Ahlfors regular of dimension Q. Thus, from Theorem 2.26 we get the desired conclusion.  We are now going to compare the Hausdorff measures HQ , SdQ , and CdQ on a d given Carnot group G = (Rn , ·) of homogeneous dimension Q, equipped with an invariant metric d. We have showed that they are Radon measures on G and, from the invariance by left translations (see (2.40)), they are Haar measures of the group like the measure Ln , as well. It is well known that, due to a general result in metric spaces as far as uniformly distributed measures, each of them differs from Ln by only a constant (see [168, Theorem 3.4]). Assuming now as reference measure the spherical Hausdorff measure SdQ , we compare it with HQ and CdQ . d Theorem 2.32. Let G = (Rn , ·) be a Carnot group of homogeneous dimension Q, equipped with an invariant distance d. (i) ([202, Proposition 2.1]) SdQ (B) = βQ diam(B) Q for each ball B ⊂ G. (ii) ([202, Proposition 2.3]) SdQ = Cd HQ , where Cd is the isodiametric constant, d that is,   SdQ ( A)   . : 0 < diam( A) < +∞ (2.41) Cd := sup    βQ diam( A) Q   (iii) CdQ (B) = βQ diam(B) Q for each ball B ⊂ G.

18

Francesco Serra Cassano

Proof. (iii) For the sake of simplicity, define SdQ = S Q and CdQ = S Q . From (i) and the definition of CdQ , we get at once

S Q (B) = βQ diam(B) Q ≤ C Q (B) for each ball B ⊂ G. Let µ be a normalized Haar measure on G such that µ(B) = βQ diam(B) Q for each closed ball B. In order to accomplish the proof it will suffice to prove that C Q (U) ≤ µ(U), (2.42) for each open ball U. Indeed, since C Q is a left-invariant Radon measure, diam(U) = diam(U), from Proposition 2.8, if B = U with U an open ball; then

C Q (B) = C Q (U), and, by (2.42), we get the desired inequality. Let us prove (2.42). Given an open ball U, E ⊂ U, and δ > 0, let ( ) F := B(x,r) : x ∈ E, B(x,r) ⊂ U, diam(B(x,r)) < δ . Since U is open, F is a fine covering of E, and, because C Q is a doubling measure, by a Vitali-type covering result we can assume there exists a countable, disjoint family of closed balls (Bi )i ⊂ F such that

C Q (E \ ∪∞ i=1 Bi ) = 0.

(2.43)

Observe now that

CδQ (∪∞ i=1 Bi ) ≤

∞ X

βQ diam(Bi ) Q = µ(∪∞ i=1 Bi ) ≤ µ(U),

(2.44)

i=1

and, by (2.43), Q ∞ Q ∞ CδQ (E \ ∪∞ i=1 Bi ) ≤ C0 (E \ ∪i=1 Bi ) ≤ C (E \ ∪i=1 Bi ) = 0.

(2.45)

On the other hand, by (2.44) and (2.45), Q ∞ CδQ (E) ≤ CδQ (E \ ∪∞ i=1 Bi ) + Cδ (∪i=1 Bi ) ≤ µ(U) ∀ E ⊂ U, δ > 0.

(2.46)

Taking first the limit as δ → 0 in (2.46) and then the supremum on all sets E ⊆ U, (2.42) follows.  Corollary 2.33. Under the same assumptions as Theorem 2.32, it holds that SdQ =CdQ .

1 Some topics of geometric measure theory in Carnot groups

19

Proof. Since there exists a positive constant β such that SdQ = β CdQ , and SdQ (B) =  CdQ (B) on all balls, we can infer that β = 1. Remark 2.34. Notice that, by choosing βQ = 2−Q Ln (Bd (0, 1)), we can get SdQ = CdQ = Ln . We also point out that the agreement between Sdm (E) and Cdm (E) may not occur for a dimension 1 < m < Q and for a general invariant metric d, even if E ⊂ G is very regular (see Remark 4.31). The constant Cd in (2.41) is related to the so-called isodiametric problem. Let us recall that the classical isodiametric problem in the Euclidean space Rn says that a ball maximizes the volume among all sets with prescribed diameter,

Ln ( A) ≤ β n diam( A) n , where β n is the constant defined in (2.22) (that is, the volume of a unit ball of Rn ) with %0 = 2 (see, for instance, [8, Proposition 2.52]). This gives nontrivial information about the geometry of the Euclidean space and about the relation between the Euclidean metric and the Lebesgue measure. In particular, a well-known consequence of the isodiametric inequality is the following agreement between the ndimensional Lebesgue measure and the n-dimensional Hausdorff measure defined with respect to the Euclidean distance:

Ln = H n .

(2.47)

More generally, when working on analysis on metric spaces it is natural to ask in what kind of setting one can have some, and what type of, isodiametric inequality, and what kinds of properties can be deduced from this information. In the setting of Carnot groups, the isodiametric problem has been studied in [202] and [149]. More precisely, [202] studied whether the sharp isodiametric inequality holds, or equivalently, whether balls realize the supremum in the right-hand side of (2.41) in Carnot groups equipped with some specific invariant distances. Observe that in this case Cd = 1 by Theorem 2.32(i). This amounts to saying that the following sharp isodiametric inequality holds:

SdQ ( A) ≤ βQ diam( A) Q

∀A ⊂ G,

(2.48)

or, equivalently, from Theorem 2.32(ii),

SdQ = HQ . d In particular, it was proved that, given a nonabelian Carnot group G, one can always find some invariant distance on G, namely the d ∞ distance as well as some CC distances, for which the sharp isodiametric inequality (2.48) does not hold (see [202,

20

Francesco Serra Cassano

Theorems 3.4, 3.5 and 3.6]). Moreover, an interesting relationship with some topics of geometric measure theory in Carnot groups was established. Indeed, as a consequence of those results, the pure Q-unrectifiability of Carnot groups can be obtained (see [201, Theorem 4.1] and Definition 4.98 below), which we will discuss later. Note that it is not difficult to see that one can always find sets achieving the supremum in (2.41), called isodiametric sets (see [201, Theorem 3.1]). In [149], for the case of the Heisenberg group Hn equipped with its CC distance d c , the regularity of the isodiametric sets has been studied, as well as their characterization under some symmetry assumption. The characterization of a general isodiametric set in a Heisenberg group is still open. Finally let us point out the paper [2] about the study of volume measure in subRiemannian manifolds by means of Hausdorff measures and [45] concerning the spherical Hausdorff measure and the Popp measure in sub-Riemannian manifolds.

2.4 Carnot groups as tangent spaces to sub-Riemannian manifolds. An important feature of Carnot groups is that they appear as “tangent spaces” to subRiemannian manifolds in an appropriate sense. Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian case, they are indeed vector spaces, that is, abelian groups with dilations. A general account of the many facets and contributions in this direction is far beyond the aim of these notes and we refer to [174, 35, 1, 180, 143]. We point out only that the understanding of many Riemannian geometric properties comes from the fact that the metric tangents of a Riemannian manifold are Euclidean spaces, and Euclidean geometry is quite well studied. Thus a good understanding of the geometry of Carnot groups could provide better knowledge of the geometry of sub-Riemannian structures.

3 Differential calculus on Carnot groups This section is taken from [105] and [153, Chapter 3]. We refer to [160] for a generalization of the content of this section.

3.1 Pansu differentiable maps between Carnot groups and some of their characterizations. The following definitions and results about intrinsic

1 Some topics of geometric measure theory in Carnot groups

21

differentiability in Carnot groups are basically due to Pansu ([190]), or are inspired by his ideas. The notion of P-differentiability for functions acting between Carnot groups was introduced by Pansu in [190]. Definition 3.1. Let G1 , G2 be Carnot groups, with homogeneous norms k · k1 , k · k2 and dilations δ1λ , δ2λ . We say that L : G1 → G2 is H-linear, or is a homogeneous homomorphism, if L is a group homomorphism such that L(δ1λ g) = δ2λ L(g)

∀g ∈ G1 and λ > 0.

Let us recall the following characterization of an H-linear map between Carnot groups (see also [153, Proposition 3.1.3 and Theorem 3.1.12]). We thank Nicolussi Golo and Le Donne for pointing out to us a more complete statement of this characterization than a previous version, as well thanking Le Donne for its proof and other suggestions on the topic. Theorem 3.2. Let (Gi , k · ki ) (i = 1, 2) be two Carnot groups. Denote by gi (i = 1, 2) their Lie algebras and by expi : gi → Gi (i = 1, 2) their exponential maps. Let g1 = V1 ⊕ · · · ⊕ Vk , G1 = V1 ⊕ V2 ⊕ · · · ⊕ Vk ,

g2 = W 1 ⊕ · · · ⊕ W l , G2 = W1 ⊕ W2 ⊕ · · · ⊕ Wl

be the stratifications defined according to, respectively, (2.1) and (2.9). Let L : G1 → G2 be a homomorphism. Then the following are equivalent: (i) L is H-linear; (ii) L : (G1 , k · k1 ) → (G2 , k · k2 ) is C∞ and Lipschitz; (iii) L is C∞ and it satisfies the contact property, that is, dL(V1 ) ⊂ W1 , where dL : T (G1 ) → T (G2 ) denotes the differential on tangent bundles; (iv) L is C∞ and L(V j ) ⊆ W j for each j. Let us recall some well-known facts about Lie groups and homomorphisms. Lemma 3.3. Under the same assumptions as Theorem 3.2, we have the following results. (i) Let ϕ : G1 → G2 be a continuous homomorphism. Then ϕ is C∞ . (ii) Let ϕ : G1 → G2 be a C∞ homomorphism. Then the differential map dϕ : g1 → g2 is a Lie algebra homomorphism, that is, it is a linear map such that dϕ([X,Y ]) = [dϕ(X ), dϕ(Y )] for each X, Y ∈ g1 . Moreover, the map ϕ˜ := exp−1 2 ◦ϕ◦exp1 : g1 ≡ Te (G1 ) → g2 ≡ Te (G2 ) agrees with the differential of ϕ at e, dϕ(e) : Te (G1 ) → Te (G2 ). In particular, ϕ˜ : g1 → g2 is also a Lie algebra homomorphism.

22

Francesco Serra Cassano

(iii) Let ϕ : G1 → G2 be H-linear. Then ϕ is a continuous homomorphism. Remark 3.4. By our notation of identifying a point in a Carnot group by means of exponential coordinates, by Lemma 3.3(ii) and (iii), it follows that we can identify each H-linear map L : G1 = (Rn , ·) → G2 = (Rm , ∗) with L˜ = dL(e) : g1 → g2 . In particular, L : Rn → Rm turns out to be a linear map. Remark 3.5. Observe that the continuity of a homomorphism ϕ is a necessary assumption in Lemma 3.3(i) in order to get that ϕ is C∞ . Otherwise the conclusion may fail. For instance, let G1 = G2 = R be meant as trivial Carnot groups (see Example 2.1). Let us consider R as a vector space on rational numbers Q. Let (vα )α be a basis of R over Q. (It is well known that the basis is uncountable and the axiom of choice needs to say that one such exists.) Choose an element α. ¯ Consider the only Q-linear map from ϕ : R → R such that ϕ(vα¯ ) := 1,

ϕ(vα¯ ) := 0 if α , α. ¯

Then, by construction, it is easy to see that ϕ : R → R is a homomorphism, that is, ϕ(x + y) = ϕ(x) + ϕ(y) for each x, y ∈ R and it is discontinuous. Proof of Lemma 3.3. (i) The fact that ϕ is C∞ follows by [224, Theorem 3.39]. (ii) Since ϕ is C∞ , by a classical result about homomorphisms between Lie groups (see, for instance, [43, Theorem 2.1.50(ii)]), it follows that dϕ : g1 → g2 is a Lie algebra homomorphism. On the other hand, because Gi ((i = 1, 2) are connected and simply connected Lie groups, we can apply [43, Theorem 2.1.61] and, for all X ∈ g1 , Y = ϕ(X ˜ ) ∈ g2 is the only left-invariant vector field on G2 defined by Y (ϕ(g)) = dϕ(g)(X (g))

∀ g ∈ G1 .

(3.1) 

(iii) See [153, Proposition 3.1.8].

Proof of Theorem 3.2. (i) ⇐⇒ (ii): It follows by Lemma 3.3 and [153, Proposition 3.1.8]. (ii)=⇒(iii): Let v ∈ V1 and let us consider the C1 curve γ : R → G2 , γ(t) :=  L exp1 (tv) . Then, by Lemma 3.3(i) and (3.1),  γ(t) = L exp1 (tv) = exp2 (dL(e)(tv)) = exp2 (t dL(e)(v))

∀t ∈ R.

Since, by assumption, γ : R → (G2 , k · k2 ) is also Lipschitz, by Proposition 2.12, it follows that it is horizontal. Thus, γ(0) ˙ = dL(e)(v(γ(0)) = dL(e)(v(e)) ∈ Hγ(0) G2 ≡ W1 .

1 Some topics of geometric measure theory in Carnot groups

23

(iii)=⇒(iv): By induction on i and since dL is a Lie algebra homomorphism (see Lemma 3.3(ii)), we get dL(Vi+1 ) = dL([V1 ,Vi ]) = [dL(V1 ), dL(Vi )] ⊂ [W1 ,Wi ] = Wi+1 . (iv)=⇒(i): We have to show only that L is homogeneous in order to prove that L is Hlinear. By Remark 3.4, it suffices to prove that dL(e) : g1 ≡ Te (G1 ) → g2 ≡ Te (G2 ) is homogeneous. Observe that, by assumption, dL(e)(Vj ) ⊂ W j and, by linearity, dL(e)(λ j v) = λ j dL(e)(v)

∀ v ∈ Vj , λ > 0. 

Therefore we get the desired conclusion.

Definition 3.6. We say that f : A ⊂ G1 → G2 is P-differentiable at g0 ∈ A if there is an H-linear function d P f g0 : G1 → G2 such that   −1  k d P f g0 (g0−1 · g) · f (g0 ) −1 · f (g)k2 = o kg0−1 · gk1 ,

as kg0−1 · gk1 → 0,

where o(t)/t → 0 as t → 0+ . The H-linear function d f g0 is called the P-differential of f in g0 . The fundamental result where P-differentiability applies is the so-called Pansu– Rademacher theorem for Lipschitz functions between Carnot groups. Theorem 3.7 (Pansu, [190]). Let G1 = (Rn , ·) and G2 = (Rm , ∗) be two Carnot groups equipped with two CC metrics denoted by, respectively, d 1 and d 2 . Let f : A ⊂ (G1 , d1 ) → (G2 , d 2 ) be a Lipschitz continuous function with A an open set. Then f is P-differentiable at p for Ln -a.e. p ∈ A. Remark 3.8. An extension of Pansu–Rademacher theorem for a Lipschitz continuous map f : A ⊂ (G1 , d 1 ) → (G2 , d 2 ) when A is supposed only to be measurable was proved in [153] and [218]. A Pansu–Rademacher-type theorem has also been proved in [186, Theorem 3.2] for Lipschitz continuous functions f : (Rn , d c ) → R, where d c is a CC distance defined by a C∞ family of vector fields with a particular structure, which includes the case of Carnot groups. Remark 3.9. An interesting survey about possible notions of differentiability in several different settings can be found in [212]. Moreover, counterexamples to the Rademacher-type theorem are also known for Lipschitz functions not acting between Carnot groups (see, for instance, [138]).

24

Francesco Serra Cassano

Given two Carnot groups (Gi , k·ki ), i = 1, 2, where k·ki is a homogeneous norm on Gi , denote by L H (G1 , G2 ) the set of H-linear functions L : G1 → G2 endowed with the norm kLkL H (G1,G2 ) := sup{kL(p)k2 : p ∈ G1 , kpk1 ≤ 1}. When G1 ≡ G and G2 ≡ R, we simply define LG = L H (G, R) and we will call a map L ∈ LG simply G-linear. 1 (Ω, G ) the set of continuous functions If Ω is an open set in G1 , we denote as CH 2 f : Ω → G2 such that d P f : Ω → L H (G1 , G2 ) is continuous. In the following we will deal with maps f : Ω ⊂ G → R, that is, G1 = G 1 (Ω, G ) as C 1 (Ω) and the Pansu and G2 = R. In this case we simply denote CH 2 G differential as d G := d P f : G → R. Given a basis X1 , . . . , X n , all G-linear maps are represented as follows. Proposition 3.10. A map L : G → R is G-linear if and only if there exists a = Pm1 (a1 , . . . , a m1 ) ∈ Rm1 such that, if x = (x 1 , . . . , x n ) ∈ G, then L(x) = i=1 ai x i . Notice that, for a function f : Ω ⊂ G → R, P-differentiability at x 0 ∈ Ω simply means that there is a G-linear map L : G → R such that lim

x→x 0

f (x) − f (x 0 ) − L(x −1 0 · x) d(x, x 0 )

=0

(see also [140]). Remark 3.11. The above definition is equivalent to the following one: there exists a homomorphism L from G to (R, +) such that lim

λ→0+

f (τx0 (δ λ v)) − f (x 0 ) = L(v) λ

uniformly with respect to v belonging to compact sets in G. In particular, L is unique and we shall write L = d G f (x 0 ). Notice that this definition of a differential depends only on G and not on the particular choice of the canonical generating vector fields. Indeed, any two Carnot–Carathéodory distances induced by different choices of (equivalent) scalar products in HG are equivalent as distances. Definition 3.12. If Ω is an open set in G, we shall denote by C1G (Ω, HG) the set of all sections φ of HG with canonical coordinates φ j ∈ C1G (Ω) for j = 1, . . . , m1 . Remark 3.13. We recall that C1 (Ω) ⊂ C1G (Ω) and that the inclusion may be strict; for an example see [102, Remark 5.9]. We point out that a function f ∈ C1G (Ω) may be very irregular from a Euclidean point of view (see [139]).

1 Some topics of geometric measure theory in Carnot groups

25

We say that f is differentiable along X j , j = 1, . . . , m1 at x 0 if the map λ 7→ f (τx0 (δ λ e j )) is differentiable at λ = 0, where e j is the jth vector of the canonical basis of Rn . Once a generating family of vector fields X1 , . . . , X m1 is fixed, then for any function f : G → R for which the partial derivatives X j f exist, we define, the horizontal gradient of f , denoted by ∇G f , as the horizontal section ∇G f :=

m1 X

(X i f )X i

i=1

whose coordinates are (X1 f , . . . , X m1 f ). Moreover, if φ = (φ1 , . . . , φ m1 ) is a horizontal section such that X j φ j ∈ L 1loc (G) for j = 1, . . . , m1 , we define divG φ as the real-valued function divG (φ) := −

m1 X

X ∗j φ j =

j=1

m1 X

Xj φj

j=1

(see also Section 3.4.1). Remark 3.14. The notation we have used for the gradient in a group is partially imprecise, indeed ∇G f really depends on the choice of the basis X1 , . . . , X m1 . If we P P choose a different base, say Y1 , . . . ,Ym1 , then in general i (X i f )X i , i (Yi f )Yi . Only if the two bases are one orthonormal with respect to the scalar product induced by the other, we have X X (X i f )X i = (Yi f )Yi . i

i

On the other hand, the notation divG used for the divergence is correct. Indeed, divG is an intrinsic notion and it can be computed using the previous formula for any fixed generating family. Finally, if x = (x 1 , . . . , x n ) ∈ Rn ≡ G and x 0 ∈ G are given, we set π x0 (x) =

m1 X

x j X j (x 0 ).

(3.2)

j=1

The map x 0 → π x0 (x) is a smooth section of HG. Proposition 3.15. If f : Ω ⊂ G → R is Pansu-differentiable at x 0 , then it is differentiable along X j at x 0 for j = 1, . . . , m1 , and d G f (x 0 )(v) = h∇G f , π x0 (v)i x0 .

(3.3)

For a proof see [186, Remark 3.3]. The following proposition can be proved via an approximation argument as in [102, Proposition 5.8].

26

Francesco Serra Cassano

Proposition 3.16. A continuous function f : Ω → R belongs to C1G (Ω) if and only if its distributional derivatives X j f are continuous in Ω for j = 1, . . . , m1 . 1 (Ω, G ), similar to the one of A nontrivial characterization for mappings f ∈ CH 2 Proposition 3.16, has been recently carried out in [160, Theorem 1.1]. In particular, as a by-product of this characterization we get that each continuously P-differentiable function between Carnot groups is locally Lipschitz.

Theorem 3.17 ([160, Corollary 5.8]). Given two Carnot groups (Gi , k·ki ), i = 1, 2, where k·ki is a homogeneous norm on Gi , let Ω ⊂ G1 be an open set and let f ∈ C1H (Ω, G2 ). Then f : Ω ⊂ (G1 , k·k1 ) → (G2 , k·k2 ) is locally Lipschitz.

3.2 Area formula between Carnot groups. The content of this section relies on [153, Chapter 4] (see also [10, 193, 223]). An important tool in Euclidean geometric measure theory is the so-called area formula for Lipschitz maps, which turns out to be true also for Lipschitz maps between Carnot groups. Definition 3.18 (H-Jacobian). Given two Carnot groups (Gi , d i ), i = 1, 2, endowed with an invariant distance d i , let L : G1 → G2 be an H-linear map. Let Q denote the homogeneous dimension of G1 . The Jacobian of L is defined by JQ (L) :=

HQ (L(B1 )) d2 HQ (B1 ) d1

,

where B1 denotes the unit open ball in (G1 , d 1 ). Theorem 3.19 (Area formula, [153, Theorem 4.3.4]). Let f : A ⊂ (G1 , d 1 ) → (G2 , d 2 ) be a Lipschitz continuous function, where A is measurable. Then Z Z Q JQ (d P f (x))d Hd1 (x) = N ( f , A, y)d HQ (y), d2 A

f (A)

where N ( f , A, y) denotes the multiplicity function of f , that is, the number of elements of set f −1 (y). The area formula turns out to be a useful tool in order to study low-dimensional surfaces in Carnot groups (see, for instance, Section 4.5) as well as for proving some unrectifiability results for Carnot groups (see Section 4.8).

3.3 Whitney theorem for maps between Carnot groups. It is well known that a useful tool for the approximation of a function in a Euclidean setting turns out to be Whitney’s extension theorem. In the setting of Carnot groups we mean the

1 Some topics of geometric measure theory in Carnot groups

27

possibility to extend a function f : F ⊂ G1 → G2 to a continuously P-differentiable map f˜ : G1 → G2 , provided that F is a closed set, and a suitable notion of differentiability on F. There is a positive answer if G2 = Rk . Indeed, the following theorem holds. Theorem 3.20 (Whitney extension theorem, [105, Theorem 5.2]). Let F ⊂ G be a closed set, and let f : F → R, k : F → HG be, respectively, a continuous real function and a continuous horizontal section. We set R(x, y) :=

f (x) − f (y) − hk (y), πy (y −1 · x)iy , d(y, x)

where πy is the section defined in (3.2), and, for every compact set K ⊂ F, % K (δ) := sup{|R(x, y)| :

x, y ∈ K, 0 < d(x, y) < δ}.

Assume % K (δ) → 0 as δ → 0 for every compact set K ⊂ F. Then there exist f˜ : G → R, f˜ ∈ C1 (G) such that G

f˜|F = f ,

∇G f˜|F = k.

To our knowledge, the only case with a target space G2 (nonabelian) Carnot group that has been studied, is when G1 = R and G2 = Hn E where E denotes the Engel group (see Example 4.7). More precisely, Speight [215] was able to give a positive answer for a curve γ : F ⊂ R → Hn horizontal by proving a C1 extension theorem like Theorem 3.20 (see [215, Proposition 2.7]) and by proving a Lusin approximation result for absolutely continuous horizontal curves by means of C1 horizontal curves (see [215, Theorem 1.2]). Meanwhile he disproved a Lusin approximation result in the setting of an Engel group by means of a very interesting example (see [215, Theorem 3.2]). Finally we have been informed that Hajłasz is also working on similar arguments.

3.4 Sobolev spaces and the Poincaré inequality 3.4.1 Vector fields. Consider a family X of vector fields X = (X1 , . . . , X m ) ∈ Lip (Rn ; Rn ) m . Since we are dealing with local properties, for the sake of simplicity, we assume X1 , . . . , X m are bounded in Rn . This assumption gives a simpler form to some statements below. Later on, when the vector fields will be associated with a Carnot group structure, we shall drop the boundedness assumption. This will not yield a contradiction or lack of coherence, since the local estimates we are dealing with for groups are easily extended to the whole space by translations and dilations.

28

Francesco Serra Cassano

As usual we shall identify vector fields and differential operators. If X j (x) =

n X

j

ci (x)∂i ,

j = 1, . . . , m,

i=1

we define the m × n matrix j

C(x) = [ci (x)] i=1, ..., n .

(3.4)

j=1, ..., m

In the case of a Carnot group G, the vector fields X1 , . . . , X m will agree with a system of generating vector fields X1 , . . . , X m1 of the group. We shall denote by X ∗j the operator formally adjoint to X j in L 2 (Rn ), that is, the operator which for all φ, ψ ∈ C0∞ (Rn ) satisfies Z Z φ(x)X j ψ(x) dx = ψ(x)X ∗j φ(x) dx. Rn

Rn

Moreover, if f ∈ L 1loc is a scalar function and φ ∈ (L 1loc ) m is an m-vector-valued function, we define the X-gradient and X-divergence as the following distributions: X f := (X1 f , . . . , X m f ),

div X (φ) := −

m X

X ∗j φ j .

j=1

Definition 3.21. Let Ω be an open subset of Rn . One can define the Sobolev space 1, p W X (Ω), 1 ≤ p ≤ ∞ associated with the family X as the space of all functions with P finite norm kukW 1, p = kuk p + k Xuk p , where |Xu| 2 = |X j u| 2 and the derivatives X X j u are understood in the sense of distributions. The L p -norms should be meant with respect to Lebesgue measure. Throughout this paper, if E ⊂ Rn , both |E| and Ln (E) denote its Lebesgue measure. Analogously, if µ is a measure in a set X, we write µ(E) or |E| µ for the µ-measure of the set E ⊂ X. 3.4.2 Sobolev spaces associated with vector fields 1, p

Proposition 3.22. Endowed with its natural norm, W X (Ω), 1 ≤ p ≤ ∞ is a Banach space, reflexive if 1 < p < ∞. Moreover, W X1,2 (Ω) is a Hilbert space. Another way to define the space for 1 ≤ p < ∞ is to take the closure of C ∞ functions in the above norm. As in the Euclidean case, the two approaches are equivalent. This was obtained independently in [100] and [118]. The method, however, goes back to Friedrichs [113]. The result can be stated as follows.

1 Some topics of geometric measure theory in Carnot groups

29

Theorem 3.23. Let X be a family of Lipschitz continuous vector fields. Then, if 1 ≤ p < ∞, we have 1, p

1, p

C∞ (Ω) ∩ W X (Ω) is dense in W X (Ω). If in addition ∂Ω is a smooth manifold, then 1, p

C∞ (Ω) is dense in W X (Ω). The following definition is natural, keeping in mind Theorem 3.23. Definition 3.24. Let X be a family of Lipschitz continuous vector fields. Then, if 1 ≤ p < ∞, we put ◦ 1, p

W X (Ω) := D (Ω)

1, p

W X (Ω)

.

3.4.3 Poincaré inequality. The starting point is the following Poincaré inequality in Carnot groups ([219, 135]). Theorem 3.25. Let U = Ud c (x,r (U)) be a Carnot–Carathéodory ball in G, that is, the (open) ball according to (2.16) with d = d c the CC distance. If f : G → R is a continuously differentiable function, we denote by f U the average of f in U. Then for any p ∈ [1, ∞) there exists C = C(p, G) independent of f and U such that Z Z | f (x) − f U | dx ≤ C r (U) |∇G f (x)| dx. (3.5) U

U

Thanks to the Poincaré inequality (3.5), by [95] and [112] it follows that we can represent a continuously differentiable function f in terms of a fractional integral of ∇G f . We have Theorem 3.26. Let U = Ud c (x,r (U)) be a Carnot–Carathéodory ball in G, that is, the (open) ball according to (2.16) with d = d c the CC distance, and let f : G → R be a continuously differentiable function. Then there exist τ > 1 and C > 0, both independent of U and f , such that Z |∇G f (y)| dy (3.6) | f (x) − f U | ≤ C d(x, y) Q−1 τU for x ∈ U, where τU is the ball concentric with U of radius τr (U) and Q denotes the homogeneous dimension of G. We can now apply a Hardy–Littlewood–Sobolev-type inequality in Carnot groups ([85]) that can be stated as follows.

30

Francesco Serra Cassano

Theorem 3.27. Suppose 0 < α < Q, and denote by Iα the Riesz potential in G of order α, i.e., for x ∈ G, put Z |g(y)| Iα g(x) := dy. d(x, y) Q−α G Let p be fixed, 1 ≤ p < Q/α, and set q−1 := p−1 − (α/Q): i) If f ∈ L p (G), then Iα f (x) < ∞ for a.e. x ∈ G. ii) Iα is a (p, q)-weak-type sublinear map, i.e., there exists C p > 0 such that !q k f kL p for λ > 0. |{x ∈ G : Iα (x) > λ}| ≤ C p λ iii) If p > 1, then there exists C p > 0 such that kIα f k L q ≤ C p k f k L p . Combining Theorems 3.26 and 3.27, we obtain pQ Proposition 3.28. Let p be fixed, 1 ≤ p < Q, and set q := Q−p . Moreover, let U = Uc (x,r (U)) be a Carnot–Carathéodory ball in G and let f : G → R be a continuously differentiable function. Then there exist c = c(p, G) and τ > 1, both independent of f and U, such that

i) if p > 1, then ! 1/q

Z q

| f (x) − f U | dx U

! 1/p

Z ≤ c r (U)

p

τU

|∇G f (x)| dx

;

(3.7)

ii) if p = 1, then |{x ∈ G : | f (x) − f U |(x) > λ}| ≤ C p

k∇G f k L 1 λ

! Q/(Q−1) ,

(3.8)

for λ > 0. Inequality (3.7) is “almost” the Sobolev–Poincaré inequality in the group for p > 1 (the nongeometric case). We say “almost” because of the presence of the dilation factor τ > 1. But we shall get rid of τ pretty soon. On the other hand, (3.8) is not a Sobolev–Poincaré-type inequality in G because of the presence of the weak-type norm on the left. Since p = 1 is an endpoint for the weak-type estimates of Theorem 3.27(ii), we cannot rely on the Marcinckiewicz interpolation theorem to pass from

31

1 Some topics of geometric measure theory in Carnot groups

a weak-type estimate to the strong-type estimate we are looking for. However, we can still derive from (3.8) a strong-type inequality, thanks to the presence on the right of the group gradient that is a local operator, and therefore enjoys a certain “stability” property under truncations. This idea was originally introduced in [150] and exploited in [210, 90, 94, 20]. Thus, Theorem 3.28 yields the following strongtype analogy. pQ Proposition 3.29. Let p be fixed, 1 ≤ p < Q, and set q := Q−p . Moreover, let U = Ud c (x,r (U)) be a Carnot–Carathéodory ball in G and let f : G → R be a continuously differentiable function. Then there exist c = c(p, G) and τ > 1, both independent of f and U, such that

! 1/q

Z q

| f (x) − f U | dx

! 1/p

Z p

≤ c r (U)

τU

U

|∇G f (x)| dx

.

(3.9)

Finally, we can derive from Proposition 3.29 the geometric Sobolev–Poincaré inequality getting rid of the constant τ. Theorem 3.30 (Geometric Sobolev–Poincaré inequality). Let p be fixed, 1 ≤ p < Q, pQ , where Q denotes the homogeneous dimension in (2.19). Moreover, and set q := Q−p let U = Ud c (x,r (U)) be a Carnot–Carathéodory ball in G and let f : G → R be a continuously differentiable function. Then there exists c = c(p, G) independent of f and U, such that ! 1/q ! 1/p Z Z q p | f (x) − f U | dx ≤ c r (U) |∇G f (x)| dx . (3.10) U

U

The proof of Theorem 3.30 can be carried out as follows. We say that an open set Ω in a quasimetric space (X, d) satisfies the Boman chain condition F (τ, M), τ ≥ 1, M ≥ 1, if there exists a covering W of Ω consisting of open balls U such that P (i) U ∈W χτU (x) ≤ M χΩ (x) ∀x ∈ X; (ii) there is a “central” ball U1 ∈ W which can be connected to every ball U ∈ W by a finite chain of balls U1 ,U2 , . . . ,Ur (U ) = U of W so that U ⊂ MU j for j = 1, . . . ,r (U); moreover, U j ∩ U j−1 contains a ball R j such that U j ∪ U j−1 ⊂ M R j for j = 2, . . . ,r (U). Theorem 3.30 is a corollary of Proposition 3.29 keeping in mind the following two results ([91]). Theorem 3.31. Let τ, M ≥ 1, 1 ≤ p ≤ q < ∞ and Ω satisfy the Boman chain condition F (τ, M) in a quasimetric space (X, d). Also, let µ and ν be Borel measures

32

Francesco Serra Cassano

and µ be doubling. Suppose that f and g are measurable functions on Ω and for each ball U with τU ⊂ Ω there exists a constant gU such that ||g − gU || L q



(U )

≤ A|| f || L p



(τU )

with A independent of U. Then there is a constant gΩ such that ||g − gΩ || L q



(Ω)

≤ c A|| f || L p



(Ω) ,

where c depends only on τ, M, q, and µ. Moreover, we may choose gΩ = gU1 where U1 is a central ball for Ω. Theorem 3.32. Carnot–Carathéodory balls satisfy the Boman chain condition.

3.5 BV -functions and sets of finite perimeter on Carnot groups. Since with any Carnot group we can associate a Hörmander family of smooth vector fields, then all our previous definitions and results still hold in this setting. In particular, within a Carnot group, we can define BV -spaces in a form equivalent to that of the previous section as follows. If Ω ⊆ Rn is open, the space of compactly supported smooth sections of HG is k denoted by C∞ 0 (Ω, HG). If k ∈ N, C0 (Ω, HG) is defined analogously. The space BVG (Ω) is the set of functions f ∈ L 1 (Ω) such that ) (Z 1 f (x)divG φ(x) dx : φ ∈ C0 (Ω, HG), |φ(x)| x ≤ 1 < ∞. ||∇G f ||(Ω) := sup Ω

(3.11) The space BVG,loc (Ω) is the set of functions belonging to BVG (U ) for each open set U ⊂⊂ Ω. Notice the use of the intrinsic fiber norm inside the previous definition. In the setting of Carnot groups, the structure theorem for BV -functions reads as follows (see [99, 48]). Theorem 3.33 (Structure of BVG functions). If f ∈ BVG,loc (Ω) then ||∇G f || is a Radon measure on Ω. Moreover, there exists a ||∇G f ||-measurable horizontal section σ f : Ω → HG such that |σ f (x)| x = 1 for ||∇G f ||-a.e. x ∈ Ω, and Z Z f (x)divG φ(x) dx = hφ, σ f i d||∇G f || Ω



for all φ ∈ C10 (Ω, HG). Finally the notion of the gradient ∇G can be extended from regular functions to functions f ∈ BVG defining ∇G f as the vector-valued measure   ∇G f := −σ f ||∇G f || = −(σ f )1 ||∇G f ||, . . . , −(σ f )m1 ||∇G f || , where (σ f ) j are the components of σ f with respect to the moving base X j .

33

1 Some topics of geometric measure theory in Carnot groups

It is well known that the usefulness of these definitions for the calculus of variations relies mainly on the validity of the two following theorems. In the context of sub-Riemannian geometries they are proved respectively in [118] and [99]. p

Theorem 3.34 (Compactness). BVG,loc (G) is compactly embedded in L loc (G) for Q 1 ≤ p < Q−1 where Q, defined in (2.19), is the homogeneous dimension of G. Theorem 3.35 (Lower semicontinuity). Let f , f k ∈ L 1 (Ω), k ∈ N be such that f k → f in L 1 (Ω); then lim inf ||∇G f k ||(Ω) ≥ ||∇G f ||(Ω). k→∞

Remark 3.36. Fine differentiability properties for BVG functions have been studied in [13, 177, 15]. Definition 3.37. A measurable set E ⊂ G is of locally finite G-perimeter in Ω (or is a G-Caccioppoli set) if the characteristic function 1 E ∈ BVG,loc (Ω). In this case we call the perimeter of E the measure |∂E|G := ||∇G 1 E ||

(3.12)

and we call the (generalized inward) G-normal to ∂E in Ω the vector νE (x) := −σ1 E (x).

(3.13)

Remark 3.38. This remark is analogous to Remark 3.14. The symbol |∂E|G is somehow incorrect, indeed the value of the G-perimeter depends on the choice of the generating vector fields X1 , . . . , X m1 precisely through the bound |φ| ≤ 1 in (3.11). The values of the perimeters induced by two different families of generating vector fields coincide only if the two families are mutually orthonormal; nevertheless the perimeters induced by different families are equivalent as measures and, as a consequence, the notion of being a G-Caccioppoli set is an intrinsic one depending only on the group G. Proposition 3.39. If E is a G-Caccioppoli set with C 1 boundary, then the G-perimeter has the following representation: v u tX Z m1 hX j , NE i2Rn d H n−1 ; |∂E|G (Ω) = ∂E∩Ω

j=1

here NE (x) is the Euclidean unit outward normal to E, X1 , . . . , X m1 is a family of generating vector fields, and H s is the Euclidean s-dimensional Hausdorff measure.

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Remark 3.40. The G-perimeter is invariant under group translations, that is, |∂E|G ( A) = |∂(τp E)|G (τp A),

∀p ∈ G,

and for any Borel set A ⊂ G;

indeed divG is invariant under group translations and the Jacobian determinant of τp : G → G equals 1. Moreover, the G-perimeter is homogeneous of degree Q − 1 with respect to the dilations of the group, that is, |∂(δ λ E)|G ( A) = λ 1−Q |∂E|G (δ λ A)

for any Borel set A ⊂ G;

(3.14)

also, this fact is elementary and can be proved by changing variables in formula (3.11). Fundamental estimates in geometric measure theory are the so-called relative and global isoperimetric inequalities for Caccioppoli sets. They were obtained by Garofalo and Nhieu (see also [90, 94] and the monograph [50]). Theorem 3.41 (Isoperimetric inequalities, [118, Theorem 1.18]). There exists a positive constant CI such that for any G- Caccioppoli set E ⊂ G, for any x ∈ G, r > 0, min{|E ∩ Uc (x,r)|, |Uc (x,r) \ E|} (Q−1)/Q ≤ CI |∂E|G (Uc (x,r))

(3.15)

and min{|E|, |G \ E|} (Q−1)/Q ≤ CI |∂E|G (G),

(3.16)

where Uc denotes the (open) ball with respect to the CC distance d c . Finally we point out that an extension of the notion of BV functions to a general metric space was introduced in [173].

4 Differential calculus within Carnot groups The notion of a rectifiable set is a central one in the calculus of variations and in geometric measure theory. To develop a theory of rectifiable sets inside Carnot groups, and more generally in metric spaces, has been the object of much research in the last twenty years (see, for instance, [76, 77, 137, 101, 102, 10, 11, 103, 105, 193, 106, 108, 156, 157, 12, 170, 121]). Rectifiable sets, in Euclidean spaces, are generalizations of C1 or of Lipschitz submanifolds. Hence, to understand the objects that, inside Carnot groups, naturally take the role of C1 or of Lipschitz submanifolds seems to be preliminary to developing a satisfactory theory of intrinsically rectifiable sets.

1 Some topics of geometric measure theory in Carnot groups

35

In Euclidean spaces, submanifolds are locally graphs. On the other hand, we stress that Carnot groups in general cannot be viewed as Cartesian products of subgroups (unlike Euclidean spaces). Therefore we need a notion of an intrinsic graph fitting the structure of the group G. Here we address this problem by considering functions acting between complementary subgroups of a (Carnot) group G (Definition 4.2) and introducing, for these functions, the notions of intrinsic Lipschitz continuity or of intrinsic differentiability. Then, Lipschitz or C1 submanifolds will be objects that, locally, are intrinsic graphs of these functions.

4.1 Complementary subgroups and graphs. The content of this subsection is taken from [106, 18, 108, 98]. 4.1.1 Homogeneous subgroups. A homogeneous subgroup of a Carnot group G (see [216, 5.2.4]) is a Lie subgroup H such that δ λ g ∈ H for all g ∈ H and for all λ > 0. Homogeneous subgroups are linear subspaces of G, when G is identified with Rn with exponential coordinates. The topological dimension of a (sub)group is the dimension of its Lie algebra. The metric dimension of a subset is its Hausdorff dimension, with respect to the Hausdorff measures Sdk . The metric dimension of a homogeneous subgroup is an integer usually larger than its topological dimension (see [174]). Definition 4.1. M is a (d t , d m )-subgroup of G if M is a homogeneous subgroup of G with linear or topological dimension d t and metric dimension d m ≥ d t . 4.1.2 Complementary subgroups. From now on G will always be a homogeneous stratified group, identified with Rn with exponential coordinates. Definition 4.2. Let M, H be homogeneous subgroups of G. We say that M, H are complementary subgroups in G if M ∩ H = {e} and if G = M · H, that is, for each g ∈ G, there are m ∈ M and h ∈ H such that g = m · h. If M, H are complementary subgroups of G and one of them is a normal subgroup then G is said to be the semidirect product of M and H. If both M and H are normal subgroups then G is said to be the direct product of M and H and in this case we will also write that G = M × H. Remark 4.3. By elementary facts in group theory (see, e.g., [132, Lemma 2.8]) if M, H are complementary subgroups in G, so that G = M · H, then it is also true that G = H · M,

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Francesco Serra Cassano

that is, each g ∈ G can be written—in a unique way—as g = h¯ m, ¯ with m¯ ∈ M, h¯ ∈ H. Rephrased differently, if M, H are complementary subgroups in G, also H, M are complementary subgroups in G. Example 4.4 ([108, Proposition 4.1]). Let G be the Heisenberg group Hn (see Example 2.2). All homogeneous subgroups of Hn are either horizontal, that is, contained in the horizontal fiber HHen , or vertical, that is, containing the subgroup T. A horizontal subgroup has linear dimension and metric dimension k, with 1 ≤ k ≤ n, and it is algebraically isomorphic and isometric to Rk . A vertical subgroup can have any dimension d, with 1 ≤ d ≤ 2n + 1, its metric dimension is d + 1, and it is a normal subgroup. All couples V, W of complementary subgroups of Hn are of the type (i) V horizontal of dimension k, 1 ≤ k ≤ n, (ii) W vertical and normal of dimension 2n + 1 − k. Remark 4.5 (suggested by Le Donne). Let us recall that it is easy to construct a Carnot group G with two complementary subgroups M and H, which are not normal. Indeed, let G1 be a Carnot group which is a semidirect product of two subgroups W and V, with W normal but V not normal. Then it is easy to see that the direct product G = G1 × G1 is still a Carnot group and, by Remark 4.3, the subgroups M := W × V and H = V × W are complementary in G. On the other hand, both M and H are not normal. Hence the product G = M · H is not semidirect. For instance, let G1 = H1 = (R3 , ·) be the first Heisenberg group, W := {(0, x 2 , x 3 ) : x i ∈ R, i = 1, 2}, and V := {(x 1 , 0, 0) : x 1 ∈ R}. Example 4.6. Splittings similar to those of Heisenberg groups exist in a general Carnot group G = (Rn , ·). Indeed, choose any horizontal homogeneous subgroup H = H1 ⊂ G1 and a subgroup M = M1 ⊕ · · · ⊕ Mκ such that H ⊕ M1 = G1 , and G j = M j for all 2 ≤ j ≤ k. Then M and H are complementary subgroups in G and the product G = M · H is semidirect because M is a normal subgroup. In particular, if X1 , . . . , X m1 , X m1 +1 , . . . , X n denote a basis of the Lie algebra G , and X1 , . . . , X m1 denote the system of group generating vector fields, chosen according to Section 2.1, and let H := {exp(sX j0 ) : s ∈ R} for a given 1 ≤ j0 ≤ m1 . For instance, choose j0 = 1. Then H is a (1, 1)-homogeneous subgroup of G, that is, according to Definition 4.1, a homogeneous subgroup with both topological and  metric dimension 1. Let M := span X2 , . . . , X m1 , . . . , X n and M := exp(M) = {(0, x 2 , . . . , x n ) : x i ∈ R, i = 2, . . . , n} ≡ Rn−1 . Notice that M is a subalgebra of G of dimension n − 1, since [M, M] ⊂ M and also an ideal of G , that is, [G , M] ⊂ M. Thus M turns out to be a normal subgroup of G of homogeneous dimension Q−1. Indeed, it is easy to see that M = Rn−1 can be

1 Some topics of geometric measure theory in Carnot groups

37

considered as Lie groups with Haar measure Ln−1 , satisfying (2.20) with Q ≡ Q − 1 and closed with respect to the family of dilations of G. Therefore (M, d, Ln−1 ) is a measure metric Ahlfors-regular space of dimension Q − 1: as a by-product, its metric dimension is Q − 1. In conclusion we get the splitting G = M · H, with M an (n − 1,Q − 1)-subgroup and H a (1, 1)-subgroup. Example 4.7. The Engel group is E = (R4 , ·, δ λ ), where the group law is defined as x 1 + y1 x y +/ *. 1 +/ *. 1 +/ *.. x 2 + y2 // x y 2 2 .. // · .. // = .. x 3 + y3 + (x 1 y2 − x 2 y1 )/2 . x 3 / . y3 / .. x + y + [(x y − x y ) + (x y − x y )]/2/// 4 4 1 3 3 1 2 3 3 2 , x 4 - , y4 - , +(x 1 − y1 + x 2 − y2 )(x 1 y2 − x 2 y1 )/12 and the family of dilation is δ λ (x 1 , x 2 , x 3 , x 4 ) = (λ x 1 , λ x 2 , λ 2 x 3 , λ 3 x 4 ). A basis of left-invariant vector fields is X1 , X2 , X3 , X4 defined as   X1 (x) := ∂x1 − (x 2 /2) ∂x3 + −x 3 /2 − (x 1 x 2 + x 22 )/12 ∂x4 ,   X2 (x) := ∂x2 + (x 1 /2) ∂x3 + −x 3 /2 + (x 21 + x 1 x 2 )/12 ∂x4 , X3 (x) := ∂x3 − ((x 1 + x 2 )/2) ∂x4 , X4 (x) := ∂x4 . The commutation relations are [X1 , X2 ] = X3 , [X1 , X3 ] = [X2 , X3 ] = X4 and all the other commutators are zero. Inside the Engel group there are two families of complementary subgroups. The first one is formed by 1-dimensional horizontal subgroups and 3-dimensional subgroups containing all the vertical directions. This family gives a semidirect splitting of E. The second family is formed by 2-dimensional subgroups that are not normal subgroups. All the computations can be easily done directly, preferably using a symbolic computation program. The homogeneous subgroups Mγ, β := {(γt, βt, 0, 0) : t ∈ R},

Nγ,δ := {(γt, δt, x 3 , x 4 ) : t, x 3 , x 4 ∈ R}

are complementary subgroups in E, provided that γδ − βγ , 0. Moreover, Nγ,δ is a normal subgroup, hence E is the semidirect product of Mγ, β and Nγ,δ . The second family, for γ + β , 0, is given by K := {(x 1 , −x 1 , x 3 , 0) : x 1 , x 3 ∈ R},

Hγ, β := {(γt, βt, 0, x 4 ) : t, x 4 ∈ R}.

One can compute directly that K and Hγ, β are complementary subgroups in E and that neither K nor Hγ, β are normal subgroups. Hence E = K · Hγ, β , but the product is not a semidirect product.

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4.1.3 Components along complementary subgroups. Given M, H, complementary subgroups of G, the elements m ∈ M and h ∈ H such that g = mh are unique because M ∩ H = {e} and are denoted as components of g along M and H or as projections of g on M and H. Proposition 4.8. If M, H are complementary subgroups in G, then there is c0 = c0 (M, H) > 0 such that for all g = mh, c0 (kmk∞ + khk∞ ) ≤ kgk∞ ≤ kmk∞ + khk∞ .

(4.1)

Proof. The right-hand side is the triangle inequality. For the left-hand side observe that M∩∂B(0, 1) and H∩∂B(0, 1) are disjoint compact sets, hence they have positive distance. The general statement follows by dilation.  The following estimates will be crucial later in the paper (see also [160, Lemma 3.9 and Remark 3.10]). Lemma 4.9 ([98, Lemma 2.13]). Let G be a step-κ Carnot group. Then there exists a positive constant C = C(G) such that   (k−1)/κ (k−1)/κ 1/κ 1/κ kp−1 q−1 pqk∞ ≤ C kpk∞ kqk∞ + kqk∞ kpk∞ ∀p, q ∈ G, (4.2) and, consequently,   (k−1)/κ (k−1)/κ 1/κ 1/κ kq−1 pqk∞ ≤ kpk∞ + C kpk∞ kqk∞ + kqk∞ kpk∞

∀p, q ∈ G. (4.3)

Proposition 4.10 ([98, Corollary 2.14]). Let M, H be complementary subgroups of a step-κ group G. If g ∈ G let m, m¯ ∈ M and h, h¯ ∈ H be the unique elements such that g = mh = h¯ m. ¯ Then,  C  1 ¯ ∞ (κ−1)/κ + k hk ¯ ∞ 1/κ k mk k mk ¯ ∞+ k mk ¯ ∞ 1/κ k hk ¯ ∞ (κ−1)/κ , c0 c0  1 ¯ C  ¯ ∞ (κ−1)/κ + k hk ¯ ∞ 1/κ k mk khk∞ ≤ k hk∞ + k mk ¯ ∞ 1/κ k hk ¯ ∞ (κ−1)/κ . c0 c0

kmk∞ ≤

Consequently, for all δ > 0 there is c(δ) = c(δ, M, H) > 0 such that for all g with kgk∞ ≤ δ, 1/κ 1/κ ¯ ∞ kmk∞ ≤ c(δ)k mk ¯ ∞ , khk∞ ≤ c(δ)k hk . (4.4) From now on, we will denote by gM and gH the components of g ∈ G. More precisely our convention is as follows: when M, H are complementary subgroups in

1 Some topics of geometric measure theory in Carnot groups

39

G, then M will always be the first “factor”, and H the second one and gM ∈ M and gH ∈ H are the unique elements such that g = gM gH .

(4.5)

We stress that this notation is ambiguous because each component gM and gH depends on both the complementary subgroups M and H and also on the order in which they are taken. The projection maps PM : G → M and PH : G → H are defined as PM (g) := gM ,

PH (g) := gH .

(4.6)

Observe that, in general, (gM ) −1 , (g −1 )M and (gH ) −1 , (g −1 )H . Proposition 4.11 ([98, Proposition 2.17]). Let M, H be complementary subgroups of G, then the projection maps PM : G → M and PH : G → H defined in (4.6) are polynomial maps. More precisely, if κ is the step of G, there are 2κ matrices A1 , . . . , Aκ , B1 , . . . , Bκ , depending on M and H, such that (i) A j and B j are (n j , n j )-matrices for all 1 ≤ j ≤ κ; and, with the notation of (2.4), (ii) PM g = A1 g 1 , A2 (g 2 − Q2 ( A1 g 1 , B1 g 1 )), . . . ,  Aκ (gκ − Qκ ( A1 g 1 , . . . , Bκ−1 gκ−1 )) ; (iii) PH g = B1 g 1 , B2 (g 2 − Q2 ( A1 g 1 , B1 g 1 )), . . . ,  Bκ (gκ − Qκ ( A1 g 1 , . . . , Bκ−1 gκ−1 )) ; (iv) A j is the identity on M j , and B j is the identity on H j , for 1 ≤ j ≤ κ. Let us stress that PM and PH are not, in general, Lipschitz maps when G, M, and H are endowed with the restriction of the distance d of G. Example 4.12 ([98, Example 2.18]). Let G be the Heisenberg group H1 = (R3 , ·) and let V and W be the subgroups V = {x = (x 1 , 0, 0) : x 1 ∈ R} ,

W = {x = (0, x 2 , x 3 ) : x 2 , x 3 ∈ R} .

V and W are complementary subgroups in H1 , W is a normal subgroup while V is not a normal subgroup. When we consider H1 = V · W the projections PV and PW are PV (x 1 , x 2 , x 3 ) = (x 1 , 0, 0), PW (x 1 , x 2 , x 3 ) = (0, x 2 , x 3 − x 1 x 2 /2). Here PW : H1 → W is not Lipschitz. Indeed, let q = (1, 1, 0) and pε = (1+ε, 1+ε, 0), then PW q = (0, 1, −1/2) and PW pε = (0, 1 + ε, −(1 + ε) 2 /2). Hence, as ε → 0+ , √ kq−1 pk∞ = k(ε, ε, 0)k∞ ≈ ε, k(PW q) −1 PW pε k∞ = k(0, ε, −ε − ε 2 /2)k∞ ≈ ε.

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When we consider H1 = W · V, then PV and PW take the form PV (x 1 , x 2 , x 3 ) = (x 1 , 0, 0),

PW (x 1 , x 2 , x 3 ) = (0, x 2 , x 3 + x 1 x 2 /2).

Let q = (1, 0, 0) and pε = (1, ε, ε/2), then PW q = (0, 0, 0) and PW pε = (0, ε, ε). Hence √ kq−1 pk∞ = k(0, ε, 0)k∞ ≈ ε, k(PW q) −1 PW pε k∞ = k(0, ε, ε)k∞ ≈ ε as ε → 0+ . In this case too, PW is not a Lipschitz map. The example shows that both the projections either on the first factor or on the second factor can be non-Lipschitz. This fact has many unpleasant consequences. One of them is related to the difficulty of controlling, in an easy way, the measure, or even the Hausdorff dimension, of the projections of sets (see, for instance, [26]). Notice that in both cases, we were considering projections on the normal factor. Indeed, the projection on the complement of a normal subgroup is always metric Lipschitz continuous. Proposition 4.13. Let M, H be complementary subgroups of G. Then (i) if H is a normal subgroup then PM is Lipschitz; (ii) if M is a normal subgroup then PH is Lipschitz. ¯ we have Proof. (i) For all g = mh and g¯ = m¯ h, ¯ = PM (m−1 m¯ m¯ −1 mh−1 m−1 m¯ h) ¯ = m−1 m. PM (g −1 g) ¯ = PM (h−1 m−1 m¯ h) ¯ Hence, for all g, g¯ ∈ G, PM (g −1 g) ¯ = (PM g) −1 PM g¯ and, by (4.1),   k(PM g) −1 PM gk ¯ ∞ ≤ kPM (g −1 g)k ¯ ∞ + kPH (g −1 g)k ¯ ∞ ≤ c0−1 kg −1 gk ¯ ∞. (ii) As before, PH (g −1 g) ¯ = (PH g) −1 PH g¯ and finally, k(PH g) −1 PH gk ¯ ∞ ≤ c0−1 kg −1 gk ¯ ∞

∀g, g¯ ∈ G. 

Even if the projections are not Lipschitz we have the following control on the measure of projected sets. Lemma 4.14 ([98, Lemma 2.20]). Let M and H be complementary subgroups of G. Denote by d t ≤ d m respectively, the topological and the metric dimensions of M. Then there is c = c(M, H) > 0 such that Ld t (PM (B∞ (p,r))) = c r d m for all balls B∞ (p,r) ⊂ G.

1 Some topics of geometric measure theory in Carnot groups

41

4.1.4 Intrinsic graphs in Carnot groups Definition 4.15. Let H be a homogeneous subgroup of G. We say that a set S ⊂ G is a (left) H-graph (or a left graph in direction H, or, also, an intrinsic graph) if S intersects each left coset of H in one point, at most. If A ⊂ G parametrizes the left cosets of H—i.e., if A itself intersects each left coset of H at most one time—and if S is an H-graph, then there is a unique function φ : E ⊂ A → H such that S is the graph of φ, that is, S = graph (φ) := {ξ · φ(ξ) : ξ ∈ E }. Conversely, for any ψ : D ⊂ A → H the set graph (ψ) is an H-graph. One has an important special case when H admits a complementary subgroup M. Indeed, in this case, M parametrizes the left cosets of H and we have S is an H-graph if and only if S = graph (φ) for φ : E ⊂ M → H. By uniqueness of the components along M and H, if S = graph (φ) then φ is uniquely determined among all functions from M to H. From now on we will consider mainly graphs of functions acting between complementary subgroups. Remark 4.16. Nevertheless, it is relevant to mention that examples of H-graphs that are not, in general, graphs of functions acting between complementary subgroups have been considered inside the Heisenberg groups Hn = R2n+1 . These sets are given as S = {(x 1 , . . . , x 2n , φ(x 1 , . . . , x 2n ))} ⊂ Hn . In our notation such an S is a T-graph (or simply t-graph), where T is the center of Hn . The left cosets of T are parametrized over the set A = HHen . We recall that the center T has no complementary subgroup in Hn (see Example 4.4), and in general there isn’t a couple of complementary subgroups M, H of Hn and a ψ : M → H such that S = graph (ψ), even locally. If a set S ⊂ G is an intrinsic graph then it stays an intrinsic graph after left translations or group dilations. Proposition 4.17. Let H be a homogeneous subgroup of G. If S is an H-graph then, for all λ > 0 and for all q ∈ G, δ λ S and q · S are H-graphs.

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If, in particular, M, H are complementary subgroups in G, if S = graph (φ) with φ : E ⊂ M → H, then for all λ > 0, δ λ S = graph (φ λ ), with φ λ : δ λ E ⊂ M → H and φ λ (m) = δ λ φ(δ1/λ m), for m ∈ δ λ E ;

(4.7)

for any q ∈ G, q · S = graph (φq ), where φq : Eq ⊂ M → H,

Eq = {m ∈ M : PM (q−1 m) ∈ E } and  φq (m) = (PH (q−1 m)) −1 · φ PM (q−1 m) , for all m ∈ Eq .

(4.8)

Proof. If x, x 0 ∈ S and x , x 0 then, by definition of an H-graph, x, x 0 belong to different left cosets of H. Then δ λ x, δ λ x 0 belong to different cosets of H, because H is a homogeneous subgroup, and also q · x, q · x 0 belong to different cosets of H, by elementary properties of cosets (see, e.g., [132, chapter 2, section 4]). By definition, these facts prove that both δ λ S and q · S are H-graphs and that there are φ λ and φq such that δ λ S = graph (φ λ ) and q · S = graph (φq ). To prove (4.7) observe that, by uniqueness of the components, δ λ (m · φ(m)) = m 0 · φ(m 0 ) implies that δ λ m = m 0 and that φ λ = δ λ ◦ φ ◦ δ1/λ . −1 = p−1 · p for all p ∈ G, then (q −1 · To prove (4.8) observe that, because pH M −1 −1 −1 m)H = m · q · (q · m)M , hence graph (φq ) = {m · φq (m) : m ∈ Eq }  = {m · (q−1 · m)H−1 · φ (q−1 · m)M : m ∈ Eq }  = {m · m−1 · q · (q−1 · m)M · φ (q−1 · m)M : (q−1 · m)M ∈ E } = q · graph (φ).



Remark 4.18. From (4.8) and the continuity of the projection maps PM and PH it follows that the continuity of a function is preserved by translations. Precisely, given q = qM qH and f : M → H, then the translated function f q is continuous in m ∈ M if and only if the function f is continuous in the corresponding point (q−1 m)M . Remark 4.19. The algebraic expression of φq in Proposition 4.17 can be made more explicit when G is a semidirect product of M, H. Precisely,  (i) if M is normal in G then φq (m) = qH φ (q−1 m)M , for m ∈ Eq = qE (qH ) −1 ; −1 m), for m ∈ E = q E . (ii) if H is normal in G then φq (m) = (q−1 m)H−1 φ(qM q M

If both M and H are normal in G—that is, if G is a direct product of M and H—then we get the well-known Euclidean formula (iii)

−1 φq (m) = qH φ(qM m),

See also [18, Proposition 3.6].

for m ∈ Eq = qM E .

43

1 Some topics of geometric measure theory in Carnot groups

4.2 Regular hypersurfaces in Carnot groups. In this section we deal with the problem of introducing a notion of intrinsic regular hypersurfaces (namely 1codimensional topological surfaces) in the setting of a Carnot group. By that we mean submanifolds which, in the geometry of the Carnot group, have the same role as C1 submanifolds have inside Euclidean spaces. Here and in what follows, “intrinsic” will denote properties defined only in terms of the group structure of G or, equivalently, of its Lie algebra g. This section relies on [103, 16, 106, 41, 161, 162, 109]. We define G-regular hypersurfaces in a Carnot group G, mimicking [102, Definition 6.1], as noncritical level sets of functions in C1G (Rn ). Definition 4.20 (G-regular hypersurfaces). Let G be a Carnot group. We shall say that S ⊂ G is a G-regular hypersurface if for every x ∈ S there exist a neighborhood U of x and a function f ∈ C1G (U ) such that (i) S ∩ U = {y ∈ U : f (y) = 0}; (ii) ∇G f (y) , 0 for y ∈ U . G-regular surfaces have a unique tangent plane at each point. This follows from a Taylor formula for functions in C1G that is basically proved in [190]. Proposition 4.21. If f ∈ C1G (U∞ (p,r)), then f (x) = f (p) +

m1 X

(X j f )(p)(x j − p j ) + o(d(x, p)),

as x → p.

(4.9)

j=1 g

If S = {x : f (x) = 0} ⊂ G is a G-regular hypersurface, the tangent group TG S(x) to S at x is m1 X ( ) g TG S(x) := v = (v1 , . . . , vn ) ∈ G : X j f (x)v j = 0 .

(4.10)

j=1 g

By (2.3), TG S(x) is a proper subgroup of G. We can define the tangent plane to S at x as g TG S(x) := x · TG S(x). (4.11) We stress that this is a good definition. Indeed, the tangent plane does not depend on the particular function f defining the surface S because of point (iii) of the implicit function theorem, Theorem 4.24 below, that yields g

TG S(x) = {v ∈ G : hνE (x), π x vi x = 0},

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Francesco Serra Cassano

where νE is the generalized inward unit normal defined in (3.13), and π x (v) = Pm j=1 v j X j (x). Notice that the map v 7→ π x (v), for x ∈ G fixed, π x (v) =

m1 X

v j X j (x),

(4.12)

j=1

is a smooth section of HG. Notice also that, once more from Theorem 4.24(iii), it follows that νE is a continuous function. Pm If v 0 = i=1 vi X i (0) ∈ HG0 we define the halfspaces SG± (0, v 0 ) as m m X X ( ) ( ) SG+ (0, v 0 ) := x ∈ G : x i vi > 0 and SG− (0, v 0 ) := x ∈ G : x i vi < 0 . i=1

i=1

Their common boundary is the vertical plane m ( X ) Π(0, v 0 ) := x : x i vi = 0 . i=1

If v =

Pm

i=1 vi X i (y)

∈ HGy , SG± (y, v) and Π(y, v) are the translated sets

SG± (y, v) := y · SG± (0, v 0 )

and

Π(y, v) = y · Π(0, v 0 ),

where v and v 0 have the same components vi with respect to the left-invariant basis X i . Hence m X ( ) SG± (y, v) = x ∈ G : (x i − yi )vi > 0(< 0) . (4.13) i=1

Clearly, TG S(x) = Π(x, νE (x)). Note also that the class of G-regular hypersurfaces is different from the class of Euclidean C 1 embedded surfaces in Rn . On the one hand, G-regular surfaces can have “ridges” because continuity of the derivatives of the defining functions f is required only in the horizontal directions; on the other hand, a Euclidean C 1 surface can have so-called characteristic points, i.e., points p ∈ S where the Euclidean tangent plane Tp S contains the horizontal fiber HG p . Definition 4.22. If S is a Euclidean C1 hypersurface in G, we define the characteristic set of S as Char(S) := {x ∈ S : HG x ⊆ Tx S}. (4.14)

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The points of Char(S) are, in many aspects, irregular points of S. Note indeed that the tangent group does not exist in these points. It is also well known that these points are “few” on smooth hypersurfaces. Precise estimates of the HQ−1 measure of c the characteristic sets of C 1 surfaces in general Carnot groups Hn have been obtained by Magnani [157], extending previous results of Balogh [22] in the Heisenberg group and Franchi, Serapioni, & Serra Cassano [105] in step-2 Carnot groups. Note that the study of the size of the characteristic set has a long history. We refer to the contributions of Derridj [80], Franchi & Wheeden [111], and Danielli, Garofalo, & Nhieu [68]. Magnani’s result reads as follows. Theorem 4.23. If S is a Euclidean C1 -smooth hypersurface in a Carnot group G with homogeneous dimension Q and equipped with an invariant distance d, then

HQ−1 (Char(S)) = 0. d

(4.15)

We can state now our implicit function theorem, holding that a G-regular hypersurface S = { f (y) = 0} boundary of the set E = { f (y) < 0} can be locally parametrized through a function Φ : Rn−1 → Rn so that the G-perimeter of E can be written explicitly in terms of ∇G f and Φ. In view of the notion of the intrinsic graph introduced in Definition 4.15, we can also say that S is locally an intrinsic graph in the direction of a 1-dimensional horizontal subgroup (see Corollary 4.25). Theorem 4.24 (Implicit function theorem, [103, Theorem 2.1]). Let Ω be an open set in G = (Rn , ·), 0 ∈ Ω, and let f ∈ C1G (Ω) be such that f (0) = 0 and X1 f (0) > 0. Define E = {x ∈ Ω : f (x) < 0}, S = {x ∈ Ω : f (x) = 0}, and, for δ > 0, h > 0, Iδ = {ξ = (ξ2 , . . . , ξ n ) ∈ Rn−1 , |ξ j | ≤ δ},

Jh = [−h, h].

If ξ = (ξ2 , . . . , ξ n ) ∈ Rn−1 and t ∈ Jh , denote now by γ(t, ξ) the integral curve of the vector field X1 at the time t issued from (0, ξ) = (0, ξ2 , . . . , ξ n ) ∈ Rn , i.e., γ(t, ξ) = exp(t X1 )(0, ξ). Then there exist δ, h > 0 such that the map (t, ξ) → γ(t, ξ) is a diffeomorphism of a neighborhood of Jh × Iδ onto an open subset of Rn , and, if we denote by U ⊂⊂ Ω the image of Int(Jh × Iδ ) through this map, we have (i) E has finite G-perimeter in U ; (ii) ∂E ∩ U = S ∩ U ; (iii) νE (x) = −

∇G f (x) ∀x ∈ S ∩ U ; |∇G f (x)| x

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Francesco Serra Cassano

where νE is the generalized inner unit normal defined by (3.13), that can be identified with a section of HG with |ν(x)| x = 1 for |∂E|G -a.e. x ∈ U . In particular, νE can be identified with a continuous function and |ν| ≡ 1. Moreover, there exists a unique function φ = φ(ξ) : Iδ → Jh such that the following parametrization holds: if ξ ∈ Iδ , put Φ(ξ) = γ(φ(ξ), ξ), then (iv) S ∩ U¯ = {x ∈ U¯ : x = Φ(ξ), ξ ∈ Iδ }; (v) φ is continuous; (vi) the G-perimeter has an integral representation (area formula) qP m 2 Z j=1 |X j f (Φ(ξ))| |∂E|G (U ) = d Ln−1 (ξ). X1 f (Φ(ξ)) Iδ By Dini’s Theorem 4.24 and Example 4.6, it follows that if H := {(x 1 , 0, . . . , 0)} ≡ R and M = {(0, x 2 , . . . , x n ) : x i ∈ R, i = 2, . . . , n} ≡ Rn−1 , then (i) M and H are, respectively, (n − 1,Q − 1)- and (1, 1)-subgroups of G, and they are complementary, that is, G = M · H; (ii) S ∩ U = graph (φ) where φ : Iδ ⊂ M → H is continuous. We can summarize this result in the following Corollary 4.25. Every G-regular hypersurface is locally an intrinsic continuous graph with topological dimension n − 1. Remark 4.26. Notice that, taking the Euclidean case into account, the approach of defining a regular hypersurface S ⊂ G = (Rn , ·) by means of a parametrization map cannot work in a nonabelian Carnot group. Namely, suppose that there is a map φ ∈ C1H (U , G), whose P-differential d P φ(p) : Rn−1 → G is injective for each p ∈ U , such that the image S = φ(U ) with U ⊂ Rn−1 . If this is the case, let us consider the injective H-linear map L˜ := d P φ(p) : Rn−1 → g read as a homomorphism of Lie algebras. In view of Theorem 3.2, since it is a contact map, it follows that ˜ n−1 ) ⊆ V1 , where V1 is the first horizontal layer of the stratification of g (see L(R (2.1)). Thus V1 must contain a (trivial) (n − 1)-dimensional algebra, which implies that g must also be trivial.

1 Some topics of geometric measure theory in Carnot groups

47

Our next theorem is a mild regularity result. Roughly speaking, it states that Gregular hypersurfaces do not have cusps or spikes if they are studied with respect to the intrinsic Carnot–Carathéodory distance, while they can be very irregular as Euclidean submanifolds. To make precise the former statement we recall the notion of essential boundary (or of measure-theoretic boundary) ∂∗ F of a set F ⊂ G: ( n )   L (F ∩ U (x,r)) Ln (F c ∩ U (x,r)) ∂∗ F := x ∈ G : lim sup min , > 0 . (4.16) Ln (U (x,r)) Ln (U (x,r)) r →0+ Notice that the definition above makes sense in any metric measure space and that the essential boundary does not change if, in the definition (4.16), the distance d is substituted by an equivalent distance d 0. Theorem 4.27. Let Ω ⊂ G be a fixed open set, and E be such that ∂E ∩ Ω = S ∩ Ω, where S is a G-regular hypersurface. Then ∂E ∩ Ω = ∂∗ E ∩ Ω.

(4.17)

4.3 Area formulas for Hausdorff measures in metric spaces: Application to Carnot groups. We first want to give some general area-type formulas in a metric space. Then we compare the perimeter measure, of a G-regular hypersurface S, and the intrinsic (Q − 1)-Hausdorff measure of S. Observe that it makes sense to speak of the perimeter measure of S given that S is locally the boundary of a finite G-perimeter set (as proved in Theorem 4.24). The next theorems give some explicit form of the density of the perimeter with respect to the intrinsic Hausdorff measures concentrated on S, namely the spherical and centered Hausdorff measures. As a consequence—as stated in the following corollary—G-regular hypersurfaces have coherently intrinsic Hausdorff dimension Q − 1. Let us recall two area formulas for a measure µ with respect to the m-dimensional spherical and centered Hausdorff measure, recently obtained respectively in [161] and [109]. According to Federer’s notation [83], we define a centered and a noncentered density of an outer measure µ on X. Definition 4.28. Let (X, d) be a metric space and assume that (2.21) holds. (i) The upper and lower m-densities of µ at x ∈ X are Θ∗ m (µ, x) := lim sup r →0

and Θ∗m (µ, x) := lim inf r →0

µ(B(x,r)) αm r m

µ(B(x,r)) . αm r m

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Francesco Serra Cassano

If they agree, their common value Θm (µ, x) = Θ∗ m (µ, x) = Θ∗m (µ, x) is called the m-density of µ at x. (ii) The m-Federer densities of µ at x ∈ X are  µ(B(y,r)) Θ∗Fm (µ, x) := inf sup : ε >0 β m diam(B(y,r)) m  x ∈ B(y,r), diam(B(y,r)) ≤ ε . It is easy to see that Θ∗ m (µ, x) ≤ Θ∗Fm (µ, x) ≤ 2m Θ∗ m (µ, x)

∀x ∈ X.

(4.18)

In a very interesting recent note [161], Magnani proved the following theorem. Theorem 4.29. Let µ be a Borel regular measure in X such that there exists a countable open covering of X whose elements have finite µ measure; let A ⊂ X be a Borel set. Suppose that Sdm ( A) < ∞ and µ A is absolutely continuous with respect to Sdm A. Then Z µ(B) =

B

Θ∗Fm (µ, x) d Sdm (x)

(4.19)

for each Borel set B ⊂ A. Here Θ∗Fm (µ, ·) denotes the m-dimensional Federer density (see Definition 4.28(ii)). Moreover, Magnani showed that in a general metric space the density Θ∗Fm (µ, ·) cannot be replaced by the “centered” density Θ∗ m (µ, ·) (see Definition 4.28(i)). Indeed, Magnani provided an example inside the Heisenberg group X = H1 ≡ 3 R , equipped with its sub-Riemannian metric d = d c . He constructed a Radon measure µ and a set A ⊂ H1 and two constants 0 < k 1 < k2 such that µ is absolutely continuous w.r.t. Sc2 , Θ2 (µ, x) = k1 < k 2 = Θ∗2 F (µ, x)

∀x ∈ A

and µ( A) > t Sc2 ( A)

∀t ∈ (k1 , k 2 ).

(4.20)

In particular, from what was said before, it follows that the general statement, given A ⊂ X and k > 0, Θm (µ, x) = k may fail.

∀x ∈ A



µ

A = k Sm

A,

(4.21)

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49

Therefore Magnani stressed the need for an alternative proof of the assertions in [102, 105, 103, 106], that the perimeter measure |∂E|G agrees, up to a constant, Q−1 with the (Q − 1)-dimensional spherical Hausdorff measure S∞ restricted to a Gregular boundary ∂E, where X = G is a Carnot group of Hausdorff dimension Q, and µ = |∂E|G is the perimeter measure of a set E. Meanwhile he proved Theorem 4.30 ([162, Theorem 5.2]). Let G be a Carnot group endowed with the invariant distance d ∞ in (2.18) and let Q be its homogeneous dimension. Let E ⊂ G be such that its topological boundary ∂E is a G-regular hypersurface; then there is a constant k = k (G) > 0 for which ΘQ−1 (|∂E|G , x) = Θ∗FQ−1 (|∂E|G , x) = k

∀x ∈ ∂E.

Remark 4.31. Theorem 4.30 actually holds for more general invariant distances than d ∞ : it also holds for each invariant distance d for which the unit closed ball Bd (e, 1) is convex. Observe also that for a general invariant distance d, ΘQ−1 (|∂E|G , ·) and Θ∗FQ−1 (|∂E|G , ·) may differ. For instance, in a very recent note [163], Magnani proved that, inside the Heisenberg group G = H1 ≡ R3 equipped with its subRiemannian metric d = d c , given E ⊂ H1 such that its topological boundary ∂E is a G-regular hypersurface, then there are two positive constants k i (i = 1, 2) for which Θ3 (|∂E|G , x) = k 1 < k2 = Θ∗F3 (|∂E|G , x)

∀x ∈ ∂E.

As a consequence, it follows that

Sc3 (∂E) < Cc3 (∂E). We thank Magnani for sharing with us his unpublished note as well as for some useful suggestions on the argument. From Theorems 4.29 and 4.30 we get the following Corollary 4.32. Let G be a Carnot group endowed with the invariant distance d ∞ . Let E ⊂ G with topological boundary ∂E a G-regular hypersurface; then there is a constant k = k (G) > 0 such that |∂E|G = k S∞ Q−1

S.

(4.22)

Corollary 4.33. If S is a G-regular hypersurface then the Hausdorff dimension of S, with respect to the Carnot–Carathéodory metric d or any other metric d 0 comparable with it, is Q − 1. Corollary 4.33 combined with Theorem 4.23 yields the following comparison result between Euclidean C1 -smooth hypersurfaces and G-regular hypersurfaces.

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Francesco Serra Cassano

Theorem 4.34. If S is a Euclidean C1 -smooth hypersurface in a Carnot group G with homogeneous dimension Q, then the Hausdorff dimension of S, with respect to the Carnot–Carathéodory metric d c or any other invariant metric d 0 comparable with it, is Q − 1. Remark 4.35. The reverse assertion is false: there exist G-regular hypersurfaces in G ≡ Rn that have Euclidean Hausdorff dimension greater than n − 1: indeed in [139] it was shown that there exist G-regular hypersurfaces in the Heisenberg group H1 (Q = 4, n = 3) with Euclidean Hausdorff dimension 2.5. By the previous Magnani example, it turns out that Federer densities play a privileged role when dealing with the spherical measures S m . On the other hand, these densities are frequently harder to compute than the centered ones. Therefore, in [109] we studied whether the aforementioned area formula stays true with “centered” densities Θ∗ m and measures Sdm replaced by equivalent ones. Centered Hausdorff measures Cdm (see Definition 2.21(iii)) actually play this role. Indeed, we have Theorem 4.36 ([109, Theorem 3.1]). Let µ be a Borel regular measure in X such that there exists a countable open covering of X whose elements have µ finite measure; let A ⊂ X be a Borel set. If Cdm ( A) < ∞ and µ A is absolutely continuous with respect to Cdm A, then Θ∗ m (µ, ·) : X → [0, +∞] is Borel measurable and, for each Borel set B ⊂ A, µ(B) =

Z B

Θ∗ m (µ, x) d Cdm (x).

(4.23)

Remark 4.37. Since Cdm and Sdm are equivalent, then Cdm ( A) < ∞ if and only if Sdm ( A) < ∞ and µ A is absolutely continuous with respect to Cdm if and only if µ A is absolutely continuous with respect to Sdm . Remark 4.38. According to Theorem 4.36, in the differentiation results for measure stated in [103, Theorem 3.4] and [106, Theorem 4.3] the centered Hausdorff measure has to be considered in place of the spherical one. Corollary 4.39. Under the same assumptions as Theorem 4.36, if, for A ⊆ X, there is a constant k > 0 such that Θ∗m (µ, x) = k

∀x ∈ A,

then µ

A = k Cm

A.

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51

As an immediate consequence of Theorem 4.30, Remark 4.31, Corollaries 4.32 and 4.39, we get Theorem 4.40. Let G be a Carnot group endowed with an invariant distance d whose unit closed ball Bd (e, 1) is convex. Let S ⊂ G be a G-regular hypersurface; then SdQ−1 S = CdQ−1 S. (4.24) Remark 4.41. Since HQ−1 , SdQ−1 , and CdQ−1 are equivalent measures, by (4.24) and d the Radon–Nikodym theorem, it follows that, given a G-regular hypersurface S ⊂ G, there exist two positive densities ζ i ∈ L 1loc (S, SdQ−1 S) (i = 1, 2) and two positive constants Ci (i = 1, 2) such that

HQ−1 S = ζ1 SdQ−1 S = ζ2 CdQ−1 S, d C1 ≤ ζ1 (x) ≤ ζ2 (x) ≤ C2 ,

SdQ−1 − a.e. x ∈ S.

If d is an invariant distance whose unit closed ball Bd (e, 1) is convex, by Theorem 4.40, ζ = ζ1 = ζ2 . It is not known whether ζ is constant. However, ζ1 and ζ2 may differ for a general invariant distance d (see [163]). Let us recall that in the Euclidean setting, H n−1 S = S n−1 S = C n−1 S, whenever S ⊂ Rn is a (Euclidean) C1 -regular hypersurface (see [209]). As another application to Carnot groups of the area formula for centered Hausdorff measure, we can provide an explicit (simple) characterization of the centered density ΘQ−1 (|∂E|G , ·) for sets E whose ∂E is G-regular. This is the corrected version of [103, Theorem 3.5], which, a priori, could not apply to the spherical Hausdorff measure for Magnani’s counterexample. Theorem 4.42 ([109]). Let G be a Carnot group and d be an invariant distance. Let Ω ⊂ G be a fixed open set and let E ⊂ G such that ∂E ∩ Ω = S ∩ Ω, where S is a G-regular hypersurface. Then   n−1 U (0, 1) ∩ T g S(x) L d G ∀x ∈ S, (4.25) ΘQ−1 (|∂E|G , x) = αQ−1 g

where TG S(x) denotes the tangent group to S at x according to (4.10), and Ud (0, 1) denotes the unit (open) ball defined in (2.16). In particular, |∂E|G Ω = ΘQ−1 (|∂E|G , ·) CdQ−1 (S ∩ Ω),

(4.26)

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Francesco Serra Cassano

where CdQ−1 denotes the (Q − 1)-dimensional centered Hausdorff measure defined in Definition 2.21(iii) with X = G equipped with metric d. Moreover, there is a constant β d > 1, depending only on the distance d, such that 0
n. Indeed, if this is the case, let us consider the injective H-linear map L˜ := d P φ(p) : Rk → h n read as a homomorphism of Lie algebras. In view of ˜ k ) ⊆ V1 , where V1 is the Theorem 3.2, since it is a contact map, it follows that L(R n first horizontal layer of the stratification of h (see (2.1)). Thus V1 must contain a (trivial) k-dimensional algebra with k > n. On the other hand, it is well known that V1 can contain at most a (trivial) algebra of dimension n.

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55

In turn, Definition 4.44(ii), for k = 1, gives the notion of a H-regular hypersurface introduced in previous section. Definition 4.44(ii)—unlike the previous one—could be formally extended to k > n, but we restrict ourselves to 1 ≤ k ≤ n because only in this situation is it possible to prove (see Remark 4.85) that a C1H surface of codimension k is locally a graph in a consistent suitable sense. Nevertheless we point out that level sets of P-differentiable maps φ : H1 → R2 with surjective P-differential have been studied (see [147] and [141]). Let us introduce the definition of a Heisenberg tangent cone to a set A at a point p, which extends the classical Euclidean one in [83, 3.1.21]. Definition 4.46. Let A ⊂ Hn . The intrinsic (Heisenberg) tangent cone to A at 0 is the set ( TanH ( A, 0) := v = lim δr h (x h ) : for some (r h )h ⊂ (0, +∞), x h ∈ A h→+∞ ) with lim x h = 0 , h→+∞

and the cone at a point p is given as TanH ( A, p) := τp TanH (τ−p ( A), 0). Theorem 4.47 ([106, Theorem 3.5]). Let S be a k-dimensional H-regular surface, 1 ≤ k ≤ n. (i) S is a Euclidean k-dimensional submanifold of R2n+1 of class C1 . (ii) Let Tan S denote the Euclidean tangent bundle of S. Then Tan(S, p) = TanH (S, p) k (iii) S∞

S is comparable with Hk

∀ p ∈ S.

S.

Remark 4.48. It is easy to check that low-dimensional H-regular surfaces are graphs because they are Euclidean C1 submanifolds and because low-dimensional intrinsic graphs in Hn turn out to be Euclidean graphs. Definition 4.49. Let S = {x : φ(x) = 0} be a k-codimensional H-regular surface with φ ∈ C1H (U , Rk ) and 1 ≤ k ≤ n and let p ∈ S. The tangent group to S at p, g denoted TH (S, p), is the subgroup defined as g

TH (S, p) := ker d P φ(p). Theorem 4.50. Let S be a k-codimensional H-regular surface with 1 ≤ k ≤ n. (i) ([106, Theorem 3.27]) S is locally an intrinsic graph. More precisely, for each p ∈ S there exist

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(i1 ) two complementary subgroups W and V of Hn , that is, Hn = W · V, where W and V are, respectively, a (2n + 1 − k, 2n + 2 − k)-normal subgroup and a (k, k)-subgroup; (i2 ) an open set U ⊂ Hn with p ∈ U , a relatively open set V ⊂ W, and a continuous function φ : V ⊂ W → V such that S ∩ U = {Φ(ξ) := ξ · φ(ξ) : ξ ∈ V } . In particular, the topological dimension of S turns out to be 2n + 1 − k. g

(ii) ([106, Proposition 3.29]) TanH (S, p) = τp (TH (S, p)) for each p ∈ S. (iii) ([106, Theorem 4.1]) By the previous claim (i) and with the notation therein, there exists a positive continuous function ζ ∈ C0 (U ), depending only on f which can be explicitly represented (see [106, formula (48)]), such that Z 2n+2−k C∞ (S ∩ U ) = ζ ◦ Φ d H2n+1−k . (4.28) V

In particular, the metric dimension of S turns out to be 2n + 2 − k. Theorem 4.50(i) follows by a suitable implicit function theorem which holds when Hn can be split into complementary subgroups (see [106, Proposition 3.13]). From claim (ii) it follows that the notion of a tangent group is independent of the defining function f of S. Area formula (4.28) was erroneously stated with respect to the spherical Haus2n+2−k in [106, formula (48)]. Indeed, taking into account the previdorff measure S∞ ous Magnani example and according to Remark 4.38, the centered Hausdorff mea2n+2−k has to be used in place of the spherical one. sure C∞ 4.4.2 Regular surfaces in a general Carnot group. A notion of regular surfaces in a general Carnot group has been recently introduced by Magnani in [160]. More precisely, as a by-product of an implicit function theorem ([160, Theorem 1.3]) and a rank theorem ([160, Theorem 1.4]) in Carnot groups, and in the same spirit as the notion of an H-regular surface, two types of (G, M)-regular sets have been introduced ([160, Definitions 10.3 and 10.4]): those contained in G, which are suitable level sets, and those in M, which are suitable images. In particular, it can be proved that they are locally intrinsic graphs ([160, Corollary 1.5]).

4.5 Intrinsic Lipschitz graphs in Carnot groups. The notion of intrinsic Lipschitz continuity, for functions acting between complementary subgroups M and H of G, was originally suggested by [106, Corollary 3.17] together with the fact

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57

that an H-regular surface stays H-regular after a (left) translation (see the definition given in [107]). We propose here a geometric definition. We say that f : M → H is intrinsic Lipschitz continuous if, at each p ∈ graph ( f ), there is an (intrinsic) closed cone with vertex p, axis H, and fixed opening, intersecting graph ( f ) only in p. The equivalence of this definition and others, more algebraic, is the content of Proposition 4.56. Notice also that M and H are metric spaces, being subsets of G, hence it makes sense to speak also of metric Lipschitz continuous functions from M to H. As usual, f : M → H is said to be (metric) Lipschitz if there is L > 0 such that, for all m, m 0 ∈ M, k f (m) −1 · f (m 0 )k∞ = d ∞ ( f (m), f (m 0 )) ≤ L d ∞ (m, m 0 ) = Lkm−1 · m 0 k∞ . (4.29) The notions of intrinsic Lipschitz continuity and of Lipschitz continuity are different ones (see Example 4.58) and we will try to convince the reader that intrinsic Lipschitz continuity seems more useful in the context of functions acting between subgroups of a Carnot group. 4.5.1 Definition and first properties of intrinsic Lipschitz functions. Now we come to the basic definitions. By intrinsic (closed) cones we mean Definition 4.51. (i) Let H be a homogeneous subgroup G, q ∈ G. The cones X (q, H, α) with axis H, vertex q, opening α, 0 ≤ α ≤ 1 are defined as X (q, H, α) = q · X (e, H, α) where X (e, H, α) = {p : dist(p, H) ≤ α kpk∞ }, where dist(p, H) := inf{kp−1 · hk∞ : h ∈ H}. (ii) If M, H are complementary subgroups in G, q ∈ G, and β ≥ 0, the cones CM,H (q, β), with base M, axis H, vertex q, opening β are defined as CM,H (q, β) = q · CM,H (e, β), where CM,H (e, β) = {p : kpM k∞ ≤ βkpH k∞ } . Remark 4.52. If 0 < β1 < β2 , CM,H (q, β1 ) ⊂ CM,H (q, β2 ), and CM,H (e, 0) = H. Moreover, for all t > 0,  δ t CM,H (e, β) = CM,H (e, β). The cones CM,H (q, β) are “equivalent” to cones X (q, H, α): Proposition 4.53 ([98, Proposition 3.1]). If M,H are complementary subgroups in G then, for any α ∈ (0, 1), there is β ≥ 1, depending on α, M, and H, such that CM,H (q, 1/ β) ⊂ X (q, H, α) ⊂ CM,H (q, β).

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Definition 4.54. Let M and H be complementary subgroups of G. We say that f : E ⊂ M → H is intrinsic L-Lipschitz in E if there is L > 0 such that CM,H (p, 1/L) ∩ graph ( f ) = {p}

∀p ∈ graph ( f ).

(4.30)

The Lipschitz constant of f in E is the infimum of the L > 0 such that (4.30) holds. An intrinsic Lipschitz (continuous) function, with Lipschitz constant not exceeding L > 0, is called an L-Lipschitz function. We will call a set S ⊂ G an intrinsic Lipschitz graph if there exists an intrinsic Lipschitz function f : E ⊂ M → H such that S = graph ( f ) for suitable complementary subgroups M and H. We will call the topological dimension of M and H, respectively, the dimension and codimension of graph ( f ). Because of Proposition 4.17 and Definition 4.51, left translations of intrinsic Lipschitz H-graphs, or of intrinsic L-Lipschitz functions, stay intrinsic Lipschitz H-graphs, or intrinsic L-Lipschitz functions. We state these facts in the following theorem. Theorem 4.55. Let G be a Carnot group; then for all q ∈ G, f : E ⊂ M → H is intrinsic L-Lipschitz if and only if f q : Eq ⊂ M → H is intrinsic L-Lipschitz,where f q is the function defined in (4.8). The geometric definition of intrinsic Lipschitz graphs has equivalent algebraic forms (see also [18, 107, 98]). Proposition 4.56. Let M, H be complementary subgroups in G, f : E ⊂ M → H and L > 0. Then(i) to (iv) are equivalent. (i) f is intrinsic L-Lipschitz in E . (ii) kPH ( q¯−1 q)k∞ ≤ LkPM ( q¯−1 q)k∞

∀q, q¯ ∈ graph ( f ).

(iii) k f q¯ −1 (m)k∞ ≤ Lkmk∞ ∀q¯ = m¯ f ( m) ¯ ∈ graph ( f ) and m ∈ Eq¯ −1 .     (iv) k f (m1 ) −1 m2 · f (m1 m2 )k∞ ≤ Lk f (m1 ) −1 m2 k∞ ∀m1 , m2 ∈ E . H

M

Moreover, f : E ⊂ M → H is intrinsic Lipschitz if and only if the distance between two points q, q¯ ∈ graph ( f ) is bounded by the norm of their projection on the domain M. Precisely, for all q, q¯ ∈ graph ( f ), (v) k q¯−1 qk∞ ≤ c0 (1 + L)kPM ( q¯−1 q)k∞ =⇒ kPH ( q¯−1 q)k∞ ≤ LkPM ( q¯−1 q)k∞ , where c0 < 1 is the constant in Proposition 4.8, and conversely, (vi) kPH ( q¯−1 q)k∞ ≤ LkPM ( q¯−1 q)k∞ =⇒ k q¯−1 qk∞ ≤ (1 + L)kPM ( q¯−1 q)k∞ .

59

1 Some topics of geometric measure theory in Carnot groups

Proof. The equivalence between (i) and (ii) follows from the definition (4.51), observing that if q¯ ∈ graph ( f ), then CM,H ( q, ¯ 1/L) ∩ graph ( f ) = { q} ¯ is equivalent to CM,H (e, 1/L) ∩ graph ( f q¯ −1 ) = {e}. The equivalence of (ii) and (iii) follows once more from the definition of a cone and from the left invariance of the definition. Indeed, we recall that f q¯ −1 (m) = ( f ( m)m) ¯ H

 −1

 f m( ¯ f ( m)m) ¯ M ;

then ( qm) ¯ H = ( m¯ f ( m)m) ¯ ¯ ¯ M = ( m¯ f ( m)m) ¯ ¯ f ( m)m) ¯ H = ( f ( m)m) H and ( qm) M = m( M. Changing variables, (iv) follows from (iii). Let m2 := ( f ( m)m) ¯ M , that is, m = −1 ( f ( m) ¯ m2 )M ; then f ( m)m ¯

 −1 H

   · f m( ¯ f ( m)m) ¯ ¯ f ( m) ¯ −1 m2 )M −1 ¯ 2 M = f ( m)( H · f mm   −1  = f ( m) ¯ f ( m) ¯ −1 m2 f ( m) ¯ −1 m2 −1 ¯ 2 H H · f mm   = f ( m) ¯ −1 m2 H · f mm ¯ 2 .

Then conclude with m¯ = m1 .  For the last two inequalities observe that, if k q¯−1 qk∞ ≤ c0 (1 + L)k q¯−1 q M k∞ then k q¯−1 q



H k∞



  1 −1 k q¯ qk∞ − k q¯−1 q M k∞ ≤ L k q¯−1 q M k∞ . c0

Conversely, k q¯−1 qk∞ ≤ k q¯−1 q



M k∞

+ k q¯−1 q



H k∞

≤ (1 + L)k q¯−1 q



M k∞ .

 Remark 4.57. If G is the semidirect product of M and H, then Proposition 4.56(ii), (iii), (iv) take a more explicit form. Indeed, recalling Remark 4.19, we get the following results. (i) If M is normal in G then f is intrinsic L-Lipschitz if and only if k f ( m) ¯ −1 f (m)k∞ ≤ Lk f ( m) ¯ −1 m¯ −1 m f ( m)k ¯ ∞

∀m, m¯ ∈ E .

(ii) If H is normal in G then   (( qm) ¯ −1 )H f ( qm) ¯ M = m−1 f ( m) ¯ −1 m f mm ¯ ,

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Francesco Serra Cassano

hence Proposition 4.56(iii) becomes  km−1 f ( m) ¯ −1 m f mm ¯ k∞ ≤ Lkmk∞

∀m, m¯ ∈ E .

Hence, changing variables we get that f is intrinsic L-Lipschitz if and only if  ¯ −1 m¯ −1 m 0 f m 0 k∞ ≤ Lk m¯ −1 m 0 k∞ ∀m 0, m¯ ∈ E . km 0−1 m¯ f ( m) (iii) If G is a direct product of M and H, that is, G = M × H, we get the well-known expression for Lipschitz functions k f ( m) ¯ −1 f (m)k∞ ≤ Lk m¯ −1 mk∞ ,

∀m, m¯ ∈ E .

Hence in this case intrinsic Lipschitz functions are the same as the usual metric Lipschitz functions from (M, d ∞ ) to (H, d ∞ ). Example 4.58 ([18, Example 3.21]). Here we show that condition (4.29) is not invariant under left translations of the graph. It follows that neither intrinsic Lipschitz continuity implies (metric) Lipschitz continuity nor the opposite. Let G = H1 = W · V where V := {v = (v1 , 0, 0)} and W = {w = (0, w2 , w3 )}. Notice that kwk∞ = max{|w2 |, |w3 | 1/2 }, if w ∈ W and kvk∞ = |v1 | if v ∈ V. (i) φ : W → V, defined as φ(w) := (1 + |w3 | 1/2 , 0, 0), satisfies (4.29) with L = 1, hence φ is Lipschitz. On the other hand, φ is not intrinsic Lipschitz. Indeed, let p := (1, 0, 0) ∈ graph (()φ); from Proposition 4.17 we have φ p −1 (w) = (|w2 + w3 | 1/2 , 0, 0). For φ p −1 , Proposition 4.56(iii) does not hold. This shows also that condition (4.29) is not invariant under graph translations. (ii) ψ : W → V, defined as ψ(w) := (1 + |w3 − w2 | 1/2 , 0, 0), is intrinsic Lipschitz; indeed, with p = (1, 0, 0) and φ(w) := (|w3 | 1/2 , 0, 0) we have ψ(w) = φ p (w), so that ψ is intrinsic Lipschitz because φ is intrinsic Lipschitz. On the other hand, ψ is not Lipschitz, in the sense of (4.29), as can be easily observed. Analogously, (iii) the constant function φ : V → W defined as φ(v) := (0, 1, 0) for all v ∈ V, is Lipschitz but it is not intrinsic Lipschitz; (iv) ψ : V → W defined as ψ(v) := (0, 1, −v1 ) for all v ∈ V, is intrinsic Lipschitz continuous but it is not Lipschitz. It is natural to ask whether intrinsic Lipschitz functions are metric Lipschitz functions provided that appropriate choices of the metrics in the domain or in the target spaces are made. The answer is almost always negative. Nevertheless, something relevant can be stated (see [98, Remark 3.6]).

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61

Given f : E ⊂ M → H, we consider the function d M, f = % f : E × E → [0, +∞) defined as % f (m1 , m2 ) :=

 1  −1 k(q1 q2 )M k∞ + k(q2−1 q1 )M k∞ 2

∀m1 , m2 ∈ E ,

(4.31)

where qi := mi · f (mi ) ∈ graph ( f ). If f is an intrinsic L-Lipschitz function then % f is a quasi distance in E (see [98]). By quasi distance we mean that (QD1 ) % f is symmetric, that is, % f (m1 , m2 ) = % f (m2 , m1 ) for each mi ∈ E (i = 1, 2); (QD2 ) % f (m1 , m2 ) = 0 if and only if m1 = m2 ; (QD3 ) there exists a constant C f > 1 such that   % f (m1 , m2 ) ≤ C f % f (m1 , m3 ) + % f (m3 , m2 )

∀ mi ∈ E (i = 1, 2, 3).

However, we will also refer to the function % f as a quasi distance for a general function f : E ⊂ M → H, even if it is not intrinsic Lipschitz. Observe that in this case % f still satisfies (QD1 ) and (QD2 ). The quasi distance % f , introduced in [16] in the setting of Heisenberg groups, has been called graph distance since in this case it is equivalent to the metric d ∞ restricted to graph ( f ). More precisely, let us define the parametric graph function Φ f induced through f , Φ f : E → G,

Φ f (m) := m · f (m)

∀m ∈ E ,

(4.32)

where f : E ⊂ M → H. Proposition 4.59. Let f : E ⊂ M → H be an L-intrinsic Lipschitz function. Then there exists a positive constant C = C(L) > 1 such that   1 % f (m, m) ¯ ≤ d ∞ Φ f (m), Φ f ( m) ¯ ≤ C % f (m, m) ¯ C

∀m, m¯ ∈ E .

(4.33)

Proof. The fact that the map Φ f : (E , % f ) → (G, d ∞ ) is a metric Lipschitz function, that is, the right inequality, has been proved in [98]. The left one immediately follows by Proposition 4.8.  Let us now give a characterization of the intrinsic Lipschitz functions in terms of the quasi distance. We thank Serapioni for pointing out the proof to us as well as for other suggestions on this topic.

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Theorem 4.60. Let f : E ⊂ M → H. Then the following are equivalent: (i) f is intrinsic L 1 -Lipschitz; (ii) there exists a positive constant L 2 such that   k Φ f ( m) ¯ −1 Φ f (m) k∞ ≤ L 2 % f (m, m) ¯ H

∀m, m¯ ∈ E .

(4.34)

Proof of Theorem 4.60. (i) ⇒ (ii): By Proposition 4.56(ii), we can equivalently assume that, if q¯ = Φ( m) ¯ and q = Φ(m),     k Φ f ( m) ¯ −1 Φ f (m) k∞ ≤ L 1 k Φ f ( m) ¯ −1 Φ f (m) k∞ ∀m, m¯ ∈ E . (4.35) H

M

Since   k Φ f ( m) ¯ −1 Φ f (m) k∞ ≤ 2 % f (m, m) ¯ M

∀m, m¯ ∈ E ,

we get (4.34) by choosing L 2 = 2L 1 . (ii) ⇒ (i): We will first need a useful estimation of the quantity % f which holds for a general f : E ⊂ M → H (even if not intrinsic Lipschitz). More precisely, let us prove that there exist two positive constants Ci = Ci (G) (i = 1, 2) such that   (κ−1)/κ (κ−1)/κ 1/κ 1/κ % f (m, m) ¯ ≤ C1 kpk∞ + C2 kpk∞ kqk∞ + kqk∞ kpk∞ (4.36)     if p = Φ f ( m) ¯ −1 Φ f (m) and q = Φ f ( m) ¯ −1 Φ f (m) for all m, m¯ ∈ E , where M H κ denotes the group step. Notice that, given g ∈ G, if g = gM gH , then g −1 = (g −1 )M (g −1 )H , and g˜ := (g −1 )M (g −1 )H gH . Then we get (g) ˜ M = (g −1 )M

and

−1 g˜ = gH−1 gM gH .

(4.37)

By (4.1), (4.3), and (4.37), it follows that there exist two positive constants C = C(G) and c0 = c0 (G) such that 1 −1 −1 1 k gk ˜ ∞= kg g gH k∞ c0 c0 H M  1 C  (κ−1)/κ (κ−1)/κ 1/κ 1/κ ≤ + kgH k∞ kgM k∞ . kgM k∞ + kgM k∞ kgH k∞ c0 c0 (4.38)

k(g −1 )M k∞ ≤

0 +1 By choosing g = Φ f ( m) ¯ −1 Φ f (m) in (4.38), (4.36) follows with C1 = c2c and 0 C C2 = 2c0 . Notice now that, by (4.36), applying Young’s inequality in a standard way, it follows that there exists a positive constant C3 = C3 (G, ε),     % f (m, m) ¯ ≤ C3 k Φ f ( m) ¯ −1 Φ f (m) k∞ + ε k Φ f ( m) ¯ −1 Φ f (m) k∞ (4.39)

M

H

1 Some topics of geometric measure theory in Carnot groups

63

for all m, m¯ ∈ E , ε > 0. By (4.34) and (4.39), by choosing ε = ε(L 2 ) small enough, there exists a positive constant C4 = C4 (G, L 2 ) such that   % f (m, m) ¯ ≤ C4 k Φ f ( m) ¯ −1 Φ f (m) k∞ ∀m, m¯ ∈ E . (4.40) M



Thus, by (4.34), (4.35) follows.

Remark 4.61. From Propositions 4.8, 4.59, and Theorem 4.60 it follows that f : E ⊂ M → H is intrinsic L-Lipschitz in E if and only if (4.33) holds. We stress that in general it is impossible to find a unique quasi distance that works for all intrinsic Lipschitz functions. Notice that this is possible exactly when H is a normal subgroup.When M is a normal subgroup then not only Φ f but f itself is a metric Lipschitz function from (M, % f ) → (H, d ∞ ). Indeed, the following proposition holds. Corollary 4.62. Let M, H be complementary subgroups of G. (i) Assume that H is a normal subgroup in G. Then f : E ⊂ M → H is intrinsic Lipschitz in E if and only if Φ f : (E , d ∞ ) → (G, d ∞ ) is metric Lipschitz, that is, if and only if there is K > 0 such that   d ∞ Φ f ( m), ¯ Φ f (m) ≤ K k m¯ −1 · mk∞ ∀m, ¯ m ∈ E. (4.41) (ii) Assume that M is a normal subgroup in G. Then f : E ⊂ M → H is intrinsic Lipschitz in E if and only if f : (E , % f ) → (G, d ∞ ) is metric Lipschitz, that is, if and only if there is L > 0 such that d ∞ ( f (m), f ( m)) ¯ ≤ L % f (m, m) ¯

∀m, ¯ m ∈ E.

(4.42)

Proof. (i) This equivalence has already been proved in [98, Proposition 3.7]. On the other hand, it is an immediate consequence of Remark 4.61, noticing that, since H is normal,   Φ f ( m) ¯ −1 Φ f (m) = m¯ −1 m. M

(ii) Since M is a normal subgroup,   Φ f ( m) ¯ −1 Φ f (m) = f ( m) ¯ −1 m¯ −1 m f ( m) ¯ M

∀m, m¯ ∈ E ,

(4.43)

and 

Φ f ( m) ¯ −1 Φ f (m)

 H

= f ( m) ¯ −1 f (m)

Therefore, by Theorem 4.60 the equivalence follows.

∀m, m¯ ∈ E .

(4.44) 

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Francesco Serra Cassano

Remark 4.63. If H is not a normal subgroup Proposition 4.62(i) can be false: even if f is very regular, the “natural” parametrization of graph ( f ) given by Φ f may not be metric Lipschitz (see [98]). Finally we notice that it is an open problem to understand if and when metric Lipschitz parametrizations of graph ( f ), different from the “natural” parametrization Φ f , exist. This problem was addressed in [67] where the authors proved that, if S is a noncharacteristic C1 hypersurface in H1 ≡ R3 , then a metric bi-Lipschitz parametrization of S exists from the Euclidean plane R2 , endowed with the so-called parabolic metric, which is the restriction of distance d ∞ to a vertical plane W := {0, y,t) : y,t ∈ R2 } ≡ R2 . On the other hand, it was proved in [42] that there are no bi-Lipschitz parametrizations for a given H-regular surface in H1 , defined on an open set of R2 , endowed with the parabolic metric. We conclude this section stressing that intrinsic Lipschitz functions, even if nonmetric Lipschitz, nevertheless are metric Hölder continuous. Proposition 4.64 ([98, Proposition 3.8]). Let M, H be complementary subgroups in a step-κ group G. Let L > 0 and f : E ⊂ M → H be an intrinsic L-Lipschitz function. Then (i) f is bounded on bounded subsets of E : precisely, for all R > 0 there is C1 = C1 (M, H, f , R) > 0 such that k f (m)k∞ ≤ C1

∀m ∈ E such that kmk∞ ≤ R;

(4.45)

(ii) f is κ1 -Hölder continuous on bounded subset of E : precisely, for all R > 0, there is C2 = C2 (G, M, H,C1 , L, R) > 0 such that 1/κ k f ( m) ¯ −1 f (m)k∞ ≤ C2 k m¯ −1 mk∞

∀m, m¯ ∈ E with kmk∞ , k mk ¯ ∞ ≤ R. (4.46)

4.5.2 Surface measure of intrinsic Lipschitz graphs. Let M, H be complementary subgroups in G and let E ⊂ M be an open set. If f : E ⊂ M → H is intrinsic Lipschitz then the metric dimension of graph ( f ) is the same as the metric dimension of the domain E . In fact, we state below a stronger statement: a Lipschitz graph parametrized on a homogeneous subgroup of dimension d m is (locally) Ahlfors d m regular. A nontrivial corollary of this estimate is that 1-codimensional intrinsic Lipschitz graphs are boundaries of sets of locally finite G-perimeter (see [96, Theorem 4.2.9]). We point out that in Euclidean spaces it is always true that the Hausdorff dimension of the graph of a Lipschitz function equals the Hausdorff dimension of its domain, even if the domain fails to be open. Notice that in Euclidean spaces intrinsic Lipschitz functions are the same as Lipschitz functions. In Carnot groups

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65

this stronger statement (stronger in the sense that it holds also for lower-dimensional domains E) is false in general as the following easy example in [98] shows: in the Heisenberg group H1 := (R3 , ·) let V and W be the complementary homogeneous subgroups defined as V := {v = (v1 , 0, 0) : v1 ∈ R} and W := {w = (0, w2 , w3 ) : w2 , w3 ∈ R}. Let f : W → V be the intrinsic Lipschitz function (constant) defined as w = (0, w2 , w3 ) 7→ f (w) := (1, 0, 0). Let E := {(0, w2 , 0) : w2 ∈ R}. Since E is a horizontal curve, its metric dimension equals 1. But graph ( f |E ) = {(1, w2 , −w2 /2) : w2 ∈ R} is not a horizontal curve anymore, hence its metric dimension is larger than 1, indeed it equals 2. Notice that the proofs of upper and lower bounds on the Hausdorff measure of a Lipschitz graph are trivially true in Euclidean spaces. Indeed, if f : Rk → Rn−k is Lipschitz then the map Φ f : Rk → Rn defined as Φ f (x) := (x, f (x)) is a Lipschitz parametrization of the Euclidean graph of f ; this gives the upper bound on the dimension of the graph. On the other hand, the projection Rn ≡ Rk × Rn−k → Rk is Lipschitz continuous, with Lipschitz constant 1, yielding the lower bound. Such a proof cannot work here. On the one hand, the projections PM or PH are not Lipschitz continuous (see Example 4.12 and [24]), on the other hand, the “natural” parametrization Φ : M → G, Φ f (m) := m · f (m) is almost never a Lipschitz continuous map between the two metric spaces M and G (see [98, Remark 3.6]). Notice that the following result contains a stronger statement: it states that a Lipschitz graph parametrized on a homogeneous subgroup of dimension d m is (locally) Alhfors d m regular. Theorem 4.65 ([98, Theorem 3.9]). Let M, H be complementary subgroups in G. Let d m denote the metric dimension of M. If f : M → H is intrinsic L-Lipschitz in M then there is c = c(M, H) > 0 such that  c  dm 0 dm r d m ≤ S∞ (graph ( f ) ∩ B(p,r)) ≤ c(1 + L) d m r d m 1+L for all p ∈ graph ( f ) and r > 0, where c0 is the constant in Proposition 4.8. In particular, graph ( f ) has metric dimension d m .

4.6 Intrinsic differentiable graphs in Carnot groups. A function f : M → H, acting between complementary subgroups of G, is intrinsic differentiable at a point m ∈ M if it can be approximated by suitable intrinsic linear functions. Intrinsic linear functions, acting between complementary subgroups, are those functions whose graphs are homogeneous subgroups. Definition 4.66. Let M and H be complementary subgroups in G. Then L : M → H is an intrinsic linear function if L is defined on all of M and if graph (L) = {mL(m) : m ∈ M} is a homogeneous subgroup of G.

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Before using intrinsic linear functions in the definition of intrinsic differentiability of functions we begin collecting some of their properties. Proposition 4.67 ([96, Proposition 3.1.5]). Let M and H be complementary subgroups in G. (i) If L : M → H is intrinsic linear then the homogeneous subgroups graph (L) and H are complementary subgroups and G = graph (L) · H. (ii) If N is a homogeneous subgroup such that N and H are complementary in G then there is a unique intrinsic linear function L : M → H such that N = graph (L). Intrinsic linear functions are not necessarily group homomorphisms between their domains and codomains, as the following example shows. Let V, W be the complementary subgroups of H1 defined as V = {v = (v1 , 0, 0)} and W = {w = (0, w2 , w3 )}. For any fixed a ∈ R, the function L : V → W defined as L(v) = (0, av1 , −av12 /2) is intrinsic linear because graph (L) = {(t, at, 0) : t ∈ R} is a 1-dimensional homogeneous subgroup of H1 . This L is not a group homomorphism from V to W. Intrinsic linear functions can be algebraically characterized as follows. Proposition 4.68 ([96, Proposition 3.1.3]). Let M and H be complementary subgroups in G. Then L : M → H is an intrinsic linear function if and only if (i) L(δ λ m) = δ λ (L(m)) ∀m ∈ M and λ > 0;   −1    (ii) L(m1 m2 ) = L(m1 ) −1 m2 L L(m1 ) −1 m2 ∀m1 , m2 ∈ M. H

M

Intrinsic linear functions are intrinsic Lipschitz. Proposition 4.69 ([96, Proposition 3.1.6]). Let M and H be complementary subgroups in G and L : M → H be intrinsic linear, then (i) L is a polynomial function; (ii) L is intrinsic C-Lipschitz continuous with C := sup{kL(m)k∞ : kmk∞ = 1}. Let us point out that, in the particular case of a Heisenberg group Hn , an explicit characterization of intrinsic linear functionals can be provided, in terms of an Hlinear functional. Indeed, we have Proposition 4.70 ([18, Proposition 3.23]). Let Hn = W · V as in Example 4.4. Then

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(i) L : V → W is intrinsic linear iff Φ L : V → Hn , defined as Φ L (v) = v · L(v), is H-linear; (ii) L : W → V is intrinsic linear iff L is H-linear. We use intrinsic linear functions to define intrinsic differentiability in a way that is formally completely similar to the usual definition of P-differentiability. First, if f (e) = e we say that f : M → H is intrinsic differentiable in e if there exists an intrinsic linear map L : M → H such that, for all m ∈ M, kL(m) −1 · f (m)k∞ = o(kmk∞ ),

as kmk∞ → 0,

(4.47)

where o(t)/t → 0 as t → 0+ . Up to this point the definition of intrinsic differentiability is the same as the definition of P-differentiability. The differences appear (see Definition 4.71 below) when we extend the previous notion to any point m¯ ∈ M using (4.47) in a translated invariant way. That is, given m¯ ∈ M we consider p¯ = m¯ · f ( m) ¯ and the translated function f p¯ −1 , that, by definition, satisfies f p¯ −1 (e) = e (see (4.8)). Now we say that f is intrinsic differentiable at m¯ iff f p¯ −1 satisfies (4.47). We also give a uniform version of Definition 4.71 in Definition 4.81. Definition 4.71. Let M and H be complementary subgroups in G and f : A ⊂ M → H with A relatively open in M. For m¯ ∈ A and p¯ := m¯ · f ( m) ¯ ∈ graph ( f ), let f p¯ −1 : A p¯ −1 ⊂ M → H be the translated function in (4.8). We say that f is intrinsic differentiable in m¯ ∈ A if f p¯ −1 is intrinsic differentiable in e, that is, if there is an intrinsic linear map d f = d f m¯ : M → H such that, for all m ∈ A p¯ −1 , (i) lim km k∞ →0

kd f m¯ (m) −1 · f p¯ −1 (m)k∞ kmk∞

= 0.

The intrinsic linear map d f m¯ is called the intrinsic differential of f at m. ¯ Writing explicitly the expression for f p¯ −1 (see (4.8)) in (i), we can equivalently state that f is intrinsic differentiable at m¯ ∈ A if −1 (ii) kd f m¯ (m) −1 · ( f ( m)m) ¯ ¯ f ( m)m) ¯ M )k∞ = o(kmk∞ ), as kmk∞ → 0. H · f ( m(

Here o(t) is such that limt→0+ o(t)/t = 0. Remark 4.72. From Proposition 4.67(ii), we immediately get that the intrinsic differential is unique, whenever it exists. Remark 4.73. If a function is intrinsic differentiable it stays intrinsic differentiable after a left translation of the graph. Precisely, let q1 = m1 f (m1 ) and q2 = m2 f (m2 ) ∈ graph ( f ); then f is intrinsic differentiable in m1 if and only if f q2 ·q −1 ≡ ( f q −1 ) is 1

1

q2

intrinsic differentiable in m2 . In particular, f is intrinsic differentiable in m1 if and only if f q −1 is intrinsic differentiable in e. 1

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Francesco Serra Cassano

Proposition 4.74 ([96, Proposition 3.2.3]). Let M, H be complementary subgroups in G and f : A ⊂ M → H with A relatively open in M. If f is intrinsic differentiable in m ∈ A, then f is continuous at m. Remark 4.75. Once more we write explicitly the form taken by Definition 4.71(i), (ii) when G is the semidirect product of M and H. If M is a normal subgroup, it easy to see that ( f ( m)m) ¯ ¯ f ( m) ¯ −1 , M = f ( m)m

( f ( m)m) ¯ ¯ H = f ( m).

Then f : A ⊆ M → H is differentiable at m¯ ∈ A if   kd f m¯ (m) −1 · f ( m) ¯ −1 · f m¯ f ( m)m ¯ f ( m) ¯ −1 k∞ = o(kmk∞ ), as kmk∞ → 0 or, changing variables, if   −1 kd f m¯ f ( m) ¯ −1 m¯ −1 m f ( m) ¯ · f ( m) ¯ −1 · f (m)k∞ = o(k f ( m) ¯ −1 m¯ −1 m f ( m)k ¯ ∞ ), as m → m. ¯ Arguing as before, if H is a normal subgroup then f : A ⊆ M → H is differentiable at m¯ ∈ A if kd f m¯ (m) −1 · m · f ( m) ¯ −1 · m−1 · f ( mm)k ¯ ∞ = o(kmk∞ ),

as kmk∞ → 0,

or, changing variables, if   −1 kd f m¯ m¯ −1 m · ( m¯ −1 m) · f ( m) ¯ −1 · ( m¯ −1 m) −1 · f (m)k∞ = o(k m¯ −1 mk∞ ), as m → m. ¯ If both M and H are normal subgroups then f : A ⊆ M → H is differentiable at m¯ ∈ M if kd f m¯ (m) −1 · f ( m) ¯ −1 · f ( mm)k ¯ ∞ = o(kmk∞ ),

as kmk∞ → 0.

We can also give a characterization of the intrinsic differentiability for a function f : A ⊆ M → H in terms of a difference quotient with respect to quasi distance % f provided that M is a normal subgroup. Proposition 4.76. Let M and H be complementary subgroups of G and assume that M is a normal subgroup. Let f : A ⊆ M → H with A relatively open in M and let m¯ ∈ A. Then the following are equivalent: (i) f is intrinsic differentiable at m, ¯ that is, if there is an intrinsic linear map d f = d f m¯ : M → H such that   −1 kd f m¯ f ( m) ¯ −1 m¯ −1 m f ( m) ¯ · f ( m) ¯ −1· f (m)k∞ = o(k f ( m) ¯ −1 m¯ −1 m f ( m)k ¯ ∞ ), as m → m; ¯

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(ii) there is an intrinsic linear map d f = d f m¯ : M → H such that   −1 kd f m¯ f ( m) ¯ −1 m¯ −1 m f ( m) ¯ · f ( m) ¯ −1 · f (m)k∞ = o( % f (m, m)), ¯ as m → m, ¯ where % f is the quasi distance defined in (4.31). Proof. (i) ⇒ (ii): This implication is trivial noticing that, M being normal, k f ( m) ¯ −1 m¯ −1 m f ( m)k ¯ ∞ ≤ % f (m, m) ¯

∀m ∈ A.

(4.48)

(ii) ⇒ (i): It is sufficient to prove that there exist two positive constants C0 and r 0 such that % f (m, m) ¯ ≤ C0 k f ( m) ¯ −1 m¯ −1 m f ( m)k ¯ ∞

∀m ∈ M ∩ B∞ ( m,r ¯ 0 ).

(4.49)

Let us first prove that there exist two positive constants C1 and r 0 such that k f ( m) ¯ −1 f (m)k∞ ≤ C1 % f (m, m) ¯

∀m ∈ M ∩ B∞ ( m,r ¯ 0 ).

(4.50)

Indeed, by the assumptions, it follows that, for each m ∈ A, f ( m) ¯ −1 f (m)     −1 = d f m¯ f ( m) ¯ −1 m¯ −1 m f ( m) ¯ d f m¯ f ( m) ¯ −1 m¯ −1 m f ( m) ¯ · f ( m) ¯ −1 · f (m) . Thus, by Proposition 4.69(ii) and (4.48), (4.50) follows. Now applying (4.39), (4.43), (4.44), and (4.50), there exists a suitable positive constant C0 such that (4.49) holds.  Remark 4.77. Let us recall Pansu’s differentiability (see Definition 3.1): a function f , acting between two nilpotent groups G1 and G2 , is P-differentiable at g¯ ∈ G1 if there is a homogeneous homomorphism L : G1 → G2 such that kL(g¯ −1 · g) −1 · f (g) ¯ −1 · f (g)kG2 = o(k g¯ −1 · gkG1 ),

as k g¯ −1 · gk → 0.

We remark here that P-differentiability and intrinsic differentiability, when both of them make sense, are in general different notions. Indeed, let V, W be complementary subgroups of H1 defined as V = {v = (v1 , 0, 0)} and W = {w = (0, w2 , w3 )}. As observed before, an intrinsic linear function L : V → W is of the form L(v) = (0, av1 , −av12 /2),

for any fixed a ∈ R.

A homogeneous homomorphism h : V → W is of the form h(v) = (0, av1 , 0),

for any fixed a ∈ R.

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Obviously, L is intrinsic differentiable in v = 0 while h is P-differentiable in v = 0. On the other hand, it is easy to check that neither L is P-differentiable nor h is intrinsic differentiable in v = 0. We remark also that P-differentiability is not conserved after graph translations. Indeed, consider once more the function h : V → W; then graph (h) = {(t, at, at 2 /2) : t ∈ R}. Let p := (0, p2 , 0) ∈ H1 , p , 0. Then p · graph (h) = {(t, at + p2 , (at 2 − p2 t)/2) : t ∈ R} is the graph of the function h p : V → W defined as h p (v) = (0, av1 + p2 , −p2 v1 ). It is easy to check that h p is not Pansu differentiable in v = 0. Finally, if G is the direct product of M and H it is easy to convince oneself that f : M → H is Pansu differentiable ⇐⇒ f is intrinsic differentiable. The analytic definition of intrinsic differentiability of Definition 4.71 has an equivalent geometric formulation. Indeed, intrinsic differentiability at one point is equivalent to the existence of a tangent subgroup to the graph. We begin with the definition of a tangent subgroup. Definition 4.78. Let M, H be complementary subgroups in G, f : A ⊂ M → H with A relatively open in M, and let T be a homogeneous subgroup in G. Let m ∈ A and p = m · f (m) ∈ graph ( f ). We say that p · T is a tangent (affine) subgroup or tangent coset to graph ( f ) at p if for all ε > 0 there is λ = λ(ε) > 0 such that ( ) graph ( f ) ∩ q ∈ G : kPM (p−1 · q)k∞ < λ(ε) ⊂ X (q, T, ε), where X (q, T, ε) is the cone introduced in Definition 4.51(i). Remark 4.79. The definition is translation invariant, that is, p·T is the tangent (affine) subgroup to graph ( f ) at p if and only if T is the tangent subgroup to graph ( f p −1 ) at e. Theorem 4.80 ([96, Theorem 3.2.8]). Let M, H be complementary subgroups in G and f : A ⊂ M → H with A relatively open in M. (i) If f is intrinsic differentiable at m ∈ A, set T := graph (d f m ). Then (i 1 ) T is a homogeneous subgroup of G; (i 2 ) T and H are complementary subgroups in G; (i 3 ) p · T is the tangent coset to graph ( f ) at p := m · f (m). (ii) Conversely, if p := m · f (m) ∈ graph ( f ) and if there is T such that (i 1 ), (i 2 ), (i 3 ) hold, then f is intrinsic differentiable at m and the differential d f m : M → H is the unique intrinsic linear function such that T = graph (d f m ).

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Let us now introduce a refinement of the notion of intrinsic differentiability that will turn out to be a characterization of intrinsic C1 notion of functions between complementary subgroups in Heisenberg groups (see Theorem 4.83). Definition 4.81 ([18, Definition 3.16]). Let M and H be complementary subgroups in G and f : A ⊂ M → H with A relatively open in M. We say that f is uniformly intrinsic differentiable in A if (i) f is intrinsic differentiable at each m¯ ∈ A; (ii) d f m¯ depends continuously on m, ¯ that is, for each compact K ⊂ A, there is η = η K : [0, +∞) → [0, +∞), with limt→0+ η(t) = 0 such that, for each m¯ 1 , m¯ 2 ∈ A,   sup kd f m¯ 1 (m) −1 · d f m¯ 2 (m)k∞ ≤ η k m¯ 1−1 · m¯ 2 k∞ ; m ∈K

(iii) for each compact K ⊂ A, there is ε = ε A, K : [0, +∞) → [0, +∞), with limt→0+ ε(t) = 0 and such that for all m ∈ K p¯ −1 , and for all m¯ ∈ K, kd f m¯ (m) −1 · f p¯ −1 (m)k∞ ≤ ε(kmk∞ ) kmk∞ , where p¯ = m¯ · f ( m). ¯ Uniform intrinsic differentiability implies intrinsic Lipschitz continuity. Proposition 4.82. Let M and H be complementary subgroups in G and f : A ⊂ M → H be uniformly intrinsic differentiable in A, with A relatively open in M. Then for all m ∈ A there is r > 0 such that f is intrinsic Lipschitz in A ∩ B(m,r). Proof. Let p¯ = m¯ · f ( m) ¯ ∈ graph ( f ). Since A is relatively open there exists r > 0 such that K := M ∩ B( m,r) ¯ ⊂ A. Observe now that

K p¯ −1 = Km¯ −1 = M ∩ B(e,r),

(4.51)

and, by Propositions 4.68(i) and 4.69(ii), there exists a positive constant C1 > 0 such that kd f m¯ (m)k∞ ≤ C1 kmk∞ ∀m ∈ M ∩ B(e,r). (4.52) Since f p¯ −1 (m) = d f m¯ (m) · d f m¯ (m) −1 · f p¯ −1 (m), then by the triangle inequality, (4.52), (4.51), Definition 4.81(iii), and Proposition 4.56(iii), we get the desired conclusion. 

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4.7 H-regular surfaces, intrinsic Lipschitz graphs in Heisenberg groups vs intrinsic differentiability. In this section we deal with some relationships among intrinsic regular surfaces, intrinsic Lipschitz graphs, and intrinsic differentiability in the framework of the Heisenberg group. More precisely we will present • a characterization of an H-regular surface as locally uniformly intrinsic differentiable graphs (Theorem 4.83); • a Rademacher-type theorem for 1-codimensional intrinsic Lipschitz graphs (Theorem 4.87); • a characterization of a function whose the graph is H-regular either as a continuous distributional solution of a suitable first nonlinear PDE, namely a balance law (see Theorem 4.90), or as a continuous function admitting continuous intrinsic (partial) derivatives (see Theorem 4.95); • a characterization of an intrinsic Lipschitz function as a continuous distributional solution of a suitable balance law (see Theorem 4.92). Theorem 4.83 ([18, Theorem 4.2]). The following are equivalent: (i) S ⊂ Hn is an H-regular surface (see Definition 4.44); (ii) for all p ∈ S there is an open set U such that p ∈ U and S ∩ U is the (intrinsic) graph of a uniformly intrinsic differentiable function φ acting between complementary subgroups of Hn . More precisely, with 1 ≤ k ≤ n, if S is k-dimensional H-regular, then φ is defined on a subset of a (k, k)-dimensional horizontal subgroup; if S is k-codimensional H-regular then φ is defined on a subset of a (2n + 1 − k, 2n + 2 − k)-normal subgroup. Remark 4.84. Theorem 4.83 is an extension of a previous result [16, Theorem 4.1] for 1-codimensional intrinsic graphs in Heisenberg groups (see also [61]). Remark 4.85. Describing H-regular submanifolds as (intrinsic differentiable) graphs is more general and flexible than using parametrizations or level sets. Indeed, differently from Rn —where d-dimensional C1 embedded submanifolds are equivalently defined as noncritical level sets of differentiable functions Rn → Rn−d or as images of injective differentiable maps Rd → Rn (or as graphs of C1 functions Rd → Rn−d ) in Hn —low-dimensional H-regular surfaces cannot be seen as noncritical level sets and low-codimensional ones cannot be seen as (bi-Lipschitz) images of open sets. The reasons for this are rooted in the algebraic structure of Hn . From one standpoint, low-dimensional horizontal subgroups of Hn are not normal subgroups, hence

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they are not kernels of homogeneous homomorphisms, while they are tangent spaces to low-dimensional submanifolds; on the other hand, injective homogeneous homomorphisms Rd → Hn do not exist, if d ≥ n + 1 (see Remark 4.45). From Proposition 4.82 and Theorem 4.83 we have Corollary 4.86. Each H-regular surface S ⊂ Hn is locally a graph of an intrinsic Lipschitz function. An interesting result as far as the calculus within Carnot groups is concerned is a Rademacher-type theorem for 1-codimensional Lipschitz graphs within the Heisenberg group. More precisely, we have Theorem 4.87 ([108, Theorem 4.29]). Let Hn = W · V with V a (1, 1)-dimensional subgroup. If U ⊂ W is relatively open, and f : U → V is intrinsic Lipschitz, then f is intrinsic differentiable in U , a.e. with respect to the Haar measure of W. Remark 4.88. It is an open problem whether a Rademacher-type theorem still holds for higher-codimensional Lipschitz graphs in Heisenberg groups. For instance, assume that H2 ≡ R5 = W · V with W := exp(span{Y1 ,Y2 ,T }) = {(0, 0, w3 , w4 , w5 )} and V := exp(span{X1 , X2 }) = {(v1 , v2 , 0, 0, 0)} complementary subgroups, according to the notation of Example 2.2. Note that the subgroup W is isomorphic to the first Heisenberg group H1 ≡ R3 and V to R2 . Given an intrinsic Lipschitz map f = ( f 1 , f 2 , 0, 0, 0) : W → V it is unknown whether it is intrinsic differentiable a.e. with respect to the Haar measure of W. Remark 4.89. Franchi, Marchi, and Serapioni recently extended Theorem 4.87 to 1-codimensional intrinsic Lipschitz graphs within a more general class of Carnot groups called type ?, which contains step-2 Carnot groups (see Definition 5.23 and [96]). For the sake of simplicity, we will restrict to the first Heisenberg group H1 in order to introduce the two remaining characterizations. Moreover, we will consider the so-called X1 -graphs in H1 , that is, we will consider H1 ≡ R3 = W · V with W := exp(span{Y1 ,T }) = {(0, y,t)} and V := exp(span{X1 }) = {(x, 0, 0)} complementary subgroups, according to the notation of Example 2.2. Note that subgroup W is a normal subgroup, isomorphic to the (abelian) group R2 and V to R. Moreover, each function φ : A ⊂ W → V can be meant as a function φ(0, y,t) = ( f (y,t), 0, 0) with f : A ⊂ R2 → R, that is, we can identify φ with f and a point w = (0, y,t) ∈ W with a point A = (y,t) ∈ R2 . Let us recall some definitions and results in this framework. • With W being a normal subgroup, an intrinsic linear function L : W ≡ R2 → V ≡ R can be represented as L(0, y,t) = (α L y, 0, 0) for a suitable unique

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α L ∈ R (see Proposition 4.70 and [16, Proposition 2.15]); thus the notion of intrinsic differentiability for a function φ : A ⊂ W → V (see Definition 4.71) can be equivalently stated in terms of the quasi distance %φ (see (4.31) and Proposition 4.76) as follows: φ is intrinsic differentiable at a point w¯ = (0, y¯ , t¯) ∈ A (or we will also say that f is intrinsic differentiable at A¯ = ( y¯ , t¯)) ¯ ∈ R such that if and only if there exists a unique α = α( A) lim

A→ A¯

¯ − α (y − y¯ ) f ( A) − f ( A) =0 ¯ %˜ f ( A, A)

(4.53)

where A = (y,t), ¯ :=%φ (0, y,t), (0, y¯ , t¯)  %˜ f ( A, A)  1 ¯ + max{|y − y¯ |, σ f ( A, ¯ A)} = max{|y − y¯ |, σ f ( A, A)} 2

(4.54)

and ¯ := |t − t¯ − f ( A)(y ¯ σ f ( A, A) − y¯ )| 1/2 ;

(4.55)

¯ the ∇ f -gradient of f at A¯ and will denote it by ∇ f f ( A). ¯ we will call α( A) • Let f : A → R be a continuous function, let us introduce the (nonlinear) vector field on A, ∇ f := ∂y + f ( A) ∂t , A ∈ A, and let A¯ ∈ A be given. We say that f admits a ∂ f -derivative at A¯ if there exists α ∈ R such that for each integral curve γ : (−δ, δ) → A of ∇ f with ¯ γ(0) = A, ∃

d f (γ(s)) − f (γ(0)) f (γ(s))|s=0 = lim =α s→0 ds s

¯ (see [39, Definition 2.11]). It can be proved that, and we define α = ∂ f f ( A) if f is intrinsic differentiable at a given A¯ ∈ A and given an integral curve γ : (−δ, δ) → A of ∇ f with γ(0) = A¯ for which (δ, δ) 3 s → f (γ(s)) is of ¯ = ∇ f f ( A) ¯ (see [16, Proposition 3.7]). class C1 , then there exists ∂ f f ( A) • (See [39, Definition 3.8].) A continuous function f : A → R is a broad solution of the first nonlinear PDE (also called a balance law) ∇f f = g

in A,

(4.56)

with g ∈ L ∞ (A), if there exists a universally measurable function gˆ : A → R such that

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(B.1) gˆ = g L2 -a.e. in A; (B.2) for every integral curve γ ∈ C1 ((−δ, δ); A) of the vector field ∇ f it holds that (−δ, δ) 3 s 7→ f (γ(s)) is absolutely continuous and d f (γ(s)) = g(γ(s)) ˆ ds

a.e. s ∈ (−δ, δ);

(B.3) if f admits a ∂ f -derivative at A ∈ A, then g( ˆ A) = ∂ f f ( A). • A continuous function f : A → R is a distributional solution of the balance law (4.56) with g ∈ L ∞ (A) if ! Z Z 1 2 f ∂y ψ + f ∂t ψ d L2 = − g ψ d L2 ∀ ψ ∈ C1c (A). 2 A A We recall that a set A ⊂ Rn is universally measurable if it is measurable with respect to every Borel measure (see [65, Chapter 8, Section 4]). Universally measurable sets constitute a σ-algebra, which includes analytic sets. A function f : Rn → R is said to be universally measurable if it is measurable with respect to this σ-algebra. In particular, it will be measurable with respect to any Borel measure. Notice that Borel counterimages of universally measurable sets are universally measurable. Then the composition f ◦ g of any universally measurable function f with a Borel function g is universally measurable. This composition would be nasty with f just Lebesgue measurable. Theorem 4.90 ([16, Theorem 5.1], [41, Theorem 1.2]). Let H1 = W · V, with W = {(0, y,t)} ≡ R2 and V = {(x, 0, 0)} ≡ R and let φ = ( f , 0, 0) : A ⊂ W → V, where A is a relatively open set in W. Then the following are equivalent: (i) S = graph (φ) = {w · φ(w) : w ∈ A} is an H-regular hypersurface; (ii) f is intrinsic differentiable in A and there exists a family ( f ε )ε ⊂ C1 (A) such that f ε → f , ∇ f ε f ε → ∇ f f , ε → 0+ uniformly on compact sets of A; (iii) there exists a continuous function g : A → R, such that f is a distributional continuous solution of the balance law (4.56). Remark 4.91. Notice that a distributional continuous solution f of (4.56) with continuous source g can be very irregular. For instance, it may occur that f < BVloc (A) (see [139]).

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Theorem 4.92 ([62], [39, Theorem 1.1]). Let H1 = W · V, with W = {(0, y,t)} ≡ R2 and V = {(x, 0, 0)} ≡ R and let φ = ( f , 0, 0) : A ⊂ W → V, where A is a relatively open set in W. Then the following are equivalent: (i) φ is (locally) intrinsic Lipschitz; (ii) there exists a measurable (locally) bounded function g : A → R such that f is a broad solution of the balance law (4.56); (iii) there exists a measurable (locally) bounded function g : A → R, such that f is a distributional continuous solution of the balance law (4.56). Example 4.93. In H1 , with the notation of Theorem 4.90, fix 1/2 < α < 1 and consider φ : W → V, φ(0, y,t) = (|t| α , 0, 0). Then, since f (y,t) = |t| α is a distributional continuous solution of (4.56) with g(y,t) = 2α |t| 2α−2 t, then S = graph (φ) is an H-regular hypersurface. Moreover, one can easily check that S is not a Euclidean graph in any neighborhood of the origin (see [106, Example 3.8]). Arguing in the same way, if α = 1/2, by Theorem 4.92, S turns out to be an intrinsic (locally) Lipschitz graph. Remark 4.94. The approximation, by means of regular functions, and a Poincaré inequality for 1-codimensional intrinsic Lipschitz graphs in Heisenberg groups has been obtained, respectively, in [62] and [63]. As a by-product of Theorems 4.90 and 4.92 we can give a characterization of Hregular graphs in terms of a ∂ f -derivative and a ∇ f -gradient, similar to the Euclidean one for C1 graphs. Theorem 4.95. Let H1 = W · V, with W = {(0, y,t)} ≡ R2 and V = {(x, 0, 0)} ≡ R and let φ = ( f , 0, 0) : A ⊂ W → V, where A is a relatively open set in W. Then the following are equivalent: (i) S = graph (φ) = {w · φ(w) : w ∈ A} is an H-regular hypersurface; (ii) for f ∈ C0 (A), there exists ∂ f f ( A) for each A ∈ A and ∂ f f : A → R is continuous; (iii) f is intrinsic differentiable in A and ∇ f f : A → R is continuous. Moreover, in cases (ii) and (iii), ∂ f f ( A) = ∇ f f ( A) for each A ∈ A. Proof. (ii) ⇒ (i): Let γ : (a, b) → A be an integral curve of the vector field ∇ f . Then the function (a, b) 3 s → f (γ(s)) is of class C1 . Indeed, since there exists ∂ f f ( A) for each A ∈ A and A 3 A 7→ ∂ f f ( A) is continuous, it follows that ∃

d f (γ(s)) = ∂ f f (γ(s)) ds

∀s ∈ (a, b)

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and (a, b) 3 s → ∂ f f (γ(s)) is continuous. Therefore f turns out to be a broad solution of the balance law (4.56) with g( A) := ∂ f f ( A) for each A ∈ A. From Theorem 4.92, it holds that f is also a continuous distributional solution of the balance law (4.56) with a continuous source g. Then, applying Theorem 4.90, we get the desired conclusion. (i) ⇒ (ii): Let us first observe that, by Theorem 4.90, f is intrinsic differentiable in A. Then there exists ∇ f f : A → R continuous. Let γ : (−δ, δ) → A be an integral curve of the vector field ∇ f with γ(0) = A. Let us prove that (−δ, δ) 3 s → f (γ(s)) is C1 and ∃

d f (γ(s))|s=0 = ∇ f f ( A). ds

(4.57)

Let ( f ε )ε denote the approximating family of functions in Theorem 4.90(ii). By the classical chain rule and the fundamental theorem of calculus, it holds that Z s f ε (γ(s)) − f ε (γ(0)) = ∇ f ε f ε (γ(r)) dr ∀s ∈ (−δ, δ). 0

Again applying Theorem 4.90(ii), we can pass to the limit as ε → 0+ in the previous identity and we get that Z s f (γ(s)) − f (γ(0)) = ∇ f f (γ(r)) dr ∀s ∈ (−δ, δ). (4.58) 0

By (4.58), we can infer that ∃ ∂ f f ( A) = ∇ f f ( A)

∀A ∈ A

(4.59)

and then the desired conclusion. (iii) ⇒ (ii): Note that f ∈ C0 (A) since f is intrinsic differentiable (see Theorem 4.74). In order to get the desired conclusion, we have only to prove that (4.59) holds, that is, f (γ(s)) − f ( A) − ∇ f f ( A)s = 0 ∃ lim (4.60) s→0 |s| for each integral curve γ : [−δ, δ] → A of ∇ f with γ(0) = A, for each A ∈ A. Given an integral curve γ = (γ1 , γ2 ) : [−δ, δ] → A ⊂ R2 of ∇ f with γ(0) = A¯ = ( y¯ , t¯) ∈ A, observe that its components satisfy Z s γ1 (s) = y¯ + s, γ2 (s) = t¯ + f (γ(z)) dz ∀s ∈ [−δ, δ]. 0

Observe that, since φ is intrinsic differentiable at (0, y¯ , t¯), there exist two positive constants C1 and r 0 such that kφ(m) − φ( m)k ¯ ∞ ≤ C1 %φ (m, m) ¯

∀m ∈ B∞ ( m,r ¯ 0 ) ∩ W ⊂ A.

(4.61)

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Arguing as in the proof of Proposition 4.76 (see (4.49)), by (4.61) there exists C0 > 1 such that kφ( m) ¯ −1 m¯ −1 mφ( m)k ¯ ∞ ≤ % f (m, m) ¯ ≤ C0 kφ( m) ¯ −1 m¯ −1 mφ( m)k ¯ ∞

(4.62)

for each m ∈ B∞ ( m,r ¯ 0 ) ∩ W. We can also read (4.61) and (4.62) in terms of f , respectively, as follows: ¯ ≤ C1 %˜ f ( A, A), ¯ | f ( A) − f ( A)|

(4.63)

¯ ≤ %˜ f ( A, A) ¯ ≤ C0 max{|y − y¯ |, σ f ( A, A)}, ¯ max{|y − y¯ |, σ f ( A, A)}

(4.64)

¯ 0 ) ⊂ A, where A = (y,t), BW ( A,r ¯ 0 ) := {(y,t) ∈ R2 : (0, y,t) ∈ for all A ∈ BW ( A,r B∞ ( m,r ¯ 0 ) ∩ W}, %˜ f and σ f are the quantities defined respectively in (4.54) and (4.55). Let us now show that, for each integral curve γ : [−δ, δ] → A of ∇ f with γ(0) = ¯ there exist two positive constants C2 = C2 (C0 , γ,r 0 ) and δ1 = δ1 (γ,r 0 ) ∈ (0, δ) A, such that ¯ ≤ C2 |s| ∀s ∈ [−δ1 , δ1 ]. %˜ f (γ(t), A) (4.65) If δ1 is small enough, we can assume that ¯ 0 ) ⊂ A; γ : [−δ1 , δ1 ] → BW ( A,r moreover, by (4.63), Z s   1/2 ¯ ¯ dz σ f (γ(s), A) = f (γ(z)) − f ( A) 0 ! 1/2 ≤

C01/2 |s| 1/2 max s ∈[−δ 1,δ 1 ]

≤ 2C0 |s| +

1 2

max

s ∈[−δ 1,δ 1 ]

(4.66)

¯ %˜ f (γ(s), A)

¯ %˜ f (γ(s), A)

∀s ∈ [−δ1 , δ1 ].

Therefore, by (4.64) and (4.66), (4.65) follows. Because of (4.65), f (γ(s)) − f ( A) f ¯ − ∇ f f ( A)s ¯ ¯ ¯ ≤ 1 f (γ(s)) − f ( A) − ∇ f ( A)s ¯ |s| C2 %˜ f (γ(t), A) ∀s ∈ [−δ1 , δ1 ] \ {0}, and, since f is intrinsic differentiable at A¯ (see (4.53)), (4.60) follows by (4.53). (i) ⇒ (iii): See [16, Theorem 1.2]. 

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4.8 Rectifiability in Carnot groups. Rectifiable sets are basic concepts of geometric measure theory. They were introduced in the 1920s in the plane by Besicovitch and in 1947 in general dimensions in Euclidean spaces by Federer. A systematic study of rectifiable sets in general metric spaces was made by Ambrosio and Kirchheim in [10] in 2000. However, the definitions they used are not always appropriate in Heisenberg groups, and many other Carnot groups, when equipped with their natural CC metric; indeed, often there are only trivial rectifiable sets of measure zero. In accordance with Federer [83, 3.2.14], let us recall Definition 4.96. A set E ⊂ (X, d) is said to be countably Hdk -rectifiable if there is a sequence of Lipschitz function f i : Ai ⊂ (Rk , k·kRk ) → (X, d) (i = 1, 2, . . . ) such that Hdk (E \ ∪∞ i=1 f i ( Ai )) = 0. Remark 4.97. It is also well known that, in the Euclidean setting (that is, when X = Rn ), the countable k-rectifiability can be also expressed in terms of k-dimensional surfaces, by the Whitney extension theorem. Namely, it holds that E is countably Hk -rectifiable if and only if there is a sequence of k-dimensional C1 surfaces (Si )i ⊂ Rn such that Hk (E \ ∪∞ i= Si ) = 0. Definition 4.98. A metric space (X, d) is said to be purely k-unrectifiable if for each Lipschitz function f : A ⊂ (Rk , k·kRk ) → (X, d),

Hdk ( f ( A)) = 0. Ambrosio and Kirchheim proved the following result. Theorem 4.99 ([10, Theorem 7.2]). The first Heisenberg group (H1 , d) is purely k-unrectifiable for k = 2, 3, 4, for each invariant distance d. A general criterion for unrectifiability was proved by Magnani by means of the area formula for Lipschitz maps between Carnot groups (see Theorem 3.19). Theorem 4.100 ([156, Theorem 1.1]). Let G be a Carnot group and let V1 denote the first horizontal layer in its stratification (see (2.1)). Then (G, d) is purely kunrectifiable iff V1 does not contain a (trivial) Lie subalgebra of dimension k. Therefore, taking the previous unrectifiability results into account, a new, suitable notion of rectifiability in Carnot groups is needed, better fitting the new geometry. 4.8.1 (Q − 1)-rectifiable sets in Carnot groups. A suitable notion of rectifiability in Carnot group for sets of metric dimension (Q − 1) was first introduced in the

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setting of Heisenberg groups [102] and then step-2 Carnot groups for studying the structure of sets of finite G-perimeter [105]. We recall that De Giorgi’s celebrated structure theorem in Euclidean spaces ([74, 75]) states that if E ⊂ Rn is a set of locally finite perimeter, then the associated perimeter measure |∂E| is concentrated on a portion of the topological boundary ∂E, the so-called reduced boundary ∂ ∗ E ⊂ ∂E. In addition, ∂ ∗ E is H n−1 -rectifiable, i.e., ∂ ∗ E, up to a set of (n − 1)-Hausdorff measure zero, is a countable union of compact subsets of C1 submanifolds and the perimeter measure is the (n − 1)-Hausdorff measure of the reduced boundary. Roughly speaking, this says that the perimeter measure is supported on a portion of the topological boundary ∂E, that can be expressed— after removing a negligible set of “bad points”—as the countable union of compact subsets of “good hypersurfaces”. If in the spirit of De Giorgi’s theorem we want to describe the structure of sets of finite intrinsic perimeter in a Carnot group G, we need a natural notion of rectifiable subsets, and in this perspective, the correct definition of “good hypersurfaces”, i.e., of intrinsic C 1 -regular submanifolds of G given in Definition 4.20. Keeping in mind this notion, the following definition is the natural counterpart of the corresponding Euclidean definition (see [103] and [96]). Definition 4.101. Let G be a Carnot group and let Q denote its homogeneous dimension. (i) E ⊂ G is said to be (Q−1)-dimensional G-rectifiable if there exists a sequence of G-regular hypersurfaces (Si )i such that ∞ [   Q−1 S∞ E\ Si = 0.

(4.67)

i=1

(ii) E ⊂ G is said to be (Q − 1)-dimensional G L -rectifiable if there exists a sequence of 1-codimensional intrinsic Lipschitz graphs (Si )i such that ∞ [   Q−1 E\ Si = 0. S∞

(4.68)

i=1

Remark 4.102. Franchi, Marchi, and Serapioni recently proved that, for Carnot groups of type ? (see Definition 5.23), (Q − 1)-dimensional G-rectifiability and (Q − 1)-dimensional G L -rectifiability are equivalent (see [96, Proposition 4.4.4]). Before we enter the study of the rectifiability of the reduced boundary (whatever this means, as we shall see below), let us point out the relationships between our definition in Carnot groups and the standard Euclidean notion. The following result proved in [105] yields that “negligible” subsets of codimension 1 in a Carnot group

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with respect to the Euclidean distance are “negligible” subsets of codimension 1 with respect to Carnot–Carathéodory distance. Proposition 4.103 ([105, Proposition 4.4]). Let G = (Rn , ·) be a Carnot group. For any α ≥ 0 and R > 0 there is a constant c(α, R) > 0 such that for any set E ⊂ G ∩ U∞ (0, R), α+Q−n H∞ (E) ≤ c(α, R) Hα (E),

α ≥ 0.

(4.69)

In particular, for all E ⊂ G, α+Q−n Hα (E) = 0 =⇒ H∞ (E) = 0,

α ≥ 0.

(4.70)

Proposition 4.103 combined with Theorem 4.23 yields Theorem 4.104. Let G = (Rn , ·) be a Carnot group. Then, if S is an (n − 1)-dimensional Euclidean rectifiable set of Rn then S is also (Q−1)-dimensional G-rectifiable. On the other hand, there are (Q − 1)-dimensional G-rectifiable sets in Carnot groups G = (Rn , ·) that are not (n − 1)-dimensional Euclidean rectifiable. Indeed, in [28] a set N ⊂ R3 is constructed, such that for an appropriate ε > 0, 3 H∞ (N ) = 0 and H2+ε (N ) > 0.

Hence N is (trivially) 3 = (Q − 1)-dimensional H1 -rectifiable, but it is not 2-dimensional Euclidean rectifiable because its Euclidean Hausdorff dimension is strictly larger than 2. As we mentioned above, a sharper result in this direction is contained in [139]: there exist G-regular hypersurfaces in the Heisenberg group H1 (Q = 4, n = 3) with Euclidean Hausdorff dimension 2.5. We recall here that relationships between Euclidean and intrinsic Hausdorff measures in Heisenberg groups have been deeply investigated in [28], where sharp results were also obtained. This study was extended more recently to a general Carnot group in [30]. Finally let us stress that each H-regular t-graph in Hn ≡ R2n+1 , that is, each graph with respect to the subgroup T = {(0, . . . , 0, p2n+1 )}, keeps the same Euclidean metric dimension. Indeed, we have Theorem 4.105 ([213, Corollary 1.2]). Let S ⊂ Hn be an H-regular hypersurface that is not (Euclidean) countably H2n -rectifiable; then S is not a t-graph. In particular, each H regular t-graph has Euclidean metric dimension 2n. 4.8.2 Rectifiability in Heisenberg groups. In the setting of Heisenberg groups, a more complete notion of rectifiability can be introduced. With the intrinsic notion of regular submanifolds proposed in Definition 4.44 and the one of intrinsic Lipschitz graphs in Definition 4.51, it is possible, following the two usual approaches, to give an (intrinsic) notion of rectifiable sets in Hn .

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Definition 4.106. (i) E ⊂ Hn is a (k, H)-rectifiable set if there is a sequence of k-dimensional H-regular surfaces Si such that km (E \ ∪∞ S∞ i=1 Si ) = 0,

where k m = k, if 1 ≤ n, and k m = k + 1, if n + 1 ≤ k ≤ 2n. (ii) E ⊂ Hn is a (k, H L )-rectifiable set if (ii1 ) for 1 ≤ k ≤ n, there exists a sequence of Lipschitz functions f i : Ai ⊂ Rk → Hn such that k S∞ (E \ ∪∞ i=1 f i ( Ai )) = 0;

(ii2 ) for n + 1 ≤ k ≤ 2n, there exists a sequence of intrinsic Lipschitz graphs Si = graph ( f i ) of dimension k, with f i : Ai ⊂ Wi → Vi , such that k+1 S∞ (E \ ∪∞ i=1 Si ) = 0,

where Wi and Vi are, respectively, (k, k + 1)- and (2n + 1 − k, 2n + 1 − k)subgroups, which are complementary. Remark 4.107. We recall that, by a combination of Proposition 4.103 with Theorem 4.23, for n + 1 ≤ k ≤ 2n, a k-dimensional Euclidean rectifiable set E ⊂ R2n+1 ≡ Hn is always a (k, H)-rectifiable set, while the converse is false. On the other hand, for 1 ≤ k ≤ n, a (k, H)-rectifiable set is k-dimensional Euclidean rectifiable while the opposite is false: for instance, the vertical T axis in H1 ≡ R3 gives the simplest example of a Euclidean 1-dimensional rectifiable set that is not 1-dimensional Hrectifiable. Remark 4.108. Definition 4.106(i) is nontrivial only if 1 ≤ k ≤ n, because, for k ( f ( A)) = 0 always (see Theorem n+1 ≤ k and f : A ⊂ Rk → Hn Lipschitz, then S∞ 4.100), hence we will consider this notion of rectifiability only if 1 ≤ k ≤ n. Remark 4.109. We explicitly observe that we do not know whether (k, H L )-rectifiability and (k, H)-rectifiability are equivalent if 1 < k ≤ n. In Euclidean spaces these two definitions are equivalent and in fact the type of definition below was Federer’s original definition, and it has also often been used in general metric spaces. We don’t know whether they are equivalent in Hn (see also [25]). The problem is that we don’t have a suitable Whitney extension theorem for maps f : F ⊂ Rk → Hn when F is closed. Speight [215] recently gave a positive answer for k = 1. In [170] we approached the problem of characterizing intrinsic k-rectifiable sets in Hn through almost everywhere existence and uniqueness of generalized tangent subgroups or tangent measures, according to the notions introduced, respectively, by Mattila [168] and Preiss [198] in the Euclidean setting. With regard to generalized tangent groups the result reads as follows.

83

1 Some topics of geometric measure theory in Carnot groups km measure. Theorem 4.110. Let E ⊂ Hn be a set which has locally finite S∞

(i) ([170, Theorem 3.14]) Let 1 ≤ k ≤ n. Then E is (k, H L )-rectifiable if and k a.e. The latter means only if it has an approximate tangent subgroup T p , S∞ k that, for S∞ a.e. p ∈ E, there exists a homogeneous subgroup T p , of topological and metric dimension k, such that  ( ) k E ∩ B(p,r) ∩ q : d (p−1 · q, T ) > s d (p, q) S∞ ∞ p ∞ lim = 0, r →0 rk for each s > 0. (ii) ([170, Theorem 3.15]) Let n + 1 ≤ k ≤ 2n. Then E is (k, H)-rectifiable if k a.e. The latter and only if it has an approximate tangent subgroup T p , S∞ k means that, for S∞ a.e. p ∈ E, there exists a homogeneous subgroup T p , of topological dimension k and metric dimension k + 1, such that  ( ) k+1 E ∩ B(p,r) ∩ q : d (p−1 · q, T ) > s d (p, q) S∞ ∞ p ∞ lim = 0, r →0 r k+1 for each s > 0, and lim inf r →0

k+1 (E ∩ B(p,r)) S∞ > 0. r k+1

(4.71)

Remark 4.111. We point out that some other notions of rectifiability in Carnot groups were introduced by Pauls [193] and, recently, by Ghezzi & Jean [121]. However, to our knowledge, there is no comparison among these notions of rectifiability.

5 Sets of finite perimeter and minimal surfaces in Carnot groups 5.1 Sets of finite perimeter and reduced boundary. Given a Carnot group G = (Rn , ·), we have already introduced the notion of a G-perimeter measure for a set E ⊂ G, written |∂E|G , (see Definition 3.37) and the one of a set of finite G-perimeter (or a G-Caccioppoli set). When the Euclidean space G = (Rn , +) is meant as a (trivial) Carnot group (see Example 2.1), we will simply denote by |∂E| the associated perimeter measure to a set E ⊂ Rn . We will call a Euclidean set of (locally) finite perimeter, a set E for which |∂E| is a Radon measure on Rn . The notion goes back to De Giorgi, who

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introduced it in the pioneering papers [74, 75], strongly inspired by some previous ideas of Caccioppoli (see [7] for an interesting account of Euclidean sets of finite perimeter). Later, De Giorgi [78] successfully used the sets of finite perimeter in order to establish existence and regularity of minimal surfaces for 1-codimensional surfaces in Rn (see [122, 8, 152]). We will apply this approach in Section 5.3 in order to get existence of the so-called nonparametric minimal surfaces in the setting of the simplest Carnot group, namely the Heisenberg group Hn . Caccioppoli’s primitive idea, then refined by De Giorgi through sets of finite perimeter, considered oriented hypersurfaces, which (at least locally) are boundaries of sets, and exploited techniques of measure theory. Thus one of the main problems studied by De Giorgi was the structure of the perimeter’s measure support, which is a typical issue of geometric measure theory. Definition 5.1. Let (X, d) be a separable metric space and let µ be a Borel outer measure on X. The set spt µ := X \ {x ∈ X : ∃ r > 0 such that µ(B(x,r)) = 0} is called the support of µ. Let us recall some simple properties for sets of (locally) finite G-perimeter, which can be proved as in the classical case (see, for instance, [8, Proposition 3.38]). Proposition 5.2. Let G be a Carnot group and let Ω ⊂ G be an open set and let E and F be measurable subsets of G. Then (i) spt |∂E|G ⊂ ∂E; (ii) |∂E|G (Ω) = |∂(G \ E)|G (Ω); (iii) (locality of G-perimeter measure) |∂E|G (Ω) = |∂(E ∩ Ω)|G (Ω); (iv) |∂(E ∪ F)|G (Ω) + |∂(E ∩ F)|G (Ω) ≤ |∂E|G (Ω) + |∂F |G (Ω). Remark 5.3. Notice that from Proposition 5.2(i) it follows that measure |∂E|G is concentrated on ∂E, that is, |∂E|G (B) = |∂E|G (B ∩ ∂E) for each Borel set B ⊂ Rn . However, the topological boundary ∂E of a set E of finite Euclidean perimeter could be very irregular (see [122, Example I.10]). For instance, there exists a set E ⊂ Rn of finite perimeter in Rn , that is, |∂E|(Rn ) < +∞, such that Ln (∂E) > 0. Remark 5.4. An interesting characterization of sets of locally G-finite perimeter by means of a polyhedral approximation, in the spirit of the original approach due to Caccioppoli, was carried out in [176].

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De Giorgi’s strategy was to single out a smaller set within the topological boundary, where the perimeter measure is concentrated, which is also regular from the viewpoint of measure theory, that is, is countably rectifiable (see Definition 4.96). De Giorgi introduced to this goal the notion of reduced boundary, which can easily be extended to a Carnot group. Definition 5.5 (De Giorgi’s reduced boundary). Let d be an invariant metric on a Carnot group G (see Definition 2.6). Let E be a G-Caccioppoli set and let νE denote the generalized inward horizontal normal defined in (3.13); we say that a point x ∈ G belongs to the G-reduced boundary of E, written x ∈ ∂G∗ E, if (i) |∂E|G (Bd (x,r)) > 0 for any r > 0; R (ii) there exists limr →0 B (x,r ) νE d|∂E|G ; d R

(iii)

limr →0 B (x,r ) νE d|∂E|G

m1 = 1. d R The limits in Definition 5.5 should be understood as a convergence of the averages of the coordinates of νE with respect to the chosen moving base of the fibers. Definition 5.5 is a straightforward extension of its Euclidean counterpart, but its utility is not obvious. Indeed, in the Euclidean setting, it is immediate to show that the perimeter measure is concentrated on the reduced boundary, since, by Lebesgue– Besicovitch differentiation lemma (see [8, Theorem 2.22]), given a Radon measure µ on Rn , for any f ∈ L 1loc (Rn , dµ), Z f (y) dµ = f (x) for µ-a.e. x ∈ Rn , lim r →0

B Eu (x,r )

where BEu (x,r) := {y ∈ Rn : |y − x| ≤ r } is the (closed) ball induced by the Euclidean distance. This implies that |∂E| = |∂E| ∂G∗ E. Unfortunately, the Besicovitch covering lemma, that is the main tool of the proof of the Lebesgue–Besicovitch differentiation lemma fails to hold in Carnot groups (see [140, 210, 200]). Recently Le Donne and Rigot [146] proved that, in Heisenberg groups, there is an invariant distance for which the Lebesgue–Besicovitch differentiation lemma holds. Nevertheless, we do not know whether the Lebesgue–Besicovitch differentiation lemma still holds in general Carnot groups, but at least it holds when µ is the perimeter measure, thanks to a deep asymptotic estimate proved by Ambrosio in [4]. The corresponding differentiation lemma reads as follows. Lemma 5.6 (Differentiation lemma, [4]). Assume E is a G-Caccioppoli set; then Z νE d|∂E|G = νE (x), for |∂E|G -a.e. x, lim r →0

B d (x,r )

that is, |∂E|G -a.e. x ∈ G belongs to the reduced boundary ∂G∗ E.

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Moreover, the following representation of the G-perimeter measure with respect to the spherical Hausdorff measure holds. Theorem 5.7 ([4]). Let G = (Rn , ·) be a Carnot group and let Q denote its homogeneous dimension (see (2.19)). Then there exists a Borel function w : ∂G∗ E → [c1 , c2 ] such that |∂E|G = w SdQ−1 ∂G∗ E, that is, Z |∂E|G (B) = w(x) d SdQ−1 (x) for each Borel set B ⊂ Rn , (5.1) B∩∂G∗ E

where SdQ−1 denotes the (Q − 1)-dimensional spherical Hausdorff measure (see Definition 2.21(ii)) and ci = ci (G) (i=1,2) are positive geometric constants. From Theorems 5.7, 4.29, and 4.36, the following consequence immediately follows. Corollary 5.8. Under the same assumptions as Theorem 5.7, (i) w(x) = Θ∗FQ−1 (|∂E|G , x) for SdQ−1 -a.e. x ∈ ∂G∗ E, where Θ∗FQ−1 (|∂E|G , ·) denotes the (Q − 1)-Federer density introduced in Definition 4.28(ii); (ii) |∂E|G = Θ∗ Q−1 (|∂E|G , ·) CdQ−1 ∂G∗ E, where Θ∗ Q−1 (|∂E|G , ·) denotes the (Q − 1)-spherical upper density introduced in Definition 4.28(i) and CdQ−1 denotes the (Q − 1)-dimensional centered Hausdorff measure (see Definition 2.21(iii)). Remark 5.9. Let us recall that Θ∗FQ−1 (|∂E|G , x) and Θ∗ Q−1 (|∂E|G , x) may differ for x ∈ ∂G∗ E, if d is a general invariant distance on G (see Remark 4.31). Remark 5.10. The intrinsic rectifiability of the reduced boundary ∂G∗ E is still an open problem if G is a Carnot group of step greater than 2. We will deal with this issue in the next section. It is well known that, if E ⊂ G := (Rn , ·) is of locally finite Euclidean perimeter (that is, it is a set of locally finite G-perimeter for the trivial Carnot group G = (Rn , +)), then it is also of locally finite G-perimeter. For such a set one can represent its G-perimeter measure both with respect to Euclidean (n − 1)-dimensional Hausdorff measure H n−1 and (sub-Riemannian) (Q − 1)-spherical Hausdorff measure SdQ−1 . This representation is well known when ∂E is regular (see Proposition 3.39) and E is a set of locally finite perimeter in G = Hn (see [213, Proposition 2.10]). We now extend the result to any Carnot group. We denote by ∂ ∗ E and NE (P), respectively, the Euclidean reduced boundary of E and the generalized Euclidean inward normal to E at P ∈ ∂ ∗ E, that is, according to Definition 5.5, NE = νE when considering the trivial Carnot group G := (Rn , +).

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Proposition 5.11. Let G = (Rn , ·) be a Carnot group and let X = (X1 , . . . , X m1 ) denote a system of generating vector fields according to Section 2.2. Let E be a set with locally finite Euclidean perimeter in Rn . Then E has locally finite G-perimeter and |∂E|G = |NEG | H n−1 ∂ ∗ E = w SdQ−1 ∂ ∗ E, (5.2) where w is the density in (5.1) and NEG := (hX1 , NE i, . . . , hX m1 , NE i) ∈ Rm1 . In particular, if we denote ( ) Char(E) := p ∈ ∂ ∗ E : NEG (p) = 0 , then

SdQ−1 (Char(E)) = 0.

(5.3)

Remark 5.12. Proposition 5.11 extends Proposition 3.39 and Theorem 4.23. Proof. It is well known that X j χ E = hX j , NE i|∂E| = hX j , NE iH n−1

∂∗ E

holds in the sense of distributions for any j = 1, . . . , m1 , |∂E| being the Euclidean perimeter of E. The first equality in (5.2) immediately follows. Moreover, from Theorem 5.7, (5.4) |NEG | H n−1 ∂ ∗ E = |∂E|G = w SdQ−1 ∂G∗ E and thus the second equality in (5.2) will follow if we show that

SdQ−1 (∂G∗ E∆∂ ∗ E) = 0.

(5.5)

Notice that, by (5.4), since 0 < c1 ≤ w(x) ≤ c2 for each x ∈ ∂G∗ E, we have

SdQ−1 (∂G∗ E \ ∂ ∗ E) = 0 and

Z F1

|NEG | d H n−1 = 0

if F1 = ∂ ∗ E \ ∂G∗ E.

(5.6)

(5.7)

From (5.6), (5.5) follows provided we show that

SdQ−1 (∂ ∗ E \ ∂G∗ E) = 0.

(5.8)

H n−1 (∂ ∗ E \ ∂G∗ E) = 0

(5.9)

By (5.7) it follows that

or NEG = 0,

H n−1 -a.e. on ∂ ∗ E \ ∂G∗ E.

(5.10)

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If (5.9) holds, because of (4.69), since d ∞ and d are equivalent, (5.8) follows at once. Thus, assume that (5.10) holds. Since ∂ ∗ E is locally (n − 1)-countably rectifiable in the Euclidean sense (see, for instance, [8, Theorem 3.59]), there exists a family (S j ) j ∈N of (Euclidean) C1 surfaces Q−1 ∗ (∂ E \∪∞ in G such that H n−1 (∂ ∗ E \∪∞ j=0 S j ) = 0, again j=0 S j ) = 0 (whence also Sd because of (4.69) ) and N E = NS j ,

H n−1 -a.e. on ∂ ∗ E ∩ S j ,

(5.11)

NS j being the Euclidean unit normal to S j . Theorem 4.23 ensures that for any j,  SdQ−1 {P ∈ S j : hNS j (P), X1 (P)i = · · · = hNS j (P), X m1 (P)i = 0} = 0. Taking into account the fact that (5.11) holds also SdQ−1 -a.e. on ∂ ∗ E ∩ S j (recall  (4.69)), we deduce that SdQ−1 {P ∈ ∂ ∗ E ∩ S j : NEG (P) = 0} = 0 for any j, i.e., (5.3). By (5.3) and using again (5.4) as before, we get that  (5.7) now holds with F1 = (∂ ∗ E \ char(E)) \ ∂G∗ E. This now implies that H n−1 (∂ ∗ E \ char(E)) \ ∂G∗ E = 0, which in combination with (4.69) yields   (5.12) SdQ−1 (∂ ∗ E \ char(E)) \ ∂G∗ E = 0. By (5.3) and (5.12), (5.8) follows.



Corollary 5.13. Under the same assumptions as Proposition 5.11, the G-reduced boundary ∂G∗ E is (Q − 1)-dimensional G-rectifiable (see Definition 4.101(i)), that is, Q−1 ∗ (∂G E \ there exists a sequence of G-regular hypersurfaces (S ∗j ) j of G such that S∞ ∞ ∗ ∪ j=1 S j ) = 0. Proof. Since d and d ∞ are equivalent, it is sufficient to prove that there exists a sequence of G-regular hypersurfaces (S ∗j ) j of G such that ∗ SdQ−1 (∂G∗ E \ ∪∞ j=1 S j ) = 0.

(5.13)

By (5.6), we have only to prove that ∂ ∗ E is (Q − 1)-dimensional G-rectifiable

(5.14)

in order to get (5.13). Let us recall that ∂ ∗ E is (n − 1)-countably rectifiable in a Euclidean sense. Thus from Theorem 4.104, (5.14) follows. 

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5.2 De Giorgi’s structure theorem for sets of finite perimeter in Carnot groups of step 2. From now on the group G will be a step-2 Carnot group endowed with the invariant distance d ∞ . Indeed, the key step for the main result of this paper, i.e., the so-called blow-up theorem stated below, fails to be true for general groups of step greater than 2, as we can see from Example 5.15. Specializing our notation, in step-2 Carnot groups, we have g = V1 ⊕ V2 ,

[V1 ,V1 ] = V2 ,

[V1 ,V2 ] = {0},

with homogeneous dimension Q = m1 + 2(n − m1 ). We can now prove the following results: (i) at each point of the reduced boundary of a G-Caccioppoli set there is a (generalized) tangent group (see also (4.10)); (ii) both the reduced boundary and the measure-theoretic boundary are (Q − 1)dimensional G-rectifiable sets; Q−1 (iii) |∂E|G = ϑ0 S∞ ∂G∗ E, i.e., the perimeter measure equals a constant times the spherical (Q − 1)-dimensional Hausdorff measure restricted to the reduced boundary;

(iv) an intrinsic divergence theorem holds for G-Caccioppoli sets. The precise meaning of statement (i) is the content of the following blow-up theorem. It is precisely point (i) that can be false in a general Carnot group. Indeed, we provide an example of a set E of (locally) finite G-perimeter in the (step-3) Engel group (see Example 4.7) such that 0 ∈ ∂G∗ E but E does not have a generalized tangent group at that point. Statement (iii) fits into the general problem of comparing different geometric measures in Carnot groups. A good reference for this problem, in Euclidean spaces, is Mattila’s book [168]. In the setting of the Heisenberg group, in [68] it is proved that the perimeter of a Euclidean C 1,1 -hypersurface is equivalent to its (Q − 1)dimensional intrinsic Hausdorff measure, whereas in [102] it was stated that on the reduced boundary of sets of finite intrinsic perimeter, the (Q − 1)-dimensional intrinsic spherical Hausdorff measure coincides—after a suitable normalization—with the perimeter measure: the correct proof was obtained later in [162]. In the setting of general Carnot groups the problem is essentially open. The equivalence of the intrinsic perimeter and of the (Q − 1)-dimensional intrinsic Hausdorff measure for G-regular hypersurfaces in a general Carnot groups was studied in Section 4.3. In addition, the perimeter measure of a smooth set in general sub-Riemannian spaces

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equals the intrinsic Minkowski content, as proved in [186]. In Ahlfors-regular metric spaces, a general representation theorem of the perimeter measure of sets of finite perimeter in terms of the Hausdorff measure is proved in [4] (see also the refined result for sub-Riemannian manifolds in [5]), showing that the intrinsic perimeter admits a density w with respect to the Hausdorff measure that is locally summable and bounded away from zero: this result in a Carnot group is Theorem 5.7. Statement (iii) says precisely that, thanks to (i) and (ii), in step-2 Carnot groups the function w is constant. Let us now fix some notation. For any set E ⊂ G, x 0 ∈ G, and r > 0 we consider the translated and dilated sets Er, x0 defined as Er, x0 = {x : x 0 · δr (x) ∈ E} = δ1/r τx −1 E. 0

If x 0 is fixed and there is no ambiguity, we shall write simply Er , and in addition we set E x0 = E1, x0 . Moreover, if v ∈ HG x0 we define the halfspaces SG+ (v) and SG− (v) as SG+ (v) := {x : hπ x0 x, vi x0 ≥ 0}, SG− (v) := {x : hπ x0 x, vi x0 ≤ 0},

(5.15)

where π x0 is the map defined in (3.2). g The common topological boundary TG (v) of SG+ (v) and of SG− (v) is the subgroup of G, g TG (v) := {x : hπ x0 x, vi x0 = 0}. Theorem 5.14 (Blow-up theorem, [105]). Let G be a step-2 Carnot group. If E is a G-Caccioppoli set, x 0 ∈ ∂G∗ E and νE (x 0 ) ∈ HG x0 is the inward normal as defined in (3.13) then lim 1 E r, x = 1SG+ (ν E (x0 )) in L 1loc (G) (5.16) r →0

0

and for all R > 0, lim |∂Er, x0 |G (U∞ (0, R)) = |∂SG+ (νE (x 0 ))|G (U∞ (0, R)).

r →0

(5.17)

Notice that, by Proposition 3.39, |∂SG+ (νE (x 0 ))|G (U∞ (0, R)) = H n−1 (TG (νE (0)) ∩ U∞ (0, R)). g

As we have already pointed out, Theorem 5.14 fails to hold in general Carnot groups of step k > 2. In fact, the core of the following example consists in showing that, in Carnot groups of step greater than 2, cones can exist (i.e., dilation-invariant sets) that are not flat (they are not of the form SG± (v) for some horizontal vector v) but nevertheless with vertex belonging to the reduced boundary. The following counterexample was inspired by Martin Reimann, and then Roberto Monti found a preliminary form of the counterexample itself.

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Example 5.15. Let us recall the definition of an Engel algebra and group. Let E = (R4 , ·) be the Carnot group whose Lie algebra is g = V1 ⊕ V2 ⊕ V3 with V1 = span {X1 , X2 }, V2 = span {X3 }, and V3 = span {X4 }, the only nonzero commutation relations being [X1 , X2 ] = −X3 ,

[X1 , X3 ] = −X4 .

Notice that the Engel algebra introduced here is different from the one introduced in Example 4.7. However, they are isomorphic, as well as their associated Carnot groups. This choice is motivated only for simplicity of calculation. In exponential coordinates the group law takes the form x·y=H

4 X

x i Xi ,

i=1

4 X

 yi X i ,

i=1

where H is given by the Campbell–Hausdorff formula 1 1 H (X,Y ) = X + Y + [X,Y ] + ([X, [X,Y ]] − [Y, [X,Y ]]) . 2 12

(5.18)

In exponential coordinates an explicit representation of the vector fields is  x3 x1 x2  x2 ∂3 + − ∂4 , 2 2 12 x1 X3 = ∂3 − ∂4 , 2

X1 = ∂1 +

X2 = ∂2 −

x2 x1 ∂3 + 1 ∂4 , 2 12

X4 = ∂4 .

Let E = {x ∈ R4 : f (x) ≥ 0}, where f (x) =

1 1 x 2 (x 21 + x 22 ) − x 1 x 3 + x 4 . 6 2

Since ∂E = {x ∈ R4 : f (x) = 0} is a smooth Euclidean manifold, then E is a G-Caccioppoli set (see Proposition 3.39). Moreover,  1  ∇E f (x) = 0, (x 21 + x 22 ) , 2 and, by the implicit function theorem (Theorem 4.24), νE (x) = −

∇E f (x) = (0, −1) |∇E f (x)|

for all x ∈ ∂E \ N, where N = {x ∈ E : x 1 = x 2 = 0}. Since |∂E|E (N ) = 0, then the origin belongs to the reduced boundary of E. On the other hand, since f (δ λ x) = λ 3 f (x) for λ > 0, it follows that Eλ,0 = δ λ E = E, so that (5.16) fails to be true since E is not a vertical halfspace.

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Even if we do not enter into the details of the proof of Theorem 5.14, we want to stress the technical point where the assumption on the step of G is used. In the ∂f ∂f Euclidean setting an elementary statement says that ∂x = · · · = ∂x = 0 implies n 2 f = f (x 1 ). In Carnot groups the corresponding statement should be that the vanishing of X2 f to X m1 f yields that f is a function of just one variable. But this is false as examples in the Heisenberg group H1 show (see Example 5.17 below). What is possible to prove in step-2 groups is that if Y1 , . . . ,Ym1 are left-invariant smooth orthonormal (horizontal) sections, if Y2 f = · · · = Ym1 f = 0 and if Y1 f is positive, then f is an increasing function of one variable. Example 5.15 shows that in groups of step 3 or larger, even this last weaker statement is false. Lemma 5.16 ([105, Lemma 3.6]). Let G be a step-2 group and let Y1 , . . . ,Ym1 be left invariant smooth orthonormal sections of HG. Assume that g : G → R satisfies Y1 g ≥ 0

and Yj (g) = 0

if

j = 2, . . . , m1 .

(5.19)

Then the level lines of g are “vertical hyperplanes orthogonal to Y1 ”, that is, sets that are group translations of S(Y1 ) := {p : hπ0 p,Y1 (0)i = 0}. Example 5.17 ([12, Remark 5.4]). The following simple example shows that the sign condition is essential for the validity of the conclusions of Lemma 5.16, even in the first Heisenberg group H1 . Let p = (x, y,t) ∈ H1 , and the vector fields X1 := ∂x − y2 ∂t and Y1 := ∂y + x2 ∂t ; the function   g(x, y,t) := γ t − 12 x y   (with γ : R → R smooth) satisfies X1 g(x, y,t) = −y γ 0 t − 12 x y and Y1 g = 0. Therefore the sets Et := {g < t} are Y1 -invariant and are not halfspaces. The same example can be used to show that there is no local version of Lemma 5.16, because the sets Et locally may satisfy X1 1 E t ≥ 0 or X1 1 E t ≤ 0 (depending on the sign of γ 0 and y), but are not locally halfspaces. We can now state our main structure theorem for G-Caccioppoli sets. Theorem 5.18 (Structure of G-Caccioppoli sets, [105]). Let G be a step-2 Carnot group, endowed with the invariant distance d ∞ . Let E ⊂ G be a G-Caccioppoli set and let ∂G∗ E denote its reduced boundary. Then (i) |∂E|G = |∂E|G

∂G∗ E;

(ii) there exists a constant ϑ0 = ϑ0 (G) > 0 such that ΘQ−1 (|∂E|G , x) = ϑ0

∀x ∈ ∂G∗ E,

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93

where the (Q − 1)-dimensional spherical density ΘQ−1 (|∂E|G , ·) must be understood according to Definition 4.28(i) with X = G and d = d ∞ ; (iii) ∂G∗ E is (Q − 1)-dimensional G-rectifiable; Q−1 Q−1 (iv) |∂E|G = ϑ0 S∞ ∂G∗ E, where S∞ denotes the (Q − 1)-dimensional spherical Hausdorff measure defined according to Definition 2.21(ii) with X = G and d = d ∞ .

Magnani [162] correctly observes that Theorem 5.18(iv)—notwithstanding being true—is not an immediate consequence of Federer’s results [83, 2.10.17(2) and 2.10.19(3)], as was claimed in [105]. This depends on the existence of examples such as the one mentioned before (see (4.20)). However Magnani himself has proved Theorem 5.18(iv) in the following result. Theorem 5.19 ([162, Theorem 1.3]). Let G be a Carnot group endowed with the invariant distance d ∞ . Under the same assumptions as Theorem 5.18, statement (iv) holds. Finally, the following divergence theorem is then an easy consequence of Theorem 5.18, but we stress the fact that the measure-theoretic boundary of E (see (4.16)), ∂∗,G E, appears in the identity (ii). As in the Euclidean space, the corresponding statement for the reduced boundary holds straightforwardly. However, the interest in the statement for the measure-theoretic boundary comes not only from the fact that—as in the Euclidean setting—the last one is sometimes easier to deal with, but mainly from the fact that the measure-theoretic boundary—unlike the reduced boundary—is independent of the choice of the metric. Theorem 5.20 (Divergence theorem, [105]). Let G be a step-2 Carnot group, endowed with the invariant distance d ∞ . Let E be a G-Caccioppoli set; then there exists a positive constant ϑ0 = ϑ0 (G) such that Q−1 (i) |∂E|G = ϑ0 S∞

∂∗,G E,

and the following version of the divergence theorem holds: R R Q−1 (ii) − E div G φ d Ln = ϑ0 ∂ E hνE , φi d S∞ ∀φ ∈ C10 (G, HG). ∗,G

Remark 5.21. Ambrosio, Kleiner, and Le Donne partially extended Theorem 5.14 in a general Carnot group [12]. More precisely, they proved that, at each point of the reduced boundary, a vertical halfspace belongs to the family of tangent spaces at that point, but, a priori, the family could be not a singleton. Remark 5.22. Marchi fully extended Theorem 5.18 to a special class of Carnot groups, called Carnot groups of type ? (see [167]).

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Definition 5.23. We say that a stratified Lie algebra g = V1 ⊕ · · · ⊕ Vk is of type ? if there exists a basis {X1 , . . . , X m1 } of V1 such that [X j , [X j , X i ]] = 0 ∀ i, j = 1, . . . , m1 . A Carnot group G is said to be a group of type ? if its Lie algebra g is of type ?. Obviously each step-2 Carnot group is of type ?, but it can be proved that there are Carnot groups of type ? of any step. Remark 5.24. Ambrosio, Ghezzi, and Magnani extended Theorem 5.14 to sets of finite perimeter in sub-Riemannian manifolds, whose tangent group is isomorphic to a step-2 Carnot group [9].

5.3 Minimal boundaries in Carnot groups 5.3.1 Existence of minimal boundaries à la De Giorgi and calibrations. Garofalo and Nhieu extended the classical De Giorgi result for minimal boundaries (see [78, Theorem 1.1, Chap. II]) from the Euclidean setting to Carnot groups. Theorem 5.25 ([118, Theorem 1.26]). Let Ω ⊂ G be a bounded open set and let L ⊂ G be a measurable set with |∂L|G (Ω) < +∞. Then there exists a set E with E = L outside of Ω and such that |∂E|G (Ω) ≤ |∂F |G (Ω) for every measurable set F ⊂ G with F = L outside of Ω. Remark 5.26. In some sense the set L determines the boundary value of E. Roughly speaking, ∂E minimizes the area among all surfaces with boundary ∂L ∩ ∂Ω. The following result is a refinement of one due to Ambrosio [6] and it extends the classical calibration method giving sufficient conditions for a Borel set E ⊂ Rn to be a minimizer of the G-perimeter. Let us recall that we say that E is a minimizer for the G-perimeter in Ω if |∂E|G (Ω0 ) ≤ |∂F |G (Ω0 )

(5.20)

for any open set Ω0 b Ω and any measurable F ⊂ Rn such that E∆F b Ω0. Theorem 5.27 ([33, Theorem 5.15]). Let G = (Rn , ·) be a Carnot group. Let E, Ω be respectively a measurable and open set of Rn , and define by νE : Ω → Rm1 the horizontal inward normal to E in Ω. Let us assume (i) E has locally finite G-perimeter in Ω;

95

1 Some topics of geometric measure theory in Carnot groups

(ii) div G (νE ) = 0 in Ω in a distributional sense; ˜ ⊂ Ω such that |∂E|G (Ω \ Ω) ˜ = 0 and νE ∈ C0 (Ω). ˜ (iii) there exists an open set Ω Then E is a minimizer of the G-perimeter in Ω. Example 5.28 (Hypersurfaces with constant horizontal normal, [33, Example 2.2]). Let E ⊂ G be a set of locally finite G-perimeter in an open set Ω ⊂ G which admits a constant inward horizontal normal νE in Ω, i.e., νE ≡ ν0 ,

|∂E|G -a.e. in Ω

for a suitable constant vector ν0 ∈ Rm1 . Then, thanks to Theorem 5.27, it is straightforward to check that E is a minimizer for the G-perimeter. Observe that many interesting questions, such as regularity and rectifiability, are open even in this quite simple class of sets: see [36, 37] and also Section 5.4. Example 5.29 (t-subgraphs in Hn , [33, Example 2.6]). Let G = Hn ≡ R2n+1 and X = (X1 , . . . , X n , X n+1 , . . . , X2n ) = (X1 , . . . , X n ,Y1 , . . . ,Yn ) (see Example 2.2). Let u ∈ C2 (ω) for a suitable open set U ⊂ R2n and let E be a t-subgraph, that is, E = E t = {(x, y,t) ∈ U × R : t < u(x, y)}.

(5.21)

Let Ω := U × R, S = ∂E ∩ Ω and let Char(S) = {(x, y,t) ∈ Ω : ∇ x u(x, y) + y/2 = ∇y u − x/2 = 0} be the set of characteristic points of S (see (4.14)). Then Char(S) is closed in Ω and it holds that H2n (Char(S)) = 0 (see Theorem 4.23). On the other 2n+1  H2n (see (4.70)) and |∂E|  S 2n+1 (see Theorem 5.20) we hand, since S∞ G ∞ get ˜ = 0, |∂E|H (Ω \ Ω) (5.22) ˜ := Ω \ Char(S). where Ω A simple calculation shows that the horizontal normal νE (x, y,t) is νE (x, y,t) = −

∇H f (x, y,t) = N (x, y) = (N1 (x, y), . . . , N2n (x, y)) |∇H f (x, y,t)|

(5.23)

for each (x, y,t) ∈ S \ Char(S), where f (x, y,t) := t − u(x, y) if (x, y,t) ∈ Ω and N (x, y) := q

(−∇ x u(x, y) − y/2, −∇y u(x, y) + x/2) |∇ x u(x, y) +

y/2|R2 n

+ | − ∇y u(x, y) +

,

˜ (x, y,t) ∈ Ω.

x/2|R2 n

In this case, when Char(S) = ∅, the minimal surface equation is simply ! n X ∂ ∂ div H νE = div N = Ni + Ni+n = 0 in ω. ∂x i ∂yi i=1

(5.24)

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In particular, whenever (5.24) is satisfied in a pointwise sense, we can at once apply Theorem 5.27 obtaining that E is a minimizer for the H-perimeter measure in Ω. The more delicate case when Char(S) , ∅ has been studied in [56, 206, 58]. In particular, in [58, Theorem 3.3], it has been proved that (5.24) holds in a weak sense, i.e., Z hN, ∇ψiR2n d L2n = 0 ∀ψ ∈ C1c (U ), (5.25) U

iff u is a minimizer of the area functional in Hn for Euclidean t-graphs (see Section 5.3.2). Equation (5.24) (respectively (5.25)) is called a minimal (respectively, weak minimal) surface equation for t-graphs in Hn . When n ≥ 2, it is enough to provide that u is a classic solution of (5.24) in Ω \ Char(S) in order that it satisfies (5.25) (see [58, Corollary F]) and counterexamples are provided when n=1 (see [58, section 7]). We can get a strong result by exploiting Theorem 5.27: in fact, if (5.25) holds, by (5.22) and (5.23) we get that E is a minimizer for the H-perimeter in Ω. In particular, E minimizes the H-perimeter not only among sets whose boundary is a Euclidean t-graph but in a very much larger class of competitors. 5.3.2 Existence of nonparametric minimal surfaces in Heisenberg groups: Minimal t- and intrinsic graphs of bounded variation. In this section we are going to deal with existence, uniqueness, and regularity of minimal t-graphs and intrinsic graphs of 1-codimensional topological dimension. The techniques exploited in this section are strongly inspired from the important work [171], where a unifying study between minimal surfaces of codimension 1 and solutions of the minimal surface equation was carried out in the Euclidean case. We also refer to [122]. Minimal t-graphs. Let us first deal with the class of t-graphs. Let us recall that S ⊂ Hn is called a t-graph in Hn (see Remark 4.16) if it is a graph with respect to the nonhorizontal vector field T, namely if there exists a function u : U → R such that S = Sut := { A · u( A) e2n+1 : A ∈ U } = {(x, y,u(x, y)) ∈ Hn : (x, y) ∈ U }. Hereafter, by U we will denote a fixed open and bounded subset of the 2n-dimensional plane Π := exp(span{X j : j = 1, . . . , 2n}) = {(x, y,t) ∈ Hn : t = 0}. When clear from the context we will canonically identify Π with R2n , and accordingly we will write (x, y) instead of (x, y, 0). By U × R we will mean the t-cylinder

U × R := {(x, y,t) ∈ Hn : (x, y) ∈ U ,t ∈ R}.

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The t-subgraph Eut of u : U → R is defined as Eut := {(x, y,t) ∈ Hn : (x, y) ∈ U ,t < u(x, y)}.

(5.26)

For maps u with Sobolev regularity, the area functional for t-graphs At : W 1,1 (U ) → R is Z

At (u) := |∂Eut |H (U × R) =

U

|∇u + X∗ |d L2n ,

(5.27)

where, following the notation in [58], X∗ : R2n → R2n is defined by X∗ (x, y) := 1 1 2 (y, −x). Formula (5.27) was proved in [48] for u ∈ C (U ). The functional At is 2n convex since, for a given z ∈ U , the Lagrangian R 3 ξ → |ξ + X∗ (z)| ∈ R is convex and it is not strictly convex. When u ∈ C2 (U ) is a local minimizer of At , a first variation of the functional yields the minimal surface equation for t-graphs, div (N (u)) = 0 We have defined N (u) :=

∇u + X∗ |∇u + X∗ |

in Unc (u).

on Unc (u),

(5.28)

(5.29)

where Unc (u) := U \ Char(u) and Char(u) is the set of characteristic points of u defined as Char(u) := {(x, y) ∈ U : ∇u(x, y) + X∗ (x, y) = 0}, (5.30) i.e., the projection on Π of Char(Sut ) (see (4.14)). The solutions of (5.28) are called H-minimal. One is not allowed to deduce that (5.28) is satisfied on U in the sense of distributions even when L2n (Char(u)) = 0; moreover, the size of Char(u) may be large even for u ∈ W 1,1 (U ) (see [22]). These problems have been studied in detail in [58] and a suitable minimal surface equation was obtained. The study of the relaxed functional At : L 1 (U ) → [0, +∞] of At with respect to the L 1 -topology and its representation formula on its domain has been studied in [213]. We therefore introduce ( ) Z At (u) := inf lim inf |∇uk + X∗ | d L2n : uk ∈ W 1,1 (U ),uk → u in L 1 (U ) . k→∞

U

Definition 5.30. We say that u ∈ L 1 (U ) belongs to the class BVt (U ) of maps with bounded t-variation if |∂Eut |H (U × R) < +∞. We say that u ∈ BVt,loc (U ) if Eut has finite H-perimeter in U 0 × R for any open set U0 b U.

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The structure of the space BVt (U ) and several different notions of “area” for the boundary of t-subgraphs Eut of u ∈ L 1 (U ) have been studied in [213, Section 3]. In particular, among them it has been proved that the perimeter |∂Eut |H (U × R) and the relaxed functional At of At agree on L 1 (U ). Theorem 5.31 ([213, Theorem 3.2]). Let U ⊂ R2n be a bounded open set. Then |∂Eut |H (U × R) = At (u)

(5.31)

holds for any u ∈ L 1 (U ). We introduce the notation Z |Du + X∗ | := |∂Eut |H (U × R) = At (u),

u ∈ L 1 (U ).

U

It turns out that BVt (U ), which is the finiteness domain of these functionals, coincides with the classical space BV (U ) of functions with bounded variation in U . Theorem 5.32 ([213, Theorem 1.2]). Let U ⊂ R2n be a bounded open set. Then BVt (U ) = BV (U ). In particular, each function in BVt (U ) can be approximated with respect to the “strict” metric (see [8, pp. 125–126]) by a sequence of C ∞ regular functions (see [213, Corollary 3.3]). Moreover, the space BVt (U ) can be compactly embedded in L 1 (U ) and the classical notion of trace u |∂U on ∂ U is well defined provided U is bounded with Lipschitz regular boundary (see [213, Theorem 3.4]). Let us now deal with the existence of t-minimizers. Definition 5.33. Let U ⊂Π be a bounded open set with Lipschitz regular boundary. We say that u∈BV (U ) is a t-minimizer of the area functional (briefly, t-minimizer) if Z Z |Du + X∗ | ≤ |Dv + X∗ | U

U

for any v ∈ BV (U ) such that v |∂U = u |∂U . Given a generic open set U ⊂ Π, we say that u ∈ BVloc (U ) is a local t-minimizer if Z Z |Du + X∗ | ≤ |Dv + X∗ |. U0

U0

for any U 0 b U and any v ∈ L 1loc (U ) with {u , v} b U 0. Equivalently (see [213, Remark 3.12]), if u is a t-minimizer on any open set U 0 b U with Lipschitz regular boundary.

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99

A t-minimizer is also a local t-minimizer (see [213, Remark 3.11]). Moreover, it is easily seen that a t-subgraph Eut that is locally H-perimeter minimizing in U × R, must be associated with a local t-minimizer u ∈ BV (U ). Vice versa, it can be proved that a local t-minimizer u ∈ BV (U ) induces a t-subgraph Eut that is a local minimizer of the H-perimeter in U × R (see [213, Corollary 3.16]). Local t-minimizers have been widely studied assuming u ∈ W 1,1 (U ), the classical Sobolev space which is strictly contained in BV (U ). The functional (5.27) has good variational properties such as convexity and lower semicontinuity with respect to the L 1 topology. On the other hand, it is not coercive and differentiable due to the presence of the characteristic points. Notice that, if u ∈ W 1,1 (U ), the set Char(u) must be understood up to an L2n -negligible set. Nevertheless, the existence of solutions to the Dirichlet problem with regular boundary conditions was obtained in [194] and [58], by means of an elliptic approximation argument, for U satisfying suitable convexity assumptions. The lack of coercivity for the functional (5.27) does not allow a first variation near the set Char(u). This and related questions have been studied in [194, 56, 53, 206, 214] for C2 minimizers of At , and in [59, 57] for C1 regular ones, also in connection with the Bernstein problem for t-graphs. A suitable minimal surface equation for t-graphs (see (5.28)) has been obtained in these papers; its solutions are called Hminimal surfaces. In particular, in [56] a deep analysis of Char(u) was carried out for local minimizers u ∈ C2 (U ) together with other regularity properties like comparison principles and uniqueness for the associated Dirichlet problem. The much more delicate case of minimizers u ∈ W 1,1 (U ) was attacked in [58]. Several examples of t-minimizers with at most Lipschitz regularity have been provided in H1 (see [194, 58, 204]). Therefore, at least in the H1 setting, the problem of regularity for t-minimizers is very different from the Euclidean case, where minimal graphs of codimension 1 are analytic regular (see [122, Theorem 4.13]). In the spirit of the previous results, we are able to establish an existence result for the Dirichlet minimum problem for the functional (5.27) on the class of t-graphs of bounded variation. Theorem 5.34 (Existence of minimal t-graphs for a penalized functional, [213, Theorem 1.4]). Let U ⊂ R2n be a bounded open set with Lipschitz regular boundary. Then, for any given ϕ ∈ L 1 (∂ U ) the functional Z Z BV (U ) 3 u 7−→ |Du + X∗ | + |u |∂U − ϕ|d H2n−1 (5.32) ∂U

U

attains its minimum and (Z ) ∗ inf |Du + X | : u ∈ BV (U ), u |∂U = ϕ U (Z ) Z ∗ 2n−1 = min |Du + X | + |u |∂U − ϕ|d H : u ∈ BV (U ) . U

∂U

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Remark 5.35. The last integral in (5.32) equals both the Euclidean and sub-Riemannian areas of that part of the cylinder ∂ U × R between the graphs of u and ϕ, hence it can be seen as a penalization for not taking the boundary values ϕ on ∂ U . See also [213, Proposition 3.7 and Remark 3.8]. Theorem 5.34 extends the existence results contained in [193] and [58] because formulation (5.32) allows more general domains U to be considered. Remark 5.36. A minimizer of the penalized functional (5.32) might not take the prescribed boundary value ϕ: this situation can be illustrated by explicitly constructing an example where t-minimizers do not exist (see [213, Example 3.6]). In particular, the existence of solutions for the Dirichlet minimum problem for At is not guaranteed even when the boundary ∂ U and the datum ϕ are very regular: in this sense, Theorem 5.34 does not extend the results in [193] and [58]. An existence result for continuous BV t-minimizers for continuous boundary data on smooth parabolically convex domains has been obtained in [53]. In [197], existence, uniqueness, and Lipschitz regularity of t-minimizers (assuming the prescribed boundary datum) is proved under the assumption that the boundary datum ϕ satisfies the so-called bounded slope condition. Definition 5.37. We say that a function ϕ : ∂Ω → R satisfies the bounded slope condition with constant Q > 0 (Q-B.S.C. for short, or simply B.S.C. when the constant Q does not play any role) if for every z0 ∈ ∂Ω there exist two affine functions wz+0 and wz−0 such that wz−0 (z) ≤ ϕ(z) ≤ wz+0 (z)

∀z ∈ ∂Ω,

wz−0 (z0 ) = ϕ(z0 ) = wz+0 (z0 ), Lip(wz−0 ) ≤ Q and Lip(wz+0 )

(5.33) ≤ Q,

where Lip(w) denotes the Lipschitz constant of w. We also recall that a set U ⊂ R2n is said to be uniformly convex if there exist a positive constant C = C(U ) and, for each z0 ∈ ∂ U , a hyperplane Πz 0 passing through z0 such that |z − z0 | 2 ≤ C dist(z, Πz 0 )

∀z ∈ ∂ U ,

where dist(z, Πz 0 ) := inf{|z − w| | w ∈ Πz 0 }. Remark 5.38. We collect here some facts on the B.S.C. (a) If ϕ : ∂Ω → R satisfies the B.S.C. and is not affine, then Ω has to be convex (see [123, p. 20]) and ϕ is Lipschitz continuous on ∂Ω. Moreover, if ∂Ω has flat faces, then ϕ has to be affine on them.

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101

This property seems to say that the B.S.C. is quite a restrictive assumption. Anyhow the following one, due to Miranda [172] (see also [123, Theorem 1.1]), shows that the class of functions satisfying the B.S.C. on a uniformly convex set is quite large. (b) Let Ω ⊂ Rn be open, bounded, and uniformly convex; then every ϕ ∈ C 1,1 (Rn ) satisfies the B.S.C. on ∂Ω. Theorem 5.39 (Existence, uniqueness, and Lipschitz regularity of minimal t-graphs under B.S.C., [197, Theorem 1.1]). Let Ω ⊂ R2n be open, bounded, and with Lipschitz regular boundary, and let ϕ : ∂Ω → R satisfy the Q-B.S.C. for some Q > 0. Then, the minimization problem ) (Z ∗ |Du + X | : u ∈ BV (Ω), u |∂Ω = ϕ (5.34) min U

admits a unique solution u. ˆ Moreover, uˆ is Lipschitz continuous and Lip(u) ˆ ≤ Q+ 4 supz ∈Ω |z|. Remark 5.40. It can be proved that the previous theorem is sharp, at least for n = 1, in the sense that minimizers might not be better than Lipschitz regular (see [197, Examples 6.6 and 6.7]). In particular, Theorem 5.39 extends Theorem 5.34 as well as some related results in [193] and [58]. Finally a result on boundedness for local t-minimizer holds. Theorem 5.41 (Local boundedness of minimal t-graphs, [213, Theorem 1.5]). Let u ∈ BV (U ) be a local t-minimizer. Then u ∈ L ∞ loc (U ). As a consequence, we also obtain a local boundedness result for the weak solutions u ∈ W 1,1 (U ) of the minimal surface equation (5.28) (see [213, Theorem 3.14]). Theorem 5.41 is sharp at least in the first Heisenberg group H1 . Indeed, a minimal 0 t-graph can be induced by a function u ∈ L ∞ loc (U ) \ C (U ) (see [213, Section 3.4]). It is an open problem whether a similar example can also be constructed in Hn , n ≥ 2. Minimal intrinsic graphs. We are now going to deal with minimal surfaces which are intrinsic graphs, that is, graphs of functions φ : ω ⊂ W → V acting between a (2n, 2n+1)-subgroup W and (1, 1)-subgroup V of Hn , which are also complementary (see Definitions 4.1 and 4.2, Example 4.4). Without loss of generality, we will always consider X1 -graphs, i.e., intrinsic graphs along the X1 -direction. Let us introduce some preliminary notation. If n ≥ 2, we identify the maximal subgroup W := exp(span{X2 , . . . , X n ,Y1 , . . . ,Yn ,T }) = {(x, y,t) ∈ Hn : x 1 = 0}

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with R2n by writing (x 2 , . . . , x n , y1 , . . . , yn ,t) instead of (0, x 2 , . . . , x n , y1 , . . . , yn ,t); similarly W := exp(span{Y1 ,T }) = {(0, y,t) ∈ H1 : y,t ∈ R} ≡ R2y, t if n = 1. We identify the 1-dimensional horizontal subgroup V := exp({sX1 : s ∈ R}) = {(s, 0, . . . , 0) ∈ Hn : s ∈ R} with R by writing s instead of (s, 0, . . . , 0). Let ω denote a fixed open bounded subset of W; the intrinsic cylinder ω · R is defined by ω · R := { A · s ∈ Hn : A ∈ ω, s ∈ R}, where, for A ∈ W and s ∈ R we write A · s to denote the Heisenberg product A · (s, 0, . . . , 0). In this way I · J = { A · s : A ∈ I, s ∈ J} for any I ⊂ W, J ⊂ R. Similarly, we will write s · A to denote (s, 0, . . . , 0) · A. Given φ, we denote by Φ : ω → Hn the corresponding X1 -graph map Φ( A) := A · φ( A),

A ∈ ω.

(5.35)

A set S ⊂ Hn is called an X1 -graph of φ : ω → R if S := Φ(ω) = { A · φ( A) : A ∈ ω} . The X1 -subgraph and the X1 -epigraph (or X1 -upper-graph) of φ are defined, respectively, as Eφ := { A · s : A ∈ ω, s < φ( A)} (5.36) and E φ := { A · s : A ∈ ω, s > φ( A)} .

(5.37)

Let Lip(ω) be the classical space of Lipschitz functions on ω ⊂ W ≡ R2n . The area functional AW : Lip(ω) → R is Z q 1 + |∇φ φ| 2 d L2n , (5.38) AW (φ) := |∂Eφ |H (ω · R) = ω

where ∇φ is the nonlinear intrinsic gradient for X1 -graphs, ( (X2 φ, . . . , X n φ,W φ φ,Y2 φ, . . . ,Yn φ) if n ≥ 2, φ ∇ φ := Wφφ if n = 1, where W φ φ := Y1 φ +

1 1 T (φ2 ) = ∂y1 φ + ∂t (φ2 ). 2 2

(5.39)

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We agree that, when φ is not regular, the differential operators appearing in (5.39) will be understood in the sense of distributions. The intrinsic gradient ∇φ was introduced and studied in [16]; see also [61, 40, 41, 62]. Observe that we can write Z L W ( A, φ( A), ∇φ( A)) d L2n ( A), AW (φ) = ω

where L W : W × R ×

R2n

→ [0, +∞) is defined by 2n X

 L W ( A, φ, ξ) := 1 + hX n+1 ( A) + φ T ( A), ξi2 +

hX j ( A), ξi2

 1/2 (5.40)

j=2, j,n+1

if n ≥ 2, while L W ( A, φ, ξ) := 1 + hY1 ( A) + φ T ( A), ξi2

 1/2

(5.41)

if n = 1. The vector fields X j ( j = 2, . . . , n),Yj ( j = 1, . . . , n), and T are tangent to W ≡ R2n and therefore can be viewed as elements of R2n . The scalar products in (5.40) and (5.41) are the usual ones between vectors in R2n . When φ ∈ C2 (ω) is a local minimizer of the functional AW , the first variation of the functional AW gives the minimal surface equation for X1 -graphs, ∇φ φ +=0 ∇φ · * p φ 2 , 1 + |∇ φ| -

in ω.

(5.42)

It was pointed out in [72] that AW is not convex for n = 1. Indeed, for any α > 0 the function 2α y t φ(y,t) := 2 + α y2 satisfies (5.42) on R2 , while (see [70]) φ is not a local minimizer for AW : v(ω) → R on a suitable bounded open set ω ⊂ R2 . In particular, AW cannot be convex on v(ω) because the stationary point φ is not a minimum. The presence of stationary points that are not minimizers for AW is an interesting open question in the case n ≥ 2. Nevertheless, the nonconvexity of AW : v(ω) → R can also occur in the higherdimensional case (see [213, Proposition 4.1]). A study of the C2 minimizers of AW was carried out in [70, 33, 69, 72] also in connection with the Bernstein problem for intrinsic graphs. First and second vari1,1 ations for minimizers in WW (ω), a suitable class of intrinsic graphs with Sobolev regularity introduced in [187] (see [213, Definition 2.8]), have been studied in [187]. The regularity of Lipschitz continuous vanishing viscosity solutions of the minimal surface equation for intrinsic graphs has been studied in [46, 47, 31]. We have to mention that, in the first Heisenberg group H1 , there are minimizers of AW whose regularity is not better than 21 -Hölder: see [187, Theorem 1.5].

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Definition 5.42. We say that φ ∈ L 1 (ω) belongs to the class BVW (ω) of functions with intrinsic bounded variation if |∂Eφ |H (ω · R) < +∞. We say that φ belongs to BVW,loc (ω) if Eφ is a set with finite H-perimeter in ω 0 · R for any open set ω 0 b ω. The class BVW (ω) is deeply different from BV (ω): for instance, it is not even a vector space (see [213, Remark 4.2]). In spite of these differences, BVW (ω) shares with BV (ω) several properties: • The functional φ 7→ |∂Eφ |H (ω · R) coincides with the relaxed one AW of AW on L 1 (ω) (see [213, Theorem 4.7]). • Each function in BVW (ω) can be approximated by a sequence of C ∞ regular functions (φ j ) j such that φ j → φ in L 1 (ω)

and

|∂Eφ j |H (ω · R) → |∂Eφ |H (ω · R)

(see [213, Theorem 4.9]). • When ω has Lipschitz regular boundary, a trace in a generalized sense exists at least for some large subclass of BVW (ω) (see [213, Proposition 4.5]). This notion of trace is related to the possibility of extending the set Eφ out of ω · R without “creating” a perimeter on the boundary ∂ω · R (see [213, Definition 4.11]). The conjecture is that any φ in BVW does have a trace in this sense. However, as shown in [213, Remark 4.10], any meaningful notion of trace in BVW cannot possess all the features of classical traces. • Define kφk BVW := k f k L 1 (ω) + |∂Eφ |H (ω · R)

if φ ∈ BVW (ω),

then any sequence (φ j ) j ⊂ BVW (ω) bounded in the k · k BVW “norm” and such that   sup |∂Eφ j |H (H+n ) + |∂E φ j |H (H−n ) < +∞ (5.43) j

is compact with respect to the L 1loc (ω · R)-convergence of its subgraphs (Eφ j ) j (see [213, Proposition 4.18]), where we have set   H+n := (x, y,t) ∈ Hn : x 1 ≥ 0 , H−n := (x, y,t) ∈ Hn : x 1 ≤ 0 . (5.44) Condition (5.43) is equivalent to   sup |∂Eφ j |H (∂ω · R+ ) + |∂E φ j |H (∂ω · R− ) < +∞, j

where R+ := [0, +∞), R− := (−∞, 0].

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We are now going to deal with the problem of an existence result for minimal X1 graphs on ω with prescribed “boundary datum”. Let ω0 c ω be a bounded open set and ϑ ∈ BVW (ω0 ) be such that |∂Eϑ |H (∂ω0 · R+ ) + |∂E ϑ |H (∂ω0 · R− ) < ∞,

|∂Eϑ |H (∂ω · R) = 0.

We consider the problem  inf |∂E∂ |H (ω · R) : ∂ ∈ BVW (ω0 ), ∂ = ϑ on ω0 \ ω .

(5.45)

(5.46)

When φ ∈ BVW (ω) has a trace in a generalized sense, then it possesses an extension ϑ ∈ BVW (ω0 ), on a suitable ω0 c ω, satisfying (5.45): if this is the case, then problem (5.46) can be viewed as that of minimizing area with boundary datum given by φ. Theorem 5.43 (Existence of minimal X1 -graphs, [213, Theorem 1.8]). Problem (5.46) attains a minimum in BVW (ω0 ). Also a local boundedness result for minimal X1 -graphs holds. Theorem 5.44 (Local boundedness of minimal X1 -graphs, [213, Theorem 1.8]). Let φ ∈ L 2n+1 loc (ω) be such that Eφ is a local minimizer of the H-perimeter in ω · R. Then φ ∈ L∞ loc (ω). This result is not the exact counterpart of Theorem 5.41 for minimal t-graphs. We do not know whether the additional (2n + 1)-summability is only a technical problem or if there exist minimal X1 -graphs φ < L ∞ loc (ω). Moreover, in Theorem 5.41 we prove the local boundedness of t-minimizers using the fact that they are also H-perimeter minimizing sets. In Theorem 5.44 we instead require the subgraph Eφ to be H-perimeter minimizing: as far as we know, there is no geometric arrangement, similar to the one for t-graphs given by [213, Theorem 3.15], ensuring that the subgraph of a minimal intrinsic graph is also H-perimeter minimizing.

5.4 Regularity of minimal boundaries in Carnot groups. The regularity of minimizers for the G-perimeter in a Carnot group G = (Rn , ·) is a challenging and very difficult open problem. Let us recall that a minimizer E ⊂ G in a given open set Ω ⊂ G is a set of locally finite G-perimeters satisfying (5.20). The problem deals with the regularity of its topological boundary ∂E ∩ Ω, or, more precisely, of a significant subset of ∂E, that is, a G-reduced boundary ∂G∗ E ∩ Ω (see Section 5.1). There are only a few (partial) results in the setting of particular Carnot groups, which we will list below. 1. Euclidean group. Here we mean that G = (Rn , +), a trivial Carnot group (see Example 2.1). The problem was fully solved by De Giorgi [78].

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One of the most important results in this setting claims that the portion of reduced boundary ∂ ∗ E ∩ Ω of a minimizer E in Ω is (locally) an analytic hypersurface (see, for instance, [122, Theorem 8.4]). However there may be minimizers with irregular topological boundary ∂E, which are cones, in dimension n ≥ 8 (see, for instance, [122, Theorems 11.7 and 11.8]). Another relevant result concerns the (global) regularity of a nonparametric minimal surface, that is, minimal boundaries of sets which are subgraphs. More precisely, assume E := {(x 1 , . . . , x n−1 , s) ∈ U ×R : s < u(x 1 , . . . , x n−1 )} is a minimizer in Ω := U × R and u is a function of Euclidean bounded variation, that is, u ∈ BVG (U ) and G = (Rn−1 , +) in accordance with Example 2.1 and Section 3.5. Then u is an analytic function and therefore the boundary ∂E ∩ Ω is an analytic hypersurface (see, for instance, [122, Theorem 14.13]). 2. Heisenberg group. Here we mean that G = Hn = (R2n+1 , ·) endowed with the sub-Riemannian structure introduced in Example 2.2. We recommend lecture notes [184, Section 5] for a detailed account of the results in this setting. However, there are no complete results as in the Euclidean case. As far as the regularity of general minimal boundaries for dimension n ≥ 2 is concerned, there are some technical steps towards a full De Giorgi regularity result: an approximation in measure of a minimizer’s boundary by means of intrinsic Lipschitz graphs [183], a so-called height estimate [189] for the Hperimeter measure of a minimizer. As far as the regularity of nonparametric minimal surfaces is concerned, there are more results and examples, mainly in the first Heisenberg group H1 . Let us recall some of the most significant ones. • For 1-codimensional t-graphs (see Section 5.3.2), it is well known that, in the setting of the first Heisenberg group H1 there may exist minimizers Eut (see (5.26)) for which u is no better than (Euclidean) Lipschitz continuous (see, for instance, [195, 58, 204]) as well as that u can be discontinuous (see, for instance, [213]). This is evidence that the issue of regularity for minimal boundaries in a sub-Riemannian setting is deeply different from the Euclidean one. However we have to point out that those examples do not seem to extend to higher dimensions n ≥ 2. For dimensions n ≥ 1, to our knowledge, only a result of local boundedness for the defining function u is proved, provided that Eut is a minimizer and u ∈ L 1loc (U ) (see Theorem 5.41). • For 1-codimensional X1 -intrinsic graphs (see Section 5.3.2), it is well known that, in the setting of the first Heisenberg group H1 , there may exist minimizers Eφ (see (5.36)) for which φ is no better than (locally)

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intrinsic Lipschitz continuous (see, for instance, [187]). For dimensions n ≥ 1, to our knowledge, only a result of local boundedness for the defining function φ is proved, provided that Eφ is a minimizer and φ ∈ L 2n+1 loc (ω) (see Theorem 5.44). 3. Engel group. Here we mean that G = E = (R4 , ·) endowed with the subRiemannian structure introduced in Example 4.7. Let us recall that G is a step-3 Carnot group and its Lie algebra g is generated by four left-invariant vector fields X j , j = 1, 2, 3, 4. Such an algebra is stratified as g = V1 ⊕ V2 ⊕ V3 with V1 = span {X1 , X2 }, V2 = span {X3 }, and V3 = span {X4 }, the only nonzero commutation relations being [X1 , X2 ] = X3 , [X1 , X3 ] = X4 . In this setting the structure and regularity of a set E of finite G-perimeter with constant generalized normal νE have been studied in [36]. Namely, we assume there exists a constant section ν0 = a X1 + b X2 ∈ V1 such that νE (x) = ν0 |∂E|G -a.e. x ∈ R4 . Let us recall that, by Example 5.28, such a set is a (global) minimizer for a G-perimeter in Ω = G. Notice also that such a set, in a step-2 Carnot group, is always equivalent to a vertical hyperplane orthogonal to ν0 (see Lemma 5.16). It is not the case in the Engel group as already shown in Example 5.15. The general regularity results proved in [36] claim that each set E ⊂ G of locally G-finite perimeter with constant normal is equivalent to a (Euclidean) ˜ = 0. Lipschitz regular set E˜ ⊂ G, that is, L4 (E∆ E) In [36] the possibility of expressing a set with constant normal as an intrinsic upper-graph is also studied, when using the model of the Engel group corresponding to the so-called exponential coordinates of the second kind, and what regularity to expect for such a graph. The authors proved that there are examples of sets with constant horizontal normal such that, if one writes the set as an intrinsic upper-graph in the direction of the normal, the function giving the graph is not even continuous. However, all constant normal sets are intrinsic Lipschitz upper-graphs in other horizontal directions. This last feature is peculiar to the Engel group. In a forthcoming paper [37], they will prove that in general Carnot groups sets with constant horizontal normal might fail to be intrinsic Lipschitz upper-graphs in every horizontal direction. We thank Le Donne for sharing his unpublished results with us.

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Chapter 2

Hypoelliptic operators and some aspects of analysis and geometry of sub-Riemannian spaces Nicola Garofalo1

Contents 1 Sub-Riemannian geometry and hypoelliptic operators . . . . . . . . . 2 Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fundamental solutions and the Yamabe equation . . . . . . . . . . . . 4 Carnot–Carathéodory distance . . . . . . . . . . . . . . . . . . . . . . 5 Sobolev and BV spaces . . . . . . . . . . . . . . . . . . . . . . . . . 6 Fractional integration in spaces of homogeneous type . . . . . . . . . 7 Fundamental solutions of hypoelliptic operators . . . . . . . . . . . . 8 The geometric Sobolev embedding and the isoperimetric inequality . . 9 The Li–Yau inequality for complete manifolds with Ricci ≥ 0 . . . . . 10 Heat semigroup approach to the Li–Yau inequality . . . . . . . . . . . 11 A heat equation approach to the volume doubling property . . . . . . . 12 A sub-Riemannian curvature-dimension inequality . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Sub-Riemannian geometry and hypoelliptic operators These lectures are about some local and global aspects of sub-Riemannian geometry. The first part will be devoted to introducing the audience to the former aspects, [email protected] Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, Via Trieste, 50, Padova, Italy. This work was partially supported by the NSF Grant DMS-1001317, and by a grant from the University of Padova, “Progetti d’Ateneo”, 2014.

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whereas the second half focuses primarily on global aspects having to do with curvature, and on the pervasive role that heat equation techniques play in connection with it (the interested reader should be aware that this second part is introductory to the lectures notes of Fabrice Baudoin, in this same volume). Having said this, it is appropriate to start with a preliminary motivational discussion. Sub-Riemannian geometry is an extension of Riemannian geometry which was created to model media with non-holonomic constraints: motion at any point is allowed only along a limited set of directions which are prescribed by the physical problem at hand. When the set of directions coincides with the whole tangent space we obtain Riemannian geometry. Examples of systems with non-holonomic constraints include the Foucault pendulum; in robotics, when the controllable degrees of freedom are less than the total degrees of freedom of a robot; a falling cat; and a rolling sphere. For instance, when studying the physics of semiflexible polymers one encounters the so-called Mumford operator on M × (0, ∞), where we have denoted M = S1 × R,

M = ∂ϑ2 + cos ϑ∂x + sin ϑ∂y − ∂t ;

(1.1)

see [M94]. If we consider the two smooth vector fields X0 = cos ϑ∂x + sin ϑ∂y − ∂t ,

X1 = ∂ϑ ,

it is clear that M = X12 + X0 . Obviously, X0 , X1 do not span the whole of R4  T (M × (0, ∞)). However, by considering the commutators X2 = [X1 , X0 ] = − sin ϑ∂x + cos ϑ∂y , X3 = [X1 , X2 ] = [X1 , [X1 , X0 ]] = − cos ϑ∂x − sin ϑ∂y , then at every point of M × (0, ∞) the determinant of the matrix formed with the coefficients of {X0 , X1 , X2 , X3 } is given by 1 *. 0 det .. .0

0 0 0 + cos ϑ sin ϑ −1// = −1 − sin ϑ cos ϑ 0 // ,0 − cos ϑ − sin ϑ 0 -

and thus at every point of R4 we have T (M × (0, ∞)) = span{X0 , X1 , X2 , X3 }  R4 .

(1.2)

1.1 The hypoellipticity theorem of Hörmander. The Mumford operator is an example of a large class of partial differential equations of order 2, known as

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125

operators of Hörmander type. Before stating the relevant result about this class let us recall that, given an open set Ω ⊂ R N , a partial differential operator P with coefficients in C ∞ (Ω) is said to be hypoelliptic in Ω if, given f ,u ∈ D 0 (Ω), such that P u = f , one has u ∈ C ∞ (ω) for any open set ω ⊂ Ω in which f ∈ C ∞ (ω). Examples of hypoelliptic operators are the Laplace operator ∆ in R N (this is the content of the celebrated Caccioppoli–Cimmino–Weyl lemma,2 the heat equation ∆ − ∂t in R N +1 , and every strictly elliptic, or parabolic, second-order partial differential operator with C ∞ coefficients. According to the next fundamental result, the class of second-order hypoelliptic operators is in fact much wider. We will need the following Definition 1.1. Let Ω ⊂ R N be an open set and consider a system of vector fields {X0 , X1 , . . . , X m } in C ∞ (Ω). We say that they satisfy the finite rank condition in Ω if at every point of Ω one has rank Lie[X0 , X1 , . . . , X m ] = N.

(1.3)

Condition (1.3) means that at every point of Ω the vector fields and a sufficiently high number of their commutators X j1 , [X j1 , X j2 ], [X j1 , [X j2 , X j3 ]], . . . , [X j1 , [X j2 , [X j3 , . . . , X j k ]]], . . . , ji = 0, 1, . . . , m, generate the whole of R N , i.e., the tangent space. In other words, at every point of Ω among such differential operators, there exist N which are linearly independent. One has the following celebrated result; see [Ho67]. Theorem 1.2 (Hörmander (1967)). Suppose that {X0 , X1 , . . . , X m } satisfy the finite rank condition (1.3) in Ω, and let c ∈ C ∞ (Ω). Then, the second-order differential operator m X L= X 2j + X0 + c (1.4) j=1

is hypoelliptic in Ω.

1.2 Mumford and the roto-translation group. Since the Mumford operator can be written as M = X12 + X0 , by virtue of the property (1.2) above and of Theorem 1.2, M is hypoelliptic on the whole of M × R. Furthermore, we can endow R4 with 2This fundamental lemma states that, given an open set Ω ⊂ R N and a distributional solution U of ∆U = 0 in Ω, there exists a u ∈ C ∞ (Ω) such that U = u as distributions. It is usually referred to as Weyl’s lemma; see [W40]. However, as pointed out by Miranda in his monograph [Mi70, p.122], “This nomenclature is inappropriate because the same proposition can be found in a more general form in Caccioppoli’s memoir [C37]. And even the extension to the case of arbitrary M given by G. Cimmino in [Ci38/1], [Ci38/2] precedes Weyl’s memoir by some years. To be accurate, Caccioppoli and Cimmino considered the case N = 2, Weyl the case N = 3.”

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a group law ◦ with respect to which the operator M is left invariant. The group law is given by (ϑ, x, y,t) ◦ (ϑ 0, x 0, y 0,t 0 ) = (ϑ + ϑ 0, x + x 0 cos ϑ − y 0 sin ϑ, y + x 0 sin ϑ + y 0 cos ϑ,t + t 0 ).

(1.5)

Note that, if we identify z = x + iy and z 0 = x 0 + iy 0, then we can write (1.5) in the following suggestive fashion: (ϑ, z,t) ◦ (ϑ 0, z 0,t 0 ) = (ϑ + ϑ 0, z + eiϑ z 0,t + t 0 ),

(1.6)

thus in the variable z we have a roto-translation. If we denote by m RT = Lie(X0 , X1 ), the Lie algebra generated by the vector fields X0 , X1 , then the corresponding Lie group, known as the roto-translation group, is given by RT = (R4 , ◦); see [M94]. Notice that m RT is not nilpotent. As a consequence, RT is not nilpotent either. As we will see, such a lack of nilpotency is closely connected to a lack of dilation invariant structure. If we let g = (ϑ, x, y,t) and we define the left translation by L g (g 0 ) = g ◦ g 0, then it is easy to check that for every g, g 0 ∈ R4 we have X0 ( f ◦ L g )(g 0 ) = ((X0 f ) ◦ L g )(g 0 ),

X1 ( f ◦ L g )(g 0 ) = ((X1 f ) ◦ L g )(g 0 ).

This shows that for every g ∈ R4 we have

M ( f ◦ L g ) = (M f ) ◦ L g , and thus M is invariant with respect to the left-translation operator L g .

1.3 Approximating Mumford with Kolmogorov. In 1934, Kolmogorov published a short visionary note [K34] on Brownian motion and the kinetic theory of gases. In the note the following second-order partial differential equation in R3 appeared for the first time:

Ku = ∂x2 u + x∂y u − ∂t u = 0.

(1.7)

This is known as Kolmogorov’s equation. Let us observe that, if we let X1 = ∂x , X0 = x∂y − ∂t , then Kolmogorov’s operator can be written as K = X12 + X0 . Since [X1 , X0 ] = ∂y , it is clear that K is an operator of Hörmander type, and thus its hypoellipticity follows from Theorem 1.2. In fact, Kolmogorov constructed an explicit fundamental solution for (1.7) which is C ∞ outside of the diagonal, thus proving its

2 Hypoelliptic operators, etc.

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hypoellipticity more than thirty years before Hörmander’s work [Ho67]. It is worth mentioning that the motivating example in the opening of [Ho67] is precisely equation (1.7), and some generalizations of it. We also mention that a scale-invariant Harnack inequality for the operator K was first proved in the work [GL90]. As we saw in Section 1.2 the roto-translation Lie group RT associated with the Mumford operator is not nilpotent. There is however a nilpotent group which, at the infinitesimal level, plays for the roto-translation group RT the same role played by the tangent space of a smooth manifold. Such a nilpotent group is generated by vector fields whose coefficients are the Taylor polynomials of order 2 of those in the expression of the Mumford operator M (note that we need commutators of order 3 to generate the Lie algebra m RT ). They are given by X 1 = ∂x ,

X0 =

x2 ∂y + x∂z − ∂t , 2

(1.8)

and the approximating operator is given by

L = X12 + X0 .

(1.9)

For a partial differential equation related to (1.9) one should see the seminal work of Kac [Kac49]. The operator in (1.9) is modeled on the Kolmogorov operator K above, but with an important difference: it contains superlinear powers of the variable x, and the highest degree of such powers is an even number. This makes the analysis of the operator L more difficult than that of K. A scale-invariant Harnack inequality for the operator (1.9) will appear in [GMP15]. Notice that one has X2 = [X1 , X0 ] = x∂y + ∂z , and all higher commutators vanish. Since  det X0 X1 X3

X3 = [X1 , X2 ] = ∂y ,

(1.10)

 X4 = −1

at every point (x, y, z,t) ∈ R4 , the vector fields {X0 , . . . , X3 } generate the Lie algebra k2 = Lie{X0 , X1 }. Thus, the operator L is of Hörmander type and hence hypoelliptic. There is a natural family of non-isotropic dilations associated with the Lie algebra k2 : δ λ (x, y, z,t) = (λ x, λ 4 y, λ 3 z, λ 2 t), λ > 0. (1.11) A direct computation shows that

L ( f ◦ δ λ ) = λ 2 (L f ) ◦ δ λ ,

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thus the operator L is homogeneous of degree 2 with respect to the non-isotropic dilations (1.11). The homogeneous dimension with respect to (1.11) is Q = 1 + 4 + 3 + 2 = 10.

1.4 The exponential map and the group law. Since the vector fields X0 , X1 in (1.8) have real analytic coefficients, according to a general result by Bonfiglioli and Lanconelli (see [BL12, Theorem 1.1]), there exists a group law g ◦ g 0 in R4 such that, if we define the left-translation operator L g (g 0 ) = g ◦ g 0, then the vector fields X0 , X1 are left invariant with respect to L g . This group law is found in (1.15) below. To actually compute it we proceed as follows. With X1 , X0 as in (1.8) and X2 , X3 as in (1.10), we set Y1 = X1 ,Y2 = X0 ,Y3 = X2 ,Y4 = X3 . Given a point u = (u1 , . . . ,u4 ) ∈ R4 , we have u · Y = u1Y1 + u2Y2 + u3Y3 + u4Y4 u1 0 0 0 1 *. +/ *. x 2 *. +/ *. x 2 +/ *. +/ + x 1 0 u + u3 x + u4 // 2 2 / . / . / . . / . 2 = u1 . / + u2 . / + u3 . / + u4 . / = . // . .1/ .0/ . u x + u .0/ .x/ 2

,0-

,0-

,−1-

,0-

,

3

−u2

-

To compute the exponential map, given g = (x, y, z,t), we must solve the system of ODEs,  γ10 (s) = u1 , γ1 (0) = x,      γ 1 (s) 2 0    γ2 (s) = u2 2 + u3 γ1 (s) + u4 , γ2 (0) = y,   γ30 (s) = u2 γ1 (s) + u3 , γ3 (0) = z,      γ 0 (s) = −u , γ4 (0) = t. 2  4 This gives  γ1 (s)         γ2 (s)   γ3 (s)      γ4 (s) 

= x + su1 ,   3 = y + u22 sx 2 + s3 u12 + s2 u1 x + u3 sx + 2 = z + su2 x + s2 u1 u2 + su3 , = t − su2 ,

u1 u3 2 2 s

+ su4 ,

thus obtaining *. y+ Exp(u · Y )(g) = γ(1) = .. . ,

u2 2



x2

x + u1 + + u1 x + u3 x + z + u3 + u2 x + 12 u1 u2 t − u2 1 2 3 u1

u1 u3 2

+ + u4 // // . -

(1.12)

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We thus find u1 *. u12 u2 u1 u3 +/ Exp(u · Y )(0) = .. 6 + u2 1 u2+ u4 // . . u3 + 2 / −u , 2 If we set v = Exp(u · Y )(0), then the inverse mapping is given by

*. Log(v) = .. .

v1 −v4 v3 + v12v4

,v2 −

v1 v3 2



v 12 v 4 12

+/ // . /

(1.13)

-

We can introduce a group law in R4 by defining for g 0 = (x 0, y 0, z 0,t 0 ), g ◦ g 0 = exp(Log(g 0 ) · X )(g). If we set u 0 = Log(g 0 ), then formula (1.13) gives x0 +/ −t 0 // . 0 0 z 0 + x2t / (x 0 ) 2 t 0 x0 z0 0 , y − 2 − 12 -

u0 *. 10 +/ *. u u 0 = .. 20 // = .. .u / . 3

0 ,u4 -

Applying (1.12) with u replaced by this u 0, we finally have after some elementary computations,

*. y+ 0 g ◦ g = ... . ,

u 20 2

x + u10 

 x 2 + 13 (u10 ) 2 + u10 x + u30 x + z + u30 + u20 x + 12 u10 u20 t − u20

x + x0 *. 0 2+ y + y 0 + xz 0 − t 2x // = .. . . z + z 0 − t 0 x // ,

t + t0

u 10 u 30 2

+ + u40 // // / -

(1.14)

-

Equation (1.14) is the sought-for group law. If we write it in the more traditional form g◦g 0 = (x, y, z,t)◦(x 0, y 0, z 0,t 0 ) = (x+ x 0, y+ y 0 + xz 0 −

t 0 x2 , z+z 0 −t 0 x,t +t 0 ), (1.15) 2

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the identity element with respect to ◦ is e = (0, 0, 0, 0), and the inverse of g with respect to ◦ is given by   x2 g −1 = − x, −y + xz + t, −z − xt, −t . (1.16) 2 If we define the left-translation operator as L g (g 0 ) = g ◦ g 0, then its differential is given by 1 0 0 0 *. 2+ 0 1 x − x2 // dL g = .. , and so det dL g ≡ 1 ∀g ∈ R4 . (1.17) .0 0 1 −x // 1 ,0 0 0 Furthermore, by (1.17) one sees that Lebesgue measure is invariant with respect to L g . Proposition 1.3. If we denote K2 = (R4 , ◦, δ λ ), with δ λ given by (1.11), then K2 is a four-dimensional nilpotent Lie group of step r = 3. Furthermore, Lebesgue measure is invariant with respect to the left translations on K2 . If k2 denotes the Lie algebra of K2 , then we have k2 = Lie{X0 , X1 }, and the operator L = X12 + X0 is left invariant on K2 and δ λ -homogeneous of degree 2. Once we have the explicit expression of the group law (1.15), we can a posteriori verify the left invariance of X0 , X1 directly. Considering in fact   t 0 x2 F (g 0 ) = F (x 0, y 0, z 0,t 0 ) = f ◦ L g (g 0 ) = f x + x 0, y + y 0 + xz 0 − , z + z 0 −t 0 x,t +t 0 , 2 we have Fx 0 = f x ,

Fy 0 = f y ,

Fz 0 = x f y + f z ,

Ft 0 = −

x2 fy − x fz + ft . 2

This gives X1 ( f ◦ L g ) = (X1 f ) ◦ L g , (x 0 ) 2

(1.18)

Fy 0 + x 0 Fz 0 − Ft 0 (1.19) 2 (x 0 ) 2 x2 = f y + x 0 (x f y + f z ) − f t + fy + x fz 2 2 (x + x 0 ) 2 = f y + (x + x 0 ) f z − f t = (X0 f ) ◦ L g . 2 Formulas (1.18) and (1.19) show that the vector fields X0 , X1 are left invariant with respect to (1.15), and therefore so is L in (1.9) above. X0 ( f ◦ L g ) =

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2 Carnot groups The above discussion of the Mumford operator (1.1) and of its “approximating” higher-order Kolmogorov operator (1.9) brings us to introduce a class of Lie groups which, besides supporting notable families of differential operators which are lefttranslation invariant, also share with Euclidean R N a so-called homogeneous structure associated with the existence of dilations. As we saw in the discussion in Section 1.4 above, such groups naturally arise as approximating “tangent spaces” to the subRiemannian spaces associated with a Hörmander-type operator. This guiding idea was in fact at the basis of a visionary program which Stein presented in his address at the 1970 International Congress of Mathematicians in Nice [S71]. The program culminated in the celebrated works of Rothschild and Stein [RS76], and of Nagel, Stein, and Wainger [NSW85]. The reader who wants to become acquainted with the theory of Carnot groups should consult the classical books [FS82], [CGr90], and the more recent exhaustive contribution [BLU07] which also contains notable applications to the potential theory associated with such groups. We recall that a Carnot group of step r is a connected, simply connected Lie group G whose Lie algebra g admits a stratification g = V1 ⊕ · · · ⊕ Vr , which is r-nilpotent, i.e., [V1 ,Vj ] = Vj+1 ,

j = 1, . . . ,r − 1,

[Vj ,Vr ] = {0},

j = 1, . . . ,r.

(2.1)

From the assumptions (2.1) on the Lie algebra one immediately sees that any basis of the first layer V1 bracket generates the whole Lie algebra g. Because of the special role played by V1 , this vector subspace of the Lie algebra is usually called the horizontal layer of the stratification. In the case in which r = 1 we are in the Abelian situation in which g = V1 , and thus G is isomorphic to Rm , where m = dim V1 . We are thus back in Rm , there is no non-commutative geometry involved, and everything is classical. The case in which r = 2 is the first genuinely non-Abelian case. We will analyze some important instances of such Carnot groups of step 2. A Carnot group of step r is also known in the literature as a stratified nilpotent Lie group of step r; see [Fo75]. As a rule, we will use letters g, g 0, g 00, g0 for points in G, whereas we will reserve the letters ξ, ξ 0, ξ 00, ξ0 , η, for elements of the Lie algebra g. We will always denote by e ∈ G the group identity. The stratification (2.1) of the Lie algebra g allows a natural one-parameter family of non-isotropic dilations to be introduced on g. To do this we assign to each element of the layer Vj the formal degree j. Accordingly, if ξ = ξ1 + · · · + ξr ∈ g, with ξ j ∈ Vj , one defines dilations on g by the rule ∆ λ ξ = λξ1 + · · · + λ r ξr . (2.2)

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We let m j = dim Vj , j = 1, . . . ,r, and denote by N = m1 + · · · + mr the topological dimension of G. We will always assume that the Lie algebra g is endowed with a scalar product h·, ·ig with respect to which the layers Vj0 s, j = 1, . . . ,r, are mutually orthogonal. We will indicate with q ||ξ ||g = hξ, ξig , ξ ∈ g, the corresponding norm. Given ξ = ξ1 + · · · + ξr , with ξ s ∈ Vs , s = 1, . . . ,r, we define in g a non-isotropic gauge in the following way: |ξ |g =

r X

||ξ s || 2r !/s

 1/2r !

.

(2.3)

s=1

Let us note right away that | · |g is homogeneous of degree 1 with respect to the group dilations (2.2) above, i.e., |∆ λ ξ |g = λ|ξ |g . (2.4) Using the exponential map exp : g → G, the anisotropic dilations (2.2) can be transferred to the group G as follows: δ λ (g) = exp ◦∆ λ ◦ exp−1 g.

(2.5)

One can also use the exponential map and (2.3) to define a so-called gauge on G by letting for g = exp ξ, |g|G = |ξ |g . (2.6) We observe that   |δ λ (g)|G = |δ λ (exp ξ)|G = exp ◦∆ λ ◦ exp−1 (exp ξ) = | exp(∆ λ ξ)|G G = (by (2.6)) |∆ λ ξ |g = (by (2.4)) λ|ξ |g = λ|g|G ,

(2.7)

and thus | · |G is homogeneous of degree 1 with respect to the non-isotropic group dilations (2.5). Definition 2.1. We define the homogeneous dimension of G by the equation Q=

r X

jm j .

(2.8)

j=1

The motivation for this name comes from the important equation (2.11) below. In the non-Abelian case r > 1, one clearly has Q > N. A fundamental tool in differential geometry is the exponential map, which sends the tangent space at a point to the manifold itself. When, in addition, the manifold

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133

is a real Lie group, then the exponential map possesses much deeper properties. For a Carnot group G a remarkable fact is that the exponential map exp : g → G defines an analytic diffeomorphism of the Lie algebra g onto G; see, e.g., the book by Varadarajan [V87, Sec. 2.10 onwards]. Perhaps, the most fundamental property of the exponential map is the Baker–Campbell–Hausdorff formula (see, e.g., [V87, Sec. 2.15])   1 1  exp(ξ) exp(η) = exp ξ + η + [ξ, η] + [ξ, [ξ, η]] − [η, [ξ, η]] + · · · , (2.9) 2 12 where the dots indicate commutators of order 4 and higher. Furthermore, since by (2.1) above all commutators of order r and higher are trivial, in every Carnot group the Baker–Campbell–Hausdorff series on the right-hand side of (2.9) is finite. It should be clear to the reader that, using (2.9), one can recover the group law g ◦ g 0 in G from the knowledge of the algebraic commutation relations between the elements of its Lie algebra. We will respectively denote by L g (g 0 ) = g ◦ g 0,

Rg (g 0 ) = g 0 ◦ g,

(2.10)

the operators of left and right translation by an element g ∈ G. Throughout these notes we will indicate by dg the bi-invariant Haar measure on G obtained by lifting via the exponential map the Lebesgue measure on g; see, e.g., [CGr90]. One easily checks that (d ◦ δ λ )(g) = λ Q dg,

(2.11)

where the number Q in (2.11) is the homogeneous dimension of G introduced in (2.8) above. The number plays an important role in the analysis of Carnot groups.

2.1 Sub-Laplacians and harmonic functions on Carnot groups. Given a Carnot group G of step r with Lie algebra g = V1 ⊕ · · · ⊕ Vr , with any orthonormal basis {e1 , . . . , em } of the horizontal layer V1 we can associate corresponding leftinvariant C ∞ vector fields on G by the formula X j (g) = (L g )∗ (e j ),

j = 1, . . . , m,

where (L g )∗ indicates the differential of L g . Let us note explicitly that, given a smooth function u on G, the derivative of u in g ∈ G along the vector field X j is given by d (2.12) X j u(g) = u(g exp se j ) s=0 . ds We assume throughout these notes that G is endowed with a left-invariant Riemannian metric with respect to which the vector fields {X1 , . . . , X m } are orthonormal.

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Given a smooth function u on G we denote by ∇H u =

m X

X j uX j

(2.13)

j=1

its horizontal gradient. We have |∇ H u| 2 =

m X

(X j u) 2 .

(2.14)

j=1

It should be clear that, if F ∈ C ∞ (G), and L g (F)(g 0 ) = F (L g (g 0 )), then X j (L g (F)) = L g (X j F),

j = 1, . . . , m,

(2.15)

i.e., the vector fields X1 , . . . , X m are invariant with respect to the left translations on G. Furthermore, they are homogeneous of degree 1 with respect to the non-isotropic dilations (2.5), i.e., X j (δ λ F) = λδ λ (X j F),

j = 1, . . . , m.

(2.16)

Definition 2.2. The sub-Laplacian associated with the basis {e1 , . . . , em } is defined by the formula m X ∆H u = X 2j u. (2.17) j=1

In view of (2.15) every sub-Laplacian is left-translation invariant, i.e., for any g ∈ G one has ∆ H (L g (u)) = L g (∆ H u). Furthermore, thanks to (2.16) it is a differential operator of order 2 with respect to the non-isotropic dilations (2.5), in the sense that ∆ H (δ λ u) = λ 2 δ λ (∆ H u). Definition 2.3. Given a sub-Laplacian ∆ H on a Carnot group G and an open set Ω ⊂ G, a distribution u ∈ D 0 (Ω) is called harmonic in Ω if ∆ H u = 0 in D 0 (Ω). Since the vector fields X1 , . . . , X m and their commutators up to step r generate the whole Lie algebra of left-invariant vector fields on G, thanks to Hörmander’s Theorem 1.2 above, ∆ H is hypoelliptic (but not, in general, real-analytic hypoelliptic). Thus, we have the following important result. Theorem 2.4. Let u ∈ D 0 (Ω) be harmonic in Ω. Then, there exists U ∈ C ∞ (Ω) such that ∆ H U = 0 in Ω and such that u = U in D 0 (Ω).

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Let π j : g → Vj denote the projection onto the jth layer of g. Since the exponential map exp : g → G is a global analytic diffeomorphism, we can define analytic maps ξ j : G → Vj , j = 1, . . . ,r by letting ξ j = π j ◦ exp−1 . The notation {e j,1 , . . . , e j, m j }, j = 1, . . . ,r will indicate a fixed orthonormal basis of the jth layer Vj . For g ∈ G, the projections of the exponential coordinates of g onto the layer Vj , j = 1, . . . ,r are defined as x j, s (g) = hξ j (g), e j, s ig ,

s = 1, . . . , m j .

(2.18)

The vector ξ j (g) ∈ Vj , j = 1, . . . ,r will be routinely identified with the point (x j,1 (g), . . . , x j, m j (g)) ∈ Rm j . Whenever convenient, we will identify g ∈ G with its exponential coordinates def

x(g) = (x 1 (g), . . . , x m (g),t 1 (g), . . . ,t k (g), . . . , x r,1 (g), . . . , x r, m r (g)) ∈ R N , (2.19) and we will ordinarily drop in the latter the dependence on g, i.e., we will write g = (x 1 , . . . , x r, m r ). Consider the orthonormal basis {e1 , . . . , em , ε 1 , . . . ., ε k , . . . , er,1 , . . . , er, m r } of g. Using (2.10) we define left-invariant vector fields on G by letting X j, s (g) = (L g )∗ (e j, s ) ,

j = 1, . . . ,r,

s = 1, . . . , m j ,

(2.20)

where (L g )∗ indicates the differential of L g . Before stating the next result we need a useful definition. As previously mentioned, the layer Vj , j = 1, . . . ,r in the stratification of g is assigned the formal degree j. Correspondingly, each homogeneous monomial ξ1α1 ξ2α2 . . . ξrα r , with multiindices α j = (α j,1 , . . . , α j, m j ), j = 1, . . . ,r, is said to have weighted degree k if mj r X X  j α j, s = k. j=1

s=1

Using the Baker–Campbell–Hausdorff formula (2.9) we can express (2.12) using the exponential coordinates (2.19), obtaining the following result. Proposition 2.5. For each i = 1, . . . , m, and g = (x 1 , . . . , x r, m r ), we have r

mj

XX ∂ ∂ Xi = + bsj, i (x 1 , . . . , x j−1, m ( j−1) ) ∂ x i j=2 s=1 ∂ x j, s r

=

mj

XX ∂ ∂ + bs (ξ1 , . . . , ξ j−1 ) , ∂ x i j=2 s=1 j, i ∂ x j, s

(2.21)

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where each bsj,i is a homogeneous polynomial of weighted degree j − 1. We notice that an immediate consequence of (2.21) is that div X i = 0,

i = 1, . . . , m,

(2.22)

where div X i indicates the divergence of X i with respect to the exponential coordinates.

2.2 Carnot groups of step r = 2. Since Carnot groups of step r = 2 often play a special role in analysis and geometry, it will be convenient to have a simplified notation for objects in the horizontal layer V1 , and in the first vertical layer V2 . For simplicity, we set m = m1 , k = m2 , and let {e1 , . . . , em } = {e1,1 , . . . , e1, m1 },

{ε 1 , . . . , ε k } = {e2,1 , . . . , e2, m2 }.

(2.23)

Notice that for a Carnot group of step r = 2 the homogeneous dimension of the group, defined in (2.8) above, is given by the number Q = m + 2k. We indicate with   i = 1, . . . , m,  z i (g) = hξ1 (g), ei ig , t s (g) = hξ2 (g), ε s >g , s = 1, . . . , k 

(2.24)

the projections of the exponential coordinates of g onto V1 and V2 respectively. For later purposes it will be useful to introduce the group constants of G. By the grading assumption on the Lie algebra, we have [V1 ,V1 ] = V2 . Therefore, if ei , e j ∈ {e1 , . . . , em }, we let def

bisj = h[ei , e j ], ε s ig , so that [ei , e j ] =

k X

bisj ε s ,

i, j = 1, . . . , m.

(2.25)

(2.26)

s=1

Since [ei , e j ] = −[e j , ei ], it should be obvious that bisj = −bsji . As in (2.23) we use special notation for the left-invariant vector fields associated with the basis {e1 , . . . , em } and {ε 1 , . . . , ε k }, and let X i (g) = (L g )∗ (ei ),

i = 1, . . . , m,

Ts (g) = (L g )∗ (ε s ),

s = 1, . . . , k, g ∈ G. (2.27) The following expressions of the vector fields X i , and of the sub-Laplacian ∆ H in exponential coordinates will be useful; see [GV01].

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2 Hypoelliptic operators, etc.

Proposition 2.6. Let G be a Carnot group of step 2. Then, in the exponential coordinates (z,t) one has k

X i = ∂z i + Ts =

k

m

∂ 1X ∂ 1 XX s bi j z j hJ (ε ` )z, ei i∂t ` = − , 2 `=1 ∂z i 2 s=1 `=1 ∂t s

∂ , ∂t s

∆ H = ∆z +

(2.28) (2.29)

k k X 1 X hJ (ε ` )z, J (ε `0 )zi∂t ` ∂t `0 + ∂t ` Θ` , 4 `,`0 =1 `=1

(2.30)

where ∆z represents the standard Laplacian in the variable z = (z1 , . . . , z m ), and Θ` =

m m X X hJ (ε ` )z, ei i∂z i = − b`i j z j ∂z i . i=1

(2.31)

i, j=1

2.3 The Heisenberg group Hn . A Carnot group of fundamental relevance is the so-called Heisenberg group. This is the simplest non-Abelian example of a Carnot group of step r = 2, whose underlying manifold is R2n+1 with coordinates g = (x, y,t), where x, y ∈ Rn , t ∈ R. We can describe its Lie algebra hn = V1 ⊕ V2 , where V1 = R2n x, y × {0}t and {0}R2n × Rt . If we adopt the canonical basis {e1 , . . . , en , en+1 , . . . , e2n } of R2n x, y , and we identify it in a natural way with a basis of V1 , and we take ε 1 = (0, . . . , 0, 1) as a basis of V2 , then the Lie algebra structure of hn is obtained by postulating that [ei , en+ j ] = ε 1 δ i j , i, j = 1, . . . , n, (2.32) all other commutators being assumed to be trivial. It is clear that (2.32) implies that [V1 ,V1 ] = V2 ,

[V1 ,V2 ] = {0},

and thus the Heisenberg algebra hn is a stratified Lie algebra which is nilpotent of step r = 2. Consequently, the Heisenberg group provides an example of a Carnot group of step r = 2. For a beautiful introduction to the group Hn we refer the reader to the book [CDPT07]. One should also see the monumental works [S93] and [BLU07].

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The subspace V1 is called the horizontal layer, whereas V2 is called the vertical layer of the Heisenberg algebra. It is clear that V2 constitutes the center of hn with respect to (2.33). If we now identify g, g 0 ∈ Hn respectively with x 1 e1 + · · · + x n en + y1 en+1 + · · · + yn e2n + tε 1 and g 0 = x 10 e1 + · · · + x n0 en + y10 en+1 + · · · + yn0 e2n + t 0 ε 1 , then from (2.32) and the Baker–Campbell–Hausdorff formula (2.9) above, which in the present case reads   1 exp(ξ) exp(η) = exp ξ + η + [ξ, η] , 2 we easily obtain that the non-Abelian group multiplication on Hn is given by g ◦ g 0 = (x, y,t) ◦ (x 0, y 0,t 0 )   1 = x + x 0, y + y 0,t + t 0 + (hx, y 0i − hx 0, yi) . 2

(2.33)

We will indicate with e = (0, 0, 0) ∈ Hn the group identity with respect to (2.33). Notice that for a given g = (x, y,t) one has g −1 = (−x, −y, −t). We let L g (g 0 ) = g ◦ g 0 denote the operator of left translation on Hn , and indicate with (L g )∗ its differential. Identifying hn with the space of left-invariant vector fields on Hn , one easily recognizes that a basis for hn is given by the 2n + 1 vector fields  (L g )∗       (L g )∗       (L g )∗

∂  def ∂ = X i (g) = ∂x∂ i − y2i ∂t , ∂x i xi ∂ ∂ ∂  def ∂y i = X n+i (g) = ∂y i + 2 ∂t , ∂ ∂  def ∂t = T (g) = ∂t ,

i = 1, . . . , n, i = 1, . . . , n,

(2.34)

and that the only non-trivial commutation relation is [X i , X n+ j ] = T δ i j ,

i, j = 1, . . . , n.

(2.35)

The sub-Laplacian on Hn with respect to the canonical basis {e1 , . . . , e2n } is given by ∆H =

n X

2 (X 2j + X n+ j ).

j=1

This operator has a non-negative characteristic form, but it fails to be elliptic at every point g = (x, y,t) ∈ Hn . This circumstance can be verified by computing the matrix A(x, y,t) = A(x, y) associated with the principal symbol of ∆ H and verifying that the eigenvalues of the matrix are all non-negative, but that one of them vanishes identically. In fact, in the real coordinates (x, y,t) we have n

∆ H = ∆ x, y +

X   |x| 2 + |y| 2 2 ∂t + ∂t x i ∂y i − yi ∂x i . 4 i=1

2 Hypoelliptic operators, etc.

139

It is easy to see that, when n = 1 and we are dealing with the three-dimensional Heisenberg group H1 , the matrix A(x, y) has eigenvalues λ 1 = 0,

λ 2 = 1,

λ3 = 1 +

|x| 2 + |y| 2 . 4

If we assign to the elements of Vj , j = 1, 2 the formal degree j, then the associated non-isotropic dilations of Hn are given by δ λ (g) = (λ x, λ y, λ 2 t).

(2.36)

The homogeneous dimension of Hn with respect to (2.36) is the number Q = 2n + 2. Since Lebesgue measure dg on R2n+1 is a left- and right-invariant Haar measure on Hn , one easily checks that (d ◦ δ λ )(g) = λ Q dg. In the analysis of Hn the number Q plays much the same role as that of the Euclidean dimension of Rn .

2.4 The four-dimensional Engel group. We next describe the four-dimensional cyclic or Engel group. This group is important in many respects since it represents the next level of difficulty with respect to the Heisenberg group and provides an ideal framework for testing whether results which are true in step 2 generalize to step 3 or higher. The reader unfamiliar with the cyclic group can consult [CGr90], or also [M02]. The Engel group E = K3 (see [CGr90, ex. 1.1.3]) is the Lie group whose underlying manifold can be identified with R4 , and whose Lie algebra is given by the grading e = V1 ⊕ V2 ⊕ V3 , where V1 = span{e1 , e2 }, V2 = span{e3 }, and V3 = span{e4 }, so that m1 = 2 and m2 = m3 = 1. We assign the bracket relations [e1 , e2 ] = e3

[e1 , e3 ] = e4 ,

(2.37)

all other brackets being assumed trivial. The relations (2.37) show that [V1 ,V1 ] = V2 ,

[V1 ,V2 ] = V3 ,

[V1 ,V3 ] = {0},

so that the Engel group E is a Carnot group of step r = 3. We observe that the homogeneous dimension of the Engel group E is Q = m1 + 2m2 + 3m3 = 7.

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We will denote with (x, y), t, and s respectively the variables in V1 , V2 , and V3 , so that any ξ ∈ e can be written as ξ = xe1 + ye2 + te3 + se4 . If g = exp(ξ), we will identify g = (x, y,t, s). The group law in E is given by the Baker–Campbell– Hausdorff formula (2.9). In exponential coordinates, if g = exp(ξ), g 0 = exp(ξ 0 ), we have 1 1  g ◦ g 0 = ξ + ξ 0 + [ξ, ξ 0] + [ξ, [ξ, ξ 0]] − [ξ 0, [ξ, ξ 0]] . 2 12 Using such formula, a computation based on (2.37) gives (see also [CGr90, ex. 1.2.5])   g ◦ g 0 = x + x 0, y + y 0,t + t 0 + P3 , s + s 0 + P4 , where  1 x y 0 − yx 0 , 2 0  1 0 1 2 0 P4 = (xt − t x 0 ) + x y − x x 0 (y + y 0 ) + yx 2 . 2 12

P3 =

(2.38) (2.39)

Once we have the group law, we define the left-translation operator L g (g 0 ) = g ◦ g 0 on E. Denoting by (L g )∗ its differential, we can associate with the basis {e1 , e2 , e3 , e4 } the following left-invariant vector fields on E: X i (g) = (L g )∗ (ei ),

i = 1, 2,

T (g) = (L g )∗ (e3 ),

S(g) = (L g )∗ (e4 ). (2.40) Computing (L g )∗ and using the fact that the vector fields X1 , X2 ,T and S are given by (2.40), we easily find the following expressions   ∂ ∂ ∂  − y2 ∂t − 2t + x12y ∂s , X1 = ∂x     2  x ∂ x ∂ ∂    X2 = ∂y + 2 ∂t + 12 ∂s , (2.41) ∂ ∂   T = ∂t + x2 ∂s ,     S = ∂ . ∂s  From the equations (2.41) we obtain the corresponding commutator relations [X1 , X2 ] = T,

[X1 ,T] = [X1 , [X1 , X2 ]] = S,

(2.42)

all other commutators being trivial. We note that the action of X1 , X2 ,T on a function on E which is independent of the variable s reduces to the action of the corresponding vector fields in H1 . The sub-Laplacian on E with respect to the basis {e1 , e2 } of the horizontal layer V1 is given by ∆ H = X12 + X22 .

2 Hypoelliptic operators, etc.

141

2.5 Groups of Metivier and groups of Heisenberg type. In this section we discuss two notable examples of Carnot groups of step r = 2 which, from the point of view of analysis, first arose in connection with questions of hypoellipticity. Such groups display, in a simplified fashion, a remarkable feature which is common to all Carnot groups: algebraic hypothesis on the Lie algebra translate into fundamental properties of the corresponding differential operators on the group. We assume throughout that G is a Carnot group of step 2, with Lie algebra g = V1 ⊕ V2 , where [V1 ,V1 ] = V2 , [V1 ,V2 ] = {0}, and we recall that g is assumed to be endowed with an inner product with respect to which {e1 , . . . , em } and {ε 1 , . . . , ε k } denote an orthonormal basis of V1 and V2 , respectively. If X i (g) = (L g )∗ (ei ), i = 1, . . . , m and T` = (L g )∗ (ε ` ), ` = 1, . . . , k are the left-invariant vector fields generated by such a basis, we assume that G is endowed with a left-invariant Riemannian tensor with respect to which the vector fields X1 , . . . , X m , T1 , . . . ,Tk are orthonormal at every point. The sub-Laplacian with respect to the basis {e1 , . . . , em } Pm 2 is given by ∆ H = i=1 X i . Consider the analytic mappings z : G → V1 , t : G → V2 uniquely defined through the equation g = exp(z(g) + t(g)). For each i = 1, . . . , m we set z i = z i (g) = hz(g), ei i, whereas for s = 1, . . . , k we let t s = t s (g) = ht(g), ε s i. We will indicate with (z,t) ∈ g the exponential coordinates of a point g ∈ G. Consider the linear mapping J : V2 → End(V1 ), defined by hJ (t)z, z 0i = h[z, z 0],ti.

(2.43)

A trivial, yet useful observation is that hJ (t)z, zi = 0,

z ∈ V1 ,t ∈ V2 .

This follows from [z, z] = 0. Definition 2.7. A Carnot group of step 2 is called a Metivier group if there exists a constant B > 0 such that | J (t)z| ≥ B|z||t|,

z ∈ V1 , t ∈ V2 .

Definition 2.8. A Carnot group of step 2 is called of Heisenberg type if for every t ∈ V2 such that |t| = 1, the mapping J (t) is orthogonal. This is equivalent to saying that for every z, z 0 ∈ V1 , and every t ∈ V2 , one has hJ (t)z, J (t)z 0i = |t| 2 hz, z 0i.

(2.44)

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In particular, when G is of Heisenberg type, then (2.44) with z = z 0 yields | J (t)z| 2 = hJ (t)z, J (t)zi = |t| 2 |z| 2 .

(2.45)

We will need the following simple, yet crucial, result about groups of Heisenberg type. Proposition 2.9. If G is of Heisenberg type, then for every t,t 0 ∈ V2 and any z ∈ V1 one has hJ (t)z, J (t 0 )zi = |z| 2 ht,t 0i. In particular, we obtain for every `, ` 0 = 1, . . . , k, hJ (ε ` )z, J (ε `0 )zi = |z| 2 δ ``0 . Proof. By polarization, ) 1( | J (t)z + J (t 0 )z| 2 − | J (t)z − J (t 0 )z| 2 4 ) 1( | J (t + t 0 )z| 2 − | J (t − t 0 )z| 2 = 4 ) |z| 2 ( |t + t 0 | 2 − |t − t 0 | 2 = 4 = |z| 2 ht,t 0i,

hJ (t)z, J (t 0 )zi =

where in the second to the last equality we have used (2.44).



Remark 2.10. Clearly, every group of Heisenberg type is a Metivier group. The opposite inclusion is false. For instance (see [BLU07, Remark 3.7.5]), consider G = R5 = R4z × Rt with the group law 1 (z,t)(z 0,t 0 ) = (z + z 0,t + t 0 + hAz, z 0i), 2 where A is the 4 × 4 skew-symmetric matrix 0 −1 0 0 *. + 1 0 0 0 // . . A=. .0 0 0 −2// ,0 0 2 0 Since det A , 0, the matrix A is non-singular and therefore G is a Metivier group. But G is not of H-type since A < O(4). The following expressions in exponential coordinates of the vector fields X i , and of the sub-Laplacian ∆ H will be useful; see Proposition 2.6 above.

2 Hypoelliptic operators, etc.

143

Lemma 2.11. Let G be a Carnot group of step 2. Then, in the exponential coordinates (z,t) one has k

X i = ∂z i + ∆ H = ∆z +

1X hJ (ε ` )z, ei i∂t ` , 2 `=1

(2.46)

k k X 1 X hJ (ε ` )z, J (ε `0 )zi∂t ` ∂t `0 + ∂t ` Θ` , 4 `,`0 =1 `=1

(2.47)

where ∆z represents the standard Laplacian in the variable z = (z1 , . . . , z m ), and Θ` =

m X

hJ (ε ` )z, ei i∂z i .

(2.48)

i=1

In particular, when G is of Heisenberg type we obtain from (2.47) and (2.44), k

∆ H = ∆z +

X |z| 2 ∆t + ∂t ` Θ` . 4 `=1

(2.49)

Remark 2.12. Groups of Metivier type are important since they are the largest known class of Carnot groups in which every sub-Laplacian is real-analytic hypoelliptic. Thus, sub-Laplacians on Metivier groups share the same fundamental property of the classical Laplace operator in R N . To better elucidate Remark 2.12 we recall that in the framework of Carnot groups of step 2, Helffer proved in [Hel] that if a sub-Laplacian in G is real-analytic hypoelliptic, then G must be a Metivier group. Thanks to a result of Metivier [Me2] (see also [Me1]), it is known that if G is a Metivier group, then a sub-Laplacian ∆ H is real-analytic hypoelliptic if and only if it is hypoelliptic. Since by Hörmander’s Theorem 1.2 above every sub-Laplacian is hypoelliptic, it follows that in Metivier groups all sub-Laplacians are real-analytic hypoelliptic. However, outside groups of step 2 there is (to the best of the writer’s knowledge) no known example of a realanalytic sub-Laplacian. It is plausible (for this conjecture see Rothschild in [R84]) that no sub-Laplacian on a Carnot group which is not a Metivier group should be real-analytic hypoelliptic, but this question seems to be open. These considerations lead us to a fundamental question. Open problem: Is it true that in a Carnot group every sub-Laplacian possesses the unique continuation property, or even the strong unique continuation property? It is known that a real-analytic function cannot vanish to infinite order at one point of a connected open set, without being identically zero. This is known as the

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strong unique continuation property (sucp). From what we said above, since in every Metivier group harmonic functions are real-analytic, they possess the sucp. What happens with the sucp beyond Metivier groups? We can ask for a weaker property: can harmonic functions in a Carnot group vanish in an open subset of a connected open set? These important questions presently constitute terra incognita!

2.6 Groups of step 2: some useful computations. Every Carnot group can be endowed with a non-isotropic gauge which respects the stratification of its Lie algebra; see, e.g., [Fo75] and also [FS82]. In the special case of groups of step r = 2 it is particularly convenient to introduce the gauge % : G → [0, ∞), %(g) = (|z| 4 + 16|t| 2 ) 1/4 ,

(2.50)

where we have denoted by (z,t) the exponential coordinates of g ∈ G. Since the group dilations are given by δ λ (z,t) = (λz, λ 2 t), it is clear that % ◦ δ(g) = λ %(g), i.e., % is positively homogeneous of degree 1. We note explicitly that, except for the normalizing factor 16, (2.50) is precisely the non-isotropic gauge defined by (2.3), (2.6). Lemma 2.13. Let G be a Carnot group of step 2, and consider the gauge (2.50). Then,  1 |∇ H %| 2 = 6 |z| 6 + 16| J (t)z| 2 ; (2.51) % see (2.14) above. In particular, if G is of Heisenberg type, then |∇ H %| 2 =

|z| 2 . %2

(2.52)

Proof. In what follows we let r = r (g) = |z|, s = s(g) = |t|, so that % = (r 4 + 16s2 ) 1/4 . Using Lemma 2.11 we find k

Xi % = =

1 ∂% 1 ∂% X zi + hJ (ε ` )z, ei it ` r ∂r 2s ∂s `=1 1 ∂% 1 ∂% zi + hJ (t)z, ei i. r ∂r 2s ∂s

(2.53)

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2 Hypoelliptic operators, etc.

Using the fact that hJ (t)z, zi = 0 for every t !2 ∂% 1 2 |∇ H %| = + 2 ∂r 4s !2 1 ∂% = + 2 ∂r 4s

∈ V2 , z ∈ V1 , this gives !2 m ∂% X hJ (t)z, ei i2 ∂s i=1 !2 ∂% | J (t)z| 2 . ∂s

Now we observe that

∂ % 8s ∂ % r3 (2.54) = 3, = 3. ∂r ∂s % % Inserting these formulas into the last equation we obtain (2.51). When G is of Heisenberg type, using (2.45) in (2.51), we obtain (2.52).  We next develop some further results which prove useful in many computations connected with Carnot groups of step r = 2. Lemma 2.14. Let G be a Carnot group of step r = 2. Then, for every i, j = 1, . . . , m and ` = 1, . . . , k, one has X i (z j ) = δ i j ,

X i (t ` ) = h[z, ei ], ε ` i.

(2.55)

From this, one obtains |∇ H (|z| 2 )| 2 = 4|z| 2 .

(2.56)

Furthermore, one has ∆ H (z j ) = 0,

∆ H (t ` ) = 0.

(2.57)

As a consequence of (2.55) and (2.57), one has ∆ H (|z| 2 ) = 2m.

(2.58)

Proof. We begin by observing that, since G has step r = 2, the Baker–Campbell– Hausdorff formula (2.9) easily gives for every j, ` = 1, . . . , m and every i = 1, . . . , k, z j (g exp sei ) = z j (g) + sδ i j ,

s t ` (g exp sei ) = t ` (g) + h[z, ei ], ε ` i. 2

(2.59)

The formulas (2.59) and (2.12) imply that Xi z j =

d z j (g exp sei ) s=0 = δ i j , ds

Xi t ` =

d 1 t ` (g exp sei ) s=0 = h[z, ei ], ε ` i. ds 2

The same formulas also trivially give X i2 z j =

d2 z j (g exp sei ) s=0 = 0, ds2

X i2 t ` =

d2 t ` (g exp sei ) s=0 = 0. ds2 

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Lemma 2.15. Let G be a group of Heisenberg type. The following formulas hold: k 2 |z| , 2 |∇ H (|t| 2 )| 2 = |z| 2 |t| 2 ,

(2.61)

h∇ H (|z| ), ∇ H (|t| )i = 0.

(2.62)

∆ H (|t| 2 ) = 2

(2.60)

2

Proof. Using the second formula in (2.57) we find ∆ H (|t| ) = 2 2

k X

t ` ∆ H (t ` ) + 2

k X

`=1

|∇ H (t ` )| = 2 2

`=1

k X

|∇ H (t ` )| 2 .

`=1

Next, equation (2.55) gives m  m m X 2 1 X 1X |∇ H (t ` )| 2 = X j (t ` ) = h[z, e j ], ε ` i2 = hJ (ε ` )z, e j i2 4 4 j=1 j=1 j=1 = 41 | J (ε ` )z| 2 = 41 |z| 2 , where in the second to the last equality we have used (2.45). Substitution of this equation into the previous one gives ∆ H (|t| 2 ) =

k 2 |z| . 2

Next, we have |∇ H (|t| 2 )| 2 =

m  X

X j (|t| 2 )

2

=

m X k X

j=1

=

j=1

m X k X j=1

t ` h[z, e j ], ε ` i

`=1

X j (t 2` )

2

=4

m X k X

`=1

2

=

`=1

j=1 m  X

h[z, e j ],ti

2

j=1

2 t ` X j (t ` )

=

m  X

hJ (t)z, e j i

2

j=1

= | J (t)z| = |z| |t| , 2

2

2

where, again, we have used (2.45). Finally, we have h∇ H (|z| 2 ), ∇ H (|t| 2 )i =

m k X m X X hX j (|z| 2 ), X j (|t| 2 )i = 4 z i t ` X j (z i )X j (t ` ) `=1 i, j=1

j=1

=2 =2

k X m X

δ i j z i t ` h[z, e j ], ε ` i = 2

`=1 i, j=1 m X

m X

j=1

j=1

z j h[z, e j ],ti = 2

k X m X

z j t ` h[z, e j ], ε ` i

`=1 j=1

z j hJ (t)z, e j i = 2hJ (t)z, zi = 0.

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2 Hypoelliptic operators, etc.



This completes the proof.

The next result shows evidence of a remarkable property of the non-isotropic gauge in (2.50) above when G is a group of Heisenberg type. Lemma 2.16. Let G be a group of Heisenberg type, then ∆H % =

Q−1 |∇ H %| 2 %2

in G \ {e}.

Proof. Consider the function K = %4 = |z| 4 + 16|t| 2 . The chain rule gives ∇ H K = 4%3 ∇ H %, and therefore ∆ H K = div H (4%3 ∇ H %) = 4%3 ∆ H % + 12%2 |∇ H %| 2 . From this formula we obtain ∆H % =

g 1 f ∆ H K − 12%2 |∇ H %| 2 . 3 4%

(2.63)

Next, we have ∆ H K = ∆ H (|z| 4 ) + 16∆ H (|t| 2 ) = 2|z| 2 ∆ H (|z| 2 ) + 2|∇ H |z| 2 | 2 + 8k |z| 2 , where in the last term we have used (2.60) above. By (2.56) and (2.58) we find ∆ H K = 4(m + 2k)|z| 2 + 8|z| 2 = 4(Q + 2)|z| 2 . Substituting this equation into (2.63), and using (2.52) above, we conclude that ∆H % =

Q − 1 |z| 2 , %2 %2

which, again by (2.52), yields the desired conclusion.



3 Fundamental solutions and the Yamabe equation In this section we prove, among other things, a beautiful result which is due to Folland for the Heisenberg group Hn (see [Fo73]), and to Kaplan for groups of Heisenberg type (see [Ka80]). Since this result is not intuitive at all, in order to acquire a flavor of it let us start with some preliminary considerations. Suppose that ω ∈

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C ∞ (G \ {e} and, given a function h ∈ C 2 ([0, ∞)), let us consider v = h ◦ ω = h(ω). The chain rule immediately tells us that ∆ H v = h 00 (ω)|∇ H ω| 2 + h 0 (ω)∆ H ω. Suppose now that ω possesses the property that ∆H ω =

Q−1 |∇ H ω| 2 . ω

(3.1)

In this case, we would obtain from the above equation # " Q−1 0 00 h (ω) |∇ H ω| 2 . ∆ H v = h (ω) + ω At this point a direct computation shows that there is a notable choice of the function h that makes the function v a harmonic function outside the identity e ∈ G, namely 0 h(t) = t 2−Q . Such a choice in fact gives h 00 (t) + Q−1 t h (t) = 0, and from the latter equation we would conclude that ∆ H v = 0 in G \ {e}. Remarkably, as we have proved in Lemma 2.16, in every group of Heisenberg type there is a choice that accomplishes (3.1), and it is provided by the non-isotropic gauge in (2.50) above. Quite possibly these were the considerations which led Folland [Fo73] to guessing that in the Heisenberg group Hn a multiple of the function %2−Q should be a fundamental solution of the sub-Laplacian ∆ H . This remarkable discovery was subsequently generalized by Kaplan [Ka80] to all groups of Heisenberg type. Theorem 3.1. Let G be a group of Heisenberg type, and define the universal constant C = C(m, k) > 0 by the formula Z dz dt −1 C = m(Q − 2)  2  (Q+2)/4 , 2 G (|z| + 1) + 16|t| 2 where Q = m + 2k denotes the homogeneous dimension of G. Then, the function Γ(z,t) =

C %(z,t) Q−2

(3.2)

is a fundamental solution with singularity at e for minus the sub-Laplacian associated with the orthonormal basis {e1 , . . . , em }. Remark 3.2. One should compare the fundamental solution in (3.2) with its Euclidean ancestor for the classical Laplace equation. We recall that when n ≥ 3 then the fundamental solution (vanishing at infinity) of the Laplacian −∆ in R N is given by CN Γ(x) = , |x| N −2

2 Hypoelliptic operators, etc.

where CN = S N −1 ⊂ R N .

1 (N −2)σ N −1 ,

149

and σ N −1 represents the surface measure of the unit sphere

The proof of Theorem 3.1 will be divided into several steps. We start as in [Fo73], but soon after we take an interesting, different path which has the advantage of uncovering a beautiful connection with the Yamabe equation from CR geometry. We emphasize that we could have presented a completely different proof of Theorem 3.1 following more classical arguments based on representation formulas. Although such an approach also has its obvious appeal, had we followed it we would not have met the beautiful Yamabe equation. As in [Fo73] for every ε > 0 we consider the function   1/4 %ε (z,t) = (ε 2 + |z| 2 ) 2 + 16|t| 2 . (3.3) Unlike the function %, which presents a singularity in e ∈ G, we have %ε ∈ C ∞ (G) and obviously %ε (g) → %(g), as ε → 0, for every g ∈ G. Hereafter, % indicates the gauge defined in (2.50). We now establish some important properties of %ε . Lemma 3.3. Let G be a group of Heisenberg type; then one has |z| 2 , %2ε Q−1 mε 2 ∆ H %ε = |∇ H %ε | 2 + 3 . %ε %ε

|∇ H %ε | 2 =

Proof. For ease of notation we let f = %ε , K = f 4 = (ε 2 + |z| 2 ) 2 + 16|t| 2 . One easily finds 1 |∇ H K | 2 , 16 f 6 " # 1 3 2 ∆H f = ∆ H K − 4 |∇ H K | . 4f3 4f

|∇ H f | 2 =

(3.4) (3.5)

Since ∇ H K = 2(ε 2 + |z| 2 )∇ H (|z| 2 ) + 16∇ H (|t| 2 ), using (2.61) and (2.62) in Lemma 2.15, we obtain |∇ H K | 2 = 4(ε 2 + |z| 2 ) 2 |∇ H (|z| 2 )| 2 + 162 |∇ H (|t| 2 )| 2 + 64(ε + |z| )h∇ H (|z| ), ∇ H (|t| )i 2

2

2

2

= 16(ε 2 + |z| 2 ) 2 |z| 2 + 162 |z| 2 |t| 2 f g = 16|z| 2 (ε 2 + |z| 2 ) 2 + 16|t| 2 = 16|z| 2 f 4 = 16|z| 2 K.

(3.6)

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Substitution in (3.4) gives the first part of the lemma. Next we compute ∆ H K. Applying Lemma 2.15 again one finds     ∆ H K = ∆ H (ε 2 + |z| 2 ) 2 + 16∆ H (|t| 2 ) = ∆ H (ε 2 + |z| 2 ) 2 + 8k |z| 2 . On the other hand, (2.58) in Lemma 2.14 gives   ∆ H (ε 2 + |z| 2 ) 2 = 2|∇ H (|z| 2 )| 2 + 2(ε 2 + |z| 2 )∆ H (|z| 2 ) = 4(m + 2)|z| 2 + 4mε 2 . Recalling that the homogeneous dimension of G is Q = m + 2k, we conclude that ∆ H K = 4(Q + 2)|z| 2 + 4mε 2 .

(3.7)

Finally, replacing (3.6) and (3.7) in (3.5) we obtain the second part of the lemma.  We now consider a function h ∈ C 2 ([0, ∞)) and form a new function v(g) = h( %ε (g)). The chain rule gives ∆ H v = h 00 ( %ε )|∇ H %ε | 2 + h 0 ( %ε )∆ H %ε . Using Lemma 3.3 we obtain mε 2 Q−1 |∇ H %ε | 2 + h 0 ( %ε ) 3 ∆ H v = h 00 ( %ε )|∇ H %ε | 2 + h 0 ( %ε ) %ε %ε # " 2 mε Q−1 0 = h 00 ( %ε ) + h ( %ε ) |∇ H %ε | 2 + h 0 ( %ε ) 3 . %ε %ε If we now choose h(t) = t 2−Q , then it is clear that h 00 (t) + obtain from the previous equation ∆H v =

Q−1 0 t h (t)

= 0, and we

mε 2 0 = −(Q − 2)mε 2 v (Q+2)/(Q−2) . h ( %ε ) = −(Q − 2)mε 2 %−(Q+2) ε %3ε

We have thus proved the following remarkable result. Theorem 3.4. Let G be a group of Heisenberg type and for any ε > 0 consider the function vε = %2−Q . Then, v solves the semilinear Yamabe equation in G, ε ∆ H vε = −(Q − 2)mε 2 vε(Q+2)/(Q−2) .

2 Hypoelliptic operators, etc.

151

We will also need the following simple Lemma 3.5. There exists a universal number ω = ω(m, k) > 0 such that for every s > 0 we have Z dH N −1 Q−1 Qωs = , %=s |∇%| where we have denoted by H N −1 the standard (N − 1)-dimensional Hausdorff measure on G and by ∇% the Riemannian gradient of % in the left-invariant Riemannian metric on G with respect to which the vector fields X1 , . . . , X m ,T1 , . . . ,Tk are an orthonormal basis. Proof. For every s > 0 we have by the group dilations, Z dg = ωsQ , % 0 we have Z Z  (Q+2)/(Q−2)  p v1 dg ≤ % >1

% >1

%−p(Q+2) dg Z



! ds 1 dH N −1 p(Q+2) |∇%| s

Z

= (by Federer’s co-area formula) 1 %=s Z ∞ ds < ∞, = (by Lemma 3.5) Qω p(Q+2)−Q+1 s 1 provided that p(Q + 2) − Q + 1 > 1

⇐⇒

p>

Q . Q+2

In particular, we conclude that v1(Q+2)/(Q−2) ∈ L 1 ( % > 1), which proves the Claim.  Since the vector fields X j are left-translation invariant (see (2.15)), from Theorem 3.1 we immediately obtain the following result. Corollary 3.6. Let G be a group of Heisenberg type, and for every g, g 0 ∈ G consider the distribution C Γ(g, g 0 ) = Γ(g 0, g) = , −1 %(g ◦ g 0 ) Q−2 where C > 0 is the universal constant defined in Theorem 3.1. Then, for every g ∈ G the function g 0 → Γ(g 0, g) is a fundamental solution of −∆ H with singularity at g, i.e., ∆ H Γ(·, g) = −δ g , in D 0 (G).

3.1 The Yamabe equation and the Sobolev embedding theorem. We saw in Theorem 3.4 that the function v1 = %2−Q 1 solves the semilinear Yamabe equation in G, ∆ H v1 = −(Q − 2)m v1(Q+2)/(Q−2) . If we choose the constant λ = (m(Q − 2)) (Q−2)/4 , then it is easy to recognize that the function u = λv1 solves the normalized Yamabe equation in G, ∆ H u = −u (Q+2)/(Q−2) . (3.13)

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Notice that if we integrate u∆ H u on G, and use the fact that u → 0 at infinity, one easily obtains by an integration by parts, Z Z |∇ H u| 2 dg. u∆ H u dg = − G

G Q+2 Q−2

On the other hand, since

Z

+1=

2Q Q−2 ,

we see from (3.13) that

u∆ H u dg = − G

Z u2Q/(Q−2) dg. G

In conclusion, the function u in (3.13) satisfies the equation Z Z 2 |∇ H u| dg = u2Q/(Q−2) dg. G

(3.14)

G

We note explicitly that the exponent 2∗ =

2Q Q−2

satisfies the equation

1 1 1 − = . 2 2∗ Q The number 2∗ constitutes the so-called Sobolev exponent relative to 2. The general question involved is the following: does there exist a universal constant S2 > 0 such that for every function ϕ ∈ C0∞ (G) one has Z  1/2∗ Z  1/2 2∗ |ϕ| dg ≤ S2 |∇ H ϕ| 2 dg ? (3.15) G

G

This question is highly non-trivial, especially since, as we have seen, the horizontal gradient ∇ H ϕ of the function ϕ is degenerate and at every point it misses a good amount of directions in the tangent space. As a consequence of a basic result of Folland and Stein [Fo75], the inequality (3.15) does hold in greater generality for every Carnot group G. However, the question of the existence of the best constant in (3.15) in much subtler, and it is intimately connected to that of the classification of all positive solutions of the Yamabe equation (3.13). In this context the partial differential equation in (3.13) arises in the study of the following problem. Definition 3.7 (CR Yamabe problem). Given a compact, strictly pseudo-convex CR manifold M, find a choice of contact form ϑ for which the Webster–Tanaka pseudohermitian scalar curvature Rϑ is constant. For the relevant notions from CR geometry we refer the reader to the beautiful monograph [DT06]. In the important papers [JL87, JL88] Jerison and Lee solved the CR Yamabe problem in the case in which n ≥ 2 and (M, ϑ) is not locally CR

155

2 Hypoelliptic operators, etc.

equivalent to the sphere S2n+1 of Cn . After more than a decade, in [GY01] Gamara and Yacoub solved the CR Yamabe problem for the case in which (M, ϑ) is locally CR equivalent to the sphere S2n+1 for all n. Subsequently, Gamara in [G01] solved the CR Yamabe problem in the remaining case n = 1, thus completing the resolution of the CR Yamabe conjecture for all dimensions. Similarly to the classical situation, a crucial step in the analysis of the CR Yamabe problem was the explicit computation of the extremal functions in the horizontal Sobolev embedding (3.15) in the special situation when G is the Heisenberg group Hn . Jerison and Lee made the deep discovery that, up to group translations and dilations, a suitable multiple of the function u(z,t) = (1 + |z| 2 ) 2 + t 2

 −(Q−2)/4

(3.16)

is the only positive entire solution of (3.13) in Hn . Here, we have denoted by (z,t), z = (x, y) ∈ R2n , t ∈ R the variable point in Hn . In this connection, it is worth mentioning the following basic question. Open problem: Generalize Jerison and Lee’s result for the Heisenberg group in [JL88] to all groups of Heisenberg type. In other words, compute the best constant in the horizontal Sobolev embedding (3.15) for all groups of Heisenberg type. Equivalently, prove that in every such group, all positive solutions of equation (3.13) are given by left translations of a suitable multiple of the function m(Q − 2)ε 2 Kε (g) = (ε 2 + |x(g)| 2 ) 2 + 16|y(g)| 2

! (Q−2)/4 ,

g ∈ G.

This problem is considerably harder than its already difficult Heisenberg group predecessor. Some interesting partial progress is contained in [GV01, BU04, IMV10].

3.2 The general Folland–Stein horizontal Sobolev embedding. In connection with (3.15) above, given a Carnot group G one may ask the following more general question: given a number 1 ≤ p < ∞, does there exist 1 ≤ q < ∞ such that for some universal constant S p,q > 0 one has for every ϕ ∈ C0∞ (G), ! 1/q

Z |ϕ| q dg G

! 1/p

Z

|∇ H ϕ| p dg

≤ S p,q

?

(3.17)

G

Similarly to the classical case, a necessary condition for the validity of (3.17) is obtained by means of the group non-isotropic dilations. Suppose in fact that (3.17) does hold, and for λ > 0 consider the function ϕ λ = ϕ ◦ δ λ ∈ C0∞ (G). Substituting

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Nicola Garofalo

this function into (3.17), and noting that ∇ H (ϕ λ ) = λδ ◦ ∇ H ϕ (see (2.16) above), we find ! 1/q ! 1/p Z Z q p |ϕ(δ λ g)| dg ≤ S p,q λ |∇ H ϕ(δ λ g)| dg . G

G

By the change of variable g 0 = δ λ g, which gives dg 0 = λ Q dg, we thus obtain λ

! 1/q

Z q

Q/q

|ϕ(g)| dg

≤ S p,q λ

! 1/p

Z

|∇ H ϕ(g)| dg p

1+(Q/p)

.

G

G

It is then clear that, in order for (3.17) to hold with a universal constant S p,q , one must have 1 1 1 − = . (3.18) p q Q Let us note immediately that the necessary condition (3.18) for (3.17) forces − q1 > 0, hence one must have the gain in integrability 1 ≤ p < q. Furthermore, since p1 − Q1 = q1 > 0, we must also have 1 ≤ p < Q. Finally, let us observe that, given a number 1 ≤ p < Q, equation (3.18) completely determines the exponent q in terms of p and Q, and in fact it results in 1 p

q=

pQ . Q−p

This number is usually denoted by p∗ and it is called the Sobolev exponent relative to p. The fact that (3.18) is also sufficient for (3.17) is the content of the following embedding due to Folland and Stein; see [Fo75]. In fact, these authors proved this result only in the range 1 < p < Q. Theorem 3.8 (Folland–Stein, 1975). Let G be a Carnot group with homogeneous dimension Q, and let 1 < p < Q. If q satisfies (3.18), there exists a universal constant S p > 0 such that (3.17) holds for every ϕ ∈ C0∞ (G). The geometric case p = 1 of Theorem 3.8 is also true, ! (Q−1)/Q

Z |ϕ| G

Q/(Q−1)

dg

Z

|∇ H ϕ|dg,

≤ S1

ϕ ∈ C0∞ (G),

(3.19)

G

but it was established later by Varopoulos, and with different techniques. It is also important to observe that the geometric inequality (3.19) implies in a trivial way the Folland–Stein inequality (3.17). In fact, if we apply (3.19) to the function |ϕ| α , for

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2 Hypoelliptic operators, etc.

α > 0 to be suitably chosen, we obtain ! (Q−1)/Q Z Z |ϕ| αQ/(Q−1) dg ≤ αS1 |ϕ| α−1 |∇ H ϕ|dg (Hölder inequality) G

G

≤ αS1

Z |ϕ|

(α−1) p 0

! 1/p0 Z

G

If we now choose α such that (α − 1)p0 = αQ Q−1

=

pQ Q−p ,

pQ Q−p ,

! 1/p |∇ H ϕ| p dg

dg

.

G

which means α =

p(Q−1) Q−p ,

we obtain

and thus (3.17) follows from the latter inequality.

3.3 The Folland–Stein embedding and quasilinear equations. Given a number 1 ≤ p ≤ ∞, the p-horizontal energy is represented by the functional Z 1 E p (u) = |∇ H u| p dg. p G This functional is well defined on functions in the (weak) horizontal Sobolev space 1, p W H (G). Given an open set Ω ⊂ G we denote by 1, p

W H (Ω) = {u ∈ L p (Ω) | X j u ∈ L p (Ω), j = 1, . . . , m}, where the notation X j u now indicates the weak (distributional) derivative of u ∈ L p (Ω) along X j . Endowed with the norm ||u||W 1, p (Ω) = ||u|| L p (Ω) + ||∇ H u|| L p (Ω) , H

1, p

the linear space W H (Ω) becomes a Banach space, which is reflexive when 1 < p < 1, p 1,2 ∞. When p = 2, the space W H (Ω) is a Hilbert space. We indicate with W H,loc (Ω) p

1, p

the space of all functions in L loc (Ω) such that for every ω ⊂⊂ Ω one has W H (ω). 1, p We say that a function u ∈ W H,loc (Ω) is a critical point of the p-energy in Ω if for any given ϕ ∈ C0∞ (Ω) we have d E p (u + tϕ) = 0. dt t=0 This is equivalent to saying that Z   p/2 d 1 |∇ H u| 2 + 2th∇ H u, ∇ H ϕi + t 2 |∇ H ϕ| 2 dg dt t=0 p Ω Z = |∇ H u| p−2 h∇ H u, ∇ H ϕidg = 0. Ω

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Nicola Garofalo 1, p

Therefore, a function u ∈ W H,loc (Ω) is a critical point of the p-energy in Ω if it satisfies the condition Z |∇ H u| p−2 h∇ H u, ∇ H ϕidg = 0, for every ϕ ∈ C0∞ (Ω). (3.20) Ω

Equation (3.20) is the weak version of the quasilinear partial differential equation ∆ H, p u = div H (|∇ H u| p−2 ∇ H u) = 0,

(3.21)

known as the horizontal p-Laplacian. Here, for a horizontal vector field ζ = ζ1 X1 + · · · + ζ m X m , we have indicated its horizontal divergence with m X div H ζ = Xj ζj. j=1

In conclusion, critical points of the p-energy are weak solutions of the horizontal p-Laplacian (3.21). In the case when p = Q the quasilinear equation (3.21) plays an important role in Mostow’s celebrated work [Mo73] on rigidity. When 1 < p < Q it is, as we have seen, intimately connected to the Folland–Stein horizontal embedding Theorem 3.8.

3.4 Fundamental solutions for horizontal p-Laplacians. We saw in Theorem 3.1 that, in groups of Heisenberg type, the fundamental solution of any subLaplacian admits an explicit formula in terms of the appropriate power of the nonisotropic gauge. It is quite surprising that such a remarkable phenomenon continues to hold for the quasilinear horizontal p-Laplacian (3.21) above. In order to state the relevant result we introduce some definitions. Let G be a group of step r = 2 with gauge % as in (2.50) above. Given 1 < p < ∞ we set Z ωp = |∇ H %| p dg, σ p = Qω p , (3.22) % 0 such that for any g, g 0 ∈ G, with g , g 0 one has α |g −1



g 0 |GQ−2

β

≤ Γ(g −1 ◦ g 0 ) ≤

|g −1

◦ g 0 |GQ−2

,

where | · |G indicates the non-isotropic gauge defined in (2.6) above. Proof. By left translation it suffices to prove that for every g ∈ G one has α |g|GQ−2

≤ Γ(g) ≤

β |g|GQ−2

,

(3.25)

for suitable universal constants α, β > 0. By Theorem 3.10 we have for every g , e, Γ(g) = Γ(δ |g |G δ |g |−1 g) = |g|G2−Q Γ(δ |g |−1 g). G

G

By (2.7) we see that |δ |g |−1 g|G = 1. G

To establish (3.25) it will thus suffice to prove that def

α = inf Γ(g) > 0, |g |G =1

def

β = sup Γ(g) < ∞. |g |G =1

Since by [Fo75, Lemma 1.2] the gauge sphere |g|G = 1 is compact with respect to the Riemannian topology of G, and from Theorem 1.2 above Γ ∈ C ∞ (G \ {e}), we conclude that the latter two inequalities must hold. 

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Theorem 3.10 above brings us to pose the following challenging question. Open problem: Let G be a Carnot group, and for 1 < p < Q let ∆ H, p be a horizontal p-Laplacian on G. Does there exist a unique fundamental solution Γp of −∆ H, p , with singularity at e ∈ G, which goes to zero at infinity, and such that Γp (δ λ g) = λ (p−Q)/(p−1) Γp (g)? We stress that, unlike Theorem 3.10, now the relevant differential operator is no longer hypoelliptic, and it is non-linear. In connection with the horizontal p-Laplacian and the Folland–Stein embedding in the case p , 2 we now state another fundamental problem which is fully open. Open problem: Understand the minimizers in the Folland–Stein embedding Theorem 3.8 when p , 2 and G = Hn , the Heisenberg group. We have mentioned that in the case in which p = 2 this problem was solved by Jerison and Lee, who proved that up to rescaling and left translations the unique minimizer is given by the function (3.16). What happens when p , 2? We recall that in the classical Sobolev embedding in Rn for 1 < p < n the minimizers were independently found by Talenti [T76] and Aubin [A76]. They are given by all translations and dilations of the function u(x) =

(1 +

Cn, p , p/(p−1) |x| ) (Q−p)/p

where Cn, p > 0 is an explicitly computed constant. By adapting the concentration of the compactness method of Lions, the existence of minimizers in the Folland–Stein embedding (3.17) above was established by Vassilev in [Va06]; see Theorem 3.1 in that paper. However, the value of the best constant is not provided by this method. Remarkably, to date no guess exists about the specific form of the minimizer. We note explicitly that, suitably normalized, a minimizer for the Folland–Stein embedding (3.17) would solve the equation ∆ H, p u = −u ((p−1)Q+p)/(Q−p) , which is a generalization of Yamabe’s equation (3.13) above. Here, ∆ H, p is the horizontal p-Laplacian in (3.21) above.

4 Carnot–Carathéodory distance In this section we develop some elementary background topological material which plays a basic role in sub-Riemannian geometry. Although for the sake of simplicity

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2 Hypoelliptic operators, etc.

we choose to place our discussion in R N , with suitable modifications everything we say applies to a connected Riemannian manifold. For an inspiring account of the theory we refer the reader to Gromov’s chapter [G96] in the book Sub-Riemannian geometry, by Bellaïche and Risler (see [Be96]). In what follows we consider a system of vector fields in R N , Xj =

N X k=1

bkj

∂ , ∂x k

j = 1, . . . , m,

0,1 (R N ), i.e., they are locally and we assume that the real-valued functions bkj ∈ Cloc Lipschitz continuous. Let B = [bkj ] ∈ MN ×m , and consider the matrix A = BB t ∈ MN ×N . It is clear that At = A, and that A ≥ 0. One has in fact, for every ξ ∈ R N ,

hAξ, ξi = hB t ξ, B t ξi = |B t ξ | 2 ≥ 0. Definition 4.1. A vector v ∈ R N is called subunit at x ∈ R N if hv, ξi2 ≤ hAξ, ξi,

for every ξ ∈ R N .

It is easy to recognize that Definition 4.1 is equivalent to the condition hv, ξi2 ≤

m X

hX j (x), ξi2

for every ξ ∈ R N .

(4.1)

j=1

It is worth observing that if v is subunit at x, then v ∈ span{X1 (x), . . . , X m (x)}. Otherwise, we could write v = v1 + v2 , with v1 ∈ span{X1 (x), . . . , X m (x)}, v2 ∈  span{X1 (x), . . . , X m (x)} ⊥ . Taking ξ = v2 in (4.1), we would reach the contradiction m X 0 < |v2 | 2 ≤ hX j (x), v2 i2 = 0. j=1

With this observation in mind we leave it as an exercise to show the following Proposition 4.2. A vector v ∈ R N is subunit at x if and only if v = P 2 with m j=1 a j ≤ 1.

Pm j=1

a j X j (x)

The following definition plays a fundamental role. Definition 4.3. A piecewise C 1 curve γ : [0,T] → R N is called subunitary if γ 0 (t) is subunit at γ(t) whenever γ 0 (t) exists.

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Let γ : [0,T] → R N be a subunitary curve; then we call the number T the subunitary length of γ and we denote it by ` s (γ) = T. For later use, let us notice that if γ is subunitary, then for every t ∈ [0,T] at which exists, one has |γ 0 (t)| ≤ |X (γ(t))|, (4.2) P  m 2 1/2 . where we have defined |X (x)| = j=1 |X j (x)| Given two points x, y ∈ R N we denote by γ 0 (t)

S (x, y) = {γ : [0,T] → R N | γ(0) = x, γ(T ) = y, γ is subunitary}. In the generality we work in we could a priori have S (x, y) = ∅. In order to develop the theory we will make the following fundamental Hypothesis 4.4. For every x, y ∈ R N , one has

S (x, y) , ∅.

(4.3)

Definition 4.5. The Carnot–Carathéodory distance associated with the system {X1 , . . . , X m } is defined by d(x, y) = inf {` s (γ) | γ ∈ S (x, y)}. The reader should prove as an exercise that d(x, y) defines a distance on R N . Given x ∈ R N and r > 0, from now on we will define B(x,r) = {y ∈ R N | d(x, y) < r } to be the balls in the metric d, whereas Euclidean balls will be denoted by Be (x,r) = {y ∈ R N | |x − y| < r }. A basic observation is the following Proposition 4.6. The inclusion i : (R N , d) → (R N , | · |) is continuous.

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2 Hypoelliptic operators, etc.

Proof. We argue by contradiction and suppose that there exists x 0 ∈ R N and a sequence {x k }k ∈N such that d(x k , x 0 ) → 0 as k → ∞, but |x k − x 0 | ≥ ε 0 , for some ε 0 > 0. For each k ∈ N we can find γk ∈ S (x 0 , x k ) such that ` s (γk ) = Tk ≤ 2d(x k , x 0 ). By connectedness, there exists t k ∈ [0,Tk ] such that γk (t) ∈ Be (x 0 , ε 0 ) for 0 ≤ t < t k , γk (t k ) = pk ∈ ∂Be (x 0 , ε 0 ). We thus have from Z t k Z t k 0 γk0 (t) dt 0 < ε 0 = |x 0 − pk | = γk (t)dt ≤ 0 0 Z tk ≤ (by (4.2)) max |X (z)|` s (γk ) |X (γk (t))| dt ≤ z ∈B e (x 0,ε 0 )

0

≤2

max

z ∈B e (x 0,ε 0 )

|X (z)|d(x k , x 0 ) → 0,

as k → ∞. We have reached a contradiction.



We next establish an elementary property of the metric balls which has several important consequences. We begin with a definition. Definition 4.7. Let (S, d) be a metric space. An open set Ω ⊂ S is said to admit an interior corkscrew at x 0 ∈ ∂Ω if there exist K, R0 > 0 such that for 0 < r < R0 one can find Ar (x 0 ) ∈ Ω for which r < d( Ar (x 0 ), x 0 ) ≤ r, K

dist( Ar (x 0 ), ∂Ω) >

r . K

(4.4)

If Ω admits an interior corkscrew at every x 0 ∈ ∂Ω with the same K, R0 > 0, then c we say that Ω satisfies the uniform interior corkscrew condition. If both Ω and Ω satisfy the uniform interior corkscrew condition, then we say that Ω satisfies the uniform corkscrew condition. Proposition 4.8. Assume that Hypothesis 4.4 holds. For every x ∈ R N and R > 0 the metric ball B(x, R) satisfies the uniform interior corkscrew condition. Proof. The proof follows along the same lines as that provided in Jerison’s paper [J86] for the metric balls associated to a system of smooth vector fields satisfying the finite rank condition.  In the generality in which we work the continuity of the opposite inclusion i : (R N , | · |) → (R N , d) is not guaranteed (see [Be96, p. 18]), therefore we will introduce it as an assumption. Hypothesis 4.9. The metric balls B(x,r), x ∈ R N , r > 0 are open in the Euclidean topology.

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4.1 The theorem of Chow–Rashevsky. When the vector fields {X1 , . . . , X m } are C ∞ and they satisfy the finite rank condition (1.3), then the following fundamental result holds. Theorem 4.10 (Chow (1939) [C39] and Rashevsky (1938) [R38]). Let Ω ⊂ R N be a connected open set and suppose that X i ∈ C ∞ (Ω), i = 1, . . . , m satisfy the finite rank condition (1.3) in Definition 1.1 above. Then, for every x, y ∈ there exists γ ∈ S(x, y) such that {γ} ⊂ Ω. In particular, Hypothesis 4.4 holds. Furthermore, from the results in [NSW85] we can see that Hypothesis 4.9 also holds. Proposition 4.11. Suppose that the vector fields X i ∈ C ∞ (R N ) and satisfy the finite rank condition (1.3). Then, the Carnot–Carathéodory space (R N , d) fulfills Hypothesis 4.9. From this result and from Proposition 4.6 we infer that the metric topology is equivalent to the underlying (Euclidean) topology of R N . Proof. Let us recall the following basic estimate obtained in [NSW85]: for every connected Ω ⊂⊂ R N there exist C, ε > 0 such that C|x − y| ≤ d Ω (x, y) ≤ C −1 |x − y| ε ,

x, y ∈ Ω.

Here, d Ω is the Carnot–Carathéodory distance defined by subunit curves whose trace lies in Ω. We note that the accessibility Theorem 4.10 above guarantees that for any x, y ∈ Ω there exists such a curve connecting x to y. Since, obviously, d(x, y) ≤ d Ω (x, y) for any x, y ∈ Ω, we conclude that d(x, y) ≤ C −1 |x − y| ε ,

x, y ∈ Ω.

This implies the continuity of the inclusion i : (R N , |·|) → (R N , d), or, equivalently, the (Euclidean) openness of the metric balls. 

4.2 Compactness matters. We return here to the general situation of a Carnot– Carathéodory space (R N , d) associated with a system {X1 , . . . , X m } of vector fields 0,1 in Cloc (R N ) for which Hypothesis 4.9 hold. Proposition 4.6 and Hypothesis 4.9 guarantee that the metric and the pre-existing Euclidean topology on R N are equivalent. In particular, compact sets are the same in either topology. A basic consequence of Hypothesis 4.9 is the following Proposition 4.12. The metric space (R N , d) is locally compact. Furthermore, for any (Euclidean) bounded set U ⊂ R N there exists R0 = R0 (U) > 0 such that the closed ball B(x,r) is compact for every x ∈ U and every 0 < r < R0 .

2 Hypoelliptic operators, etc.

165

Proof. For every x ∈ R N and r > 0 let M (x,r) =

max

|X (z)|.

z ∈B e (x,r )

Suppose we can prove ! r ⊂ Be (x,r). B x, M (x,r)

(4.5)

Keeping in mind that the closure of a set with respect to either topology is the same, we would have that ! r V = B x, ⊂ B e (x,r) M (x,r) is a compact neighborhood of x in the Euclidean topology. By Hypothesis 4.9 the set i(V ) = V would be compact in (R N , d), thus proving that (R N , d) is locally compact. To establish the second part, let U be a (Euclidean) bounded set. If δ = sup |x − y| < ∞, x, y ∈U

define R = inf

x ∈U

δ . M (x, δ)

If R = 0, then by the (Euclidean) compactness of U we can find a sequence {x j } j ∈N and a point x 0 ∈ U such that |x j − x 0 | → 0 and M (x j , δ) → ∞. But this is impossible since M (x j , δ) ≤ M (x 0 , 2δ) < ∞. Therefore, it must be that R > 0 and then (4.5) gives, for any x ∈ U and 0 < r < R, ! δ B(x,r) ⊂ B x, ⊂ B e (x, δ), M (x, δ) which shows that B(x,r) is Euclidean compact, hence d-compact.   r To complete the proof we are thus left to establish (4.5). Let y ∈ B x, M (x,r ) . For any ε > 0 such that r d(x, y) + ε < , M (x,r) there exists γε ∈ S(x, y) such that d(x, y) ≤ ` s (γε ) < d(x, y) + ε. We claim that γε ⊂ B e (x,r).

(4.6)

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To see this suppose that there exists y0 ∈ {γε } such that |y0 − x| > r. By connectedness there would exist t ε ∈ [0, ` s (γε )) such that γε (t) ∈ Be (x,r) for 0 ≤ t < t ε and for which γε (t ε ) ∈ ∂Be (x,r). We thus have Z tε Z tε 0 |X (γε (t))|dt |γε (t)|dt ≤ r = |x − γε (t ε )| ≤ 0

0

≤ M (x,r)` s (γε ) < M (x,r)(d(x, y) + ε) < M (x,r)

r = r. M (x,r)

This contradiction proves (4.6). But then we have Z ` s (γ ε ) |x − y| ≤ |γε0 (t)|dt ≤ (by (4.6)) M (x,r)` s (γε ) 0

< M (x,r)(d(x, y) + ε) < M (x,r)

r = r. M (x,r)

This shows that y ∈ Be (x,r), thus establishing (4.5).



Concerning Proposition 4.12, if the system {X1 , . . . , X m } is globally Lipschitz, then the local information concerning the compactness of small balls can be converted to global. Proposition 4.13. Suppose the vector fields {X1 , . . . , X m } have coefficients in C 0,1 (R N ). Then, for every x ∈ R N and every r > 0 the closed ball B(x,r) is compact. Proof. By assumption there exists M > 0 such that for every x ∈ R N one has |X (x)| ≤ M (1 + |x|).

(4.7)

Fix x ∈ R N , r > 0. Our objective is to show that there exists a constant C = C(M) > 0 such that ! q B(x,r) ⊂ Be 0,

eCr (1 + |x| 2 ) .

(4.8)

From (4.8) we would conclude that B(x,r) is Euclidean compact, hence d-compact. In order to prove (4.8) let y ∈ B(x,r). By Hypothesis 4.4 there exists γ ∈ S (x, y). Letting ξ (t) = |γ(t)| 2 , we find ξ 0 (t) = 2hγ(t), γ 0 (t)i ≤ 2|γ(t)||γ 0 (t)| ≤ 2|γ(t)||X (γ(t))| ≤ (by (4.7)) 2M |γ(t)|(1 + |γ(t)|) ≤ C(1 + |γ(t)| 2 ) = C(1 + ξ (t)). Integrating this differential inequality we have Z ξ (t ) Z t dτ ≤C ds, ξ (0) 1 + τ 0

2 Hypoelliptic operators, etc.

which gives log

167

! 1 + ξ (t) ≤ Ct, 1 + ξ (0)

and therefore ξ (t) ≤ (1 + ξ (t)) ≤ eC t (1 + ξ (0)).

(4.9)

Since by assumption d(y, x) < r, for any 0 < ε < r − d(y, x) there exists γε ∈ S (x, y) such that d(y, x) ≤ ` s (γε ) < d(y, x) + ε. Applying (4.9) with γ = γε , letting t → ` s (γε ) in it, and recalling that ξ (0) = |γε (0)| 2 = |x| 2 , ξ (` s (γε )) = |γε (` s (γε ))| 2 = |y| 2 , we conclude that       |y| 2 ≤ eC ` s (γ ε ) 1 + |x| 2 ≤ eC (d(y, x)+ε) 1 + |x| 2 < eC (r +ε) 1 + |x| 2 . Letting ε → 0 we reach the conclusion that   |y| 2 ≤ eCr 1 + |x| 2 , 

which establishes (4.8).

Remark 4.14. In the general setting of Carnot–Carathéodory metrics Proposition 4.13 is sharp as far as the growth of the vector fields at infinity is concerned. The next example, which was found in [GN96], shows that, even for a system of Hörmander type, if the vector fields are allowed to grow more than linearly at infinity the compactness of the metric balls may fail. However, when there is also a homogeneous structure, like for instance in the case of Carnot groups, this negative phenomenon does not present itself. d which of Example 4.15. Consider in R the smooth vector field X = (1 + x 2 ) dx course satisfies the finite rank condition (1.3) in Definition 1.1 above. The Carnot– Carathéodory distance associated with X is given by

d(x, y) = | tan−1 y − tan−1 x|.

(4.10)

To verify (4.10), given points x, y ∈ R, x < y, a subunit curve joining x to y is given by γ(t) = tan(t + tan−1 x), 0 ≤ t ≤ T, with T = tan−1 y − tan−1 x. We thus obtain for the corresponding Carnot–Carathéodory distance, d(x, y) ≤ T = | tan−1 y − tan−1 x|.

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We next show that the opposite inequality holds. To this end it is enough to prove that if γ : [0,T] → R is such that γ(0) = x, γ(T ) = y, and γ 0 (t) = α(γ(t))X (γ(t)) for some |α(t)| ≤ 1, then T ≥ | tan−1 y − tan−1 x| = tan−1 y − tan−1 x. Now, any such curve must have the form γ(t) = tan( β(t) + C),

with β(t) =

Z

t

α(s)ds,

| β 0 (t)| ≤ 1.

t0

The conditions γ(0) = x, γ(T ) = y force C = tan−1 x − β(0) and β(T ) = tan−1 y − tan−1 x + β(0). Since Z T Z T β 0 (s)ds ≤ | β 0 (s)|ds ≤ T, | β(T ) − β(0)| = 0 0 we have ` s (γ) = T ≥ | β(T ) − β(0)| = | tan−1 y − tan−1 x|. By taking the infimum on all γ’s we obtain d(x, y) ≥ | tan−1 y − tan−1 x|. In conclusion, we have proved (4.10). If now we choose r ≥ latter identity, B(0,r) = R,

π 2,

we obtain from the

which shows that the metric balls B(0,r), with r ≥ π2 , are not Euclidean bounded. We observe explicitly that (R, d) is not a complete metric space. In fact, the sequence x k = k is Cauchy with respect to d, but of course it admits no limit point in (R, d).

4.3 Equivalent distances. In sub-Riemannian geometry different notions of distances are often employed. Besides the Carnot–Carathéodory distance which we have introduced, there is another particularly useful one which originates in the theory of optimal control. It is important to recognize that the various notions of distance are equivalent, or even identical, and thus we turn our attention to this task. Let {X1 , . . . , X m } be a system of locally Lipschitz vector fields in R N as before. Definition 4.16. A piecewise C 1 curve σ : [0, 1] → R N is called horizontal if there exist piecewise continuous functions a j : [0, 1] → R N , j = 1, . . . , m such that σ 0 (t) =

m X

a j (t)X j (σ(t)),

j=1

whenever σ 0 (t) exists. We let aσ (t) = (a1 (t), . . . , a m (t)).

(4.11)

169

2 Hypoelliptic operators, etc.

Given points x, y ∈ R N we define

H (x, y) = {σ : [0, 1] → R N | σ(0) = x, σ(1) = y, σ is horizontal}. Proposition 4.17. Assume that Hypothesis 4.4 holds. Then, H (x, y) , ∅ for every x, y ∈ R N . Proof. Let γ ∈ S (x, y) (by Hypothesis 4.4 at least one such γ does exist), and define σ : [0, 1] → R N by letting σ(t) = γ(tT ). Since γ 0 (t) is subunit whenever it exists, by Proposition 4.2 we infer that for all t ∈ P 2 [0, 1], but a finite number of them, there exist a1 (t), . . . , a m (t), with m j=1 a j (t) ≤ 1, such that m X 0 γ (t) = a j (t)X j (γ(t)). j=1

The functions a j are piecewise continuous, since γ 0 (t) is. If we let a˜ j (t) = T a j (tT ), then σ (t) = T γ (tT ) = T 0

0

m X

a j (tT )X j (γ(tT )) =

j=1

m X

a˜ j (t)X j (σ(t)).

j=1

This proves that σ ∈ H (x, y).



For 1 ≤ p ≤ ∞ we define the pth horizontal length of σ ∈ H (x, y) as ` h, p (σ) = ||aσ ||

L p (0,1)

=

! 1/p

1

Z

p

|aσ (t)| dt

,

0

with the obvious modification when p = ∞. We then introduce a new distance, % p (x, y) =

inf

σ ∈H (x, y)

` h, p (σ).

The proof of the next result can be accomplished by adapting the ideas in [JSC87]. Proposition 4.18. The Carnot–Carathéodory distance d(x, y), and the distances % p (x, y), 1 ≤ p ≤ ∞, are all equal.

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4.4 Length-spaces. In this section we will see that the situation depicted in the above Example 4.15 should not be surprising. What we mean by this is that the lack of completeness of the metric space and the failure of compactness of the metric balls are two equivalent properties in view of a powerful generalization of the classical theorem of Hopf–Rinow which is due to Cohn-Vossen. We recall that given a continuous curve γ : [a, b] → S in a metric space (S, d), its metric length is defined by `(γ) = sup σ

p X

d(γ(t i ), γ(t i+1 )),

i=1

where the supremum is taken over all the partitions σ = {a = t 1 < · · · < t p+1 = b} of the interval [a, b]. The curve γ is said to be rectifiable if `(γ) < ∞. We denote by R (x, y) the collection of all continuous, rectifiable curves joining x to y. Definition 4.19. A metric space (S, d) is called a length-space if for any x, y ∈ S one has d(x, y) = inf `(γ). (4.12) γ ∈R (x, y)

Given points x, y ∈ S for any γ ∈ R (x, y) and any partition σ = {a = t 1 < · · · < t p+1 = b} of the interval [a, b], one trivially has from the triangle inequality, d(x, y) ≤

p X

d(γ(t i ), γ(t i+1 )).

i=1

Taking the infimum in the latter inequality we find d(x, y) ≤

inf

γ ∈R (x, y)

`(γ).

A simple, yet basic observation is that, when the metric d is a Carnot–Carathéodory metric, the latter inequality is in fact an equality. Proposition 4.20. Any Carnot–Carthéodory metric space (R N , d) is a length-space. Proof. In view of the above observation, we need only to prove that d(x, y) ≥

inf

γ ∈R (x, y)

`(γ).

(4.13)

Fix x, y ∈ R N . To prove (4.13) it suffices to show that for every γ ∈ S (x, y) one has `(γ) ≤ ` s (γ).

(4.14)

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2 Hypoelliptic operators, etc.

In fact, since clearly S (x, y) ⊂ R (x, y), one would have inf

γ ∈R (x, y)

`(γ) ≤

inf

γ ∈S (x, y)

`(γ) ≤ (by (4.14))

inf

γ ∈S (x, y)

` s (γ) = d(x, y),

and (4.13) would be established. We are thus left with proving (4.14). Let γ ∈ S (x, y), with γ : [0,T] → R N , and consider a partition σ = {0 = t 1 < · · · < t p+1 = T } of [0,T]. For each i = 1, . . . , p let x i = γ(t i ). For 0 ≤ t ≤ t i+1 − t i define γi (t) = γ(t i + t). Obviously, γi (0) = x i and γ(t i+1 − t i ) = x i+1 . It is easy to recognize that γi ∈ S (x i , x i+1 ), i.e., that γi is subunitary. Therefore, d(x i , x i+1 ) ≤ ` s (γi ) = t i+1 − t i . This gives p X i=1

d(γ(t i ), γ(t i+1 )) =

p X i=1

d(x i , x i+1 ) =

p X

(t i+1 − t i ) = t p+1 − t 1 = T = ` s (γ).

i=1

Taking the supremum over all partitions σ of the interval [0,T] we obtain (4.14).



4.5 The theorem of Cohn-Vossen. This section is devoted to recalling a basic generalization of the classical theorem of Hopf–Rinow from geometry. This result, when combined with Proposition 4.20, has some important consequences, and we will present some of them. Theorem 4.21 (Cohn-Vossen (1935)). In any locally compact length-space (S, d) the compactness of the closed metric balls B(x,r) is equivalent to the completeness of (S, d). In light of Theorem 4.21 the negative phenomenon in Example 4.15 should now be completely clear. Since (R, d) is not complete, the metric balls cannot possibly all be compact. To state a first consequence of Theorem 4.21 we introduce a definition. Definition 4.22 (Segment property). Let (S, d) be a metric space. Given points x, y ∈ S one says that a continuous curve γ : [a, b] → S is a segment joining x to y if γ(a) = x, γ(b) = y, and for every z ∈ {γ} one has d(x, z) + d(z, y) = d(x, y).

(4.15)

If for any x, y ∈ S there exists a segment joining x to y, then (S, d) is said to have the segment property.

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For a proof of the next result we refer the reader to the classical, beautiful book by Busemann [Bu70]. Theorem 4.23. Let (S, d) be a metric space which is also a length-space. Then, the following are true. (i) If for x ∈ S the closed metric ball B(x,r) is compact, then for any y ∈ B(x,r) there exists a segment contained in B(x,r) joining x to y. (ii) If for any x ∈ S and r > 0 the closed ball B(x,r) is compact, then (S, d) possesses the segment property. (iii) In particular, if (S, d) is complete, then by Theorem 4.21 the closed balls are compact, and therefore thanks to (ii) (S, d) has the segment property. Proposition 4.24. Let (R N , d) be a Carnot–Carathéodory metric for which Hypothesis 4.9 holds. Then, for every (Euclidean) bounded set U ⊂ R N there exists R0 = R0 (U) > 0 such that for any x ∈ U and 0 < r < R0 , given any y ∈ B(x,r) there exists a segment joining x to y contained in B(x,r). Proof. It follows from Propositions 4.12, 4.20 and from Theorem 4.23(i).



The next result provides us with a basic consequence of the theory developed so far. Proposition 4.25. Suppose the vector fields {X1 , . . . , X m } are C 0,1 (R N ), and that they generate a Carnot–Carathéodory metric d for which Hypothesis (4.9) holds. Then, (R N , d) is complete and satisfies the segment property. Proof. According to Proposition 4.13 the closed metric balls B(x,r) are compact for every x ∈ R N and r > 0. Since by Proposition 4.20 (R N , d) is a length-space, we can apply Theorem 4.21 to conclude that (R N , d) is complete. By of Theorem 4.23(ii) we conclude that (R N , d) possesses the segment property. 

4.6 Distances in Carnot groups. Proposition 4.25 clearly applies to groups of step r = 2, since by (2.28) in Proposition 2.6 above every orthonormal basis of the horizontal layer V1 generates a system of globally Lipschitz vector fields {X1 , . . . , X m }. For groups of step r ≥ 3 this global Lipschitz character is no longer true since at infinity the vector fields grow at least quadratically; see Proposition 2.5 and also (2.41). Nonetheless, the homogeneous structure of a Carnot group makes up for the superlinear growth of the vector fields at infinity, as the next result shows.

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173

Proposition 4.26. Let G be a Carnot group with Carnot–Carathéodory distance d. Then, (i) for every g ∈ G and r > 0, the closed metric ball B(g,r) is compact; (ii) (G, d) is a complete metric space; (iii) (G, d) possesses the segment property. We postpone the proof of Proposition 4.26 until later in this section. It should be obvious from the definition of Carnot–Carathéodory distance that if g, g 0, g 00 ∈ G, one has d(L g (g 0 ), L g (g 00 )) = d(g 0, g 00 ). (4.16) We will need the following self-similarity property of d. Lemma 4.27. For every g, g 0 ∈ G and every λ > 0 one has d(δ λ (g), δ λ (g 0 )) = λd(g, g 0 ). Proof. Using (4.16) we see that proving the lemma is equivalent to showing d((δ λ (g)) −1 ◦ (δ λ (g 0 )), e) = λd(g −1 ◦ g 0, e). Since (δ λ (g)) −1 = δ λ (g −1 ) and since {δ λ } λ >0 is a one-parameter family of group automorphisms, the latter identity is equivalent to d(δ λ (g), e) = λd(g, e).

(4.17)

To establish (4.17) we invoke Proposition 4.18 and work with horizontal, rather than subunitary, curves. For any curve σ ∈ H (0, g) consider the new curve defined by σ λ = δ λ ◦ σ. Claim: σ λ ∈ H (e, δ λ (g)), and furthermore ` h,1 (σ λ ) = λ` h,1 (σ). If the claim were true we would infer %1 (δ λ (g), e) = λ %1 (g, e), and the desired conclusion (4.17) would follow from Proposition 4.18 again. Let then m X 0 a j (t)X j (σ(t)), σ (t) = j=1

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with a j : [0, 1] → R piecewise continuous. It is clear that σ λ = δ λ ◦ σ satisfies σ λ (0) = δ λ (σ(0)) = δ λ (e) = e, σ λ (1) = δ λ (σ(1)) = δ λ (g). Denoting by (δ λ )∗ the differential of δ λ , we obtain σ λ0 (t) = (δ λ )∗ σ 0 (t) =

m X

a j (t)(δ λ )∗ X j (σ(t)) = λ

j=1



m X

m X

a j (t)X j (δ λ ◦ σ(t))

j=1

a j (t)X j (σ λ (t)).

j=1

This equation shows that σ λ ∈ H (e, δ λ (g)), and that moreover, ` h,1 (σ λ ) =

1

Z

|aσ λ (t)|dt = λ

0

1

Z

|aσ (t)|dt = λ` h,1 (σ).

0

This establishes the claim, thus completing the proof of the lemma.



Proposition 4.28. Let G be a Carnot group. There exist universal constants C1 ,C2 > 0 such that for g, g 0 ∈ G one has C1 |g −1 ◦ g 0 |G ≤ d(g, g 0 ) ≤ C2 |g −1 ◦ g 0 |G . Proof. In view of (4.16) it suffices to prove that for every g ∈ G one has for suitable constants C1 ,C2 > 0, C1 |g|G ≤ d(g, e) ≤ C2 |g|G . (4.18) Thanks to Lemma 4.27, inequality (4.18) is equivalent to   C1 ≤ d δ |g |−1 (g), e ≤ C2 . G

Therefore, it suffices to show that there exist constants C1 ,C2 > 0 such that for every g ∈ G, with |g|G = 1, one has C1 ≤ d(g, e) ≤ C2 .

(4.19)

At this point we observe that in light of Proposition 4.11 the metric topology is equivalent to the underlying one generated by the Riemannian distance on G. Therefore, in particular, the function g → d(g, e) is continuous with respect to the underlying topology of G. Since by [Fo75, Lemma 1.2] the gauge sphere |g|G = 1 is compact with respect to the Riemannian topology of G, we conclude that (4.19) must hold. 

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175

Remark 4.29. We emphasize that if we let %(g, g 0 ) = |g −1 ◦ g 0 |G , then % is not a distance in general, but only a pseudo-distance. However, it was shown by Cygan in [Cy81] that in every group of Heisenberg type the gauge in (2.50) above induces a true distance. For the proof of the next result we refer the reader to [Fo75, Lemma 1.2]. Proposition 4.30. Let G be a Carnot group. For every g ∈ G and r > 0 the gauge closed pseudo-ball BG (g,r) = {g 0 ∈ G | |g −1 ◦ g 0 |G ≤ r } is compact with respect to the Riemannian topology of G. The following result is a direct consequence of Proposition 4.11 above. Proposition 4.31. In a Carnot group G every Carnot–Carthéodory topology is equivalent to the underlying Riemannian topology. We are finally able to provide the Proof of Proposition 4.26. According to Proposition 4.28 one has BG (g,C2 r) ⊂ B(g,r) ⊂ B G (g,C1 r). Since by Lemma 4.30 the pseudo-ball BG (g,C1 r) is compact with respect to the Riemannian topology, therefore, according to Proposition 4.31, with respect to the d-topology, we conclude that B(g,r) is d-compact. This proves (i). By Propositions 4.12 and 4.20, (G, d) is a locally compact length-space. Since by (i) the closed metric balls are compact, we conclude from Cohn-Vossen’s Theorem 4.21 that (G, d) must be complete, thus proving (ii). Finally, by Theorem 4.23(iii) we infer that (G, d) possesses the segment property. This establishes (iii). 

5 Sobolev and BV spaces In this section we collect various basic properties of some functional spaces of Sobolev type which naturally arise in connection with subelliptic operators. We also introduce a suitable space of functions of bounded variation (BV functions) which plays an important role in geometric measure theory, and discuss its main properties.

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At first, we are going to be mainly concerned with matters which, remarkably, at least with hindsight, have nothing to do with the geometric aspects of the Carnot– Carathéodory metric associated with the system {X1 , . . . , X m }. In this connection a basic result, whose localized version was discovered by Friedrichs in 1944 (see [F44]), is that a theorem of Meyers–Serrin type holds, i.e., the space of weak Sobolev functions coincides with that of strong ones (see [MS64] for this classical result). This phenomenon has various important applications to partial differential equations. We consider the following general situation. Let Ω ⊂ R N be an open set. For `−1,1 ` ∈ N we denote by Cloc (Ω) the class of functions which have partial derivatives PN k ∂ 0,1 up to order ` − 1 in Cloc (Ω). We assume that the vector fields X j = k=1 b j ∂x k have `−1,1 (Ω). If r ∈ N we let I = (i 1 , . . . ,i r ), where 1 ≤ i j ≤ m, and coefficients bkj ∈ Cloc j = 1, . . . ,r. We set |I | = r, and let

X I = Xi1 . . . Xi r . If f ∈ L 1loc (Ω) the distributional or weak derivative X j f is defined by the identity Z hX j f , ϕi = − f X ∗j ϕ dx, ϕ ∈ C0∞ (Ω), Ω

where X ∗j = −

N X ∂  k  bj · ∂x k k=1

denotes the formal adjoint of X j . We notice that X ∗j is well defined under the mere 0,1 assumption that bkj ∈ Cloc (Ω). Once the definition of the weak derivative X j is given, we then define the weak derivative of order r, X I , inductively. `, p

Definition 5.1. Let 1 ≤ p ≤ ∞, ` ∈ N. The weak Sobolev space W X with the system X = {X1 , . . . , X m } is defined as

associated

`, p

W X (Ω) = { f ∈ L p (Ω) | X I f ∈ L p (Ω), |I | ≤ `}. `, p

We endow W X (Ω) with the norm || f ||W `, p (Ω) = X

X

||X I f || L p (Ω) .

| I | ≤`

We also define `, p

W X,0 (Ω) = C0∞ (Ω) The strong Sobolev space defined as `, p

`, p S X (Ω)

||·||

W

`, p (Ω) X

.

associated with the system X = {X1 , . . . , X m } is `, p

S X (Ω) = C ∞ (Ω) ∩ W X (Ω)

||·||

W

`, p (Ω) X

.

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2 Hypoelliptic operators, etc. `, p

`, p

The local spaces W X,loc (Ω), S X,loc (Ω) are defined in the usual way. For instance, `, p

`, p

u ∈ W X,loc (Ω) if for any ω ⊂⊂ Ω one has u ∈ W X (ω). As in the classical case one can show the following Proposition 5.2. Let 1 ≤ p ≤ ∞. Endowed with the norm || · ||W `, p (Ω) the spaces `, p

X

`, p

W X (Ω) and S X (Ω) are Banach spaces. Furthermore, they are Hilbert spaces `, p when p = 2. When 1 < p < ∞ the space W X (Ω) is a reflexive Banach space. It should be obvious from Definition 5.1 that `, p

`, p

S X (Ω) ,→ W X (Ω). It is quite remarkable that such an inclusion can be reversed in the generality in which we work. For classical Sobolev spaces the next result is known as the theorem of Meyers and Serrin; see [MS64]. Theorem 5.3. For 1 ≤ p < ∞ and ` ∈ N one has `, p

`, p

W X (Ω) = S X (Ω). As we have mentioned above, the local version of this result, i.e., `, p

`, p

S X,loc (Ω) = W X,loc (Ω), was established by Friedrichs in [F44], a fact which is not generally known, and which was rediscovered (more than fifty years later!) in [GN96] (see Theorem A.2 in that paper) and independently in [FSS97]. A first basic application of Theorem 5.3 is the following chain rule for the space W X1,1 (Ω) which plays a basic role in the analysis of regularity properties of variational PDEs arising from vector fields. We will assume in this result that the vector fields 0,1 have coefficients in Cloc (Ω). 1, p

Proposition 5.4. Let f ∈ C 1 (R) with | f 0 | ≤ M. For any u ∈ W X (Ω) with 1 ≤ p < 1, p ∞ one has f ◦ u ∈ W X (Ω) and moreover, X j ( f ◦ u) = ( f 0 ◦ u)X j u

in D 0 (Ω).

Furthermore, one has u+ ,u− , |u| ∈ W X (Ω) and 1, p

  Xju Xju =   0  +

a.e. in {x ∈ Ω | u(x) > 0}, otherwise;

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   −X j u X j u− =   0 

a.e. in {x ∈ Ω | u(x) < 0},

 Xju      X j |u| =  −X j u     0 

a.e. in {x ∈ Ω | u(x) > 0},

otherwise;

a.e. in {x ∈ Ω | u(x) < 0}, a.e. in {x ∈ Ω | u(x) = 0}.

1, p

Proof. Let u ∈ W X (Ω) and let f be as in the statement of the proposition. By 1, p Theorem 5.3 it will suffice to show that f ◦ u ∈ S X (Ω). Now, by Theorem 5.3 there 1, p exists a sequence uk ∈ C ∞ (Ω) ∩ W X (Ω) such that as k → ∞ one has uk → u,

in L p (Ω).

X j uk → X j u,

By passing to a subsequence, if necessary, we can assume that uk (x) → u(x) at a.e. point x ∈ Ω. Now the mean value theorem and the assumption on f easily give Z Z p p | f ◦ uk − f ◦ u| dx ≤ M |uk − u| p dx → 0. Ω



Furthermore, Z | f 0 ◦ uk X j uk − f 0 ◦ uX j u| p dx Ω Z Z 0 0 p ≤ Cp | f ◦ uk X j uk − f ◦ uk X j u| dx + C p | f 0 ◦ uk X j u − f 0 ◦ uX j u| p dx Ω Ω Z Z 0 p p ≤ Cp | f (uk )| |X j uk − X j u| dx + C p | f 0 (uk ) − f 0 (u)| p |X j u| p dx Ω Ω Z Z p p ≤ Cp M |X j uk − X j u| dx + C p | f 0 (uk ) − f 0 (u)| p |X j u| p dx. Ω

We clearly have x ∈ Ω,



R Ω

|X j uk − X j u| p dx → 0 as k → ∞. Furthermore, we have for a.e. | f 0 (uk (x)) − f 0 (u(x))| p |X j u(x)| p → 0,

and moreover, | f 0 (uk (x)) − f 0 (u(x))| p |X j u(x)| p ≤ 2 p M p |X j u(x)| p ∈ L 1 (Ω). By Lebesgue dominated convergence we conclude that Z | f 0 (uk ) − f 0 (u)| p |X j u| p dx → 0. Ω

2 Hypoelliptic operators, etc.

179

1, p

This proves at once that f ◦ uk ∈ C 1 (Ω) ∩ W X (Ω) and moreover, f ◦ uk → f ◦ u in 1, p L p (Ω), and X j ( f ◦uk ) → X j ( f ◦u) in L p (Ω). This establishes that f ◦u ∈ S X (Ω), and that its weak derivative along X j is given by X j ( f ◦ u) = f 0 (u)X j u. In view of Theorem 5.3 this proves the first part of the lemma. √ To establish the second part we consider the function f ε (t) = ε 2 + t 2 − ε if t ≥ 0, f ε (t) = 0 if t ≤ 0. Clearly, f ε ∈ C 1 (R) and | f ε0 | ≤ 1. By the first part we conclude that f ε ◦ u ∈ W X1,1 (Ω). Moreover, for every ϕ ∈ C0∞ (Ω) we have Z Z f ε0 (u)X j uϕ dx = f ε (u)X ∗j ϕ dx. {x ∈Ω |u(x)>0}

Letting ε → 0 we obtain Z



X j uϕ dx =

{x ∈Ω |u(x)>0}

Z Ω

u+ X ∗j ϕ dx.

From the arbitrariness of ϕ ∈ C0∞ (Ω) we conclude that X j u+ = {u > 0}X j u in D 0 (Ω). The statements about u− and |u| are proved similarly. 

5.1 BV functions. In this section we introduce a generalization of the classical space of BV (bounded variation) functions which, like its predecessor, is relevant in geometric measure theory, in the study of minimal surfaces and in questions of isoperimetry. In such contexts the space BV becomes the appropriate replacement for the Sobolev space W X1,1 (Ω). 0,1 Let {X1 , . . . , X m } be a system of vector fields in Cloc (R N ), and suppose that Hypothesis 4.4 is satisfied so that d(x, y) defines a Carnot–Carathéodory metric in R N . Let Ω ⊂ R N be an open set and consider the family of test functions m  X  1/2  F (Ω) = ζ ∈ C01 (Ω; Rm ) ||ζ ||∞ = sup ζ 2j ≤1 . Ω j=1

Definition 5.5. The total variation of a function u ∈ L 1 (Ω) is defined as Var(u; Ω) = sup

ζ ∈F (Ω)

Z u Ω

m X

X ∗j ζ j dx.

j=1

A function u ∈ L 1 (Ω) is said to be of bounded variation if Var(u; Ω) < ∞. Endowed with the norm ||u||BV(Ω) = ||u|| L 1 (Ω) + Var(u; Ω),

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the space BV(Ω) = {u ∈ L 1 (Ω) | Var(u, Ω) < ∞} 1,1 becomes a Banach space. If u ∈ S X (Ω), then by definition there exists uk ∈ C 1 (Ω)∩ W X1,1 (Ω) such that uk → u and X j uk → X j u in L 1 (Ω) as k → ∞. If now ζ ∈ F (Ω), then an integration by parts gives Z Z m m X X uk X ∗j ζ j dx = X j uk ζ j dx. Ω



j=1

j=1

Taking the supremum over all ζ ∈ F (Ω), we thus conclude that Z Var(uk ; Ω) = |Xuk | dx. Ω

Letting k → ∞ we conclude that Var(u; Ω) =

Z |Xu| dx. Ω

This observation, combined with Theorem 5.3, proves that W X1,1 (Ω) ,→ BV X (Ω), but the inclusion is strict. A useful, yet almost obvious property of the total variation is its lower semicontinuity. Proposition 5.6. Let {uk }k ∈N be a sequence in BV X (Ω) such that uk → u in L 1loc (Ω). Then, Var(u; Ω) ≤ lim inf Var(uk ; Ω). k→∞

Remarkably, for the space BV X (Ω) one has an approximation theorem similar to Theorem 5.3 above. Theorem 5.7 (See [GN96, Theorem 1.14]). For any function u ∈ BV X (Ω) there exists a sequence {uk }k ∈N in C ∞ (Ω) such that the following hold: (a) limk→∞ kuk − uk L 1 (Ω) = 0. (b) limk→∞ Var X (uk ; Ω) = Var X (u; Ω). Given a set A ⊂ R N we denote by 1 A its indicator function. Definition 5.8. Given a measurable set E ⊂ R N its X-perimeter with respect to Ω is defined as PX (E; Ω) = Var X (1 E ; Ω).

2 Hypoelliptic operators, etc.

181

The above definition was introduced in [CDG96] and represents a generalization of the notion of perimeter set forth by De Giorgi in [DG54]; see also [DG61]. For later purposes we record the following fact, for whose proof we refer to [CDG96]. Proposition 5.9. Suppose that E ⊂ R N is a bounded C 1 domain, and denote by ν the outward unit normal to ∂E. Then, 1/2 m X 2+ * PX (E; Ω) = hX i , νi dH N −1 , Ω∩∂E , i=1 -

Z

where H N −1 indicates the standard (N − 1)-dimensional Hausdorff measure in R N . An important consequence of Theorem 5.7 is the following extension of Federer’s classical co-area formula to functions in BV X (Ω). For its proof, see [GN96, Theorem 1.14]. Theorem 5.10 (Co-area formula). Let u ∈ BV X (Ω), and for t ∈ R define Et = {x ∈ Ω : u(x) > t}. Then, the following hold: (I) PX (Et ; Ω) < ∞ for a.e. t ∈ R. R∞ (II) Var X (u; Ω) = −∞ PX (Et ; Ω)dt. R∞ (III) Conversely, if u ∈ L 1 (Ω) and −∞ PX (Et ; Ω)dt < ∞, then u ∈ BV X (Ω). The generalized co-area formula in (II) is a remarkable tool. It plays an important role in the proof of a basic endpoint result: the geometric embedding theorem. This aspect will be discussed in Section 8.

5.2 Metric Lipschitz functions and the space W X1,∞ (Ω). In this subsection

we present a remarkable sufficient condition for a function to be a member of the space W X1,∞ (Ω), namely that the function be Lipschitz continuous with respect to the Carnot–Carathéodory metric d. We need some preliminary material. In what follows 0,1 we consider a vector field Y in R N with coefficients in Cloc (R N ). Given a bounded open set Ω ⊂ R N we consider the flow Θ(t, x) = ΘY (t, x) associated with Y . In other words, given any x ∈ Ω we consider the unique solution of the Cauchy problem d    dt Θ(t, x) = Y (Θ(t, x)),  Θ(0, x) = x. 

A solution always exists for sufficiently small |t|. In fact, given ω ⊂⊂ Ω there exists T = T (ω) > 0 such that Θ(t, x) is well defined as a C 0,1 map, Θ(·, ·) : (−T,T ) × ω → R N .

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The uniqueness of the solution of the Cauchy problem shows that Θ(s, Θ(t, x)) = Θ(s + t, x),

x ∈ ω, |s| + |t| < T,

and that ΘY (λt, x) = Θ λY (t, x),

x ∈ ω, |λt| < T.

In particular, the map Θ(t, ·) is invertible and we have Θ(t, ·) −1 = Θ(−t, ·),

|t| < T.

We will need the following Lemma 5.11. Let ω ⊂⊂ Ω. There exist T = T (ω) > 0 and M = M (ω, Lip(Y,ω)) > 0 such that |Θ(t, x) − Θ(t, y)| ≤ M |x − y|, x, y ∈ ω, |t| ≤ T. (5.1)   i denotes the Jacobian of Θ, then for a.e. x ∈ ω Furthermore, if JΘ = ∂Θ ∂x j i, j=1, ..., N

and for |t| ≤ T one has JΘ(t, x) = I + η(t, x),

(5.2)

where I = [δ i j ] is the identity matrix in and η(t, x) = [η i j (t, x)]i, j=1, ..., N is a matrix-valued function such that for a.e. x ∈ ω and |t| ≤ T, RN

||η(t, x)|| =

N X

η i, j (t, x) 2

 1/2 ≤ C|t|,

(5.3)

i, j=1

for some C = C(ω, Lip(Y,ω)) > 0. In particular, one has det J (t, x) = 1 + σ(t, x),

(5.4)

where |σ(t, x)| ≤ C|t|,

a.e. x ∈ ω, |t| ≤ T.

(5.5)

Before proving Lemma 5.11 we recall an elementary classical inequality. Lemma 5.12 (Gronwall). Let f , g, K ∈ C([a, b]) with f , g, K ≥ 0, and such that Z t f (t) ≤ g(t) + K (s) f (s)ds, a ≤ t ≤ b. a

Then, f (t) ≤ g(t) +

t

Z

K (s) exp{ a

R

t s

K (τ)dτ}g(s)ds.

In particular, if g(t) ≡ c and K (t) ≡ K, then we obtain f (t) ≤ ce K (t−a) ,

a ≤ t ≤ b.

2 Hypoelliptic operators, etc.

183

0,1 Proof of Lemma 5.11. Since Y ∈ Cloc (Ω) and ω ⊂⊂ Ω, there exists L = L(ω) > 0 such that |Y (x) − Y (y)| ≤ L|x − y|, x, y ∈ ω.

Since for every x ∈ ω and |t| ≤ T we have Z t Θ(t, x) = x + Y (Θ(s, x))ds, 0

we thus find Z t |Θ(t, x) − Θ(t, y)| ≤ |x − y| + |Y (Θ(s, x)) − Y (Θ(s, y))|ds 0Z t ≤ |x − y| + L |Θ(s, x) − Θ(s, y)|ds . 0 If we let f (t) = |Θ(t, x) − Θ(t, y)|, then the latter inequality reads Z t f (s)ds . f (t) ≤ |x − y| + L 0 By Lemma 5.12 we conclude that f (t) ≤ |x − y|e Lt ≤ e LT |x − y|, which establishes (5.1). In particular, (5.1) shows that for every fixed |t| ≤ T the map x → Θ(t, x) is Lipschitz continuous in ω. By the theorem of Rademacher–Stepanov, the function is thus differentiable at a.e. x ∈ ω. Fix a point x of differentiability. Then, for |h| small we have Z tf g Θ(t, x + he j ) − Θ(t, x) 1 = ej + Y (Θ(s, x + he j )) − Y (Θ(s, x)) ds, (5.6) h h 0 where {e1 , . . . , e N } denotes the standard basis of R N . For j = 1, . . . , N set Z tf g 1 η j (t, x; h) = Y (Θ(s, x + he j )) − Y (Θ(s, x)) ds. h 0 Since lim

h→0

Θ(t, x+he j )−Θ(t, x) h

exists, then from (5.6) we conclude that def

lim η j (t, x; h) = η j (t, x)

h→0

also exists. Moreover, L |η j (t, x; h)| ≤ |h|

Z t |Θ(s, x + he j ) − Θ(s, x)|ds ≤ (by (5.1)) LM |t|. 0

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Passing to the limit as h → 0 from the latter inequality we infer |η j (t, x)| ≤ LM |t|,

|t| ≤ T.

(5.7)

If we now pass to the limit as h → 0 in (5.6) we conclude that ∂Θ (t, x) = e j + η j (t, x), ∂x j with η j (t, x) satisfying (5.7). This establishes (5.2) with (5.3). From these latter equations and linear algebra, the equations (5.4), (5.5) follow.  Definition 5.13 (Hölder and Lipschitz spaces). Given a system {X1 , . . . , X m } of vec0,1 tor fields in Cloc (R N ), suppose that Hypothesis 4.4 is satisfied, so that d(x, y) defines a Carnot–Carathéodory metric in R N . If Ω ⊂ R N is an open set, and α ∈ (0, 1], we denote by Γ0,α X (Ω) the linear space of all functions f : Ω → R for which there exists L ≥ 0 such that | f (x) − f (y)| ≤ Ld(x, y) α , x, y ∈ Ω. (5.8) If f ∈ Γ0,α X (Ω) we introduce the seminorm [ f ]Γ0, α (Ω) = X

| f (x) − f (y)| . d(x, y) α x, y ∈Ω, x,y sup

When f is also bounded on Ω, setting || f ||Γ0, α (Ω) = [ f ]Γ0, α (Ω) + || f || L ∞ (Ω) X

X

0,1 provides a norm on the space Γ0,α X (Ω). When α = 1 and f ∈ Γ X (Ω), we say that f is d-Lipschitz in Ω.

The reader should note that, in this generality, it is not automatically true that a function f ∈ Γ0,α X (Ω) is also continuous with respect to the underlying Euclidean metric of R N . However, if we assume in addition Hypothesis 4.9, then this is true. We notice that if f ∈ Γ0,α X (Ω) then it is not automatically true that f is bounded on Ω. This is guaranteed if one assumes that Ω is d-bounded. 1,∞ Theorem 5.14 (Γ0,1 X (Ω) ,→ W X (Ω)). Let {X1 , . . . , X m } be a system of vector fields 0,1 in Cloc (R N ), and suppose that Hypothesis 4.4 is satisfied, so that d(x, y) defines a Carnot–Carathéodory metric in R N , and that Hypothesis 4.9 is also satisfied. Let Ω ⊂ R N be a (Euclidean) bounded open set such that

d Ω = sup d(x, y) < ∞. x, y ∈Ω

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2 Hypoelliptic operators, etc.

If for a given f : Ω → R there exists L > 0 such that | f (x) − f (y)| ≤ L d(x, y),

x, y ∈ Ω,

(5.9)

then f ∈ W X1,∞ (Ω), and we have the estimate ||X f || L ∞ (Ω) ≤ L.

(5.10)

Proof. Let us note that by (5.9) the function f is d-continuous. By Hypothesis 4.9 we thus see that f is also continuous with respect to the Euclidean distance, and therefore it is measurable. To show that f ∈ L ∞ (Ω) we fix x 0 ∈ Ω. We can assume without restriction that f (x 0 ) = 0, otherwise we consider g = f − f (x 0 ). Then, for every x ∈ Ω we have from (5.9), | f (x)| = | f (x) − f (x 0 )| ≤ Ld(x, x 0 ) ≤ Ld Ω , and so || f || L ∞ (Ω) ≤ Ld Ω .

(5.11)

We next fix j = 1, . . . , m. If we show that the distribution X j f , hX j f , ϕi = h f , X ∗j ϕi, defines a continuous linear functional on L 1 (Ω), then we conclude that X j f ∈ L ∞ (Ω). We fix a point x ∈ Ω and let ε > 0 be such that B e (x, ε) ⊂ Ω. Since Z ||X j f || L ∞ (B e (x,ε)) = sup X j f ϕ dy , ϕ ∈C 0∞ (B e (x,ε)), B e (x,ε) | |ϕ | | L 1 (B e (x, ε))≤1

it will suffice to show that for every ϕ ∈ C0∞ (Be (x, ε)) one has Z f X ∗j ϕ dy ≤ C||ϕ|| L 1 (B e (x,ε)) . B e (x,ε)

(5.12)

Inequality (5.12) would prove that X j f ∈ L ∞ (B(x,r)). From the arbitrariness of x ∈ Ω we see that (5.12) would establish X j f ∈ L ∞ (Ω), and ||X j f || L ∞ (Ω) ≤ C. In order to prove (5.12) we fix a function ϕ ∈ C0∞ (Be (x, ε)) and let ω ⊂⊂ Be (x, ε) be such that supp ϕ ⊂ ω. For a fixed j = 1, . . . , m we write for simplicity Y = X j , and consider the flow Θ(t, x) = ΘY (t, x) as in Lemma 5.11. By compactness we may assume that for |t| sufficiently small we have ω ⊂ Be (x, ε) ∩ Θ(t, Be (x, ε)). We have Z Z Z ∗ hY f , ϕi = f Y ϕ dy = − f Y ϕ dy − f (divY )ϕ dy. B e (x,ε)

B e (x,ε)

B e (x,ε)

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Now, Z B

e (x,ε)

f (div Y )ϕ dy ≤ || f || L ∞ (B(x,ε)) || div Y || L ∞ (Ω) ||ϕ|| L 1 (B e (x,ε)) .

Next, we want to show Z  f Y ϕ dy ≤ L + C|| f || L ∞ (B e (x,ε)) ||ϕ|| L 1 (B e (x,ε)) . B (x,ε)

(5.13)

e

Assume for a moment that we have proved (5.13). Then, by combining (5.12) with (5.13) we would conclude that Y f ∈ L ∞ (B(x, ε)) and that ||Y f || L ∞ (B(x,ε)) ≤ L + C|| f || L ∞ (B e (x,ε)) .

(5.14)

We assume now, as it is not restrictive, that f (x) = 0 and that furthermore |X f (x)| > 0 and |X f | is approximately continuous at x. Consider the vector field defined by Z (y) =

m X X j f (x) X j (y). |X j f (x)| j=1

Clearly, Z is subunitary and Z f (x) =

Pm j=1

|X j f (x)|. Applying (5.14) we obtain

|X f (x)| = | Z f (x)| = lim+ || Z f || L ∞ (B(x,ε)) ≤ L. ε→0

This estimate would establish (5.10), thus completing the proof of the theorem. We are thus left with obtaining (5.14). By Lebesgue dominated convergence we find Z Z ϕ(Θ(t, x)) − ϕ(x) f (x)lim f (x)Y ϕ(x)dx = dx t→0 t B e (x,ε) B e (x,ε) Z ϕ(Θ(t, x)) − ϕ(x) = lim f (x) dx t t→0 B e (x,ε) Z ϕ(Θ(t, x)) − ϕ(x) f (x) = lim dx . t→0 B e (x,ε) t To establish (5.13) it will thus suffice to show that there exists C > 0 such that Z  ϕ(Θ(t, x)) − ϕ(x) lim f (x) dx ≤ L + C|| f || L ∞ (B e (x,ε)) ||ϕ|| L 1 (B e (x,ε)) . t→0 B e (x,ε) t (5.15) For t , 0 we now have Z Z Z ϕ(Θ(t, x)) − ϕ(x) 1 1 f (x) dx = f (x)ϕ(Θ(t, x))dx − f (x)ϕ(x)dx. t t Ω t Ω Ω

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In the first integral on the right-hand side we make the change of variable y = Θ(t, x), for which dy = | det J (Θ(t, x))|dx = (by (5.4)) [1 + σ(t, x)]dx, with σ(t, x) verifying (5.5). Since Θ−1 (t, y) = Θ(−t, Θ(t, x)) = x, we thus obtain Z Z f (x)ϕ(Θ(t, x))dx = f (Θ(−t, y))ϕ(y)[1 + σ(t, ˜ y)]dy, Ω



with | σ(t, ˜ y)| ≤ C|t|. This gives Z Z ϕ(Θ(t, x)) − ϕ(x) f (Θ(−t, y))[1 + σ(t, ˜ y)] − f (y) f (x) dx = ϕ(y)dy t t Ω Ω Z Z f (Θ(−t, y)) − f (y) σ(t, ˜ y) = ϕ(y)dy + f (Θ(−t, y))ϕ(y) dy. t t Ω Ω From this equation we conclude that Z ϕ(Θ(t, x)) − ϕ(x) f (x) dx lim t→0 Ω t Z Z f (Θ(−t, y)) − f (y) σ(t, ˜ y) = lim ϕ(y)dy + lim f (Θ(−t, y))ϕ(y) dy. t→0 Ω t→0 Ω t t def

We now observe that for any given y ∈ ω the curve t → Θ(t, y) = γ(t) is a subunitary curve joining y to Θ(t, y). As a consequence, we have ` s (γ) ≤ t. This gives d(y, Θ(t, y)) ≤ ` s (γ) ≤ t. Moreover, for every y ∈ ω and |t| ≤ T, one has f (Θ(−t, y)) − f (y) d(Θ(−t, y), y) ≤ L ≤ L. t t By Lebesgue dominated convergence we thus obtain Z f (Θ(−t, y)) − f (y) lim ϕ(y)dy ≤ L||ϕ|| L 1 (Ω) . t→0 Ω t To establish (5.15) it will thus suffice to prove that Z σ(t, ˜ y) lim f (Θ(−t, y))ϕ(y) dy ≤ C|| f || L ∞ (B e (x,ε)) ||ϕ|| L 1 (B e (x,ε)) . t→0 Ω t

(5.16)

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With this objective in mind we observe that Z σ(t, ˜ y) f (Θ(−t, y))ϕ(y) dy t Ω Z Z f (Θ(−t, y)) − f (y) σ(t, ˜ y) = ϕ(y) σ(t, ˜ y)dy + f (y)ϕ(y) dy. t t Ω Ω Since | σ(t, ˜ y)| ≤ C|t|, we see that Z f (Θ(−t, y)) − f (y) lim ϕ(y) σ(t, ˜ y)dy = 0, t→0 Ω t whereas, in a similar way, we have Z σ(t, ˜ y) lim f (y)ϕ(y) dy ≤ C|| f || L ∞ (B(x,ε)) ||ϕ|| L 1 (B(x,ε)) . t→0 Ω t This shows that (5.16) does hold, thus completing the proof.



Remark 5.15. A interesting situation is when the set Ω in Theorem 5.14 is a metric ball B(x,r). If we are in the situation in which the metric ball is bounded in the underlying Euclidean distance (remember that this is not always the case; see Example 4.15), then Theorem 5.14 applies. A case of special relevance is when the d-Lipschitz function is f (y) = d(y, x). Notice that in such case || f || L ∞ (B(x,r )) ≤ r.

5.3 Existence of Lipschitz cut-off functions. In this section we prove a result which has fundamental relevance in partial differential equations: the existence of cut-off functions. Theorem 5.16 (See [GN98]). Let {X1 , . . . , X m } be a system of vector fields in 0,1 Cloc (R N ), and suppose that Hypothesis 4.4 is satisfied so that d(x, y) defines a Carnot–Carathéodory metric in R N . Assume in addition that Hypothesis 4.9 is satisfied. Let B(x,r) be a bounded metric ball. There exists an absolute constant C > 0 such that for every 0 < s < t < r one can find a d-Lipschitz continuous function ϕ : R N → [0∞) such that (i) ϕ ≡ 1 on B(x, s), ϕ ≡ 0 on R N \ B(x,t); (ii) |X ϕ| ≤

C t−s

1, p

a.e. in R N ;

(iii) ϕ ∈ W X (R N ) for every 1 ≤ p < ∞.

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2 Hypoelliptic operators, etc.

Proof. Let h ∈ C ∞ ([0, ∞)) be such that 0 ≤ h ≤ 1, h ≡ 1 on [0, s], h ≡ 0 on C [t, ∞), and |h 0 (τ)| ≤ t−s , for some absolute constant C > 0. Consider the function ϕ(y) = h(d(y, x)). Since |d(y, x) − d(z, x)| ≤ d(y, z), by Theorem 5.14 the function y → d(y, x) belongs to W X1,∞ (B(x,r)), with ||X (d(·, x))|| L ∞ (R N ) ≤ 1. 1, p

By Proposition 5.4 the composition ϕ = h ◦ d(·, x) belongs to W X (B(x,r)) for any 1 ≤ p < ∞, and moreover (i) obviously holds. To establish (ii) we observe that the chain rule gives |X j ϕ| = |h 0 (d(·, x))||X j d(·, x)| ≤

C C ||X (d(·, x))|| L ∞ (B(x,r )) ≤ . t−s t−s 

6 Fractional integration in spaces of homogeneous type In this section we analyze the main properties of a generalization of the classical Riesz fractional integration operators to metric spaces with a measure. We study the mapping properties of some generalized fractional integration operators between suitable extensions of the notion of Morrey spaces. All the results will be of a purely metric nature. We consider a triple (S, d, ν), where (S, d) is a metric space, and ν is a nonnegative Borel measure on S satisfying the following Hypothesis 6.1 (Doubling condition). There exists a constant C1 > 0 such that for every x ∈ S and r < 0 one has ν(B(x, 2r)) ≤ C1 ν(B(x,r)),

(6.1)

where B(x,r) = {y ∈ S | d(y, x) < r }. This assumption will be in force throughout this section. It is also worth noting that all the results in this section continue to hold if we assume that d is only a pseudo-metric. In this case the triple (S, ν, d) is called a space of homogeneous type according to Coifmann and Weiss [CW71]. This means that d has all the properties of a distance, except the triangle inequality, which is instead replaced by the pseudo triangle inequality: there exists K > 0 such that for every x, y, z ∈ S, d(x, y) ≤ K[d(x, z) + d(z, x)].

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However, in order to simplify the exposition we will assume that d is a true metric. Let us note the following simple, yet important, fact. Proposition 6.2. Hypothesis 6.1 implies the following: for every x ∈ S, r > 0 and 0 < t ≤ 1 one has ν(B(x,tr)) ≥ C2 t Q ν(B(x,r)), (6.2) with C2 = C1−1 , and Q=

log C1 . log 2

(6.3)

Proof. Suppose that (6.1) holds. For 0 < t ≤ 1 pick k ∈ N such that 2−k < t ≤ 2−k+1 . This gives ν(B(x,r)) = ν(B(x,t −1 tr)) ≤ ν(B(x, 2k tr)) ≤ C1k ν(B(x,tr)) = C1 e (k−1) log C1 ν(B(x,tr)) ≤ C1 t −Q ν(B(x,tr)).  Definition 6.3. The number Q in (6.3) is called the homogeneous dimension of the space (S, d, ν). We observe explicitly that (6.2) implies  s Q ν(B(x, sr)) ≥ C2 , ν(B(x,tr)) t

(6.4)

for every x ∈ S, r > 0, 0 < s < t. In what follows, given a set Ω ⊂ S we let d Ω = d(x, y) . x, y ∈Ω

Clearly, Ω is d-bounded if and only if d Ω < ∞. Proposition 6.4. Let Ω ⊂ S be such that d Ω < ∞. Then, for every x ∈ Ω and 0 < r ≤ d Ω one has !Q ν(B(x,r)) r ≥ C2 . ν(Ω) dΩ Definition 6.5. Given a number A > 0, a measurable set Ω ⊂ S is called of A-type if for every x ∈ Ω and 0 < r < d Ω one has ν(Ω ∩ B(x,r)) ≥ A ν(B(x,r)).

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If Ω is of A-type, then from (6.1) one infers for every x ∈ Ω and 0 < r < d Ω , ν(Ω ∩ B(x, 2r)) ≤ C1 ν(B(x,r)) ≤ A−1 C1 ν(Ω ∩ B(x,r)).

(6.5)

Definition 6.6. An open set Ω ⊂ S is said to admit an interior corkscrew at x 0 ∈ ∂Ω if there exist K, R0 > 0 such that for 0 < r < R0 one can find Ar (x 0 ) ∈ Ω for which r < d( Ar (x 0 ), x 0 ) ≤ r, K

dist( Ar (x 0 ), ∂Ω) >

r . K

(6.6)

If Ω admits an interior corkscrew at every x 0 ∈ ∂Ω with the same K, R0 > 0, then c we say that Ω satisfies the uniform interior corkscrew condition. If both Ω and Ω satisfy the uniform interior corkscrew condition, then we say that Ω satisfies the uniform corkscrew condition. The notion of corkscrew was introduced in the fundamental paper of Jerison and Kenig on NTA (non-tangentially accessible) domains; see [JK82]. This notion guarantees the existence of regions of non-tangential accessibility for a given domain and is tailor made for the application to boundary value problems in partial differential equations. We note in particular that if Ar (x 0 ) is a corkscrew for x 0 ∈ ∂Ω, then for every 0 < r < R0 one has B(x 0 ,r) ⊂ B( Ar (x 0 ), 2r). (6.7) In fact, if d(y, x 0 ) < r, from the triangle inequality and from the right-hand side of the former inequality in (6.6), one has d(y, Ar (x 0 )) ≤ d(y, x 0 ) + d(x 0 , Ar (x 0 )) < r + r = 2r. It is also easy to see that for every 0 < r
0 such that Ω is of A-type.

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Proof. By the assumption there exist K, R0 > 0 such that given any x 0 ∈ ∂Ω, condition (6.6) is fulfilled for any 0 < r < R0 . By (6.8) we have   r  ν(Ω ∩ B(x 0 , 2r)) ≥ ν B Ar (x 0 ), 2K ≥ (by the doubling condition (6.1)) C(C1 , K )ν(B( Ar (x 0 ), 2r)) ≥ (by (6.7)) C(C1 , K )ν(B(x 0 ,r)) ≥ (again by the doubling condition (6.1)) C(C1 , K )ν(B(x 0 , 2r)). Changing r into r/2 in the above inequality, we reach the desired conclusion.



From Proposition 6.7 and Proposition 4.8 we obtain the following result which will prove important in the study of the Sobolev embedding. 0,1 Corollary 6.8. Given a system {X1 , . . . , X m } of vector fields in Cloc (R N ), suppose that Hypothesis 4.4 is satisfied, so that d(x, y) defines a Carnot–Carathéodory metric in R N . Assume the doubling condition (6.1) above. Then, every metric ball B(x,r) is of A-type.

6.1 The Hardy–Littlewood maximal operator. In this subsection we introduce a generalization of the classical Hardy–Littlewood operator and we recall its basic continuity properties between the Lebesgue spaces. We begin with a covering lemma which generalizes to a space of homogeneous-type classical results that go back to Vitali and Wiener. For its proof we refer the reader to the book by Coifman and Weiss [CW71, Theorem III.1.3 and the proof of Theorem III.2.1]. Lemma 6.9. Let (S, d, ν) be a space of homogeneous type. Given a d-bounded set S E ⊂ S, suppose that r : E → [0, ∞) is such that E ⊂ B(x,r (x)). Then, there x ∈E

exists a (finite or infinite) sequence {x j } j ∈N in E such that the balls B(x j ,r (x j )) are disjoint and for some κ = κ(C1 ) > 1 one has [ E⊂ B(x j , κr (x j )). j ∈N

We next introduce a generalization of the classical Hardy–Littlewood (centered) maximal operator. Let f ∈ L 1loc (S, dν). For every x ∈ S we let Z 1 M f (x) = sup | f (y)|dν(y). (6.9) r >0 ν(B(x,r) B(x,r ) Lemma 6.9 plays a crucial in the following generalization, due to Calderón, of the Hardy–Littlewood maximal theorem. We recall that, given a measurable function

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2 Hypoelliptic operators, etc.

f on S, we say that f belongs to weak-L p , and write f ∈ L p,∞ (S, dν), if sup t ν ({x ∈ S | | f (x)| > t}) 1/p < ∞. t >0

One easily verifies that L p (S, dν) ,→ L p,∞ (S, dν) but that the inclusion is strict. Theorem 6.10 (Calderón [Ca76]). Suppose that f ∈ L 1 (S, dν). Then, M f ∈ L 1,∞ (S, dν) and there exists a constant CM,1 > 0, depending only on the doubling constant C1 in (6.1), such that sup t ν ({x ∈ S | M f (x) > t}) ≤ CM,1 || f || L 1 (S, dν) . t >0

Given 1 < p ≤ ∞ there exists CM, p > 0, depending on p and C1 , such that for any f ∈ L p (S, dν) one has Z Z p M f (x) dν(x) ≤ CM, p | f (x)| p dν(x). S

S

6.2 Morrey spaces. In this subsection we introduce a class of spaces which generalize L p spaces and which, in the classical case of R N were introduced by Morrey. The spaces of Morrey were subsequently modified by Campanato in order to provide a useful integral characterization of Hölder continuity (in this respect the reader should also consult the interesting article by Spanne [Sp65]). Definition 6.11. Let p ≥ 1, λ ≥ 0. Given an open set Ω ⊂ S, a function f ∈ L p (Ω, dν) is said to belong to the Morrey space M p, λ (Ω, dν) if || f || M p, λ (Ω, dν) = || f ||Ω, p, λ =

 sup

x ∈Ω,0t }

(6.12)



where C = C(C1 ) > 0. Proof. We prove only (6.11). If Mϕ = ∞ ν-a.e. in Ω, then there is nothing to prove. We can thus assume that Mϕ is the density of a positive measure dµ = Mϕ dν, and similarly that dζ = ϕ dν is a positive measure. Proving the lemma amounts to showing that M : L p (dµ) → L p (dζ ), continuously. This will follow from the theorem of real interpolation of Marcinckiewicz (see [SW71, Theorem 2.4 in Chap. 5]) if we can prove that

M : L 1 (Ω, dµ) → L 1,∞ (Ω, dζ ),

(6.13)

M : L ∞ (Ω, dµ) → L ∞ (Ω, dζ ).

(6.14)

and Now, the proof of (6.14) is easy. Indeed, if Mϕ(x) = 0 for some x ∈ Ω, then ϕ = 0 ν-a.e. in Ω, and therefore L ∞ (Ω, dζ ) = {0}, so that there is nothing to prove. Suppose then that Mϕ(x) > 0 for every x ∈ Ω, and let f ∈ L ∞ (Ω, dµ). Extend f with zero outside Ω and let t > || f || L ∞ (Ω, dµ) . Then, Z µ({x ∈ Ω | | f (x)| > t}) = dµ = 0. {x ∈Ω| | f (x) |>t }

This shows that | f (x)| ≤ t for µ-a.e. x ∈ S (recall, f = 0 in Ωc ), and hence the same conclusion holds for ν-a.e. x ∈ S. Therefore M f (x) ≤ t for every x ∈ Ω, which implies || M f || L ∞ (Ω, dζ) ≤ t. By the arbitrariness of t > || f || L ∞ (Ω, dµ) , we conclude that || M f || L ∞ (Ω, dζ) ≤ || f || L ∞ (Ω, dµ) . This proves (6.14). We next establish (6.13). To this end, it suffices to show that there exists a constant C = C(C1 ) > 0 such that for every f ∈ L 1 (Ω, dµ) one has Z sup t ζ ({x ∈ Ω| M f (x) > t}) ≤ C | f |dµ. (6.15) t >0



Without restriction we can assume that f ∈ L 1 (Ω, dν), and setting f = 0 in Ωc , we can in fact suppose that f ∈ L 1 (S, dµ). Let x ∈ Ω be such that M f (x) > t. There exists r x > 0 such that Z 1 | f |dν > t. ν(B(x,r x )) B(x,r x )

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By Lemma 6.9 there exists a sequence of points x j ∈ Ω such that the balls B(x j , r (x j )) are pairwise disjoint, and for which [ {x ∈ Ω| M f (x) > t} ⊂ B(x j , κr (x j )), j ∈N

for some κ = κ(C1 ) > 1. We thus have ζ ({x ∈ Ω| M f (x) > t}) ∞ ∞ Z X X ≤ ζ (B(x j , κr (x j ))) = j=1 ∞ X

≤C

C ≤ t

j=1 ∞ X j=1

j=1

ν(B(x j ,r (x j ))) ν(B(x j , κr (x j )))

Z

1 ν(B(x j , κr (x j )))

Z

B(x j ,κr (x j ))

B(x j ,κr (x j ))

B(x j ,κr (x j ))

ϕ dν

ϕ dν ϕ(y) dν(y)

Z B(x j ,r (x j ))

| f (x)|dν(x).

Next, for any x ∈ B(x j ,r (x j )) the triangle inequality gives B(x j , κr (x j )) ⊂ B(x, 2κr (x j )). We thus have Z 1 ϕ(y)dν(y) ν(B(x j , κr (x j ))) B(x j ,κr (x j )) Z 1 ≤ ϕ(y)dν(y) ν(B(x j , κr (x j ))) B(x,2κr (x j )) Z ν(B(x, 2κr (x j ))) 1 ϕ(y)dν(y) ≤ C Mϕ(x), = ν(B(x j , κr (x j ))) ν(B(x, 2κr (x j ))) B(x,2κr (x j )) where C = C(C1 ) > 0. We have thus shown that ∞ Z CX ζ ({x ∈ Ω| M f (x) > t}) ≤ | f (x)| Mϕ(x)dν(x) t j=1 B(x j ,r (x j )) ∞ Z CX = | f (x)|dµ(x) t j=1 B(x j ,r (x j )) Z Z C C = | f (x)|dµ(x) ≤ | f |dµ. t S B(x j ,r (x j )) t Ω j ∈N

This proves (6.15), and thus (6.13). As we have mentioned above, the desired conclusion (6.11) now follows by the theorem of Marcinckiewicz. 

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Lemma 6.15. Let Ω ⊂ S be d-bounded. Then, for every fixed x ∈ Ω one has

M1Ω∩B(x,r ) (y) ≤

ν(Ω ∩ B(x,r)) ν(B(y, d(x, y) − r))

for every y ∈ Ω such that d(x, y) > r. Proof. It suffices to observe that if d(x, y) > r, then B(y, %) ∩ B(x,r) , ∅ if and only if % ≥ d(x, y) − r. Hence, ν(Ω ∩ B(x,r)) ν(B(y, %) ∩ B(x,r) ∩ Ω) ≤ . ν(B(y, %)) ν(B(y, d(x, y) − r)) % >0

M1Ω∩B(x,r ) (y) = sup

 The next theorem provides the main result about the mapping properties of the maximal operator between Morrey spaces. Theorem 6.16. Let Ω ⊂ S be a d-bounded set. For every λ > 0 there exists C = C(C1 , λ) > 0 such that for every y ∈ Ω, r < d Ω , and any t > 0 one has, for f ∈ M 1, λ (Ω, dν), ν(Ω ∩ B(y,r)) || f || M 1, λ (Ω, dν) . rλ t >0 (6.16) Given 1 < p < ∞, λ > 0, there exists C = C(C1 , p, λ) > 0 such that for any f ∈ M p, λ (Ω, dν) one has sup t ν ({x ∈ Ω | M f (x) > t} ∩ B(y,r)) ≤ C

|| M f || M p, λ (Ω, dν) ≤ C|| f || M p, λ (Ω, dν) .

(6.17)

Proof. We first analyze the case 1 < p < ∞. Consider f ∈ M p, λ (Ω), and extend it with zero in Ωc . For x ∈ Ω and 0 < r < d Ω we let R = d Ω , and fix k0 ∈ N such that 2k0 ε ≤ 2R < 2k0 +1 ε. One has Z

M f (y) p dν(y) Ω∩B(x,r )

=

Z

M f (y) p 1Ω∩B(x,r ) (y)dν(y) (by Lemma 6.14) Ω Z ≤C | f (y)| p M1Ω∩B(x,r ) (y)dν(y) Ω Z =C | f (y)| p dν(y) Ω∩B(x,r )

+

k0 Z X Ω∩ B(x,2k +1 r )\B(x,2k r ) 

k=0



| f (y)| p

 ν(Ω ∩ B(x,r)) dν(y) , ν(B(y, d(y, x) − r))

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where in the estimate of the second term in the right-hand side we have used Lemma 6.15. In order to establish (6.17) we multiply both sides of the latter inequality by rλ the quantity ν (Ω∩B(x,r )) . The first term in the right-hand side is trivially estimated p by || f || M p, λ (Ω) . The second term, instead, is estimated by k0 X (2k+1 r) λ ν(Ω ∩ B(x, 2k+1 r)) ν(Ω ∩ B(x, 2k+1 r)) (k+1)λ k+1 r)) 2 ν(Ω ∩ B(x, 2 k=0 Z | f (y)| p × dν(y).   k−1 r)) Ω∩ B(x,2k +1 r )\B(x,2k r ) ν(B(y, 2

By the doubling condition the latter term is in turn majorized as follows: Z k0 X (2k+1 r) λ ν(Ω ∩ B(x, 2k+1 r)) C | f (y)| p dν(y) (k+1)λ k+1 r)) k +1 r ) 2 ν(Ω ∩ B(x, 2 Ω∩B(x,2 k=0 p

≤ C|| f || M p, λ (Ω) , where C = C(C1 , λ) > 0. This proves (6.17). Finally, (6.16) follows from the weak inequality (6.12) in Lemma 6.14. Taking in fact ϕ = M1Ω∩B(y,r ) in (6.12) we find Z t ν({x ∈ Ω ∩ B(y,r) | M f (x) > t}) = t 1Ω∩B(y,r ) (x)dν(x) {x ∈Ω | M f (x)>t } Z ≤C | f (x)| M1Ω∩B(y,r ) (x)dν(x) ZΩ =C | f (x)|dν(x) Ω∩B(y,r )

+C

k0 Z X k=0

Ω∩



B(y,2k +1 r )\B(y,2k r )

ν(Ω ∩ B(y,r)) ≤C || f || M 1, λ (Ω) . rλ



| f (x)|

ν(Ω ∩ B(y,r)) dν(x) ν(B(x, d(y, x) − r)) 

6.3 Fractional integration. In this section we establish a result on the continuity properties of a generalization of the classical Riesz fractional operator between Morrey spaces. This result will play an important role subsequently. Definition 6.17. Let Ω ⊂ S be a measurable set and f : Ω → R be a measurable function. Let Q be as in (6.3) above. If 0 < α < Q the fractional integration operator of order α is defined in x ∈ Ω by Z d(x, y) α | f (y)| Iα ( f )(x) = dν(y). ν(B(x, d(x, y))) Ω

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The following theorem is the central result of this section. Theorem 6.18. Let 0 < α < Q, 1 < p < Q α , αp < λ ≤ Q. Given a d-bounded set Ω ⊂ S there exists a constant C = C(C1 , p, α, λ) > 0 such that for any f ∈ M p, λ (Ω, dν) one has ||Iα f || M q, λ (Ω, dν) ≤ C|| f || M p, λ (Ω, dν) ,

(6.18)

provided that

1 1 α = − . (6.19) q p λ If instead p = 1, then for any α < λ ≤ Q there exists C = C(C1 , α, λ) > 0 such that λ for any y ∈ Ω and r > 0 one has with q = λ−α , sup t q ν ({x ∈ Ω ∩ B(y,r) | Iα f (x) > t}) ≤ C t >0

Proof. We consider first the case 1 < p < ε > 0 such that B(x, ε) ⊂ Ω. We write Iα f (x) Z =

d(x, y) α dν(y) + | f (y)| ν(B(x, d(x, y))) B(x,ε)

Q α,

ν(B(y,r)) q || f || M 1, λ (Ω, dν) . λ r

(6.20)

αp < λ ≤ Q. Fix x ∈ Ω and pick

Z | f (y)| Ω∩B(x,ε) c

d(x, y) α dν(y) ν(B(x, d(x, y)))

= Iα1 f (x) + Iα2 f (x). We now have Iα1 f (x) =

∞ Z X k=0

ε 2k +1

≤d(y, x)
0. Next, we estimate Iα2 f (x). We extend f to all S by setting f ≡ 0 in Ωc . Let R = d Ω , and fix k0 ∈ N such that 2k0 ε ≤ 2R < 2k0 +1 ε.

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For k = 0, 1, . . . , k0 denote   Rk (x) = Ω ∩ B(x, 2k+1 ε) \ B(x, 2k ε) . For 0 < σ < 1 we then have Iα2 f (x) Z ≤

d(x, y) α dν(y) ν(B(x, d(x, y))) Ω∩(B(x,2R)\B(x,ε)) k0  Z  1/p X d(x, y) ασ p dν(y) ≤ | f (y)| p ν(B(x, d(x, y))) σ p R k (x) k=0 0  1/p0 Z d(x, y) α(1−σ) p × dν(y) (1−σ) p 0 R k (x) ν(B(x, d(x, y))) Z k0 (  1/p X (2k+1 ε) λ (2k+1 ε) ασ  p ≤ | f (y)| dν(y) ν(B(x, 2k ε)) σ ν(Ω ∩ B(x, 2k+1 ε)) Ω∩B(x,2k +1 ε) k=0 ) ν(Ω ∩ B(x, 2k+1 ε)) 1/p (2k+1 ε) α(1−σ) k+1 1/p 0 × ν(Ω ∩ B(x, 2 ε)) (2k+1 ε) λ/p ν(B(x, 2k ε) 1−σ k0 X λ ν(Ω ∩ B(x, 2k+1 ε)) ≤ (2k+1 ε) −( p −α) || f || M p, λ (Ω, dν) (by (6.1)) ν(B(x, 2k ε) k=0

≤ C1 2

| f (y)|

−( λp −α) −( λp −α)

ε

|| f || M p, λ (Ω, dν)

∞ X

λ

λ

2−k ( p −α) = C ∗∗ ε −( p −α) || f || M p, λ (Ω, dν) ,

k=0

where C ∗∗ = C ∗∗ (C1 , p, α, λ). Combining the estimates for Iα1 f (x) and Iα2 f (x), we find for every ε > 0, λ

Iα f (x) ≤ C ∗ ε α M f (x) + C ∗∗ ε −( p −α) || f || M p, λ (Ω, dν) . We next consider the function g(ε) = Aε α + Bε −β with α+ β= so that β =

λ p

λ , p

− α > 0. Since g 0 (ε) = 0 if and only if ε α+β =

βB , αA

(6.21)

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i.e., ε=

βB αA

! 1/(α+β) =

βB αA

! p/λ ,

we obtain gmin

βB =A αA

! α p/λ

βB +B αA

! λp ( λp −α)

.

Comparing with (6.21) we see that taking A=C ∗ M f (x) and B=C ∗∗ || f || M p, λ (Ω, dν) , then the previous calculation gives with a C = C(C1 , p, α, λ) > 0, α p/λ

Iα f (x) ≤ C|| f || M p, λ (Ω, dν) M f (x) 1−(α p/λ) ,

x ∈ Ω.

(6.22)

We emphasize that (6.22) continues to be true also when p = 1. At this point, for every 0 < r < d Ω we obtain from (6.22), rλ ν(Ω ∩ B(x,r)) ≤

! 1/q

Z q

|Iα f | dν Ω∩B(x,r )

α p/λ C|| f || M p, λ (Ω, dν)

Since by (6.19) we have

rλ ν(Ω ∩ B(x,r))

Z |M f |

q(1− αλp )

! 1/q dν

.

Ω∩B(x,r )

 αp  q 1− = p, λ

from the previous inequality we conclude that there exists C = C(C1 , p, α, λ) > 0 for which α p/λ

p/q

||Iα f || M q, λ (Ω) ≤ C|| f || M p, λ (Ω, dν) || M f || M p, λ (Ω, dν) . If we now invoke (6.17) in Theorem 6.16, we reach the desired conclusion (6.18) from the latter inequality by simply observing that αp p + = 1. λ q To prove (6.20) we use the case p = 1 of (6.22) which gives for t > 0 (note that we can assume without loss of generality that || f || M 1, λ (Ω, dν) , 0), t q ν ({x ∈ Ω ∩ B(y,r) | Iα f (x) > t})   α/λ q 1−(α/λ) ≤ t ν x ∈ Ω ∩ B(y,r) | C|| f || M 1, λ (Ω, dν) M f (x) >t

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( =t ν q



x ∈ Ω ∩ B(y,r) | M f (x) > tC

−1

|| f || −α/λ M 1, λ (Ω, dν)

 λ/(λ−α) )!

(by the p = 1 case of Theorem 6.16)   −λ/(λ−α) ν(Ω ∩ B(y,r)) ≤ Ct q t|| f || −α/λ || f || M 1, λ (Ω, dν) M 1, λ (Ω, dν) rλ  α λ , and thus we also have 1 + λ−α =q keeping in mind that q = λ−α ν(Ω ∩ B(y,r)) q =C || f || M 1, λ (Ω, dν) . rλ 

This completes the proof of the theorem.

It is worth stating explicitly the following basic corollary of Theorem 6.18, Proposition 6.12, and Corollary 6.13. Theorem 6.19. Let 0 < α < Q, 1 < p < Q α . Given a d-bounded set of A-type Ω ⊂ S, there exists a constant C = C(C1 , A, p, α) > 0 such that for any f ∈ L p (Ω, dν) one has !α dΩ ||Iα f || L q (Ω, dν) ≤ C || f || L p (Ω, dν) , (6.23) ν(Ω) 1/Q provided that

1 1 α = − . q p Q

(6.24)

If instead p = 1, then there exists C = C(C1 , A, d Ω ) > 0 such that for any y ∈ Ω and Q , r > 0 one has with q = Q−α sup t ν ({x ∈ Ω ∩ B(y,r) | Iα f (x) > t})

1/q

t >0

ν(B(y,r)) ≤C rQ

! 1/q

Q dΩ

ν(Ω)

|| f || L 1 (Ω, dν) . (6.25)

In particular, when Ω = B(x,r) ⊂ S one has sup t ν ({x ∈ B(x,r) | Iα f (x) > t}) t >0

1/q

r ≤C ν(B(x,r)) 1/Q

!α || f || L 1 (B(x,r ), dν) . (6.26)

7 Fundamental solutions of hypoelliptic operators This section differs substantially from the previous three in that the results that will be discussed are not mostly topological (as in Section 4) or (as in Section 5) based

2 Hypoelliptic operators, etc.

203

on delicate, yet qualitative differential properties, or of a purely metric nature (as in Section 6). The theorems in this section involve a deep quantitative analysis of the local differential geometric properties of exponential maps associated with a system {X1 , . . . , X m } of C ∞ vector fields in R N satisfying the finite rank condition (1.3) in Definition 1.1 above. Henceforth in this section we assume that N ≥ 3. Given an open set Ω ⊂ R N we consider a system {X1 , . . . , X m } of vector fields in C ∞ (Ω) satisfying the condition (1.3) in Definition 1.1. We are interested in some basic local estimates of fundamental solutions of second partial differential operators of the type ∆X = −

m X

X ∗j X j ,

(7.1)

j=1

where X ∗j denotes the adjoint of the vector field X j . When the vector fields X j are those associated with a basis of the horizontal layer of a Carnot group, we have X ∗j = −X j (see [Fo75]), and the operator in (7.1) coincides with (2.17) in Definition 2.2 above. Definition 7.1. A distribution u ∈ D 0 (Ω) is called harmonic in Ω if ∆ H u = 0. Since by Hörmander’s Theorem 1.2 the operator ∆ H is hypoelliptic, if u ∈ D 0 (Ω) is harmonic there exists U ∈ C ∞ (Ω) such that u = U in D 0 (Ω).

7.1 The theorem of Nagel, Stein, and Wainger. We denote by Y1 , . . . ,Y` the collection of the X j ’s and of those commutators which are needed to generate R N . A “degree” is assigned to each Yi , namely the corresponding order of the commutator. If I = (i 1 , . . . ,i N ), 1 ≤ i j ≤ ` is an N-tuple of integers, following [NSW85] one defines N X d(I) = deg(Yi j ) and a I (x) = det[Yi 1 , . . . ,Yi N ]. j=1

Definition 7.2. The Nagel–Stein–Wainger polynomial is defined by X Λ(x,r) = |a I (x)|r d(I ) , r > 0. I

For a given bounded open set U ⊂ R N , we let Q = sup {d(I) | |a I (x)| , 0, x ∈ U},

Q(x) = inf {d(I) | |a I (x)| , 0}, (7.2)

and notice that from the work in [NSW85] we know 3 ≤ N ≤ Q(x) ≤ Q.

(7.3)

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It is immediate that for every x ∈ U, and every r > 0, one has t Q Λ(x,r) ≤ Λ(x,tr) ≤ t Q(x) Λ(x,r),

0 ≤ t ≤ 1.

(7.4)

The numbers Q and Q(x) are respectively called the local homogeneous dimension of U, and the pointwise homogeneous dimension at x, with respect to the system {X1 , . . . , X m }. Remark 7.3. In a Carnot group G the Nagel–Stein–Wainger polynomial is simply a monomial, and it does not depend on the point. In other words, Λ(g,r) ≡ Cr Q ,

g ∈ G, r > 0,

where C > 0 is an absolute constant, and Q is the homogeneous dimension of the group G. The following fundamental result is due to Nagel, Stein, and Wainger [NSW85]. Theorem 7.4 (Local doubling condition). For every bounded set U ⊂ R N there exist constants C, R0 > 0 such that for any x ∈ U, and 0 < r ≤ R0 , one has CΛ(x,r) ≤ |B(x,r)| ≤ C −1 Λ(x,r).

(7.5)

As a consequence, with C1 = 2Q , one has for every x ∈ U, and any 0 < r ≤ R0 , |B(x, 2r)| ≤ C1 |B(x,r)|.

(7.6)

Henceforth, the numbers C1 , R0 in (7.6) will be referred to as the characteristic local parameters of U with respect to the system {X1 , . . . , X m }. One important consequence of the doubling condition (7.6) is the following estimate which justifies the definition of the number Q as the local homogeneous dimension of the bounded set U (see (6.4) above):  r Q |B(x,r)| ≤ C1 , x ∈ U, 0 < r < s ≤ R0 . (7.7) s |B(x, s)| The following observation is often useful; see [DGN98, Cor 2.10]. Corollary 7.5. Let U ⊂ R N be a connected, bounded set with |U | > 0, and let R0 be as in Theorem 7.4 above. For any 0 < r ≤ R0 one has Cr = inf |B(x,r)| > 0. x ∈U

In view of Corollary 7.5 we obtain from (7.7) with C ∗ = C1 CR−10 > 0, rQ ≤ C ∗ R0Q , |B(x,r)|

x ∈ U, 0 < r ≤ R0 .

(7.8)

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Proposition 7.6. The polynomial function Λ(x,r) in Definition 7.2 satisfies the following property. Given a bounded set U ⊂ R N one has Q(x)

Λ(x,r) Λ(x,r 2 ) − Λ(x,r 1 ) Λ(x,r) ≤ ≤Q r r2 − r1 r

for any x ∈ U, 0 < r 1 < r 2 < R0 , and some r = r (x) ∈ (r 1 ,r 2 ). Here, R0 is the characteristic local parameter of U and Q is its local homogeneous dimension (7.2). Proof. We begin by observing that, from Definition 7.2, for any bounded set U ⊂ R N one has Q(x) ≤

rΛ0 (x,r) ≤ Q, Λ(x,r)

for every x ∈ U, 0 < r < R0 ,

(7.9)

where Q and Q(x) are as in (7.2). We fix x ∈ U and 0 < r 1 < r 2 ≤ R0 , and apply the mean value theorem to the function Λ(x, ·) to reach the conclusion from (7.9).  Finally, we recall the following definition from [NSW85, p. 123]. For x ∈ R N , and r > 0, we set Box(r) = {x ∈ R

N

| x = exp(

` X

u j Yj )

with |u j | < r d j },

(7.10)

j=1

where we have let d j = deg(Yj ). Here, exp denotes the exponential mapping associated with the vector fields Y1 , . . . ,Y` . For its definition and main properties we refer the reader to the appendix of [NSW85]. The following result is contained in [NSW85, Theorem 7]. Theorem 7.7 (Ball-box). Given a bounded set U ⊂ R N there exist η ∈ (0, 1), and R0 > 0, such that for any x ∈ U, and 0 < r < R0 , one has B(x, ηr) ⊂ exp x (Box(r)) ⊂ B(x,r). Remark 7.8. One can be more precise about the shape of the sets B(x,r). They have size r in the directions of the X j ’s, whereas they have size r 2 in the directions of the commutators [X i , X j ], and so on (see [NSW85]).

7.2 Size estimates of the fundamental solution. Let G be a Carnot group and let Γ be the unique fundamental solution of a sub-Laplacian ∆ H on G. By Corollary 3.11 we have the estimate α |g −1 ◦

g 0 |GQ−2

≤ Γ(g −1 ◦ g 0 ) ≤

β |g −1 ◦ g 0 |GQ−2

,

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for some universal constants α, β > 0. Combining this estimate with Proposition 4.28 we obtain α∗ β∗ −1 0 ≤ Γ(g ◦ g ) ≤ , d(g, g 0 ) Q−2 d(g, g 0 ) Q−2 where α ∗ , β ∗ > 0 are appropriate universal constants, and d(g, g 0 ) indicates the Carnot–Carathéodory distance on G. On the other hand, Lemma 4.27 and a change of variable imply the existence of a universal constant ω > 0 such that |B(g,r)| = ωr Q ,

g ∈ G, r > 0.

(7.11)

Using (7.11) in the above estimate, we see that the latter can be recast in the form C

d(g, g 0 ) 2 d(g, g 0 ) 2 −1 0 −1 ≤ Γ(g ◦ g ) ≤ C , |B(g, d(g, g 0 ))| |B(g, d(g, g 0 ))|

(7.12)

for a certain universal constant C > 0. The following fundamental result which was established independently by Nagel, Stein, and Wainger in [NSW85], and by Sánchez-Calle in [SCa84], represents a (local) generalization of (7.12) to operators ∆ X of the form (7.1) above. Given such an operator on a bounded open set Ω ⊂ R N , we indicate with Γ(x, y) its (positive) fundamental solution in Ω, as constructed in [RS76] and [NSW85]. This means that for every x ∈ Ω we have Γ(·, x) ∈ D 0 (Ω) ∩ C ∞ (Ω \ {x}), and ∆ X Γ(·, x) = δ x in D 0 (Ω), where δ x indicates the Dirac delta function concentrated in x. Since ∆ X is self-adjoint, one can prove that Γ(x, y) = Γ(y, x),

x, y ∈ Ω.

Theorem 7.9. Given a bounded set U ⊂ R N , there exists R0 > 0, depending on U and on X, such that for x ∈ U, 0 < d(x, y) ≤ R0 , one has for s ∈ N ∪ {0}, and for some constant C = C(U, X, s) > 0, d(x, y) 2−s , |B(x, d(x, y))| d(x, y) 2 Γ(x, y) ≥ C . |B(x, d(x, y))|

|X j1 X j2 . . . X j s Γ(x, y)| ≤ C −1

(7.13)

In the first inequality in (7.13), one has ji ∈ {1, . . . , m} for i = 1, . . . , s, and X j i is allowed to act on either x or y.

7.3 X-balls and representation formulas. The Nagel–Stein–Wainger polynomial Λ(x,r) in Definition 7.2 above and the size estimates in Theorem 7.9 suggest the introduction of a family of sets which are equivalent to the Carnot–Carathéodory

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2 Hypoelliptic operators, etc.

balls, but display some distinguishing features which make them better suited than the metric balls to the study of various fine questions which specifically involve the intrinsic geometry of operators such as (7.1). Given a bounded open set Ω ⊂ R N with characteristic local parameters C1 , R0 > 0, for every x ∈ Ω and 0 < r ≤ R0 we introduce the quantity E(x,r) =

Λ(x,r) . r2

(7.14)

We observe that, because of (7.3), for every fixed x ∈ Ω the function r → E(x,r) is strictly increasing on (0, R0 ). We denote by F (x, ·) the inverse function of E(x, ·), so that F (x, E(x,r)) = r. We remark explicitly that in a Carnot group one has E(g,r) = Cr Q−2 ,

F (g,r) = C −1/(Q−2) r 1/(Q−2) ,

g ∈ G, r > 0,

(7.15)

where C > 0 is a universal constant. Let Γ(x, y) denote the (positive) fundamental solution of −∆ X in Ω. For any x ∈ Ω and 0 < r ≤ R0 we consider the level sets ( ) 1 n Ω(x,r) = y ∈ R Γ(x, y) > . r Definition 7.10. For every x ∈ Rn , and r > 0, the set ( ) 1 n B X (x,r) = y ∈ R Γ(x, y) > E(x,r) will be called the X-ball, centered at x with radius r. We note explicitly that B X (x,r) = Ω(x, E(x,r))

and that

Ω(x,r) = B X (x, F (x,r)).

One of the main geometric properties of the X-balls is that they are equivalent to the Carnot–Carathéodory balls. To see this, we observe that in view of (7.6), (7.13), it is easy to recognize that there exists a > 1, depending on Ω and X, such that for x ∈ Ω, 0 < r ≤ R0 , B(x, a−1 r) ⊂ B X (x,r) ⊂ B(x, ar). (7.16) We observe that, as a consequence of (7.5), and of (7.13), one has ! 1 Cd(x, y) ≤ F x, ≤ C −1 d(x, y), Γ(x, y) for all x ∈ Ω, 0 < d(x, y) ≤ R0 .

(7.17)

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Another key property of the X-balls is that, since Γ(·, x) ∈ C ∞ (Ω\{x}), by Sard’s theorem for a.e. r ∈ (0, R0 ] the set ∂B X (x,r) is a C ∞ manifold of codimension 1 in R N . Therefore, we can use the tools of calculus on these sets. For instance, we have the following basic representation formula for a smooth function which will prove quite useful in the sequel. In what follows, if D ⊂ R N denotes a bounded open set, with characteristic local parameters C1 , R0 , we will let [ D R0 = {Ω(x,r) | x ∈ D, 0 < r < R0 } . Theorem 7.11 (Representation formula; see [CGL93]). Let ψ ∈ C ∞ (D R 0 ). For every x ∈ D and 0 < t ≤ R0 , one has Z Z f |X y Γ(x, y)| 2 1g ψ(x) = ψ(y) dH N −1 (y) + ∆ X ψ(y) Γ(x, y) − dy. |DΓ(x, y)| t ∂Ω(x, t ) Ω(x, t ) (7.18) The same approach as that of the proof of Theorem 7.11 leads to the following result which, among other things, plays a crucial role in establishing an important geometric embedding and an isoperimetric inequality. For these aspects the reader should consult Section 8. Corollary 7.12. Let U ⊂ R N be a bounded open set. Then, there exists a constant C = C(U, X ) > 0 such that for any function f ∈ C00,1 (U), and any x ∈ U one has Z d(x, y) dy = CI1 (|X f |)(x). | f (x)| ≤ C |X f (y)| |B(x, d(x, y))| U Proof. Using the fact that supp f is a compact subset of U, we can proceed as in the proof of Theorem 7.11 and, after some integration by parts, obtain the following identity: Z f (x) = −

hX f (y), X Γ(x, y)idy,

x ∈ U.

U

From this identity the desired conclusion follows if one invokes the basic estimate (7.13) in Theorem 7.9 above. 

7.4 Subelliptic mollifiers. We next define a one-parameter family of smooth mollifiers introduced in [CDG97] which are tailor made on the intrinsic geometry of ∆ X . RWe choose a non-negative function f ∈ C0∞ (R), with supp f ⊂ [1, 2], and such that R f (s)ds = 1, and let f R (s) = R−1 f (R−1 s). We define the kernel ! |X y Γ(x, y)| 2 1 K R (x, y) = f R , Γ(x, y) Γ(x, y) 2 whenever the right-hand side makes sense.

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209

Definition 7.13. Given a function u ∈ L 1loc (R N ), we define the subelliptic mollifier of u by the equation Z JR u(x) = u(y)K R (x, y)dy, R > 0. (7.19) Rn

We note that for any fixed x ∈ R N , supp K R (x, ·) ⊂ Ω(x, 2R) \ Ω(x, R).

(7.20)

One of the important features of JR u is expressed by the following theorem. Theorem 7.14 (See [CDG97]). Let D ⊂ R N be bounded and open and suppose that ∆ X u = 0 (u is harmonic) in D. There exists R0 > 0, depending on D and X, such that for any x ∈ D, and every 0 < R ≤ R0 for which Ω(x, 2R) ⊂ D, one has u(x) = JR u(x). Proof. Let u and Ω(x, R) be as in the statement of the theorem. Taking ψ = u in (7.18), we find u(x) =

Z ∂Ω(x, t )

u(y)

|X y Γ(x, y)| 2 dH N −1 (y). |DΓ(x, y)|

(7.21)

We are now going to use (7.21) to complete the proof. The idea is to start from the definition of JR u(x), and then to use Federer’s co-area formula [Fe69]. One finds Z  |X y Γ(x, y)| 2 f R (t)  u(y) dH N −1 (y)  dt. |DΓ(x, y)|  ∂Ω(x, t )  0 R This equality, (7.21), and the fact that R f R (s)ds = 1, imply the conclusion. JR u(x) =

Z





Our main a priori estimate is contained in the following theorem. Theorem 7.15 (See [CDG97]). Fix a bounded open set D ⊂ R N . There exists a constant R0 > 0, depending only on D and on the system X = {X1 , . . . , X m }, such that for any u ∈ L 1loc (R N ), x ∈ D, 0 < R ≤ R0 , and s ∈ N, one has for some C = C(D, X, s) > 0, Z C 1 |X j1 X j2 . . . X j s JR u(x)| ≤ |u(y)|dy. R F (x, R) 2+s Ω(x, R)

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Proof. We first consider the case s = 1. From (7.13), and from the support property (7.20) of K R (x, ·), we can differentiate under the integral sign in (7.19), to obtain Z |X JR u(x)| ≤ |u(y)||X x K R (x, y)|dy. B(x,2R)

By the definition of K R (x, y) and the chain rule it is easy to recognize that the components of its subgradient X x K R (x, y) are estimated as follows: |X j K R (x, y)| ≤ C R−2 |X Γ(x, y)| 3 Γ(x, y) −4 m X + C R−1 Γ(x, y) −2 |X j X k Γ(x, y)||X k Γ(x, y)| k=1 3

+ C R |X Γ(x, y)| Γ(x, y) −3 −1

= I R1 (x, y) + I R2 (x, y) + I R3 (x, y). To control the three terms in the right-hand side of the above inequality we use the size estimates (7.13), along with the observation that, due to the fact that on the support of K R (x, ·) one has 1 1 < Γ(x, y) ≤ , 2R R then Theorem 7.9 and (7.17) give for all x ∈ D, 0 < R ≤ R0 , and y ∈ Ω(x, 2R) \ Ω(x, R), d(x, y) ≤ C −1 . (7.22) C≤ F (x, R) Using (7.13), (7.22), one obtains that for i = 1, 2, 3, sup y ∈Ω(x,2R)\Ω(x, R)

|I Ri (x, y)| ≤

C RF (x, R) 3

for any x ∈ D, provided that 0 < R ≤ R0 . This completes the proof in the case s = 1. The case s ≥ 2 is handled recursively by similar considerations based on Theorem 7.9, and we omit details. It may be helpful for the interested reader to note that Theorem 7.9 implies |X j1 X j2 . . . X j s Γ(x, y)| ≤ Cd(x, y) −s Γ(x, y), so that by (7.22) one obtains sup y ∈Ω(x,2R)\Ω(x, R)

|X j1 X j2 . . . X j s Γ(x, y)| ≤

C . RF (x, R) s

(7.23) 

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7.5 Interior Schauder estimates. In this subsection we establish some basic interior Schauder-type estimates that, besides playing an important role in the sequel, also have an obvious independent interest. Such estimates are tailored to the intrinsic geometry of the system X = {X1 , . . . , X m }, and are obtained by means of Theorem 7.15 above. For convenience, we state the relevant estimates in terms of the X-balls B X (x,r) introduced in Definition 7.10, but we stress that, thanks to (7.16), we could have as well employed the metric balls B(x,r). Since our focus is on harmonic functions, we do not explicitly treat the non-homogeneous equation ∆ X u = f with a non-zero right-hand side. Estimates for solutions of the latter equation can, however, be obtained by relatively simple modifications of the arguments in the homogeneous case. The following is the main result in this subsection. Theorem 7.16 (See [CDG97]). Let D ⊂ Rn be a bounded open set and suppose that u is harmonic in D. There exists R0 > 0, depending on D and X, such that for every x ∈ D and 0 < r ≤ R0 for which B X (x,r) ⊂ D, one has for any s ∈ N |X j1 X j2 . . . X j s u(x)| ≤

C max |u|, r s B X (x,r )

for some constant C = C(D, X, s) > 0. In the above estimate, for every i = 1, . . . , s, the index ji runs in the set {1, . . . , m}. Remark 7.17. We emphasize that Theorem 7.16 cannot be established similarly to its classical ancestor for harmonic functions, where one uses the mean value theorem coupled with the trivial observation that any derivative of a harmonic function is harmonic. In the present non-commutative setting, derivatives of harmonic functions are no longer harmonic! Proof of Theorem 7.16. We observe explicitly that the assumption states that with R = E(x,r)/2, then Ω(x, 2R) = B X (x,r) ⊂ D. By Theorem 7.14, and by (7.23), we find |X j1 X j2 . . . X j s u(x)| = |X j1 X j2 . . . X j s (JR u)(x)| Z C |Ω(x, R)| ≤ |u(y)|dy ≤ C max |u|. RF (x, R) 2+s Ω(x, R) RF (x, R) 2+s Ω(x, R) To complete the proof we need to observe only that Ω(x, R) = B X (x,r), and that, thanks to Theorem 7.9, (7.17), one has C |B(x,r)| C −1 ≤ ≤ . rs rs RF (x, R) 2+s



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Remark 7.18. We observe explicitly that when G is a Carnot group with X1 , . . . , X m being a fixed basis of the horizontal layer of its Lie algebra, then the constant C in Theorem 7.16 can be taken independent of the open set D. Corollary 7.19 (Liouville-type theorem). Let G be a Carnot group, and suppose that u is harmonic in G with respect to a given sub-Laplacian ∆ H . If there exists C > 0 and 0 < ε < 1 such that |u(g)| ≤ C(1 + |g|Gε ),

g ∈ G,

(7.24)

then u must be a constant in G. Proof. Our objective is proving that for every g ∈ G one has ∇ H u(g) = 0. Once this is accomplished we can invoke the assumption on the Lie algebra (2.1) to prove that every derivative of order 1 of u vanishes identically, and therefore u must be constant. By considering the function v(g 0 ) = u(g −1 ◦g 0 ) we have ∇ H v(g 0 ) = L g −1 (∇ H u(g 0 )). Therefore, ∇ H v(g) = L g −1 (∇ H u(g)) = ∇ H u(e). To prove that ∇ H u ≡ 0 in G it will thus suffice to show that ∇ H u(e) = 0. By Theorem 7.16 one has for some absolute constant C > 0 and for every r > 0, |∇ H u(e)| ≤

C C∗ max |u| ≤ (by (7.24)) (1 + r ε ) → 0, r B(e,r ) r

as r → ∞. 

This completes the proof.

8 The geometric Sobolev embedding and the isoperimetric inequality In this section we derive some results which play a fundamental role in the study of the regularity of weak solutions of PDEs in sub-Riemannian geometry. The following result, which is taken from [CDG94], is the main theorem of the section. Theorem 8.1 (Geometric Sobolev embedding). Given a system {X1 , . . . , X m } of vec0,1 tor fields in Cloc (R N ), suppose that Hypothesis 4.4 is satisfied, so that d(x, y) defines a Carnot–Carathéodory metric in R N . Assume that Lebesgue measure on Rn satisfies the doubling condition (6.1) above, and let Q be as in (6.3). Furthermore, suppose that for any given bounded set U ⊂ R N there exist C = C(U, X ) > 0 and R0 = R0 (U, X ) > 0 such that for every x ∈ U and 0 < r < R0 , one has for every f ∈ C00,1 (B(x,r)), | f (y)| ≤ CI1 (|X f |)(y),

y ∈ B(x,r).

(8.1)

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Then, there exists S1 > 0, depending on C1 in (6.1) and on C in (8.1), such that for 1,1 (B(x,r)) one has every f ∈ W X,0 1 |B(x,r)|

! (Q−1)/Q

Z |f|

Q/(Q−1)

dy

B(x,r )

1 ≤ S1 r |B(x,r)|

!

Z

|X f |dy . (8.2) B(x,r )

We note explicitly that inequality (8.2) in Theorem 8.1 cannot be derived directly from the hypothesis (8.1) since, as we saw in Theorem 6.18, the fractional integration operator I1 does not map L 1 (B(x,r), dν) into L q (B(x,r), dν), with q = Q/(Q − 1), but it maps instead L 1 (B(x,r), dν) into L q,∞ (B(x,r), dν). Our strategy for proving Theorem 8.1 consists, in fact, in first establishing the following weak version of it. Theorem 8.2 (Weak geometric Sobolev embedding). Under the assumptions of The0,1 (B(x,r)) one has with orem 8.1, there exists S1 > 0 such that for every f ∈ W X,0 q = Q/(Q − 1), Z r sup t |{y ∈ B(x,r) | | f (y)| > t}| 1/q ≤ S1 |X f |dy. (8.3) |B(x,r)| 1/Q B(x,r ) t >0 Proof. It follows immediately from the assumption (8.1), by appealing to (6.26) in Theorem 6.19.  We next show how Theorem 8.2 implies a remarkable, sharp isoperimetric inequality. Theorem 8.3 (Isoperimetric inequality). Under the assumptions in Theorem 8.2, let U and B(x,r) be as in its statement. Let E ⊂ E ⊂ B(x,r) be a C 1 open set. Then, with the same constant S1 > 0 as in equation (8.3) above, one has |E| (Q−1)/Q ≤ S1

r PX (E; B(x,r)), |B(x,r)| 1/Q

(8.4)

where PX (E; B(x,r)) denotes the relative X-perimeter of E with respect to B(x,r) (see Definition 5.8 above). Proof. Let r (x) = inf |x − y| = diste (x, E). y ∈E

Set δ0 =

1 2

diste (E, B(x,r)), and for 0 < δ ≤ δ0 consider the function r (x) uδ (x) = 1 − δ

!+

.

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Clearly, uδ ∈ C00,1 (B(x,r)) and uδ ≡ 1 on E. Applying Theorem 8.2 to uδ we obtain for every 0 < t < 1, |E| (Q−1)/Q ≤ |{y ∈ B(x,r) | uδ (y) > t}| (Q−1)/Q Z Z C∗ r S1 r 1 ≤ |Xuδ |dy = |Xr |dy, t |B(x,r)| 1/Q B(x,r ) t |B(x,r)| 1/Q δ Aδ where we have let Aδ = {y ∈ B(x,r) | 0 < r (x) < δ}. Letting t → 1 in the above inequality and applying Federer’s co-area formula (see [Fe69, Theorem 3.2.3]), we find Z δZ r |Xr | S1 dH N −1 ds, |E| (Q−1)/Q ≤ δ |B(x,r)| 1/Q 0 |Dr | {y ∈B(x,r ) |r (y)=s } where we have denoted by |Dr | the Euclidean length of the ordinary gradient of the function r, and H N −1 indicates the standard (N − 1)-dimensional Hausdorff measure in R N . We note explicitly that the application of the co-area formula is justified since, 0,1 given the fact that the vector fields X1 , . . . , X m have coefficients in Cloc (R N ), there exists a constant C(U) > 0 such that |X ϕ| ≤ C(U)|Dϕ| for any Lipschitz function |Xr | ∞ on U. Thereby, |Dr | ∈ L (U). Indicating with ν(x) the outward unit normal to the level set {y ∈ B(x,r) | r (y) = t} we now have on this set, 1/2 m |Xr | *X = hX i , νi2 + . |Dr | , i=1 -

Letting δ → 0+ in the last inequality, we thus find |E|

(Q−1)/Q

r ≤ S1 |B(x,r)| 1/Q

Z

1/2 m X * hX i , νi2 + dH N −1 . ∂E , i=1 -

Since E has a C 1 boundary, the desired conclusion follows by invoking Proposition 5.9 above.  We next show that Theorem 8.3 implies the geometric Sobolev embedding Theorem 8.1. Proof of Theorem 8.1. We begin with a function f ∈ C01 (B(x,r)), f ≥ 0. By Sard’s theorem for a.e. 0 < t ≤ max f , the level set Et = {y ∈ B(x,r) | f (y) > t} is a ¯ B(x,r )

relatively compact C 1 subdomain of B(x,r). By Federer’s co-area formula we easily have (see [CDG96, (4.1)]) Z Z ∞ |X f |dy = PX (Et ; B(x,r))dt. B(x,r )

0

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2 Hypoelliptic operators, etc.

Applying formula (8.4) in Theorem 8.3 to the set Et we conclude from the last equation, Z 1/Q Z ∞ −1 |B(x,r)| |X f |dy ≥ (S1 ) |Et | (Q−1)/Q dt. (8.5) r B(x,r ) 0 We now claim that Q Q−1



Z

t

! Q/(Q−1)



Z 1/(Q−1)

|Et |dt ≤

|Et |

(Q−1)/Q

dt

.

0

0

The proof of this inequality is a simple consequence of the fact that the function t → |Et | is non-increasing. Using this inequality in (8.5), and the real analysis result which states that Z ∞ Z Q Q/(Q−1) t 1/(Q−1) |Et |dt, |f| dy = Q−1 0 B(x,r ) 

we finally obtain the desired conclusion (8.2) by a density argument.

Remark 8.4. We stress that the above proofs show that the weak geometric Sobolev embedding is in fact equivalent to the strong one since, trivially, we have that Theorem 8.1 =⇒ Theorem 8.2, and we have proved that Theorem 8. 2 =⇒ Theorem 8. 3 =⇒ Theorem 8. 1 . An important consequence of the geometric Sobolev embedding Theorem 8.1 is the following result which constitutes a generalization of the celebrated Sobolev embedding theorem.3 Theorem 8.5 (Non-geometric Sobolev embedding). Under the assumptions of Theorem 8.1, given any number 1 < p < Q there exists a constant S p > 0, depending on 1, p C1 in (6.1), C in (8.1), and p, such that for any function f ∈ W X,0 (B(x,r)) one has 1 |B(x,r)|

! 1/q

Z | f | q dy B(x,r )

≤ Sp r

1 |B(x,r)|

! 1/p

Z |X f | p dy

,

(8.6)

B(x,r )

where the number q is determined by the equation 1 1 1 − = . p q Q 3This does not reflect the historical development of the subject. The non-geometric case 1 < p < n was first proved by Sobolev in 1938 in [So38]. His result did not include the geometric case p = 1. This was first established independently by Gagliardo [Ga58] and Nirenberg [Ni59] in the late fifties by means of an interpolation argument. The remarkable equivalence between the geometric Sobolev embedding theorem and the isoperimetric inequality was first discovered independently by Maz’ya [Ma85] and by Fleming and Rishel [FR60]. A delightful account of the theory of Sobolev spaces is provided by Ziemer’s classical book [Z89]

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Remark 8.6. As a consequence of Theorem 7.9 and Corollary 7.12 we see that when X1 , . . . , X m ∈ C ∞ (R N ) is a system satisfying the finite rank condition (1.3) above, then the assumption (8.1) is fulfilled. Furthermore, thanks to Theorem 7.4 above, the doubling condition (7.6) is also satisfied. It follows that Theorems 8.1 and 8.3 are both valid for such systems. These facts play an important role in the study of the regularity of (weak) solutions of partial differential equations of subelliptic type. In closing we mention that, under very general assumptions, a comprehensive theory of isoperimetric and Sobolev inequalities for BV functions was developed in the paper [GN96], which also contains an existence result for minimal surfaces in Carnot–Carathéodory spaces. The interested reader should consult that source for all the relevant results.

9 The Li–Yau inequality for complete manifolds with Ricci ≥ 0 With this section the second part of these lecture notes begins: the one devoted to global aspects of sub-Riemannian geometry connected with curvature. Here, the notion of curvature that we intend to introduce is that of Ricci lower bounds. The approach that we plan to discuss finds its origin in a beautiful circle of ideas introduced, in the Riemannian case, by Bakry, Ledoux, and collaborators (the reader is encouraged to consult the recent monograph [BGL14] for a remarkable account of these developments). Inspired by these ideas a sub-Riemannian approach to Ricci curvature was first introduced in the joint work [BG09] with Baudoin, a part of which was subsequently published in [BG15]. The remaining sections of these lecture notes also provide the first part of the notes of Fabrice Baudoin in this same volume.

9.1 Introduction. Our first main objective is discussing an a priori inequality due to Li and Yau. To motivate their main result let us consider a special positive solution of the heat equation Hu = ∆u − ut = 0 on Rn , namely, the heat kernel given by ! |x| 2 − n2 u(x,t) = (4πt) exp − , (x,t) ∈ R+n+1 = Rn × (0, ∞). 4t The entropy is defined by n |x| 2 f (x,t) = log u(x,t) = − log(4πt) − . 2 4t From this formula one immediately recognizes that n in R+n+1 . |∇ f | 2 − f t ≡ 2t

2 Hypoelliptic operators, etc.

217

9.1.1 The Li–Yau inequality. A fundamental result of Li and Yau (see [LY86]), states that in every complete, boundaryless, n-dimensional Riemannian manifold M with Ricci ≥ 0, for any positive solution u of the heat equation on M one has for f = log u, n |∇ f | 2 − f t ≤ in M × (0, ∞). (9.1) 2t The proof of this result follows easily from the following deep a priori gradient estimate. Theorem 9.1 (See [LY86]). Let M be a complete non-compact Riemannian manifold with ∂M = ∅ and Ric(M) ≥ 0. There exists a constant C(n) > 0 such that if u > 0 is a solution of ∆u − ut = 0 in B2R × (0, ∞), where B2R is a geodesic ball of radius 2R centered at O ∈ M, then for any α > 1 one has ! nα 2 |∇u| 2 ut α4 − α + ≤ C(n) . (9.2) sup u 2t u2 (α 2 − 1)R2 BR Let us see how the Li–Yau inequality (9.1) can be derived from Theorem 9.1. But let us first give the formal statement. Theorem 9.2 (See [LY86]). Suppose that M is a complete non-compact Riemannian manifold with ∂M = ∅ and satisfying Ric(M) ≥ 0. If u > 0 is a solution of ∆u −ut = 0 in M × (0, ∞), then n |∇u| 2 ut − ≤ . (9.3) 2 u 2t u Proof. Since M is non-compact we can let R → ∞ in (9.2) obtaining for every fixed t > 0, ! nα 2 |∇u| 2 ut ≤ . sup − α u 2t u2 M At this point we let α → 1 to reach the desired conclusion.



Before discussing the original proof of the Li–Yau inequality (9.3) we briefly recall the two fundamental tools from Riemannian geometry which constitute its key ingredients: (i) the identity of Bochner; (ii) the Laplacian comparison theorem.

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9.1.2 The Levi-Civita connection. On a Riemannian manifold (M, g) there exists a unique Levi-Civita connection ∇. Let us indicate with X(M) the collection of all C ∞ vector fields on M. In other words, a C ∞ vector field on M is a smooth mapping X : M → TM. Recall that by a connection on M we mean a mapping ∇ which associates with every X,Y ∈ X(M) another element of X(M), denoted by ∇ X Y , and called the covariant derivative of Y with respect to X. A connection is specified by the properties (i) ∇a X +bY Z = a∇ X Z + b∇Y Z; (ii) ∇ X (aY ) = a∇ X Y + X (a)Y . The torsion tensor with respect to ∇ is defined by T (X,Y ) = ∇ X Y − ∇Y X − [X,Y ]. Definition 9.3. The connection ∇ is called Levi-Civita if (i) the metric tensor is parallel, i.e., ∇g = 0; and (ii) the connection is torsion free, i.e., T (X,Y ) = 0,

for every X,Y ∈ X(M).

These conditions are equivalent to saying that (i0) Zg(X,Y ) = g(∇ Z X,Y ) + g(X, ∇ Z Y ), for all X,Y, Z ∈ X(M); (ii0) ∇ X Y − ∇Y X = [X,Y ], for every X,Y ∈ X(M). The Riemann curvature tensor is defined by R(X,Y ) Z = ∇ X ∇Y Z − ∇Y ∇ X Z − ∇[X,Y ] Z. Definition 9.4. The Ricci tensor is defined by Ric(X,Y ) =

n X

g(R(Ei , X )Y, Ei ),

X,Y ∈ X(M),

i=1 n is a local basis in a given coordinate neighborhood. In this local basis, where {Ei }i=1 we set n X Ri j = Ric(Ei , E j ) = g(R(Ek , Ei )E j , Ek ). k=1

Given % ∈ R we say that M satisfies Ric(M) ≥ % provided that for every x ∈ M one has Ric(v, v) ≥ % g(v, v), ∀v ∈ Tx M.

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2 Hypoelliptic operators, etc.

In local coordinates this means n X

Ri j vi v j ≥ %

i, j=1

if v =

n X

gi j vi v j ,

i, j=1

Pn

i=1 vi Ei .

9.1.3 The Bochner identity. Henceforth in these notes we will adopt the summation convention over repeated indices. Given a function f ∈ C ∞ (M) we define its gradient as the vector field ∇ f ∈ X(M) defined by g(∇ f ,V ) = d f (V ),

∀V ∈ X(M),

where d f is the one-form on M defined by d f (V ) = V f . Henceforth, given two tangent vectors v, w ∈ Tx M, we adopt the convenient notation hv, wi for the inner product gx (v, w). We have the following celebrated identity. Proposition 9.5 (Bochner’s identity). Let f ∈ C 3 (M), then ∆(|∇ f | 2 ) = 2||∇2 f || 2 + 2h∇ f , ∇(∆ f )i + 2 Ric(∇ f , ∇ f ).

(9.4)

Proof. The proof follows easily from the following commutation identity known as a special case of Ricci’s formulas (see, e.g., [CLN06, Lemma 1.36]), ∆∇i f = ∇i ∆ f + Ri j ∇ j f .

(9.5)

We also observe that for a function F we have ∆F 2 = 2F∆F + 2|∇F | 2 . From this formula and from (9.5) (applied to F = ∇ j f ) we obtain ∆|∇ f | 2 =

n X j=1

∆(∇ j f ) 2 = 2

n X

∇ j f ∆(∇ j f ) + 2

n X

j=1

= 2∇ j f ∇ j (∆ f ) + 2∇ j f R j k ∇k f + 2

|∇(∇ j f )| 2

j=1 n X

|∇(∇ j f )| 2

j=1

= 2h∇ f , ∇(∆ f )i + 2 Ric(∇ f , ∇ f ) + 2||∇2 f || 2 .



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Nicola Garofalo

9.1.4 Laplacian comparison theorem. Let M be a complete, n-dimensional Riemannian manifold. If we fix a point x ∈ M it is natural to consider the distance function r (y) = dist(y, x). The function r (y) is Lipschitz continuous on M, and therefore differentiable at a.e. y ∈ M. Given the exponential map exp x : Tx M → M, for any given vector v ∈ Tx M, with ||v|| = gx (v, v) = 1, denote by γ(t) the unique geodesic starting at x along the direction v, i.e., γ(0) = x, γ 0 (0) = v. This geodesic is given by γ(t) = exp x (tv) for t > 0. When t is small, γ is the unique length-minimizing geodesic joining x and exp x (tv), and also d exp x t v : Tt v (Tx M) → Tγ(t ) M is an isomorphism. However, as t increases the property may no longer be true. We define t x = sup{t > 0 | γ is the unique length-minimizing geodesic joining x and γ(t)}. Definition 9.6. If t x < ∞, then γ(t x ) is called a cut point of x. The cut locus of x is defined as Cut(x) = {y ∈ M | y is a cut point of x}. The next result plays a key role in the original proof of Theorem 9.1. Theorem 9.7 (Laplacian comparison theorem). Let M be an n-dimensional complete Riemannian manifold with Ric(M) ≥ −(n − 1)κ 2

(κ ≥ 0),

and denote by N the n-dimensional simply connected space of constant sectional curvature −κ 2 . Let r M and r N respectively denote the geodesic distances from some fixed points in M and N. If x ∈ M is such that r M is differentiable at x, then for any y ∈ N at which r M (x) = r N (y), we have ∆M r M (x) ≤ ∆N r N (y).

(9.6)

In particular, if κ = 0, then N = Rn and r N (y) = |y|. Since in this case ∆N r N (y) =

n−1 , r N (y)

∆M r M (x) ≤

n−1 . r M (x)

we conclude from (9.6) that (9.7)

It is worth emphasizing at this moment that comparison theorems in Riemannian geometry are intimately connected to the existence of a rich supply of Jacobi fields, and therefore to the fact that the exponential mapping is a local diffeomorphism.

2 Hypoelliptic operators, etc.

221

9.1.5 The Li–Yau inequality. Let M be an n-dimensional complete Riemannian manifold with Laplace–Beltrami operator ∆. Throughout this subsection we assume that M has non-negative Ricci tensor, i.e., for any x ∈ M and any v ∈ Tx M, Ric(v, v) ≥ 0.

(9.8)

Let u > 0 be a solution of the heat equation on M, i.e., ∆u − ut = 0,

in M × (0, ∞).

Since u is positive we can consider the function f = log u. With h(t) = log t, and by means of the formula ∆(h ◦ u) = h 00 (u)|∇u| 2 + h 0 (u)∆u, we immediately recognize that f satisfies the equation ∆ f − ft =

∆u − ut |∇u| 2 − 2 = −|∇ f | 2 . u2 u

(9.9)

One has the following crucial result. Lemma 9.8. For any α ≥ 1, let F = t(|∇ f | 2 − α f t ). Then, ∆F − Ft ≥ −2h∇ f , ∇Fi +

2t (|∇ f | 2 − f t ) 2 − (|∇ f | 2 − α f t ). n

(9.10)

Proof. The proof of this lemma is based exclusively on the Bochner identity (9.4). For its details the reader should see the original paper of Li and Yau, or the upcoming book [BG16].  We can now establish the main result of Li and Yau. Proof of Theorem 9.1. Let f = log u as before, and let F have the same meaning as in Lemma 9.8. Since |∇u| 2 ut F −α = , 2 u t u proving the theorem is equivalent to showing that sup BR

α4 nα 2 F ≤C 2 + . t 2t (α − 1)R2

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Nicola Garofalo

Let ψ ∈ C ∞ ([0, ∞)) be such that 0 ≤ ψ ≤ 1, with ψ ≡ 1 on [0, 1] and ψ ≡ 0 on [2, ∞). We also request that there exist constants C1 ,C2 > 0 such that ψ 0 ≤ 0,

(ψ 0 ) 2 ≤ C2 . ψ

ψ 00 ≥ −C1 ,

With %(x) = dist(x,O), set φ(x) = ψ( %(x) R ), so that supp φ ⊂ B2R and φ ≡ 1 on B R . We want to apply the maximum principle to φF. The problem is that φ may fail to be smooth at the cut-locus of O ∈ M, but only Lipschitz continuous. Now suppose that φF attains its maximum at a point (x 0 ,t 0 ) ∈ B2R × [0,T]. We can assume that F (x 0 ,t 0 ) > 0, otherwise we would have F (x 0 ,t 0 ) ≤ 0, and therefore sup B R Ft ≤ 0, which would finish the proof. Notice that x 0 ∈ B2R , and 0 < t 0 ≤ T (note that if t 0 = 0, we would have F (x 0 ,t 0 ) = 0, against the hypothesis). We have two possibilities: (1) x 0 < Cut(O); (2) x 0 ∈ Cut(O). We sketch case (1) only. For discussion of case (2) we refer the reader to [LY86] or [BG16]. The function % is smooth in a sufficiently small neighborhood of x 0 , and then from calculus we must have at (x 0 ,t 0 ), 0 = ∇(φF) = F∇φ + φ∇F, 0 ≤ (φF)t = φFt , ∆(φF) ≤ 0. Henceforth, all computations will be performed at the point (x 0 ,t 0 ). We have ∇F = −F

∇φ , φ

and this gives 0 ≥ ∆(φF) = φ∆F + F∆φ + 2h∇φ, ∇Fi = φ∆F + F∆φ − 2F Now ∇φ =

ψ0 ∇%, R

|φ| 2 . φ

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2 Hypoelliptic operators, etc.

and so (ψ 0 ) 2 |φ| 2 C2 = |∇%| 2 ≤ 2 . φ ψ R2 R In order to estimate ∆F from below we are going to use Lemma 9.8 (which ultimately relies on the Bochner formula), whereas to estimate the term ∆φ from below the Laplacian comparison Theorem 9.7 plays a crucial role. We recall that, under the assumption Ric(M) ≥ 0, the latter gives ∆% ≤

n−1 . %

Since the support of ψ 0 is concentrated on the interval [1, 2], from the formula ∆φ =

ψ0 ψ 00 2 |∇%| + ∆%, R R2

we conclude that ∆φ ≥ −

C1 (n − 1)|ψ 0 | − . R2 R2

We thus find |φ| 2 φ C1 + 2C2 + (n − 1)|ψ 0 | F. ≥φ∆F − R2

0 ≥φ∆F + F∆φ − 2F

Using Lemma 9.8 we now obtain (

2t 0 ≥ φ −2h∇ f , ∇Fi + (|∇ f | 2 − f t ) 2 − (|∇ f | 2 − α f t ) + Ft n C1 + 2C2 + (n − 1)|ψ 0 | − F R2 ( ) 2t h∇ f , ∇φi 2 2 2 ≥φ 2 F + (|∇ f | − f t ) − (|∇ f | − α f t ) φ n C1 + 2C2 + (n − 1)|ψ 0 | − F, R2

)

since φFt = (φF)t ≥ 0 at (x 0 ,t 0 ). At this point we set for notational simplicity A=

C1 + 2C2 + (n − 1)max |ψ 0 | R

R2

.

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Nicola Garofalo

From this point on, after several clever algebraic manipulations one arrives at the inequality 2 nα 2 C2 t 0 ≥ − [1 + At](φF) + (φF) 2 − (φF) 2 nα 4(α − 1)R2 ( " #) 2 nα 2 C2 t =(φF) . (φF) − 1 + At + nα 2 4(α − 1)R2 This inequality implies that we must have at (x 0 ,t 0 ), # " nα 2 nα 2 C2 t φF ≤ . 1 + At + 2 4(α − 1)R2 Since (x 0 ,t 0 ) ∈ B2R × (0,T], for any x ∈ BR we must have ! nα 2 nα 2 nα 2 C2 + A+ T. (φF)(x,T ) ≤ 2 2 4(α − 1)R2 Since φ ≡ 1 on BR , recalling that A = sup BR

C 1 +2C 2 +(n−1)max |ψ 0 | R

R2

, we conclude that

! F (·,T ) nα 2 α 2 C5 α2 ≤ + 1 + . T 2T α−1 R2

(9.11)

By the arbitrariness of T > 0 we finally obtain the desired conclusion (9.2) in case (1). 

10 Heat semigroup approach to the Li–Yau inequality In this section we provide a completely different approach to Theorem 9.1 which is based solely on some monotonicity properties of the heat semigroup and its associated entropy. An interesting feature of this approach is that, ultimately, it relies only on the Bochner identity and makes no use of the delicate Laplacian comparison theorem. This is an attractive feature which, as we will see in these lectures, has the potential of being generalized to situations, such as that of sub-Riemannian manifolds, in which the exponential mapping is not a local diffeomorphism, and consequently there is not a rich supply of Jacobi fields.

10.1 The curvature-dimension inequality. Following Bakry and Émery (see [BE83]), with the Laplacian ∆ we can associate the two differential bilinear forms

2 Hypoelliptic operators, etc.

225

defined on the space of smooth functions f , g : M → R: Γ( f , g) =

 1 ∆( f g) − f ∆g − g∆ f = h∇ f , ∇gi 2

(10.1)

and

 1 ∆Γ( f , g) − Γ( f , ∆g) − Γ(g, ∆ f ) . 2 Henceforth, we will simply write Γ2 ( f , g) =

Γ( f ) = Γ( f , f ) = |∇ f | 2 , Γ2 ( f ) = Γ2 ( f , f ) = =

1 ∆|∇ f | 2 − h∇ f , ∇(∆ f )i 2

(10.2)

(10.3)

(10.4)

1 ∆Γ( f ) − Γ( f , ∆ f ). 2

By means of the forms Γ and Γ2 we now introduce the following definition. Definition 10.1. We say that a Riemannian manifold (M, g) satisfies the curvaturedimension inequality CD( %, n), for % ∈ R and n > 0, if for every f ∈ C ∞ (M) one has 1 (10.5) Γ2 ( f ) ≥ (∆ f ) 2 + %Γ( f ), f ∈ C ∞ (M). n It is remarkable that the curvature-dimension inequality (10.5) perfectly captures the notion of the Ricci curvature lower bound. This will be evident after we state the following result. Theorem 10.2. On a complete n-dimensional Riemannian manifold M the inequality CD( %, n) is equivalent to Ric(M) ≥ %. Proof. We begin with proving the simple direction, namely that Ric(M) ≥ % =⇒ CD( %, n). For this direction, in fact, we do not need M to be complete. We apply Bochner’s formula (9.4), which we can rewrite as ∆Γ( f ) = 2||∇2 f || 2 + 2Γ( f , ∆ f ) + 2 Ric(∇ f , ∇ f ).

(10.6)

Using (10.4) one obtains from (10.6), Γ2 ( f ) = k∇2 f k 2 + Ric(∇ f , ∇ f ). The Schwarz inequality now gives k∇2 f k 2 ≥

1 (∆ f ) 2 . n

(10.7)

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Nicola Garofalo

If we assume that the Ricci tensor on M is bounded from below by % ∈ R, then we have in particular for any f ∈ C ∞ (M), Ric(∇ f , ∇ f ) ≥ %|∇ f | 2 = %Γ( f ). Using the latter two inequalities in (10.7) we conclude that the curvature-dimension inequality CD( %, n) holds. To prove the opposite direction, we assume that CD( %, n) holds and we want to show that, given x ∈ M, for any v ∈ Tx M we have Ric(v, v) ≥ %|v| 2 . We can find a sufficiently small neighborhood of x, Ux ⊂ M, and a function f ∈ C ∞ (Ux ) such that ∇ f (x) = v and ∇2 f (x) = 0. By suitably modifying the function f outside Ux we can assume that it is smooth in all of M. From the Bochner identity written in the form (10.7), and the choice of f , we obtain Ric(v, v) = Ric(∇ f (x), ∇ f (x)) = Γ2 ( f )(x) − 2||∇2 f (x)|| 2 1 ≥ (by CD( %, n)) ∆ f (x) 2 + %Γ( f )(x) − 2||∇2 f (x)|| 2 n 2 ≥ %|∇ f (x)| = %|v| 2 . We conclude that Ric(M) ≥ %.



10.2 The main variational inequality. If we fix T > 0 and for a function g ∈ C0∞ (M) we consider

v(x,t) = PT −t g(x),

then it should be clear that v solves the backward Cauchy problem   in M × (−∞,T ),  ∆v + vt = 0  v(x,T ) = g(x), x ∈ M. 

(10.8)

In what follows we consider a fixed x ∈ M and a fixed T > 0. For f ∈ C0∞ (M), f ≥ 0, we introduce the functionals Ψ(t) = Pt (PT −t f ln PT −t f )(x),

0≤t 0 we have for every α > 1, Γ(ln Pt f )(x) − ∂t (ln Pt f )(x) ≤

nα 2 (10.18) 8t(α − 1) (  )  n %t 2t +% −1 − ∂t (ln Pt f )(x) . 2 α+1 α+1

Proof. Let x ∈ M and T > 0 be arbitrarily fixed. Consider the entropy functional Φ(t) defined in (10.10). For any non-negative C 1 function a on the interval [0,T] we choose the function γ in a such a way that the coefficient of Φ(t) in (10.16) vanishes, i.e., 4aγ a0 − + 2%a = 0. n Solving for γ in this equation we obtain ! n a0 γ= + 2% . 4 a

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2 Hypoelliptic operators, etc.

With this choice of γ the inequality (10.16) thus reduces to (aΦ) 0 (t) ≥

4a(t)γ(t) 2a(t)γ 2 (t) ∆PT f (x) − PT f (x). n n

(10.19)

Now suppose that a is such that a(0) = 1 and a(T ) = 0 (recall that γ is determined by the above equation). Integrating (10.19) on [0,T], and keeping (10.12) in mind, after simple algebraic manipulations we find Z Z ∆PT f (x) T 0 n T (a 0 (t) + 2%a(t)) 2 dt − (a (t) + 2%a(t))dt Γ(ln PT f )(x) ≤ 8 0 a(t) PT f (x) 0 (10.20) ! Z T Z T 0 ∆PT f (x) n (a (t) + 2%a(t)) 2 = 1 − 2% a(t)dt + dt. PT f (x) 8 0 a(t) 0 In what follows it will prove useful to linearize the functional inequality (10.20) with respect to the function a. With this objective in mind we write a = V 2 , obtaining from (10.20), ! Z T ∆PT f (x) V 2 dt − 1 Γ(ln PT f )(x) + 2% (10.21) PT f (x) 0 ! Z T Z T n 0 2 2 2 ≤ (V ) dt + % V dt − % . 2 0 0 We now choose a = V 2 as  t α , a(t) = 1 − T

α > 1.

Clearly, a ∈ C 1 ([0,T]), and a(0) = 1, a(T ) = 0. Noting that such a choice gives Z T Z T T α2 2 V dt = , (V 0 ) 2 dt = , α+1 4(α − 1)T 0 0 from inequality (10.21) we obtain ! 2%T ∆PT f (x) α + 1 PT f (x) ! n %T nα 2 + % −1 + . 2 α+1 8T (α − 1)

Γ(ln PT f )(x) ≤ 1 −

By the arbitrariness of T > 0 and the fact that ∆Pt f (x) = ∂t Pt f (x), we reach the desired conclusion (10.18). 

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We can now establish the celebrated Li–Yau inequality. Corollary 10.6. Let M be a complete n-dimensional Riemannian manifold with Ric(M) ≥ %, % ∈ R. For any f ∈ C0∞ (M), f ≥ 0, x ∈ M, and t > 0, we have (  )  2t n %t n +% − 1 − ∂t (ln Pt f )(x) . Γ(ln Pt f )(x) − ∂t (ln Pt f )(x) ≤ 2t 2 3 3 (10.22) In particular, when % = 0 we obtain |∇(ln Pt f )(x)| 2 −

∂ n (ln Pt f )(x) ≤ . ∂t 2t

Proof. It is enough to take α = 2 in (10.18).

(10.23) 

A fundamental consequence of Corollary 10.6 is the following Harnack inequality for the heat kernel. In its proof we will need the following basic lemma, which is a standard fact in Riemannian geometry. Lemma 10.7. Let M be a complete Riemannian manifold. There exists a nondecreasing sequence hk ∈ C0∞ (M) such that 0 ≤ hk % 1 on M, and ||∇hk || L ∞ (M) → 0 as k → ∞. Proof. See for instance [GW79].



Theorem 10.8 (Harnack inequality). Let M be an n-dimensional complete Riemannian manifold with Ric(M) ≥ %, where % ≤ 0, and denote by p(x, y,t) the heat kernel on it. For every x, y, z ∈ M and every 0 < s < t < ∞ one has ! !  t  n/2 d(y, z) 2 2| %|t 3n| %|(t − s) exp 1+ + . p(x, y, s) ≤ p(x, z,t) s 4(t − s) 3 4 Proof. For a fixed x ∈ M, consider the function p(x, ·, τ) ∈ C ∞ (M). If hk is the sequence in Lemma 10.7, the functions f k = hk p(x, ·, τ) are non-negative and in C0∞ (M). We now set uk (y,t) = log Pt f k (y). For such functions we have from Corollary 10.6, ( 2 ) n% t n| %| 2| %|t n Γ(uk ) − ∂t uk ≤ + + + ∂t uk . 2t 6 2 3 We rewrite this inequality in the following way: ! 2| %|t n n%2 t n| %| − 1+ ∂t uk ≤ −Γ(uk ) + + + . 3 2t 6 2

(10.24)

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2 Hypoelliptic operators, etc.

We now fix two points y, z ∈ M, and fix 0 < s < t < ∞. Since d(y, z) = inf{T > 0 |there exists a C 1 path γ : [0,T] → M, such |γ 0 | ≤ 1, γ(0) = z, γ(T ) = y}, we pick such a path γ : [0,T] → M, and define   τ α(τ) = γ(τ),t + (s − t) , T

0 ≤ τ ≤ T.

Notice that α(0) = (z,t),

α(T ) = (y, s).

We let φ(τ) = uk (α(τ)), so that Z T Pt f k (y) d log = φ(T ) − φ(0) = φ(τ)dτ Ps f k (z) dτ 0 Z Z T t−s T 0 ∂t uk (α(τ))dτ = h∇uk (α(τ)), γ (τ)idτ − T 0 0 Z T Z t−s T ≤ |∇uk (α(τ))|dτ − ∂t uk (α(τ))dτ. T 0 0 Now, (10.24) gives − β(τ)∂t uk (α(τ)) ≤ −Γ(uk (α(τ))) + +

n%2 (t +

τ T

2(t +

(s − t))

6

+

n (s − t))

τ T

n| %| , 2

where we have let β(τ) = 1 +

2| %|(t +

τ T

(s − t))

3

! ,

0 ≤ τ ≤ T.

Keeping in mind that β is non-increasing, that β ≥ 1 on [0,T], and also that β(τ) ≥

2| %|(t +

τ T

3

(s − t))

,

we obtain from (10.25), −∂t uk (α(τ)) ≤ −

1 n 3n| %| Γ(uk (α(τ))) + + . τ β(0) 2(t + T (s − t)) 4

(10.25)

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We thus have log

Z T t−s |∇uk (α(τ))|dτ − |∇uk (α(τ))| 2 dτ T β(0) 0 0 Z t−s T t − s 3n| %| n + dτ + T τ T T 4 0 2(t + T (s − t)) Z Z T T t−s ε T 2 |∇uk (α(τ))| dτ + − |∇uk (α(τ))| 2 dτ ≤ 2 0 2ε T β(0) 0  t  3n| %|(t − s) n + log + . 2 s 4

Pt f k (y) ≤ Ps f k (z)

Z

T

At this point we choose ε > 0 such that t−s ε = . 2 T β(0) Recalling that β(0) = 1 +

2| % |t 3 ,

this gives

! T2 2| %|t Pt f k (y) ≤ 1+ log Ps f k (z) 4(t − s) 3  t  3n| %|(t − s) n + . + log 2 s 4 If we now take the infimum on all competing T > 0, and we let k → ∞, we obtain ! !  t  n/2 d(y, z) 2 2| %|t 3n| %|(t − s) p(x, y, s + τ) ≤ p(x, z,t + τ) exp 1+ + . s 4(t − s) 3 4 Letting τ → 0+ completes the proof.



Corollary 10.9. Let M be an n-dimensional complete Riemannian manifold with Ric(M) ≥ %, where % ≤ 0. For every x ∈ M and every 0 < r 0 < r < ∞ one has ! r n 3n| %|r 2 2 2 p(x, x,r 0 ) ≤ p(x, x,r ) exp . r0 4 Corollary 10.10 (Harnack inequality for the heat kernel). Let M be an n-dimensional complete Riemannian with Ric(M) ≥ 0, and denote by p(x, y,t) the heat kernel on it, i.e., the fundamental solution of the heat operator ∆ − ∂t on M × (0, ∞). For every x, y, z ∈ M and every 0 < s < t < ∞ one has !  t  n/2 d(y, z) 2 p(x, y, s) ≤ p(x, z,t) exp . s 4(t − s)

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233

11 A heat equation approach to the volume doubling property In Section 6 we saw that the so-called doubling condition (6.1) plays a pervasive role in the development of analysis. Let us recall the relevant framework. Consider a metric space (S, d) endowed with a Borel measure µ. Denote by B(x,r) the metric ball centered at x with radius r and let V (x,r) = µ(B(x,r)). The doubling condition postulates the existence of a constant C1 > 0 such that for every x ∈ S and every r > 0 one has V (x, 2r) ≤ C1V (x,r); see Hypothesis (6.1) in Section 6 and the ensuing developments. In Riemannian geometry the growth of the volume of the geodesic balls is intimately connected to lower bounds on the Ricci tensor. In the next subsection we discuss a fundamental result in this respect whose proof, like that of the Laplacian comparison theorem, is traditionally accomplished by means of the theory of Jacobi fields. We will instead present a different proof which uses only Theorem 10.4 above, and thus is ultimately based only on the Bochner identity.

11.1 The Bishop–Gromov volume comparison theorem. In what follows, given κ ∈ R, we will indicate with Mκ the simply connected space form of constant sectional curvature κ, and with Vκ (r) the volume of the geodesic ball Bκ (r) in Mκ . We recall (see, e.g., [Ch93, Section 3.4]) that   the sphere Sn of constant sectional curvature κ    Mκ =  Rn     the hyperbolic space Hn of constant sectional curvature κ  Furthermore, we have (see [Ch93, (3.25)]) Z Vκ (r) = σ n−1

r

Sκ (t) n−1 dt, 0

where σ n−1 =

2π n/2 Γ(n/2)

is the area of the unit sphere in Rn , and √  √1 sin κt, κ > 0,   κ    Sκ (t) = t, κ = 0,    √1 sinh √ −κt, κ < 0.  −κ

when κ > 0, when κ = 0, when κ < 0.

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Theorem 11.1 (Bishop–Gromov). Let M be a complete n-dimensional Riemannian manifold such that Ric(M) ≥ (n − 1)κ, κ ∈ R. Then, for every x ∈ M and every r > 0 the function V (x,r) r → Vκ (r) is non-increasing. Since we always have lim

r →0+

V (x,r) = 1, Vκ (r)

we also have the following volume growth estimate: V (x,r) ≤ Vκ (r),

x ∈ M, r > 0.

(11.1)

Corollary 11.2. Let M be a complete n-dimensional Riemannian manifold with Ric(M) ≥ 0. Then, for every x ∈ M and every r > 0 the function r →

V (x,r) rn

is non-increasing. As a consequence, one has x ∈ M, r > 0.

V (x, 2r) ≤ 2n V (x,r),

(11.2)

We also have the following maximum volume growth estimate: V (x,r) ≤ ω n r n ,

x ∈ M, r > 0.

(11.3)

Theorem 11.1 and Corollary 11.2 play an important role in studying the spectrum of the Laplace–Beltrami on a manifold, Gaussian bounds on the heat kernel, isoperimetric theorems, etc. In this subsection we intend to illustrate how a weaker form of Corollary 11.2 can be deduced from the entropy inequality in Theorem 10.4. This approach shows, in particular, that remarkably, (11.2) can be exclusively derived from the Bochner identity on functions without direct use of the theory of Jacobi fields. Recall the definition of the function V (x,r) = µ(B(x,r)), where now µ indicates the Riemannian volume. Here is the main result that we intend to establish. Theorem 11.3. Let M be a complete n-dimensional Riemannian manifold with Ric(M) ≥ 0. There exists a constant C(n) > 0 such that, for every x ∈ M and every r > 0, one has V (x, 2r) ≤ C(n)V (x,r),

x ∈ M, r > 0.

(11.4)

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2 Hypoelliptic operators, etc.

11.1.1 Preliminary reductions. From the semigroup property and the symmetry of the heat kernel we have for any y ∈ M and t > 0, Z p(y, z,t) 2 dµ(z). p(y, y, 2t) = M

√ Consider now a function h ∈ C0∞ (M) such that 0 ≤ h ≤ 1, h ≡ 1 on B(x, t/2), and √ h ≡ 0 outside B(x, t). We thus have Z Pt h(y) = p(y, z,t)h(z)dµ(z) M Z  1/2  1/2  Z  2 h(z) 2 dµ(z) ≤ √ p(y, z,t) dµ(z) B(x,

t) 1/2

≤ p(y, y, 2t)

V (x,

M

√ t)

1/2

.

By taking y = x, and t = r 2 in the latter inequality, we obtain  Pr 2 1 B(x,r ) (x) 2 ≤ Pr 2 h(x) 2 ≤ p(x, x, 2r 2 ) V (x,r). Applying Corollary 10.10 to (y,t) → p(x, y,t), for every y ∈ B(x,

(11.5) √

t) we find

p(x, x,t) ≤ C(n)p(x, y, 2t). √

t) with respect to the variable y gives Z √ p(x, x,t)V (x, t) ≤ C(n) √ p(x, y, 2t)dµ(y) ≤ C(n),

Integration over B(x,

B(x,

t)

where we have used Pt 1 ≤ 1. Letting t = 4r 2 , we obtain from this the on-diagonal upper bound C(n) V (x, 2r) ≤ . (11.6) p(x, x, 4r 2 ) At this point we combine (11.5) with (11.6) to obtain V (x,r) p(x, x, 2r 2 )  p(x, x, 4r 2 ) Pr 2 1 B(x,r ) (x) 2 V (x,r) ≤ C ∗ (n) ,  Pr 2 1 B(x,r ) (x) 2

V (x, 2r) ≤ C(n)

for every x ∈ M and every r > 0.

(11.7)

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Nicola Garofalo

11.1.2 The main estimate. From inequality (11.7) it is clear that we would be finished with the proof of (11.2) if we could show that there exist universal constants A ∈ (0, 1), and K > 0, such that one has for every x ∈ M and r > 0,  P Ar 2 1 B(x,r ) (x) ≥ K. Note: The Harnack inequality in Corollary 10.10 gives   Pr 2 1 B(x,r ) (x) ≥ C(n)P Ar 2 1 B(x,r ) (x). We thus turn to the central step in this section. Theorem 11.4. Let M be a complete n-dimensional Riemannian manifold such that Ric(M) ≥ 0. There exists an absolute constant 0 < A < 1, with A = A(n), such that with K = (1 − e−1 )/2, one has for every x ∈ M and every r > 0, P Ar 2 (1 B(x,r ) )(x) ≥ K.

(11.8)

Proof. We use Theorem 10.4 in which we choose a(t) = τ + T − t, n γ(t) = − , 4(τ + T − t) where τ > 0 will later be optimized. Noting that we presently have a(0) = τ + T, a(T ) = τ, and that a 0 ≡ −1,

n aγ ≡ − , 4



2aγ 2 n =− , n 8(τ + T − t)

from (10.16) in Theorem 10.4 we obtain τPT f Γ(ln f )) − (T + τ)PT f Γ(ln PT f ) n  T ≥ −T∆PT f − ln 1 + PT f . 8 τ We now apply (11.9) with f replaced by e λ f , where λ > 0, and f ∈ Lipschitz(M) ∩ L ∞ (M),

Γ( f ) ≤ 1.

(We will in fact choose in the end   −d(y, x), y ∈ B(x,r), f (y) =   −r, y ∈ B(x,r) c , 

(11.9)

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2 Hypoelliptic operators, etc.

x ∈ M, and r > 0 fixed.) Set ψ(λ,t) =

 1 log Pt e λ f , λ

or alternatively

 Pt e λ f = e λψ .

With this choice we obtain from (11.9),  λ 2 τPT e λ f Γ( f ) − λ 2 (T + τ)e λψ Γ(ψ)   n T ≥ −T∆PT e λ f − e λψ ln 1 + . 8 τ The assumption Γ( f ) ≤ 1 implies   PT e λ f Γ( f ) ≤ PT e λ f = e λψ . Using this observation in combination with the fact that ∆ Pt e λ f



=

 ∂e λψ ∂ ∂ψ = λe λψ , Pt e λ f = ∂t ∂t ∂t

and switching notation from T to t, we infer  λ  n t  ∂ψ ≥− τ + 2 ln 1 + . (11.10) ∂t t τ 8λ Now optimize the right-hand side of (11.10) with respect to τ. Since the maximum value of the right-hand side is attained at r ! t n τ0 = 1+ 2 −1 , 2 2λ t substituting this value into (11.10) we find r ! ∂ψ λ n − ≤ 1+ 2 −1 ∂t 2 2λ t ! ! n 2 1 + ln 1 + q = λG 2 , 8λt λ t 1 + 2λn2 t − 1

(11.11)

where we have set r

! * +/ n n 2 1 + s − 1 + s ln ..1 + q / , s > 0. 2 8 1 + n2 s − 1 , √ Notice that G(s) → 0 as s → 0+ , and that G(s)  s as s → +∞ (such behavior at infinity will be important). 1 G(s) = 2

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Nicola Garofalo

We now integrate the inequality (11.11) between s and t, obtaining ! Z t 1 ψ(λ, s) ≤ ψ(λ,t) + λ G 2 dτ. λ τ s Notice that Jensen’s inequality gives    λψ = log e λψ = log Pt e λ f ≥ Pt log e λ f = λPt f , and so we have Pt f ≤ ψ.

(11.12)

Using (11.12), we infer Ps (λ f ) ≤ λψ(λ,t) + λ

t

Z 2 s

! 1 G 2 dτ. λ τ

Letting s → 0+ we conclude that λ f ≤ λψ(λ,t) + λ 2

t

Z

G 0

! 1 dτ. λ2 τ

(11.13)

At this point consider the function   −d(y, x), d(y, x) < r, f (y) =   −r, d(x, y) ≥ r.  With B = B(x,r) = {x ∈ M | d(y, x) < r } we have e λ f ≤ e−λr 1 B c + 1 B . Therefore, for every t > 0 one has   e λψ (λ, t )(x) = Pt e λ f (x) ≤ e−λr + Pt 1 B (x). This gives the lower bound  Pt 1 B (x) ≥ e λψ (λ, t )(x) − e−λr . To estimate the first term in the right-hand side of the latter inequality, we use (11.13) which gives 1 = e λ f (x) ≤ e λψ (λ, t )(x) eΦ(λ, t ) , where we have set Φ(λ,t) = λ

t

Z 2 0

! 1 G 2 dτ. λ τ

2 Hypoelliptic operators, etc.

239

This gives Pt (1 B )(x) ≥ e−Φ(λ, t ) − e−λr . To make use of this estimate, we now choose λ = r1 , t = Ar 2 , obtaining 2

P Ar 2 (1 B )(x) ≥ e−Φ(1/r, Ar ) − e−1 . We want to show that we can choose A > 0 sufficiently small, depending only on n, and a K > 0 (we can in fact take K = (1 − e−1 )/2) such that 2

e−Φ(1/r, Ar ) − e−1 ≥ K,

for every x ∈ M,r > 0.

(11.14)

Consider the function ! Z Ar 2 Z ∞  1 1 r2 G(t) 2 G dτ = dt. Φ , Ar = 2 2 r τ r 0 A−1 t As noted above, a direct examination shows that G(t) = O(t 1/2 ) as t → +∞, and ) therefore that G(t ∈ L 1 (1, ∞). It is then clear that t2 Z ∞ G(t) dt = 0, lim+ A→0 −1 t2 A and therefore there exists A > 0 sufficiently small such that (11.14) holds with, say, K = (1 − e−1 )/2 > 0. 

12 A sub-Riemannian curvature-dimension inequality In the previous sections we have considered the situation of a smooth manifold M endowed with the Levi-Civita connection ∇ and we analyzed the deep link which exists between curvature, expressed as lower bounds on the Ricci tensor, and global properties of solutions of the corresponding heat semigroup. Inspired by the ideas presented so far we now introduce a generalization of the curvature-dimension inequality (10.5) in Definition 10.1 which can be successfully used in sub-Riemannian geometry. This final section should be seen as merely introductory to the lecture notes of Fabrice Baudoin in this same volume. To set the relevant framework we recall that on a smooth manifold M with a smooth measure µ, a second-order differential operator L is called locally subelliptic if for any Ω b M there exist C, ε such that the following subelliptic estimate holds for any u ∈ C0∞ (Ω): f p g kuk H 2ε (dµ) ≤ C k Γ(u)k L 2 (dµ) + kuk L 2 (dµ) , (12.1)

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Nicola Garofalo

where, in analogy with (10.1), (10.3) above, we have let g 1f Γ(u) = L(u2 ) − 2uLu . 2 In (12.1) we have denoted by H s (dµ) the Sobolev space of fractional order s (for a precise definition and properties of subelliptic operators see [FSC86] and [JSC87], besides of course Hörmander’s original paper [Ho67]). We assume that on M a second-order diffusion operator L with real coefficients is given. We also suppose that L is locally subelliptic, and that it satisfies Z M

L1 = 0, Z f Lg dµ = gL f dµ, M Z f L f dµ ≤ 0,

(12.2) (12.3) (12.4)

M

for every f , g ∈ C0∞ (M). In a local chart U ⊂ M with local coordinates x = (x 1 , . . . , x n ), the operator L can be written as L=

n X

n

σi j (x)

i, j=1

X ∂2 ∂ + bi (x) , ∂x i ∂x j i=1 ∂x i

(12.5)

where bi and σi j are smooth functions and the matrix (σi j (x))1≤i, j ≤n is symmetric and positive semidefinite. There is a natural gradient (or rather, a natural square of the length of a gradient) canonically associated with L, and it is given by the quadratic functional Γ( f ) = Γ( f , f ), where Γ( f , g) =

 1 L( f g) − f Lg − gL f , 2

f , g ∈ C ∞ (M).

(12.6)

The functional

 1 L( f 2 ) − 2 f L f (12.7) 2 is known as le carré du champ. Notice that (12.2) implies Γ(1) = 0. From (12.5) it is easily recognized that in the above local chart U, the differential bilinear form Γ is given by n X ∂f ∂f . Γ( f ) = σi j (x) ∂x i ∂x j i, j=1 Γ( f ) =

This shows that Γ( f ) ≥ 0 on M and that Γ( f ) only involves differentiation of order 1. It is also worth noting that, by definition, L( f 2 ) = 2 f L f + 2Γ( f ).

2 Hypoelliptic operators, etc.

241

Furthermore, if ϕ ∈ C 2 (R) we have L(ϕ ◦ f ) = ϕ 00 ◦ f Γ( f ) + ϕ 0 ◦ f L f , and therefore Γ(ϕ ◦ f ) = (ϕ 0 ◦ f ) 2 Γ( f ). If

n

n X

X ∂ ∂2 L= σi j (x) + bi (x) ∂x ∂x ∂x i j i i, j=1 i=1

and L=

n X

n

σ ˜ i j (x)

i, j=1

X ∂2 ∂ + b˜ i (x) ∂x i ∂x j i=1 ∂x i

denote local representation of L in the charts U and U 0 respectively, then we must have σi j (x) = σ ˜ i j (x), bi (x) = b˜ i (x), x ∈ U ∩ U 0 .

12.1 Canonical distances on M. With the operator L, one can associate several canonical distances. The most basic one hinges on the following definition, which is based on Definitions 4.1 and 4.3 in Section 4. Definition 12.1. A tangent vector v ∈ Tx M is called subunit for L at x if in a local representation (12.5) of L at x, one has vi v j ≤ σi j (x) as matrices, where Pn v = i=1 vi ∂x∂ i . This amounts to having for every ξ ∈ Rn , n X i=1

vi ξ i

2



n X

σi j (x)ξ i ξ j .

(12.8)

i, j=1

We emphasize that the notion of a subunit vector at x does not depend on the local representation (12.5) of L. Definition 12.2. A Lipschitz path γ : [0,T] → M is called subunit for L if γ 0 (t) is subunit for L at γ(t) at every point of differentiability of γ, and therefore at a.e. t ∈ [0,T]. For future use let us note one important consequence of (12.8). Suppose that γ : [0,T] → M is a subunit curve for L. Given a C 1 function u on M define ϕ(t) = u(γ(t)). If we fix a point t 0 ∈ [0,T] of differentiability for ϕ(t), then in a local chart around γ(t 0 ) we have from (12.8), s X n ∂u ∂u ∂u |ϕ 0 (t)| = (γ(t))γi0 (t) ≤ σi j (γ(t)) (γ(t)) (γ(t)). ∂x i ∂x i i=1 ∂x i

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Nicola Garofalo

This shows the following important estimate: d p u(γ(t)) ≤ Γ(u(γ(t))) dt

(12.9)

at a.e. every point t ∈ [0,T]. Given x, y ∈ M, we indicate with SL (x, y) = {γ : [0,T] → M | γ is subunit for L, γ(0) = x, γ(T ) = y}. Henceforth in this section we make the following basic connectivity assumption. Hypothesis 12.3. SL (x, y) , ∅, for every x, y ∈ M. For a given subunit path γ : [0,T] → M we define the subunit length of γ as ` s (γ) = T. Under Hypothesis (12.3) it is easy to verify that d L (x, y) = inf{` s (γ) | γ ∈ SL (x, y)}

(12.10)

defines a true distance on M. This distance is canonically associated with the diffusion operator L. We next introduce another distance on M which is canonically associated with the operator L. For a function g on M we let ||g||∞ = ess sup |g|. Given two points M

x, y ∈ M we define ( ) % L (x, y) = sup | f (x) − f (y)| | f ∈ C0∞ (M), kΓ( f )k∞ ≤ 1 ,

x, y ∈ M. (12.11)

Let us notice the following trivial, yet useful, fact: for every f ∈ C ∞ (M) such that ||Γ( f )||∞ < ∞, we have | f (x) − f (y)| ≤ ||Γ( f )||∞ % L (x, y),

x, y ∈ M.

(12.12)

We have the following basic result. Proposition 12.4. Under Hypothesis 12.3 the function % L in (12.11) defines a metric on M and for every x, y ∈ M we have d L (x, y) = % L (x, y). Henceforth, we assume Hypothesis 12.5. The metric space (M, d L ) is complete.

2 Hypoelliptic operators, etc.

243

12.2 An extrinsic “carré du champ”. As we have seen, given a diffusion operator L we can canonically associate with it a “carré du champ” given by (12.7). However, to introduce a satisfactory notion of Ricci lower bounds in sub-Riemannian geometry we need also to assume that there exists on M another symmetric, firstorder differential bilinear form Γ Z : C ∞ (M) × C ∞ (M) → R, satisfying Γ Z ( f g, h) = f Γ Z (g, h) + gΓ Z ( f , h). Setting Γ Z ( f ) = Γ Z ( f , f ), we notice that the latter assumption implies Γ Z (1) = 0. We assume that for every f ∈ C ∞ (M) one has Γ Z ( f ) ≥ 0. Besides Hypotheses 12.3 and 12.5 above, we will work with three general assumptions. The former two will be listed as Hypotheses 12.6 and 12.7 below, the third one will be introduced in Definition 12.9 below. Hypothesis 12.6. There exists an increasing sequence hk ∈ C0∞ (M) such that hk % 1 on M, and ||Γ(hk )||∞ + ||Γ Z (hk )||∞ → 0, as k → ∞. We will also assume that the following commutation relation is satisfied. Hypothesis 12.7. For any f ∈ C ∞ (M) one has Γ( f , Γ Z ( f )) = Γ Z ( f , Γ( f )). Let us note explicitly that when M is a complete Riemannian manifold, µ is the Riemannian volume on M, and L = ∆, then d(x, y) in (12.11) is equal to the Riemannian distance on M. In this situation if we take Γ Z ≡ 0, then Hypotheses 12.6 and 12.7 are fulfilled. In fact, Hypothesis 12.7 is trivially satisfied, whereas 12.6 is equivalent to assuming that (M, d) is a complete metric space, which we are doing in light of Hypothesis 12.5. Actually, more generally, in all the examples of [BG09, Chapter 2], Hypothesis 12.6 is equivalent to assuming that (M, d) is a complete metric space. Before we proceed with the discussion, we pause to stress that, in the generality in which we work the bilinear differential form Γ Z , unlike Γ, is not a priori canonical. Whereas Γ is determined once L is assigned, the form Γ Z in general is given extrinsically and not associated with L. However, in the geometric examples described in these lectures the choice of Γ Z will be natural and even canonical, up to a constant. This is the case, for instance, for the important example of CR Sasakian manifolds. The reader should think of Γ Z as an orthogonal complement of Γ: the bilinear form Γ represents the square of the length of the gradient in the horizontal directions, whereas Γ Z represents the square of the length of the gradient along the

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vertical directions. We also would like to mention at this point that the existence of differential forms Γ, Γ Z satisfying Hypotheses 12.6 and 12.7 does not necessarily imply that the sub-Riemannian structure has rank 2. Example 12.8. Consider in M = R2 the C ∞ vector fields X1 = ∂x , X2 = x k ∂y , with k ∈ N and k > 2. Then, the Martinet distribution ∆ = {X1 , X2 } is bracket generating of rank k, and yet the differential forms Γ( f , g) = ∂x f ∂x g + x 2k ∂y f ∂y g, Γ Z ( f , g) = ∂y f ∂y g fulfill Hypotheses 12.6 and 12.7. Given the sub-Laplacian L and the first-order bilinear forms Γ and Γ Z on M, we now introduce the following second-order differential forms:  1 LΓ( f , g) − Γ( f , Lg) − Γ(g, L f ) , 2  1 Γ2Z ( f , g) = LΓ Z ( f , g) − Γ Z ( f , Lg) − Γ Z (g, L f ) . 2 Γ2 ( f , g) =

(12.13) (12.14)

Observe that if Γ Z ≡ 0, then Γ2Z ≡ 0 as well. As for Γ and Γ Z , we will use the notation Γ2 ( f ) = Γ2 ( f , f ), Γ2Z ( f ) = Γ2Z ( f , f ).

12.3 The generalized curvature-dimension inequality. We are ready to introduce the central character of this section, a generalization of the above mentioned curvature-dimension inequality (10.5). Definition 12.9. We shall say that M satisfies the generalized curvature-dimension inequality CD( %1 , %2 , κ, d) with respect to L and Γ Z if there exist constants %1 ∈ R, %2 > 0, κ ≥ 0, and d > 0 such that the inequality Γ2 ( f ) +

νΓ2Z ( f )

 κ 1 2 Γ( f ) + %2 Γ Z ( f ) ≥ (L f ) + %1 − d ν

(12.15)

holds for every f ∈ C ∞ (M) and every ν > 0. The essential aspect of the theory developed in [BG09] and [BG15] is that, with Hypotheses 12.3, 12.5, 12.6, and 12.7 in place, all the results are solely deduced from the curvature-dimension inequality CD( %1 , %2 , κ, d) in (12.15). It is worth observing explicitly that, as we have mentioned above, the Riemannian case corresponds to the situation Γ Z = 0 and κ = 0 in (12.15). Hence in this case with L = ∆, and d = n = dimension of M, our (12.15) is nothing but the Riemannian curvature-dimension inequality CD( %1 , n) in (10.5) above. We also remark that, changing Γ Z into aΓ Z , where a > 0, changes the inequality CD( %1 , %2 , κ, d) into CD( %1 , a %2 , aκ, d). We express this fact by saying that the quantity %κ2 is intrinsic.

2 Hypoelliptic operators, etc.

245

Hereafter, when we say that M satisfies the curvature-dimension inequality CD( %1 , %2 , κ, d), we will routinely avoid repeating at each occurrence the sentence “for some %2 > 0, κ ≥ 0, and d > 0”. Instead, we will explicitly mention whether %1 = 0, or > 0, or simply %1 ∈ R. The reason for this is that the parameter %1 in inequality (12.15) has special relevance since in geometric examples it represents the lower bound on a sub-Riemannian generalization of the Ricci tensor. Thus, %1 = 0 is, in our framework, the counterpart of the Riemannian Ric(M) ≥ 0, whereas when %1 > 0 (< 0) we are dealing with the counterpart of the case Ric(M) > 0 (< 0).

12.4 The three-dimensional Sasakian models. The purpose of this subsection is providing a first basic non-Riemannian (in fact, genuinely sub-Riemannian) example which fits the framework of Definition 10.1. Given a number %1 ∈ R, we consider the three-dimensional Lie group G( %1 ) whose Lie algebra g has a basis {X,Y, Z } which, by hypothesis, satisfy (i) [X,Y ] = Z, (ii) [X, Z] = −%1Y , (iii) [Y, Z] = %1 X. It should be clear to the reader that assumptions (i)–(iii) imply that the Lie algebra is bracket generated by the two elements X and Y . If we identify elements of the Lie algebra g with vector fields on G( %1 ), then a sub-Laplacian on G( %1 ) is the left-invariant, second-order differential operator given by L = X 2 + Y 2.

(12.16)

In view of (i)–(iii), Hörmander’s Theorem 1.2 implies that L is hypoelliptic, although it fails to be elliptic at every point of G( %1 ). From (12.6) we find in the present situation, Γ( f ) =

 1 L( f 2 ) − 2 f L f = (X f ) 2 + (Y f ) 2 . 2

If we define Γ Z ( f , g) = Z f Zg, then from (i)–(iii) we easily verify that Γ( f , Γ Z ( f )) = Γ Z ( f , Γ( f )). We conclude that Hypothesis 12.7 is satisfied. It is not difficult to show that Hypothesis 12.6 is also fulfilled.

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Using (i)–(iii) we now leave it to the reader to verify that [L, Z] = 0.

(12.17)

By means of (12.17) we easily find that 1 L(Γ Z ( f )) − Γ Z ( f , L f ) = Z f [L, Z] f + (X Z f ) 2 + (Y Z f ) 2 2 = (X Z f ) 2 + (Y Z f ) 2 .

Γ2Z ( f ) =

Finally, from definition (12.13) and from (i)–(iii) we obtain  1 L Γ( f ) − Γ( f , L f ) 2 = %1 Γ( f ) + (X 2 f ) 2 + (Y X f ) 2 + (XY f ) 2 + (Y 2 f ) 2 + 2Y f (X Z f ) − 2X f (Y Z f ).

Γ2 ( f ) =

We now notice that if we define the symmetrized Hessian of f with respect to the horizontal distribution generated by X,Y in the following way: ! 1 X2 f (XY f + Y X f ) 2 2 ∇H f = 1 , Y2 f 2 (XY f + Y X f ) then 1 (X 2 f ) 2 + (Y X f ) 2 + (XY f ) 2 + (Y 2 f ) 2 = ||∇2H f || 2 + Γ Z ( f ). 2 Substituting this information into the above formula we find that  1 Γ2 ( f ) = ||∇2H f || 2 + %1 Γ( f ) + Γ Z ( f ) + 2 Y f (X Z f ) − X f (Y Z f ) . 2 By the above expression for Γ2Z ( f ), using the Cauchy–Schwarz inequality, we obtain, for every ν > 0, 1 |2Y f (X Z f ) − 2X f (Y Z f )| ≤ νΓ2Z ( f ) + Γ( f ). ν Similarly, one easily recognizes that ||∇2H f || 2 ≥

1 (L f ) 2 . 2

Combining these inequalities, we conclude that we have proved the following result.

247

2 Hypoelliptic operators, etc.

Proposition 12.10. For every %1 ∈ R the Lie group G( %1 ), with the sub-Laplacian L in (12.16), satisfies the curvature-dimension inequality CD( %1 , 12 , 1, 2). Precisely, for every f ∈ C ∞ (G( %1 )) and any ν > 0 one has ! 1 1 1 Γ2 ( f ) + νΓ2Z ( f ) ≥ (L f ) 2 + %1 − Γ( f ) + Γ Z ( f ). 2 ν 2 Proposition 12.10 provides basic motivation for Definition 12.9. It is also important to observe at this point that the Lie group G( %1 ) can be endowed with a natural CR structure. In fact, denoting by H the subbundle of TG( %1 ) generated by the vector fields X and Y , the endomorphism J of H, defined by J (Y ) = X,

J (X ) = −Y,

satisfies J 2 = −I, and thus defines a complex structure on G( %1 ). By choosing ϑ as the form such that Ker ϑ = H and dϑ(X,Y ) = 1, we obtain a CR structure on G( %1 ) whose Reeb vector field is −Z. Thus, the above choice of Γ Z is canonical. The pseudo-hermitian Tanaka–Webster torsion of G( %1 ) vanishes, and thus (G( %1 ), ϑ) is a Sasakian manifold. It is also easy to verify that for the CR manifold (G( %1 ), ϑ) the Tanaka–Webster horizontal sectional curvature is constant and equals %1 . For all the above notions from CR geometry we refer the reader to the monograph [DT06] .

12.5 The model spaces. In this final subsection we consider three model spaces which correspond to the Riemannian space forms of constant sectional curvature: the unit sphere, flat Rn , and hyperbolic space. These three model spaces correspond respectively to the cases %1 = 1, %1 = 0, and %1 = −1 of the Sasakian three manifolds discussed in Section 12.4. For more general situations the reader should see the papers [BG09, BG15], and also the subsequent lecture notes of Fabrice Baudoin in this volume. Example 12.11 (%1 = 1). The Lie group SU(2) is the group of 2 × 2, complex, unitary matrices of determinant 1. Its Lie algebra su(2) consists of 2 × 2, complex, skew-hermitian matrices with trace 0. A basis of su(2) is formed by the following matrices X = 2i σ1 , Y = 2i σ2 , Z = 2i σ3 , where σk , k = 1, 2, 3, are the Pauli matrices: X=

1 2

0 −1

1 0

! ,

Y=

1 2

0 i

i 0

! ,

Z=

1 2

i 0

0 −i

! .

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One easily verifies [X,Y ] = Z,

[X, Z] = −Y,

[Y, Z] = X,

(12.18)

and thus %1 = 1. Example 12.12 (%1 = 0). The Heisenberg group H is the group of 3 × 3 matrices 1 *. 0 , 0

x 1 0

z y +/ , x, y, z ∈ R. 1 -

The Lie algebra of H is spanned by the matrices 0 * . X= 0 , 0

1 0 0

0 0 0 +/ , Y = *. 0 0 , 0

0 0 0

0 0 1 +/ , Z = *. 0 0 , 0

0 0 0

1 0 +/ , 0 -

for which the following commutation relations hold: [X,Y ] = Z,

[X, Z] = [Y, Z] = 0.

We thus have %1 = 0 in this case. Example 12.13 (%1 = −1). The Lie group SL(2) is the group of 2 × 2, real matrices of determinant 1. Its Lie algebra sl(2) consists of 2 × 2 matrices of trace 0. A basis of sl(2) is formed by the matrices ! ! ! 1 1 1 0 1 1 0 0 1 X= ,Y= , Z= , 2 0 −1 2 1 0 2 −1 0 for which the following commutation relations hold: [X,Y ] = Z,

[X, Z] = Y,

[Y, Z] = −X.

(12.19)

We thus have %1 = −1 in this case.

Acknowledgments The author is deeply grateful to the organizers of the semester for the gracious invitation: Andrei Agrachev, Davide Barilari, Ugo Boscain, Yacine Chitour, Frédéric Jean, Ludovic Rifford, Mario Sigalotti. He also gratefully acknowledges the most gracious hospitality and the financial support of the Institut Henri Poincaré, the CNRS, and the ERC StG 2009 “GeCoMethods”, contract number 239748 (PI: Ugo Boscain).

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Chapter 3

Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations

Fabrice Baudoin1

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Riemannian foliations and their Laplacians . . . . . . . . . . . . . . . 3 Horizontal Laplacians and heat kernels on model spaces . . . . . . . . 4 Transverse Weitzenböck formulas . . . . . . . . . . . . . . . . . . . . 5 The horizontal heat semigroup . . . . . . . . . . . . . . . . . . . . . . 6 The horizontal Bonnet–Myers theorem . . . . . . . . . . . . . . . . . 7 Riemannian foliations and hypocoercivity . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

259 261 268 282 292 302 308 318

1 Introduction It is a fact that many interesting hypoelliptic diffusion operators may be studied by introducing a well-chosen Riemannian foliation. In particular, several sub-Laplacians on sub-Riemannians manifolds often appear as horizontal Laplacians of a foliation and several of the Kolmogorov-type hypoelliptic diffusion operators which are used in the theory of kinetic equations appear as the sum of the vertical Laplacian of a foliation and of a first-order term. The goal of the present notes is to survey some geometric analysis tools to study this kind of diffusion operator. We specially would like to stress the importance of subelliptic Bochner-type identities in this framework and show how they can be used to deduce a variety of results ranging from topological information on a subRiemannian manifold to hypocoercive estimates and convergence to equilibrium for [email protected] Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, United States.

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kinetic Fokker–Planck equations. As an illustration of these methods we give a proof of a sub-Riemannian Bonnet–Myers-type compactness theorem (Section 6) and study a version of the Bakry–Émery criterion for Kolmogorov-type operators (Section 7). For the proof of the sub-Riemannian Bonnet–Myers theorem we adapt an approach developed in a joint program with Nicola Garofalo. The object of this program initiated in [13, 14] has been to propose a generalized curvature-dimension inequality that fits a number of interesting subelliptic situations including the ones considered in these notes. While some of them will be discussed here, the numerous applications of the generalized curvature-dimension inequality are beyond the scope of these notes and we will only give the relevant pointers to the literature. We focus here more on the Bonnet–Myers theorem and the geometric framework in which this curvature-dimension estimate is available. Concerning Section 7, most of the material is actually new, though the main ideas originate from [9]. These notes are organized as follows. Section 2 We introduce the concept of Riemannian foliation and define the horizontal and vertical Laplacians. Basic theorems like the Bérard-Bergery–Bourguignon commutation theorem will be proved. Section 3 We study in detail some examples of Riemannian foliations with totally geodesic leaves that can be seen as model spaces. Besides the Heisenberg group, these examples are associated to the Hopf fibrations on the sphere. We give explicit expressions for the radial parts of the horizontal and vertical Laplacians and for the horizontal heat kernels of these model spaces. Section 4 We prove a transverse Weitzenböck formula for the horizontal Laplacian of a Riemannian foliation with totally geodesic leaves. It is the main geometric analysis tool for the study of the horizontal Laplacian. As a first consequence of this Weitzenböck formula, we prove that if natural assumptions are satisfied, then the horizontal Laplacian satisfies the generalized curvature-dimension inequality. As a second consequence, we will prove sharp lower bounds for the first eigenvalue of the horizontal Laplacian. Section 5 In this section, we introduce the horizontal semigroup of a Riemannian foliation with totally geodesic leaves and discuss fundamental questions like essential self-adjointness for the horizontal Laplacian and stochastic completeness. We also prove Li–Yau gradient bounds for this horizontal semigroup. Section 6 By using semigroup methods, we prove a sub-Riemannian Bonnet–Myers theorem in the context of Riemannian foliations with totally geodesic leaves. Section 7 This last section is an introduction to the analysis of hypoelliptic Kolmogorov-type operators on Riemannian foliations. We mainly focus on the problem of convergence to equilibrium for the parabolic equation associated to the operator and

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 261

on methods to prove hypocoercive estimates. The example of the kinetic Fokker– Planck equation is given as an illustration.

2 Riemannian foliations and their Laplacians We review first some basic facts about the geometry of Riemannian foliations that will be needed in the sequel. In particular, we define the horizontal and vertical Laplacians on such foliations and show that they commute if the metric is bundlelike and the foliation totally geodesic. For further details about the geometry of Riemannian submersions we refer to [23, Chapter 9] and for more information about general Riemannian foliations, we refer to the book by Tondeur [49].

2.1 Riemannian submersions. Let (M, g) and (B, j) be smooth and connected Riemannian manifolds. Definition 2.1. A smooth surjective map π : (M, g) → (B, j) is called a Riemannian submersion if its derivative maps Tx π : Tx M → Tπ(x) B are orthogonal projections, i.e., for every x ∈ M, the map Tx π(Tx π) ∗ : Tp(x) B → Tp(x) B is the identity. Example 2.2 (Warped products). Let (M1 , g1 ) and (M2 , g2 ) be Riemannian manifolds and f be a smooth and positive function on M1 . Then the first projection (M1 × M2 , g1 ⊕ f g2 ) → (M1 , g1 ) is a Riemannian submersion. Example 2.3 (Quotient by an isometric action). Let (M, g) be a Riemannian manifold and G be a closed subgroup of the isometry group of (M, g). Assume that the projection map π from M to the quotient space M/G is a smooth submersion. Then there exists a unique Riemannian metric j on M/G such that π is a Riemannian submersion. If π is a Riemannian submersion and b ∈ B, the set π −1 ({b}) is called a fiber. For x ∈ M, V x = Ker(Tx π) is called the vertical space at x. The orthogonal complement of H x shall be denoted H x and will be referred to as the horizontal space at x. We have an orthogonal decomposition Tx M = H x ⊕ V x and a corresponding splitting of the metric g = gH ⊕ gV . The vertical distribution V is of course integrable since it is the tangent distribution to the fibers, but the horizontal distribution is in general not integrable. Actually, in

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all the situations we will consider the horizontal distribution is everywhere bracket generating in the sense that for every x ∈ M, Lie(H)(x) = Tx M. In that case it is natural to study the sub-Riemannian geometry of the triple (M, H, gH ). As we will see, many interesting examples of sub-Riemannian structures arise in this framework and this is really the situation which is interesting for us. We shall mainly be interested in submersion with totally geodesic fibers. Definition 2.4. A Riemannian submersion π : (M, g) → (B, j) is said to have totally geodesic fibers if for every b ∈ B, the set π −1 ({b}) is a totally geodesic submanifold of M. Example 2.5 (Quotient by an isometric action). Let (M, g) be a Riemannian manifold and G be a closed one-dimensional subgroup of the isometry group of (M, g) which is generated by a complete Killing vector field X. Assume that the projection map π from M to M/G is a smooth submersion. Then the fibers are totally geodesic if and only if the integral curves of X are geodesics, which is the case if and only if X has a constant length. Example 2.6 (Principal bundle). Let M be a principal bundle over B with fiber F and structure group G. Then, given a Riemannian metric j on B, a G-invariant metric k on F and a G connection form ϑ, there exists a unique Riemannian metric g on M such that the bundle projection map π : M → B is a Riemannian submersion with totally geodesic fibers isometric to (F, k) and such that the horizontal distribution of ϑ is the orthogonal complement of the vertical distribution. We refer to [51, page 78] for a proof. In the case of the tangent bundle of a Riemannian manifold, the construction yields the Sasaki metric on the tangent bundle. As we will see, for a Riemannian submersion with totally geodesic fibers, all the fibers are isometric. The argument, due to Hermann [38] relies on the notion of a basic vector field that we now introduce. Let π : (M, g) → (B, j) be a Riemannian submersion. A vector field X ∈ Γ∞ (TM) is said to be projectable if there exists a smooth vector field X on B such that for every x ∈ M, Tx π(X (x)) = X (π(x)). In that case, we say that X and X are π-related. Definition 2.7. A vector field X on M is called basic if it is projectable and horizontal. If X is a smooth vector field on B, then there exists a unique basic vector field X on M which is π-related to X. This vector is called the lift of X. Notice that if X is a basic vector field and Z is a vertical vector field, then Tx π([X, Z](x)) = 0 and thus [X, Z] is a vertical vector field. The following result is due to Hermann [38].

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 263

Proposition 2.8. The submersion π has totally geodesic fibers if and only if the flow generated by any basic vector field induces an isometry between the fibers. Proof. We denote by D the Levi-Civita connection on M. Let X be a basic vector field. If Z1 , Z2 are vertical fields, the Lie derivative of g with respect to X can be computed as (L X g)(Z1 , Z2 ) = hD Z 1 X, Z2 i + hD Z 2 X, Z1 i. Because X is orthogonal to Z2 , we now have hD Z 1 X, Z2 i = −hX, D Z 1 Z2 i. Similarly hD Z 2 X, Z1 i = −hX, D Z 2 Z1 i. We deduce (L X g)(Z1 , Z2 ) = −hX, D Z 1 Z2 + D Z 2 Z1 i = −2hX, D Z 1 Z2 i. Thus the flow generated by any basic vector field induces an isometry between the fibers if and only if D Z 1 Z2 is always vertical, which is equivalent to the fact that the fibers are totally geodesic submanifolds. 

2.2 The horizontal and vertical Laplacians. Let π : (M, g) → (B, j) be a Riemannian submersion. If f ∈ C ∞ (M) we define its vertical gradient ∇V as the projection of its gradient onto the vertical distribution and its horizontal gradient ∇H as the projection of the gradient onto the horizontal distribution. We then define the vertical Laplacian ∆V as the generator of the Dirichlet form Z EV ( f , g) = − h∇V f , ∇V gidµ, M

where µ is the Riemannian volume measure on M. Similarly, we define the horizonal Laplacian ∆H as the generator of the Dirichlet form Z EH ( f , g) = − h∇H f , ∇V gidµ. M

If X1 , . . . , X n is a local orthonormal frame of basic vector fields and Z1 , . . . , Z m a local orthonormal frame of the vertical distribution, then we have ∆H = −

n X

X i∗ X i

i=1

and ∆V = −

m X i=1

Zi∗ Zi ,

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where the adjoints are understood in L 2 (µ). Classically, we have X i∗

= −X i +

n X

hD X k X k , X i i +

m X

hD Z k Zk , X i i,

k=1

k=1

where D is the Levi-Civita connection. As a consequence, we obtain ∆H =

n X

X i2 −

i=1

n X

(D X i X i )H −

i=1

m X

(D Z i Zi )H ,

i=1

where (·)H denotes the horizontal part of the vector. In a similar way we obviously have m n m X X X ∆V = Zi2 − (D X i X i )V − (D Z i Zi )V . i=1

i=1

i=1

We can observe that the Laplace–Beltrami operator ∆ of M can be written ∆ = ∆H + ∆V . It is worth noting that, in general, ∆H is not the lift of the Laplace–Beltrami operator ∆B on B. Indeed, let us denote by X 1 , . . . , X n the vector fields on B which are π-related to X1 , . . . , X n . We have ∆B =

n X

2

Xi −

i=1

n X

DX i X i .

i=1

Since it is easy to check that D X i X i is π-related to (D X i X i )H , we deduce that ∆H lies above ∆B , i.e., for every f ∈ C ∞ (B), ∆H ( f ◦ π) = (∆B f ) ◦ π if and only if the vector m X T= D Z i Zi i=1

is vertical. This condition is equivalent to the fact that the mean curvature of each fiber is zero, or in other words that the fibers are minimal submanifolds of M. This happens for instance for submersions with totally geodesic fibers. We also note that from Hörmander’s theorem, the operator ∆H is subelliptic if the horizontal distribution is bracket generating. Of course, the vertical Laplacian is never subelliptic because the vertical distribution is always integrable. The following result, though simple, will turn out to be extremely useful in the sequel when dealing with curvature-dimension estimates and functional inequalities. Theorem 2.9. The Riemannian submersion π has totally geodesic fibers if and only if for every f ∈ C ∞ (M), h∇H f , ∇H k∇V f k 2 i = h∇V f , ∇V k∇H f k 2 i.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 265

Proof. If X1 , . . . , X n is a local orthonormal frame of basic vector fields and Z1 , . . . , Z m a local orthonormal frame of the vertical distribution, then we easily compute that h∇H f , ∇H k∇V f k 2 i − h∇V f , ∇V k∇H f k 2 i = 2

n X m X

(X i f )(Z j f )([X i , Z j ] f ).

i=1 j=1

As a consequence, h∇H f , ∇H k∇V f k 2 i = h∇V f , ∇V k∇H f k 2 i if and only if for every basic vector field X, m X

(Z j f )([X, Z j ] f ) = 0.

j=1

This condition is equivalent to the fact that the flow generated by X induces an isometry between the fibers, and so from Hermann’s Proposition 2.8 this is equivalent to the fact that the fibers are totally geodesic.  The second commutation result that characterizes totally geodesic submersions is due to Bérard-Bergery and Bourguignon [22]. Theorem 2.10. The Riemannian submersion π has totally geodesic fibers if and only if any basic vector field X commutes with the vertical Laplacian ∆V . In particular, if π has totally geodesic fibers, then for every f ∈ C ∞ (M), ∆H ∆V f = ∆V ∆H f . Proof. Assume that the submersion is totally geodesic. Let X be a basic vector field and ξ t be the flow it generates. Since ξ induces an isometry between the fibers, we have ξ t∗ (∆V ) = ∆V . Differentiating at t = 0 yields [X, ∆V ] = 0. Conversely, assume that for every basic field X, [X, ∆V ] = 0. Let X1 , . . . , X n be a local orthonormal frame of basic vector fields and Z1 , . . . , Z m be a local orthonormal frame of the vertical distribution. The second-order part of the operator [X, ∆V ] must be zero. Given the expression of ∆V , this implies m X i=1

[X, Zi ]Zi = 0.

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So X leaves the symbol of ∆V invariant which is the metric on the vertical distribution. This implies that the flow generated by X induces isometries between the fibers. Finally, as we have seen, if the submersion is totally geodesic then in a local basic orthonormal frame n n X X X i2 − (D X i X i )H . ∆H = i=1

i=1

Since the vectors (D X i X i )H are basic, from the previous result ∆H commutes with ∆V . 

2.3 Riemannian foliations. In many interesting cases, we do not actually have a globally defined Riemannian submersion but a Riemannian foliation. Definition 2.11. Let M be a smooth and connected (n + m)-dimensional manifold. An m-dimensional foliation F on M is defined by a maximal collection of pairs {(Uα , πα ), α ∈ I} of open subsets Uα of M and submersions πα : Uα → Uα0 onto open subsets of Rn satisfying • ∪α ∈I Uα = M; • if Uα ∩ Uβ , ∅, there exists a local diffeomorphism Ψα β of Rn such that πα = Ψα β π β on Uα ∩ Uβ . The maps πα are called disintegrating maps of F . The connected components of the sets πα−1 (c), c ∈ Rn are called the plaques of the foliation. A foliation arises from an integrable subbundle of TM, to be denoted by V and referred to as the vertical distribution. These are the vectors tangent to the leaves, the maximal integral submanifolds of V . Foliations have been extensively studied and numerous books are devoted to them. We refer in particular to the book by Tondeur [49]. In the sequel, we shall be interested only in Riemannian foliations with bundlelike metric. Definition 2.12. Let M be a smooth and connected (n + m)-dimensional Riemannian manifold. An m-dimensional foliation F on M is said to be Riemannian with a bundle-like metric if the disintegrating maps πα are Riemannian submersions onto Uα0 with its given Riemannian structure. If moreover the leaves are totally geodesic submanifolds of M, then we say that the Riemannian foliation is totally geodesic with a bundle-like metric. Observe that if we have a Riemannian submersion π : (M, g) → (B, j), then M is equipped with a Riemannian foliation with bundle-like metric whose leaves are the fibers of the submersion. Of course, there are many Riemannian foliations with bundle-like metric that do not come from a Riemannian submersion.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 267

Example 2.13 (Contact manifolds). Let (M, ϑ) be a (2n + 1)-dimensional smooth contact manifold. On M there is a unique smooth vector field T, the so-called Reeb vector field, that satisfies ϑ(T ) = 1, LT (ϑ) = 0, where LT denotes the Lie derivative with respect to T. On M there is a foliation, the Reeb foliation, whose leaves are the orbits of the vector field T. As is well known (see for instance [47]), it is always possible to find a Riemannian metric g and a (1, 1)-tensor field J on M so that for every vector field X,Y, g(X,T ) = ϑ(X ),

J 2 (X ) = −X + ϑ(X )T,

g(X, JY ) = (dϑ)(X,Y ).

The triple (M, ϑ, g) is called a contact Riemannian manifold. We see then that the Reeb foliation is totally geodesic with bundle-like metric if and only if the Reeb vector field T is a Killing field, i.e.,

LT g = 0. In that case (M, ϑ, g) is called a K-contact Riemannian manifold. Example 2.14 (Sub-Riemannian manifolds with transverse symmetries). The concept of sub-Riemannian manifold with transverse symmetries was introduced in [14]. Let M be a smooth, connected manifold with dimension n + m. We assume that M is equipped with a bracket-generating distribution H of dimension n and a fiberwise inner product gH on that distribution. It is said that M is a sub-Riemannian manifold with transverse symmetries if there exists an m- dimensional Lie algebra V of sub-Riemannian Killing vector fields such that for every x ∈ M, Tx M = H (x) ⊕ V (x), where

V (x) = {Z (x), Z ∈ V (x)}. The choice of an inner product gV on the Lie algebra V naturally endows M with a Riemannian metric that makes the decomposition Tx M = H (x) ⊕ V (x) orthogonal: g = gH ⊕ gV . The subbundle of M determined by vector fields in V gives a foliation on M which is easily seen to be totally geodesic with bundle-like metric. Since Riemannian foliations with a bundle-like metric can locally be described by a Riemannian submersion, we can define a horizontal Laplacian ∆H and a vertical Laplacian ∆V . Observe that they commute on smooth functions if the foliation is totally geodesic. More generally, all the local properties of a Riemannian submersion extend to Riemannian foliations.

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3 Horizontal Laplacians and heat kernels on model spaces We discuss concrete examples of Riemannian foliations with totally geodesic leaves and bundle-like metric. We focus in particular on the study of horizontal Laplacians and of the corresponding heat kernels for which we show that explicit expressions can be given. The examples we cover are the Heisenberg group and the Hopf fibrations on the sphere. They can respectively be seen as the models of flat and positively curved sub-Riemannian spaces. The negatively curved sub-Riemannian spaces come from totally geodesic pseudo-Riemannian foliations on the anti-de Sitter space and for more details we refer to the thesis of Michel Bonnefont [24] and Jing Wang [54] and their papers [25] and [55]. Besides the Hopf fibrations, there are of course many other situations where sub-Riemannian heat kernels may computed more-or-less explicitly. We mention in particular the reference [1] which deals with the case of unimodular Lie groups.

3.1 Heisenberg group. One of the simplest non-trivial Riemannian submersions with totally geodesic fibers and bracket-generating horizontal distribution is associated to the Heisenberg group. The Heisenberg group is the set  H2n+1 = (x, y, z), x ∈ Rn , y ∈ Rn , z ∈ R endowed with the group law (x 1 , y1 , z1 ) ? (x 2 , y2 , z2 ) = (x 1 + x 2 , y1 + y2 , z1 + z2 + hx 1 , y2 iRn − hx 2 , y1 iRn ). The vector fields

∂ ∂ − yi , ∂x i ∂z ∂ ∂ Yi = + xi , ∂yi ∂z

Xi =

and

∂ ∂z form an orthonormal frame of left invariant vector fields for the left invariant metric on H2n+1 . Note that the following commutations hold: Z=

[X i ,Yj ] = 2δ i j Z,

[X i , Z] = [Yi , Z] = 0.

The map π:

H2n+1 → R2n , (x, y, z) → (x, y)

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 269

is then a Riemannian submersion with totally geodesic fibers. The horizontal Laplacian is the left invariant operator ∆H =

n X

(X i2 + Yi2 )

i=1

! n n 2 X X ∂2 ∂2 ∂ ∂ ∂ 2 2 ∂ = + + 2 x − y + (k xk + k yk ) i i ∂yi ∂x i ∂z ∂z 2 ∂x 2i ∂yi2 i=1 i=1 and the vertical Laplacian is the left invariant operator ∆V =

∂2 . ∂z 2

The horizontal distribution

H = span{X1 , . . . , X n ,Y1 , . . . ,Yn } is bracket generating at every point, so ∆H is a subelliptic operator. The operator ∆H is invariant by the action of the orthogonal group of R2n on the variables (x, y). Introducing the variable r 2 = k xk 2 + k yk 2 , we see then that the radial part of ∆H is given by 2n − 1 ∂ ∂2 ∂2 ∆H = 2 + + r2 2 . r ∂r ∂r ∂z This means that p if f : R ≥0 × R → R is a smooth map and % is the submersion (x, y, z) → ( k xk 2 + k yk 2 , z) then ∆H ( f ◦ %) = (∆H f ) ◦ %. From this invariance property in order to study the heat kernel and fundamental solution of ∆H at 0 it suffices to study the heat kernel and the fundamental solution of ∆H at 0. We denote by pt (r, z) the heat kernel at 0 of ∆H . It was first computed explicitly by Gaveau [33], building on previous works by Paul Lévy. Proposition 3.1. For r ≥ 0 and z ∈ R, !n Z 2 1 λ iλz pt (r, z) = e e−(λr /2) coth(2λt ) dλ. sinh(2λt) (2π) n+1 R Proof. Since see then that

∂ ∂z

commutes with ∆H , the idea is to use a Fourier transform in z. We Z 1 pt (r, z) = ei λ z Φt (r, λ)dλ, 2π R

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where Φt (r, z, λ) is the fundamental solution at 0 of the parabolic partial differential equation ∂Φ ∂ 2 Φ 2n − 1 ∂Φ = − λ 2 r 2 Φ. + ∂t r ∂r ∂r 2 We thus want to compute the semigroup generated by the Sch¨rodinger operator

Lλ =

∂2 2n − 1 ∂ + − λ 2r 2 . 2 r ∂r ∂r

The trick is now to observe that for every f ,  2  2 L λ e λr /2 f = e λr /2 (2nλ + G λ ) f , where ! 2n − 1 ∂ ∂2 . G λ = 2 + 2λr + r ∂r ∂r The operator G λ turns out to be the radial part of the Ornstein–Uhlenbeck operator ∆R2n + 2λhx, ∇R2n i whose heat kernel at 0 is a Gaussian density with mean 0 and 1 (e4λt − 1). This means that the heat kernel at 0 of G λ is given by variance 2λ qt (r) =

1 2λ (2π) n e4λt − 1

!n e−λr

2 /(e 4λ t −1)

.

We conclude that e2nλt 2λ Φt (r, z, λ) = (2π) n e4λt − 1

!n e−λr

2 /2

e−λr

2 /(e 4λ t −1)

. 

As a straightforward corollary, we deduce the heat kernel at 0 of ∆H . Corollary 3.2. The heat kernel at 0 of ∆H is 1 pt (x, y, z) = (2π) n+1

Z e R

iλz

λ sinh(2λt)

!n e−(λ ( k x k

2 + ky k 2 )/2) coth(2λt )

dλ.

Though it does not seem very explicit, this representation of the heat kernel has many applications and can be used to get very sharp estimates and small-time asymptotics (see [21] and [41, 42]).

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 271

3.2 The Hopf fibration. The second simplest and geometrically relevant example is given by the celebrated Hopf fibration. The horizontal heat kernel was first computed in [19]; we follow that, but simplify it, since the CR structure of the sphere is not relevant for us. Let us consider the odd-dimensional unit sphere S2n+1 = {z = (z1 , . . . , z n+1 ) ∈ Cn+1 , kzk = 1}. There is an isometric group action of S1 = U(1) on S2n+1 which is defined by (z1 , . . . , z n ) → (eiϑ z1 , . . . , eiϑ z n ). The generator of this action shall be denoted by T. We thus have for every f ∈ C ∞ (S2n+1 ), d T f (z) = f (eiϑ z) |ϑ=0 , dϑ so that ! n+1 X ∂ ∂ − zj . T =i zj ∂z j ∂z j j=1 The quotient space S2n+1 /U(1) is the projective complex space CPn and the projection map π : S2n+1 → CPn is a Riemannian submersion with totally geodesic fibers isometric to U(1). The fibration U(1) → S2n+1 → CPn is called the Hopf fibration. To study the geometry of the Hopf fibration, in particular the horizontal Laplacian ∆H , it is convenient to introduce a set of coordinates that reflects the action of the isometry group of CPn on S2n+1 . Let (w1 , . . . , wn , ϑ) be the local inhomogeneous coordinates for CPn given by w j = z j /z n+1 , and ϑ be the local fiber coordinate. That is, (w1 , . . . , wn ) parametrizes the complex lines passing through the north pole,2 while ϑ determines a point on the line that is of unit distance from the north pole. More explicitly, these coordinates are given by the map   (w, ϑ) −→ weiϑ cos r, eiϑ cos r , (3.1) qP n n 2 where r = arctan j=1 |w j | ∈ [0, π/2), ϑ ∈ R/2πZ, and w ∈ CP . In these coordinates, it is clear that T =

∂ ∂ϑ

and that the vertical Laplacian is ∆V =

∂2 . ∂ϑ2

2We will call north pole the point with complex coordinates z 1 = 0, . . . , z n+1 = 1.

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Our goal is now to compute the horizontal Laplacian ∆H . This operator is invariant by the action on the variables (w1 , . . . , wn ) of the group of isometries of CPn that fix the north pole of S2n+1 (this group is SU(n)). Therefore the heat kernel at the north pole depends only on the variables (r, ϑ) and can be computed through the heat kernel of the radial part ∆H of ∆H . Proposition 3.3. Consider the submersion %:

S2n+1 → [0, π/2) × R/2πZ, (w, ϑ) → (r, ϑ),

qP n 2 where we recall that r = arctan j=1 |w j | . Then for every smooth map f : [0, π/2) × R/2πZ → R, ∆H ( f ◦ %) = (∆H f ) ◦ %, where ∆H =

∂ ∂2 ∂2 2 + ((2n − 1) cot r − tan r) . + tan r ∂r ∂r 2 ∂ϑ2

Proof. The easiest route is to compute first the radial part of the Laplace–Beltrami operator ∆ and then to use the formula ∆H = ∆ − ∆V = ∆ −

∂ . ∂ϑ2

In our parametrization of S2n+1 we have z n+1 = eiϑ cos r. Therefore if δ1 denotes the Riemannian distance based at the north pole, we have cos δ1 = cos r cos ϑ and if δ2 denotes the Riemannian distance based at the point with real coordinates (0, . . . , 0, 1) then we have cos δ2 = cos r sin ϑ. The formula for the Laplace–Beltrami operator acting on functions depending on the Riemannian distance based at a point is well known and we deduce from it that ∆ acts on functions depending only on δ1 , δ2 as ∂2 ∂ ∂2 ∂ + 2n cot δ1 + 2 + 2n cot δ2 . 2 ∂δ1 ∂δ2 ∂δ2 ∂δ1 In the variables (r, ϑ) this last operator can be written as ∂ ∂2 1 ∂2 + ((2n − 1) cot r − tan r) + . ∂r cos2 r ∂ϑ2 ∂r 2

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 273

Thus, we conclude that ∆H =

∂2 ∂ 1 ∂2 ∂ + ((2n − 1) cot r − tan r) + − . 2 2 2 ∂r cos r ∂ϑ ∂r ∂ϑ2 

We can observe that ∆H is symmetric with respect to the measure dµ =

2π n (sin r) 2n−1 cos r dr dϑ, Γ(n)

where the normalization is chosen in such a way that Z π Z π/2 2π n+1 d µ = µ(S2n+1 ) = . Γ(n + 1) −π 0 As mentioned above, the heat kernel  at the north pole of ∆H depends only on (r, ϑ), that is, p weiϑ cos r, eiϑ cos r = pt (r, ϑ), where pt is the heat kernel at 0 of ∆H . Proposition 3.4. For t > 0, r ∈ [0, π/2), ϑ ∈ [−π, π], Γ(n) 2π n+1 ! +∞ X +∞ X m + |k | + n − 1 −λ m, k t+ikϑ n−1, |k | × (2m + |k | + n) e (cos r) |k | Pm (cos 2r), n − 1 k=−∞ m=0

pt (r, ϑ) =

where λ m,k = 4m(m + |k | + n) + 2|k |n and n−1, |k | (x) = Pm

dm (−1) m ((1 − x) n−1+m (1 + x) |k |+m ) 2m m!(1 − x) n−1 (1 + x) |k | dx m

is a Jacobi polynomial. Proof. Similarly to the Heisenberg group case, we observe that ∆H commutes with ∂ ∂ϑ , so the idea is to expand pt (r, ϑ) as a Fourier series in ϑ. We can write pt (r, ϑ) =

+∞ 1 X ikϑ e φk (t,r), 2π k=−∞

where φk is the fundamental solution at 0 of the parabolic equation ∂φk ∂φk ∂ 2 φk = + ((2n − 1) cot r − tan r) − k 2 tan2 r φk . ∂t ∂r ∂r 2

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By writing φk (t,r) in the form φk (t,r) = e−2n |k |t (cos r) |k | gk (t, cos 2r), we get

∂gk = 4Lk (gk ), ∂t

where

∂2 ∂ + [(|k | + 1 − n) − (|k | + 1 + n)x] . ∂x ∂x 2 The eigenvectors of Lk solve the Jacobi differential equation, and are thus given by the Jacobi polynomials

Lk = (1 − x 2 )

n−1, |k | Pm (x) =

dm (−1) m ((1 − x) n−1+m (1 + x) |k |+m ), 2m m!(1 − x) n−1 (1 + x) |k | dx m

which satisfy n−1, |k | n−1, |k | Lk (Pm )(x) = −m(m + n + |k |)Pm (x). n−1, |k | By using the fact that the family (Pm (x)(1 + x) |k |/2 )m ≥0 is an orthogonal 2 n−1 basis of L ([−1, 1], (1 − x) dx), such that Z 1 n−1, |k | Pm (x)Pln−1, |k | (x)(1 − x) n−1 (1 + x) |k | dx −1

=

2n+ |k | Γ(m + n)Γ(m + |k | + 1) δ ml , 2m + |k | + n Γ(m + 1)Γ(m + n + |k |)

we easily compute the fundamental solution of the operator

∂ ∂t

−4Lk and thus pt . 

Note that as a by-product of the previous result we obtain that the L 2 spectrum of −∆H is given by Sp(−∆H ) = {4m(m + k + n) + 2kn, k ∈ N, m ∈ N} .

(3.2)

We can give another representation of the heat kernel pt (r, ϑ) which is easier to ∂ commute, we handle analytically. The key idea is to observe that since ∆ and ∂ϑ formally have e t∆H = e−t (∂

2 /∂ϑ 2 )

e t∆ .

(3.3)

This gives a way to express the horizontal heat kernel in terms of the Riemannian one. Let us recall that the Riemannian heat kernel on the sphere S2n+1 is given by qt (cos δ) =

+∞ Γ(n) X n (cos δ), (m + n)e−m(m+2n)t Cm 2π n+1 m=0

(3.4)

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 275

where δ is the Riemannian distance based at the north pole and n Cm (x) =

Γ(m + 2n)Γ(n + 1/2) 1 dm (−1) m (1 − x 2 ) n+m−1/2 , 2m Γ(2n)Γ(m + 1)Γ(n + m + 1/2) (1 − x 2 ) n−1/2 dx m

is a Gegenbauer polynomial. Another expression of qt (cos δ) is !n 1 ∂ n2 t qt (cos δ) = e − V, (3.5) 2π sin δ ∂δ P −(δ−2k π) 2 /4t is a theta function. where V (t, δ) = √ 1 k ∈Z e 4πt Using the commutation (3.3) and the formula cos δ = cos r cos ϑ, we then infer the following proposition which is easy to prove (see [19] for the details). Proposition 3.5. For t > 0, r ∈ [0, π/2), ϑ ∈ [−π, π], Z +∞ 2 1 e−(y+iϑ) /4t qt (cos r cosh y)dy. pt (r, ϑ) = √ 4πt −∞

(3.6)

Applications of this formula are given in [19]. We can, in particular, deduce from it small asymptotics of the kernel when t → 0. Interestingly, these small-time asymptotics allow the sub-Riemannian distance to be computed explicitly. For a study of the distance and related geodesics, we refer to [26] and [44].

3.3 The quaternionic Hopf fibration. We study now a second example of Riemannian submersion with totally geodesic fibers and compact base: the quaternionic Hopf fibration. Up to exotic examples, the Hopf fibration and the quaternionic Hopf fibration are the only Riemannian submersions of the sphere with totally geodesic fibers (see [31]). The computation of the horizontal heat kernel was first done in [20]. Let H = {q = t + xI + y J + zK, (t, x, y, z) ∈ R4 } be the field of quaternions, where I, J, K are the Pauli matrices: ! ! ! i 0 0 1 0 i , J= , K= . I= 0 −i −1 0 i 0 The quaternionic norm is given by kqk 2 = t 2 + x 2 + y 2 + z 2 . Consider now the quaternionic unit sphere which is given by n+1   X S4n+3 = q = (q1 , . . . , qn+1 ) ∈ Hn+1 , kqi k 2 = 1 . i=1

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There is an isometric group action of the Lie group SU(2) on S4n+3 which is given by g · (q1 , . . . , qn+1 ) = (gq1 , . . . , gqn+1 ). The three generators of this action are given by ! n+1 X ∂f ∂f ∂f d ∂f Iϑ −x i , f (e q) |ϑ=0 = + ti − zi + yi dϑ ∂t i ∂x i ∂yi ∂z i i=1 ! n+1 X d ∂f ∂f ∂f ∂f f (e Jϑ q) |ϑ=0 = + zi + ti − xi , −yi dϑ ∂t i ∂x i ∂yi ∂z i i=1 and ! n+1 X d ∂f ∂f ∂f ∂f Kϑ f (e q) |ϑ=0 = − yi + xi + ti . −z i dϑ ∂t i ∂x i ∂yi ∂z i i=1 The quotient space S4n+3 /SU(2) is the projective quaternionic space HPn and the projection map π : S4n+3 → HPn is a Riemannian submersion with totally geodesic fibers isometric to SU(2). The fibration SU(2) → S4n+3 → HPn is called the quaternionic Hopf fibration. As for the classical Hopf fibration, the first task is to introduce a convenient set of coordinates. Let (w1 , . . . , wn ) be the local inhomogeneous coordinates for HPn −1 q and ϑ , ϑ , ϑ be the local exponential coordinates on the given by w j = qn+1 j 1 2 3 SU(2) fiber. We can locally parametrize S4n+3 by the coordinates   (w, ϑ1 , ϑ2 , ϑ3 ) −→ (cos r)e I ϑ1 +Jϑ2 +K ϑ3 w, (cos r)e I ϑ1 +Jϑ2 +K ϑ3 ,

(3.7)

qP n 2 where r = arctan j=1 |w j | . The horizontal Laplacian ∆H is invariant by the action on the variable w of the group of isometries of HPn that fix the north pole of S4n+3 and by the action on the variables ϑ1 , ϑ2 , ϑ3 of the group of isometries of SU(2) that fix the identity. Thus qP n 2 the heat kernel of ∆H depends only on the variables r = arctan j=1 |w j | and q η = ϑ21 + ϑ22 + ϑ23 . Observe that η is the distance based at the identity in SU(2) because sin η e I ϑ1 +Jϑ2 +K ϑ3 = cos η + (Iϑ1 + Jϑ2 + Kϑ3 ) . η

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 277

Proposition 3.6. Let us denote by % the submersion from S4n+3 to [0, π/2) × [0, π) such that   % (cos r)e I ϑ1 +Jϑ2 +K ϑ3 w, (cos r)e I ϑ1 +Jϑ2 +K ϑ3 = (r, η) , q qP n 2 and η = |w | ϑ21 + ϑ22 + ϑ23 . Then for every smooth where r = arctan j j=1 function f : [0, π/2) × [0, π) → R, ∆H ( f ◦ %) = (∆H f ) ◦ %,

∆V ( f ◦ %) = (∆V f ) ◦ %,

where ∆H

∂ ∂2 ∂ ∂2 + 2 cot η + tan2 r = 2 + ((4n − 1) cot r − 3 tan r) 2 ∂r ∂η ∂r ∂η

!

and ∆V =

∂ ∂2 + 2 cot η . 2 ∂η ∂η

Proof. The formula for ∆V is clear because SU(2) is isometric to the sphere S3 . For the horizontal Laplacian, the proof follows the same lines as the case of the classical Hopf fibration. Let δ1 be the distance based at the point (0, 1) ∈ Hn × H, δ2 be the distance based at the point (0, I) ∈ Hn × H, δ3 be the distance based at the point (0, J) ∈ Hn × H, and δ4 be the distance based at the point (0, K ) ∈ Hn × H. The Laplace–Beltrami operator ∆ acts on functions depending only on δ1 , δ2 , δ3 , δ4 as 4 2 X * ∂ + (4n + 2) cot δ i ∂ + . ∂δ i ∂δ2i i=1 , Observing now that p  cos r = cos2 δ1 + cos2 δ2 + cos2 δ3 + cos2 δ4 ,     p   cos2 δ2 + cos2 δ3 + cos2 δ4    tan η = cos δ1  finishes the proof after a simple, but tedious, change of variables.



As a consequence of the previous result, we can check that the Riemannian measure of S4n+3 in the coordinates (r, η), which is the symmetric measure for ∆H , is given by 8π 2n+1 dµ = (sin r) 4n−1 (cos r) 3 (sin η) 2 dr dη. Γ(2n) As before, we denote by pt the heat kernel at 0 of ∆H .

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Proposition 3.7. For t > 0, r ∈ [0, π2 ), η ∈ [0, π], pt (r, η) =

∞ +∞ X X

α k, m e−λ k, m t

m=0 k=0

sin(m + 1)η (cos r) m Pk2n−1, m+1 (cos 2r), sin η

(3.8)

where α k, m λ k, m

! k + m + 2n Γ(2n) = 2n+2 (2k + m + 2n + 1)(m + 1) , 2n − 1 2π = 4 [k (k + 2n + m + 1) + nm] ,

and Pk2n−1, m+1 (x) =

2k k!(1

 (−1) k dk  (1 − x) 2n−1+k (1 + x) m+1+k 2n−1 m+1 k − x) (1 + x) dx

is a Jacobi polynomial. Proof. The idea is to expand the subelliptic kernel in spherical harmonics as follows: pt (r, η) =

+∞ X sin(m + 1)η φ m (t,r) sin η m=0

∂ + 2 cot η ∂η which is associated ∂p t ˜ to the eigenvalue −m(m + 2). To determine φ m , we use ∂t = Lpt and find that

where

sin(m+1)η sin η

is the eigenfunction of ∆˜ SU (2) =

∂2 ∂η 2

∂ 2 φm ∂φ m ∂φ m = + ((4n − 1) cot r − 3 tan r) − m(m + 2) tan2 r φ m . 2 ∂t ∂r ∂r Let φ m (t,r) = e−4nmt (cos r) m ϕ m (t,r); then ϕ m (t,r) satisfies the equation ∂ϕ m ∂ 2 ϕm ∂ϕ m = + [(4n − 1) cot r − (2m + 3) tan r] . 2 ∂t ∂r ∂r We now change the variable and denote by ϕ m (t,r) = gm (t, cos 2r); then we have that gm (t, x) satisfies the equation ∂gm ∂ 2 gm ∂gm = 4(1 − x 2 ) + 4[(m + 2 − 2n) − (2n + m + 2)x] . 2 ∂t ∂x ∂x ∂ ∂ We define Ψm = (1 − x 2 ) ∂x 2 + [(m + 2 − 2n) − (2n + m + 2)x] ∂x , and find that 2

∂gm = 4Ψm (gm ). ∂t

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 279

The equation Ψm (gm ) + k (k + 2n + m + 1)gm = 0 is a Jacobi differential equation for all k ≥ 0. We denote the eigenvector of Ψm corresponding to the eigenvalue −k (k + 2n + m + 1) by Pk2n−1, m+1 (x); then it is known that Pk2n−1, m+1 (x) =

 (−1) k dk  2n−1+k m+1+k (1 − x) (1 + x) . 2k k!(1 − x) 2n−1 (1 + x) m+1 dx k

At the end we can therefore write the spectral decomposition as pt (r, η) =

+∞ X ∞ X

α k, m e−4[k (k+2n+m+1)+nm]t

m=0 k=0

sin(m + 1)η (cos r) m Pk2n−1, m+1 (cos 2r) sin η

where α k, m are determined by considering the initial condition. Note that (Pk2n−1, m+1 (x)(1 + x) (m+1)/2 )k ≥0 is an orthogonal basis of the Hilbert space L 2 ([−1, 1], (1 − x) 2n−1 dx); more precisely, Z

1 −1

Pk2n−1, m+1 (x)Pl2n−1, m+1 (x)(1 − x) 2n−1 (1 + x) m+1 dx =

22n+m+1 Γ(k + 2n)Γ(k + m + 2) δ kl . 2k + m + 2n + 1 Γ(k + 1)Γ(k + 2n + m + 1)

For a smooth function f (r, ϑ), we can write f (r, η) =

+∞ X +∞ X

bk, m

m=0 k=0

sin(m + 1)η 2n−1, m+1 Pk (cos 2r) · (cos r) m sin η

where the bk, m ’s are constants. We then obtain f (0, 0) =

+∞ X +∞ X

bk, m (m + 1)Pk2n−1, m+1 (1),

m=0 k=0

and we observe that Pk2n−1, m+1 (1) = cal coordinates by dµr =

2n−1+k  k

. The measure dµ is given in cylindri-

8π 2n+1 (sin r) 4n−1 (cos r) 3 (sin η) 2 dr dη. Γ(2n)

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Fabrice Baudoin

Moreover, since Z πZ π 2 pt (r, η) f (−r, −η)dµr 0

= =

0 +∞ X +∞ 2n+2 X 4π

Γ(2n)

m=0 k=0 +∞ X +∞ 2n+2 X 2π

Γ(2n)

m=0 k=0

α k, m bk, m e

−λ k, m t

Z π 2 * (cos r) 2m+3 |Pk2n−1, m+1 | 2 (sin r) 4n−1 dr + 0 ,

α k, m bk, m e−λ m, k t Γ(k + 2n)Γ(k + m + 2) , 2k + m + 2n + 1 Γ(k + 1)Γ(k + 2n + m + 1)

where λ k, m = 4k (k + 2n + m + 1) + nm, we obtain that Z πZ π 2 lim pt f dµr = f (0, 0) t→0

as soon as α k, m =

Γ(2n) (2k 2π 2n+2

0

0

+ m + 2n + 1)(m + 1)

 k+m+2n  2n−1

.



As a by-product of the spectral expansion of pt we obtain the spectrum of −∆H , Sp(−∆H ) = {4 [k (k + 2n + m + 1) + nm] , k ≥ 0, m ≥ 0}.

(3.9)

Comparing this expansion with the result we obtained in Proposition 3.4, we obtain a very nice formula relating pt to the horizontal kernel of the usual Hopf fibration. Proposition 3.8. Let pˆt (r, ϑ) be the radial horizontal kernel of the usual Hopf fibration S4n+1 → CP2n ; then for r ∈ [0, π2 ), ϑ ∈ [0, π], e4nt ∂ pˆt (r, ϑ). (3.10) 2π sin ϑ cos r ∂ϑ As in the case of the usual Hopf fibration, we can obtain an alternative representation pt (r, ϑ) which we derive from the decomposition ∆ = ∆H + ∆V . ∂2 ∂ We denote by qt (cos δ) the heat kernel at 0 of the operator ∂δ 2 + (4n +2) cot δ ∂δ . We recall that +∞ Γ(2n + 1) X 2n+1 qt (cos δ) = (m + 2n + 1)e−m(m+4n+2)t Cm (cos δ), (3.11) 2π 2n+2 m=0 pt (r, ϑ) = −

where δ is the Riemannian distance based at the north pole and 2n+1 Cm (x) =

(−1) m Γ(m + 4n + 2)Γ(2n + 3/2) 2m Γ(4n + 2)Γ(m + 1)Γ(2n + m + 3/2) 1 dm (1 − x 2 ) 2n+m+1/2 × (1 − x 2 ) 2n+1/2 dx m

is a Gegenbauer polynomial.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 281

If we define ∆SL(2) = ∂ 2 cot η ∂η ,

∂2 ∂η 2

∂ + 2 coth η ∂η , then from the fact that ∆V =

∂2 ∂η 2

+

it is not hard to see that pt (r, η) = (e t∆SL(2) f t )(r, −iη),

(3.12)

where f t (η) = qt (cos r cos η). Therefore, an integral representation of pt , can be ˜ obtained from an explicit expression of the heat semigroup e t ∆SL(2) . Lemma 3.9. Let ∆SL(2) =

∂2 ∂η 2

∂ + 2 coth η ∂η . For every f : R ≥0 → R in the domain

of ∆SL(2) , we have e−t (e t∆SL(2) f )(η) = √ πt

+∞

Z 0

sinh r sinh (ηr/(2t)) −(r 2 +η 2 )/(4t ) e f (r)dr, sinh η

(3.13)

t ≥ 0, η ≥ 0. Proof. Let us take a function f which is smooth and compactly supported on R ≥0 . We observe that 1 ∆SL(2) f = (∆R3 − 1)(h f ), h where 2 ∂ ∂2 sinh η + ∆ R3 = , h(η) = . 2 η ∂η η ∂η As a consequence, we have (e t∆SL(2) f )(η) =

e−t t∆ 3 e R (h f )(η). h(η)

We are thus left with the computation of e t∆R3 . The operator ∆R3 is the radial part of the Laplacian ∆R3 ; thus after a routine computation, for x ∈ R3 , Z +∞  ηr  2 2 1 r t∆R3 f (η) = √ sinh e−(r +η )/(4t ) f (r)dr. e η 2t πt 0  As a consequence, we get the integral representation of pt . Proposition 3.10. For t > 0, r ∈ [0, π/2), η ∈ [0, π], Z +∞ sinh y sin  ηy  2 2 e−t 2t pt (r, η) = √ e−(y −η )/(4t ) qt (cos r cosh y)dy. sin η πt 0 We refer to [20] for applications of this formula to the computation of the smalltime asymptotics of the kernel and, as a by-product, of the sub-Riemannian distance.

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4 Transverse Weitzenböck formulas In this section we establish a Weitzenböck formula for the horizontal Laplacian of a totally geodesic foliation. As a consequence, we prove a generalized curvaturedimension inequality for the horizontal Laplacian. In a joint program with Nicola Garofalo it has been proved in a very general and abstract framework that the generalized curvature-dimension inequality implies several results: • Li–Yau-type gradient bounds for the heat kernel and associated scale-invariant parabolic Harnack inequalities [14]; • upper and lower Gaussian bounds for the heat kernel [11, 12, 14]; • boundedness of the Riesz transform [15]; • Sobolev embeddings and isoperimetric inequalities [13, 16]; • log-Sobolev and transport inequalities [10]; • Bonnet–Myers-type compactness theorem [14]. We shall not discuss all of these applications here because it would go beyond the scope of these notes, but we will focus on the Bonnet–Myers compactness result in a later section. In the next section, we will also say some words about the Li–Yau estimates since they are a crucial ingredient in the proof of the Bonnet–Myers-type result. As a second application of the transverse Weitzenböck formula we obtain sharp lower bounds for the first eigenvalue of the horizontal Laplacian.

4.1 The Bott connection. Let M be a smooth, connected manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation F with bundle-like metric g and totally geodesic m-dimensional leaves. As usual, the subbundle V formed by vectors tangent to the leaves will be referred to as the set of vertical directions and the subbundle H which is normal to V will be referred to as the set of horizontal directions. The metric g can be split as g = gH ⊕ gV . We define the canonical variation of g as the one-parameter family of Riemannian metrics 1 gε = gH ⊕ gV , ε > 0. ε On the Riemannian manifold (M, g) there is the Levi-Civita connection that we denote by D, but this connection is not adapted to the study of foliations because the

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 283

horizontal and the vertical bundle may not be parallel. More adapted to the geometry of the foliation is the Bott connection that we now define. It is an easy exercise to check that there exists a unique affine connection ∇ such that • ∇ is metric, that is, ∇g = 0; • for X,Y ∈ Γ∞ (H), ∇ X Y ∈ Γ∞ (H); • for U,V ∈ Γ∞ (V ), ∇U V ∈ Γ∞ (V ); • for X,Y ∈ Γ∞ (H), T (X,Y ) ∈ Γ∞ (V ) and for U,V ∈ Γ∞ (V ), T (U,V ) ∈ Γ∞ (H), where T denotes the torsion tensor of ∇; • for X ∈ Γ∞ (H),U ∈ Γ∞ (V ), T (X,U) = 0. In terms of the Levi-Civita connection, the Bott connection can be written as  (D X Y )H ,       [X,Y ]H , ∇X Y =    [X,Y ]V ,      (D X Y )V ,

X,Y X∈ X∈ X,Y

∈ Γ∞ (H ), Γ∞ (V ),Y ∈ Γ∞ (H), Γ∞ (H),Y ∈ Γ∞ (V ), ∈ Γ∞ (V ),

where the subscript H (resp. V ) denotes the projection on H (resp. V ). Observe that for horizontal vector fields X,Y the torsion T (X,Y ) is given by T (X,Y ) = −[X,Y ]V . Also observe that for X,Y ∈ Γ∞ (V ) we actually have (D X Y )V = D X Y because the leaves are assumed to be totally geodesic. Finally, it is easy to check that for every ε > 0, the Bott connection satisfies ∇gε = 0. Example 4.1. Let (M, ϑ, g) be a K-contact Riemannian manifold. The Bott connection coincides with the Tanno connection that was introduced in [47] and which is the unique connection that satisfies 1. ∇ϑ = 0; 2. ∇T = 0; 3. ∇g = 0; 4. T (X,Y ) = dϑ(X,Y )T for any X,Y ∈ Γ∞ (H); 5. T (T, X ) = 0 for any vector field X ∈ Γ∞ (H).

284

Fabrice Baudoin

We now introduce some tensors and definitions that will play an important role in the sequel. For Z ∈ Γ∞ (TM), there is a unique skew-symmetric endomorphism JZ : H x → H x such that for all horizontal vector fields X and Y , gH (JZ (X ),Y ) = gV (Z,T (X,Y )),

(4.1)

where T is the torsion tensor of ∇. We then extend JZ to be 0 on V x . If Z1 , . . . , Z m Pm is a local vertical frame, the operator `=1 JZ ` JZ ` does not depend on the choice of the frame and shall concisely be denoted by J2 . For instance, if M is a K-contact manifold equipped with the Reeb foliation, then J is an almost complex structure, J2 = −IdH . The horizontal divergence of the torsion T is the (1, 1) tensor which is defined in a local horizontal frame X1 , . . . , X n by δHT (X ) = −

n X

(∇ X j T )(X j , X ),

X ∈ Γ∞ (M).

j=1

The g-adjoint of δHT will be denoted δHT ∗ . Definition 4.2. We say that the Riemannian foliation is of Yang–Mills type if δHT = 0. Example 4.3. Let (M, ϑ, g) be a K-contact Riemannian manifold. It is easy to see that the Reeb foliation is of Yang–Mills type if and only if δH dϑ = 0. Equivalently this condition can be written as δH J = 0. If M is a strongly pseudo-convex CR manifold with pseudo-Hermitian form ϑ, then the Tanno connection is the Tanaka– Webster connection. In that case, we have then ∇J = 0 (see [28]) and thus δH J = 0. CR manifolds of K-contact type are called Sasakian manifolds (see [28]). Thus the Reeb foliation on any Sasakian manifold is of Yang–Mills type. Example 4.4. Let (M, g) be a smooth Riemannian manifold.We endow the tangent bundle TM with the Sasaki metric so that the bundle projection π : TM → M is a Riemannian submersion with totally geodesic fibers. In that case the torsion of the Bott connection is given by T (X,Y ) = R(X,Y ),

X,Y ∈ Γ(H),

where R is the curvature of the connection form. By using the second Bianchi identity, the Yang–Mills condition is equivalent to the fact that the Ricci tensor of the connection form is a Codazzi tensor, that is, for any vector fields X,Y, Z in Γ∞ (H ), (∇ X Ric)(Y, Z ) = (∇Y Ric)(X, Z ).

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 285

In the sequel, we shall need to perform computations on one-forms. For that purpose we introduce some definitions and notation on the cotangent bundle. We say that a one-form is horizontal (resp. vertical) if it vanishes on the vertical bundle V (resp. on the horizontal bundle H). We thus have a splitting of the cotangent space Tx∗ M = H∗ (x) ⊕ V ∗ (x). The metric gε induces a metric on the cotangent bundle which we still denote gε . By using similar notation and conventions as before we have for every η in Tx∗ M, 2 kη kε2 = kη kH + εkη kV2 .

By using the duality given by the metric g, (1, 1) tensors can also be seen as linear maps on the cotangent bundle T ∗ M. More precisely, if A is a (1, 1) tensor, we will still denote by A the fiberwise linear map on the cotangent bundle which is defined as the g-adjoint of the dual map of A. The same convention will be used for any (r, s) tensor. We define the horizontal Ricci curvature Ric H as the fiberwise symmetric linear map on one-forms such that for all smooth functions f , g, hRic H (d f ), dgi = Ricci(∇H f , ∇H g), where Ricci is the Ricci curvature of the connection ∇. A simple computation (see for instance [23, Chapter 9, Theorem 9.70]) gives the following result for the Riemannian Ricci curvature of the metric gε . Lemma 4.5. Assume that the foliation is of Yang–Mills type. Let us denote by Ricciε the Ricci curvature tensor of the Levi-Civita connection of the metric gε and by RicciV the Ricci curvature of the leaves; then for every X ∈ Γ∞ (H) and Z ∈ Γ∞ (V ), Ricciε (Z, Z ) = RicciV (Z, Z ) +

1 Tr(JZ∗ JZ ), 4ε 2

Ricciε (X, Z ) = 0, Ricciε (X, X ) = RicciH (X, X ) −

1 kJX k 2 . 2ε

We explicitly note that Ricciε (X, Z ) = 0 is due to the fact that the foliation is assumed to be of Yang–Mills type. If V is a horizontal vector field and ε > 0, we consider the fiberwise linear map from the space of one-forms into itself which is given for η ∈ Γ∞ (T ∗ M) and Y ∈ Γ∞ (TM) by   ε1 η(JY V ), Y ∈ Γ∞ (V ), TVε η(Y ) =   −η(T (V,Y )), Y ∈ Γ∞ (H ). 

286

Fabrice Baudoin

We observe that TVε is skew-symmetric for the metric gε so that ∇−T ε is a gε -metric connection. If η is a one-form, we define the horizontal gradient of η in a local frame as the (0, 2) tensor n X ∇ Xi η ⊗ ϑi . ∇H η = i=1

We denote by ∇#H η the symmetrization of ∇H η. Similarly, we will use the notation ε TH η=

n X

T εX i η ⊗ ϑ i .

i=1

Finally, we will still denote by ∆H the covariant extension on one-forms of the horizontal Laplacian. In a local horizontal frame, we have thus ∆H = −∇∗H ∇H =

n X

∇ X i ∇ X i − ∇∇ X i X i .

i=1

4.2 Bochner–Weitzenböck formulas for the horizontal Laplacian. For ε > 0, we consider the following operator which is defined on one-forms by 1 1 ε ε ∗ ) − J2 + δHT − Ric H , ) (∇H − TH ε = −(∇H − TH ε ε where the adjoint is understood with respect to the metric gε . It is easily seen that, in a local horizontal frame, ε ∗ ε −(∇H − TH ) (∇H − TH )=

n X

(∇ X i − T εX i ) 2 − (∇∇ X i X i − T∇ε X

i=1

i

X i ).

(4.2)

Observe that if the foliation is of Yang–Mills type then 1 ε ∗ ε ε = −(∇H − TH ) (∇H − TH ) − J2 − Ric H . ε As a consequence, in the Yang–Mills case the operator ε is seen to be symmetric for the metric gε . The following theorem that was proved in [18] is the main result of the section. Theorem 4.6. For every f ∈ C ∞ (M), we have d∆H f = ε d f .

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 287

Proof. We only sketch the proof and refer to [18] for the details. If Z1 , . . . , Z m is a local vertical frame of the leaves, we define J(η) = −

m X

JZ ` (ι Z ` dη V ),

`=1

where η V is the projection of η to the vertical cotangent bundle. It does not depend on the choice of the frame and therefore defines a globally defined tensor. Also, let us consider the map T : Γ∞ (∧2T ∗ M) → Γ∞ (T ∗ M) which is given in a local coframe ϑ i ∈ Γ∞ (H∗ ), νk ∈ Γ∞ (V ∗ ),

T (ϑ i ∧ ϑ j ) = −γi`j ν` ,

T (ϑ i ∧ νk ) = T (νk ∧ ν` ) = 0.

A direct computation then shows that 2 1 1 ε ∗ ε −(∇H − TH ) (∇H − TH ) = ∆H + 2J − T ◦ d + δHT ∗ − δHT + J2 . ε ε ε Thus, we just need to prove that if ∞ is the operator defined on one-forms by ∞ = ∆H + 2J − Ric H + δHT ∗ , then for any f ∈ C ∞ (M), d∆H f = ∞ d f . A computation in local frame shows that d∆H f − ∆H d f = 2J(d f ) − Ric H (d f ) + δHT ∗ (d f ), which completes the proof. We now state the following Bochner-type identity. Theorem 4.7. For any η ∈ Γ∞ (T ∗ M), 1 ε ∆H kη kε2 − hε η, ηiε = k∇H η − TH η kε2 + hRic H (η), ηiH − hδHT (η), ηiV 2 1 + hJ2 (η), ηiH . ε Proof. From the very definition of ε , we have ε ∗ ε −hε η, ηiε = h(∇H − TH ) (∇H − TH )η, ηiε + hRic H (η), ηiH − hδHT (η), ηiV 1 + hJ2 (η), ηiH . ε



288

Fabrice Baudoin

The idea is now to multiply this by any g ∈ C0∞ (M) and integrate over M. For that, observe that Z Z ε ∗ ε ε ε gh(∇H − TH ) (∇H − TH )η, ηiε dµ = h(∇H − TH )η, (∇H − TH )(gη)iε dµ. M

M

We have now ε ε (∇H − TH )(gη) = g(∇H − TH )(η) + η ⊗ ∇H g

and Z h(∇H −

ε TH )η, η

⊗ ∇H giε dµ =

M

Z

h∇H η, η ⊗ ∇H giε dµ Z 1 = h∇H g, ∇H kη k 2 iε dµ. 2 M M

Putting things together we deduce that Z Z ε ε ε ∗ η kε2 dµ gk∇H η − TH gh(∇H − TH ) (∇H − TH )η, ηiε dµ = M M Z 1 − g∆H kη kε2 dµ. 2 M Since it is true for every g, we deduce 1 ε ∗ ε ε h(∇H − TH ) (∇H − TH )η, ηiε = k∇H η − TH η kε2 − ∆H kη kε2 . 2  Let us observe that if η = d f for some f ∈ C ∞ (M), then an easy computation shows that 1 ε ε k∇H η − TH η kε2 = k∇#H η kε2 − TrH (Jη2 ) + εk∇H η − TH η kV2 ; 4 thus by the Cauchy–Schwarz inequality we have 2 1 1 1 ∆H kη kε2 − hε η, ηiε ≥ TrH ∇#H η − TrH (Jη2 ) + hRic H (η), ηiH 2 n 4 1 − hδHT (η), ηiV + hJ2 (η), ηiH . ε

(4.3)

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 289

4.3 Generalized curvature-dimension inequality. Let M be a smooth, connected manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation F with bundle-like metric g and totally geodesic m-dimensional leaves for which the horizontal distribution is Yang–Mills. We also assume that M is complete and that globally on M, for every η 1 ∈ Γ∞ (H∗ ) and η 2 ∈ Γ∞ (V ∗ ), 2 hRic H (η 1 ), η 1 iH ≥ %1 kη 1 kH ,

2 −hJ2 η 1 , η 1 iH ≤ κkη 1 kH ,

1 − TrH (Jη22 ) ≥ %2 kη 2 kV2 , 4 for some %1 ∈ R, κ, %2 > 0. The third assumption can be thought as a uniform bracket-generating condition of the horizontal distribution H and from Hörmander’s theorem, it implies that the horizontal Laplacian ∆H is a subelliptic diffusion operator. We insist that for the following results to be true, the positivity of %2 is required. We introduce the following operators defined for f , g ∈ C ∞ (M): 1 (∆H ( f g) − g∆H f − f ∆H g) = h∇H f , ∇H giH , 2 ΓV ( f , g) = h∇V f , ∇V giV , Γ( f , g) =

and their iterations which are defined by 1 (∆H (Γ( f , g)) − Γ(g, ∆H f ) − Γ( f , ∆H g)), 2 1 Γ2V ( f , g) = (∆H (ΓV ( f , g)) − ΓV (g, ∆H f ) − ΓV ( f , ∆H g)). 2 Γ2 ( f , g) =

As a consequence of Theorem 4.6, we obtain the curvature-dimension inequality introduced with Nicola Garofalo in [14]. Theorem 4.8. For every f , g ∈ C ∞ (M), and ε > 0,  κ 1 Γ( f , f ) + %2 ΓV ( f , f ), Γ2 ( f , f ) + εΓ2V ( f , f ) ≥ (∆H f ) 2 + %1 − n ε and Γ( f , ΓV ( f )) = ΓV ( f , Γ( f )). Proof. From the inequality (4.3), we have for every η = d f ∈ Γ∞ (T ∗ M), 2 1 1 1 ∆H kη kε2 − hε η, ηiε ≥ TrH ∇#H η − TrH (Jη2 ) + hRic H (η), ηiH 2 n 4 1 2 + hJ (η), ηiH . ε

290

Fabrice Baudoin

Using this inequality and taking into account the assumptions 2 hRic H (η 1 ), η 1 iH ≥ %1 kη 1 kH ,

2 −hJ2 η 1 , η 1 iH ≤ κkη 1 kH ,

1 − TrH (Jη22 ) ≥ %2 kη 2 kV2 4 immediately yields the expected result. The intertwining Γ( f , ΓV ( f )) = ΓV ( f , Γ( f )) is proved in Theorem 2.9. 

4.4 Sharp lower bound for the first eigenvalue of the horizontal Laplacian. In this section, as a second application of the transverse Weitzenböck formula proved in the previous chapter, we obtain a sharp lower for the first non-zero eigenvalue of the horizontal Laplacian. Let M be a compact, smooth, connected manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation F with bundle-like metric g and totally geodesic m-dimensional leaves. We also assume that M is of Yang–Mills type. We prove the following result that was first obtained in [17] in a less general setting. Let us point out that this bound may not be obtained as a consequence of the generalized curvature-dimension inequality only. Theorem 4.9. Assume that for every smooth horizontal one-form η, D E 2 2 hRic H (η), ηiH ≥ %1 kη kH , −J2 (η), η ≤ κkη kH , H

and that for every vertical one-form η, Tr(Jη∗ Jη ) ≥ %2 kη kV2 , with %1 , %2 > 0 and κ ≥ 0. Then the first eigenvalue λ 1 of the horizontal Laplacian −∆H satisfies λ1 ≥

%1 1−

1 n

+

3κ %2

.

To put things in perspective, we give examples where this bound is sharp. • Let us consider the Hopf fibration U(1) → S2d+1 → CPd . As we know, the horizontal Laplacian ∆H is the lift of the Laplace–Beltrami operator on CPd and in that case λ 1 = 2d (see 3.2). On the other hand, for this example, %1 = 2(d + 1), κ = 1, %2 = 2d. Thus the bound of Theorem 4.9 is sharp.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 291

• Consider now the quaternionic Hopf fibration SU(2) → S4d+3 → HPd . The sub-Laplacian ∆H is then the lift of the Laplace–Beltrami operator on HPd and in that case, λ 1 = 4d (see 3.9). For this example, %1 = 4(d + 2), κ = 3, %2 = 4d. Thus the bound of Theorem 4.9 is still sharp in this example. We also mention that it has even been proved in [17] that for some Riemannian foliations the equality λ 1 = %1 /(1 − n1 + 3κ % 2 ) actually implies that the foliation is equivalent to the classical or the quaternionic Hopf fibration. Proof. As for the classical Lichnerowicz estimate on Riemannian manifolds, the idea is to integrate on the manifold the Bochner–Weitzenböck equality in Theorem 4.7 but some tricks are needed. Let f ∈ C ∞ (M). Let us first observe that Z Z − hε d f , d f iε dµ = − hd∆H f , d f iε dµ M ZM Z =− hd∆H f , d f iH dµ − ε hd∆H f , d f iV dµ. M

M

Thus, by integrating the Bochner–Weitzenböck equality in Theorem 4.7, we obtain Z Z Z ε d f kε2 dµ (4.4) (∆H f ) 2 dµ − ε hd(∆H f ), d f iV dµ ≥ k∇H d f − TH M M M Z  κ 2 + %1 − kd f kH dµ. ε M We now compute Z Z Z ε 2 ε 2 ε k∇H d f − TH d f kε dµ = k∇H d f − TH d f kH dµ + ε k∇H d f − TH d f kV2 dµ M M M Z Z ε 2 = k∇H d f − TH d f kH dµ + ε k∇H d f kV2 dµ M M Z Z ε ε − 2ε h∇H d f , TH (d f )iV dµ + ε kTH d f kV2 dµ. M

M

(4.5) ε TH

Using the definition of together with the Yang–Mills assumption, we see that Z Z 1 ε Tr(J∇∗ V f J∇V f )dµ. (4.6) h∇H d f , TH (d f )iV dµ = ε M M By using (4.6), the trick is now to write Z Z Z 3 1 ε ε ε h∇H d f , TH (d f )iV dµ = h∇H d f , TH (d f )iV dµ − h∇H d f , TH (d f )iV dµ 2 M 2 M M Z Z 3 1 ε = h∇H d f , TH (d f )iV dµ − Tr(J∇∗ V f J∇V f )dµ. 2 M 4ε M

292

Fabrice Baudoin

Coming back to (4.5) and completing the squares gives Z k∇H d f −

ε TH d f kε2 dµ

M

Z Z 2 3 ε

ε 2 d f

dµ = k∇H d f − TH d f kH dµ + ε

∇H d f − TH 2 M M

V Z Z 1 5 ε 2 ∗ + kT d f kV dµ. Tr(J∇V f J∇V f )dµ − ε 2 M 4 M H

This yields the lower bound Z

ε d f kε2 dµ ≥ k∇H d f − TH M

1 n

Z

3 (∆H f ) 2 dµ + %2 4 M Z 5 2 κ kd f kH dµ. − 4ε M

Z kd f kV2 dµ M

We thus deduce !Z Z Z n−1 9κ 2 2 (∆H f ) dµ − ε hd(∆H f ), d f iV dµ ≥ %1 − kd f kH dµ n 4ε M M M Z 3 kd f kV2 dµ. + %2 4 M Now if f is an eigenfunction that satisfies ∆H f = −λ 1 f , we get n−1 2 λ1 n

Z

f dµ + ελ 1

Z

2

M

kd f

kV2 dµ

ελ 1 =

3 %2 , 4

M

! Z 9κ ≥ %1 − λ1 f 2 dµ 4ε M Z 3 kd f kV2 dµ. + %2 4 M

Choosing ε such that

yields the desired lower bound on λ 1 .



5 The horizontal heat semigroup We introduce here a fundamental tool in the geometric analysis of Riemannian foliations: the horizontal heat semigroup. We study some of its properties,such as stochastic completeness, and briefly discuss the Li–Yau estimates for this semigroup.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 293

5.1 Essential self-adjointness of the horizontal Laplacian. Let M be a smooth, connected manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation with a bundle-like metric g and totally geodesic mdimensional leaves. We assume that the metric g is complete and denote by C0∞ (M) the space of smooth and compactly supported functions on M. We will also assume that the horizontal distribution H of the foliation is bracket generating. From Hörmander’s theorem, the bracket-generating condition implies that the horizontal Laplacian ∆H is hypoelliptic. An important consequence of the completeness assumption is the fact that there exists an increasing sequence hn ∈ C0∞ (M) such that hn % 1 on M, and ||∇H hn ||∞ + ||∇V hn ||∞ → 0,

(5.1)

as n → ∞. We refer to Strichartz ([46]) for a proof of this fact. It will be convenient to introduce the following operators defined for f , g ∈ C ∞ (M) by Γ( f , g) =

1 (∆H ( f g) − g∆H f − f ∆H g) = h∇H f , ∇H giH 2

and ΓV ( f , g) = h∇V f , ∇V giV . As shorthand notation, we will use Γ( f ) = Γ( f , f ) and ΓV ( f ) = ΓV ( f , f ). Proposition 5.1. The horizontal Laplacian ∆H is essentially self-adjoint on the space C0∞ (M). Proof. According to Reed–Simon [45, page 137], it is enough to prove that if ∆∗H f = λ f with λ > 0, then f = 0. Since ∆H is given on the domain C0∞ (M), this means that ∆H f = λ f in the sense of distributions. From the hypoellipticity of ∆H , we first deduce that f has to be a smooth function. Now, for h ∈ C0∞ (M), Z

Γ( f , h f )dµ = −

Z

f ∆H (h f )dµ = −

2

M

2

M

Z

(∆∗H f )(h2 f )dµ Z = −λ f 2 h2 dµ ≤ 0. M

M

Since Γ( f , h2 f ) = h2 Γ( f , f ) + 2 f hΓ( f , h), we deduce that

Z

h Γ( f )dµ + 2

Z

hΓ( f , h)dµ ≤ 0.

2

M

M

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Therefore, by the Schwarz inequality, Z h2 Γ( f )dµ ≤ 4k f |22 kΓ(h)k∞ . M

If we now use a sequence hn that satisfies (5.1) and let n → ∞, we obtain Γ( f ) = 0 and therefore f = 0, as desired.  R +∞ If ∆H = − 0 λdEλ is the spectral resolution of the Friedrichs extension of ∆ in L 2 (M, µ), then by definition, the heat semigroup (Pt )t ≥0 is given by Pt = RH +∞ −λt e dEλ . It is a symmetric Markov semigroup on L 2 (M, µ). That is, it satisfies 0 the following properties: • P0 = Id; • Pt+s = Pt Ps , s,t ≥ 0; • for f ∈ L 2 (M, µ), limt→0 kPt f − f k2 = 0; • kPt f k2 ≤ k f k2 ; • if f ∈ L 2 (M, µ) is non-negative, then Pt f ≥ 0; • if f ∈ L 2 (M, µ) is less than 1, then Pt f ≤ 1. By using the Riesz–Thorin interpolation theorem, (Pt )t ≥0 induces a contraction semigroup on all the L p (M, µ)’s, 1 ≤ p ≤ ∞. Due to the hypoellipticity of ∆H , (t, x) → Pt f (x) is smooth on M × (0, ∞) and Z Pt f (x) = p(x, y,t) f (y)dµ(y), f ∈ C0∞ (M), M

where p(x, y,t) > 0 is the so-called heat kernel associated to Pt . Such a function is smooth and it is symmetric, i.e., p(x, y,t) = p(y, x,t). By the semigroup property, for every x, y ∈ M and 0 < s,t we have Z Z p(x, y,t + s) = p(x, z,t)p(z, y, s)dµ(z) = p(x, z,t)p(y, z, s)dµ(z) M

M

= Ps (p(x, ·,t))(y). For a more analytic view of (Pt )t ≥0 , we recall that it can be seen as the unique solution of a parabolic Cauchy problem in L p (M, µ), 1 < p < +∞.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 295

Proposition 5.2. The unique solution of the Cauchy problem ∂u     ∂t − ∆H u = 0,  u(x, 0) = f (x), 

f ∈ L p (M, µ), 1 < p < +∞,

that satisfies ku(·,t)k p < ∞ for every t ≥ 0, is given by u(x,t) = Pt f (x). We stress that without further conditions, this result fails when p = 1 or p = +∞. The case p = +∞ is equivalent to stochastic completeness (Pt 1 = 1) and will be discussed in a later section.

5.2 Horizontal heat semigroup on one-forms. Throughout the section, we work under the same assumptions as the previous section and we moreover assume that for every horizontal one-form η, 2 hRic H (η), ηiH ≥ −K kη kH ,

2 −hJ2 η, ηiH ≤ κkη kH ,

with K, κ ≥ 0. We also assume that the horizontal distribution H is Yang–Mills, which means that δHT = 0. We recall that if we consider the operator defined on one-forms by the formula 1 ε ε ∗ ) − J2 − Ric H , ) (∇H − TH ε = −(∇H − TH ε then for any smooth function f , d∆H f = ε d f and for any smooth one-form η, 1 2 ε ∆H kη k2ε − hε η, ηiε = k∇H η − TH η kε2 + 2  κ 2 ≥ %− kη kH . ε

*

! + 1 Ric H + J2 η, η ε ε

The operator ε is symmetric for the metric gε = gH ⊕

1 gV . ε

Owing to our assumptions we can say even more. Lemma 5.3. The operator ε is essentially self-adjoint on the space of smooth and compactly supported one-forms for the Riemannian metric gε .

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Proof. We consider an increasing sequence hn ∈ C0∞ (M), 0 ≤ hn ≤ 1, such that hn % 1 on M, and ||Γ(hn )||∞ → 0, as n → ∞. To prove that ε is essentially self-adjoint, once again it is enough to prove that for some λ > 0, ε η = λη with η ∈ L 2 implies η = 0. So, let λ > 0 and η ∈ L 2 such that ε η = λη. We have then Z Z 2 2 λ hn kη kε = hh2n η, ε ηiε M M Z ε ε =− h∇H (h2n η) − TH (h2n η), ∇H η − TH ηiε M ! + * Z 1 + h2n − J2 − Ric H (η), η ε Z ε ZM 2 2 ε =− hn k∇H η − TH η kε − 2 hn hη, ∇∇H h n ηiε M M ! + * Z 1 2 2 + hn − J − Ric H (η), η . ε M ε From our assumptions, the symmetric tensor − ε1 J2 − Ric H is bounded from above, thus by choosing λ big enough, we have Z Z 2 ε 2 hn k∇H η − TH η kε + 2 hn hη, ∇∇H h n ηiε ≤ 0. M

M

ε η k 2 = 0 which implies ∇ η − By letting n → ∞, we easily deduce that k∇H η − TH H ε ε TH η = 0. If we come back to the equation ε η = λη and the expression of ε , we see that it implies ! 1 2 − J − Ric H (η) = λη. ε

Our choice of λ then forces η = 0.



Since ε is essentially self-adjoint, it admits a unique self-adjoint extension which generates, thanks to the spectral theorem, a semigroup Qεt = e tε . We recall that Pt = e t∆H is the semigroup generated by ∆H . We have the following commutation property. Lemma 5.4. If f ∈ C0∞ (M), then for every t ≥ 0, dPt f = Qεt d f . Proof. Let η t = Qεt d f . By essential self-adjointness, it is the unique solution in L 2 of the heat equation ∂η = ε η, ∂t

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 297

with initial condition η 0 = d f . From the fact that dL = ε d, we see that α t = dPt f solves the heat equation ∂α = ε α ∂t with the same initial condition α0 = d f . In order to conclude, we thus just need to prove that for every t ≥ 0, dPt f is in L 2 . As usual, we denote by ∆H the vertical Laplacian. The Laplace–Beltrami operator of M is therefore ∆ = ∆H +∆V . Since the leaves are totally geodesic, ∆ commutes with ∆H on C 2 functions. Moreover from the spectral theorem, ∆H e t∆ maps C0∞ (M) into L 2 (M, µ). We deduce by essential self-adjointness that ∆H e t∆ = e t∆ ∆H . Similarly we obtain e s∆H e t∆ = e t∆ e s∆H which implies ∆e s∆H = e s∆H ∆. As a consequence we have that for every t ≥ 0, dPt f is in L 2 . 

5.3 Stochastic completeness. We can now give an important corollary of the commutation of Lemma 5.4. Theorem 5.5. For every ε > 0, t ≥ 0, x ∈ M, and f ∈ C0∞ (M), kdPt f (x)kε ≤ e (K +(κ/ε))t Pt kd f kε (x). Proof. The idea is to use the Feynman–Kac stochastic representation of Qεt . We denote by (X t )t ≥0 the symmetric diffusion process generated by 12 L and denote by e its lifetime. Consider the process τtε : TX∗ t M → TX∗ 0 M which is the solution of the following covariant Stratonovitch stochastic differential equation: ! !  ε  1 1 2 ε ε J + Ric H dt α(X t ), τ0ε = Id, d τt α(X t ) = τt ∇◦dX t − T◦dX t − 2 ε (5.2) where α is any smooth one-form. By using Gronwall’s lemma, we have for every t ≥ 0, kτtε α(X t )kε ≤ e (K +(κ/ε))t /2 kα(X t )kε . By the Feynman–Kac formula, we have for every smooth and compactly supported one-form, Q t /2 η(x) = E x (τt η(X t )1t 0. Let u, v : M × [0,T] → R be smooth functions such that for every T > 0, supt ∈[0,T ] ku(·,t)k∞ < ∞, supt ∈[0,T ] kv(·,t)k∞ < ∞. If the inequality ∂u ∆H u + ≥v ∂t holds on M × [0,T], then we have Z T PT (u(·,T ))(x) ≥ u(x, 0) + Ps (v(·, s))(x)ds. 0

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 299

5.4 Li–Yau estimates. We show in this section how to obtain the Li–Yau estimate which is a crucial ingredient to prove the Bonnet–Myers theorem. Henceforth, we will indicate Cb∞ (M) = C ∞ (M) ∩ L ∞ (M). A key lemma is the following. Lemma 5.8. Let f ∈ Cb∞ (M), f > 0, and T > 0, and consider the functions φ1 (x,t) = (PT −t f )(x)Γ(ln PT −t f )(x), φ2 (x,t) = (PT −t f )(x)ΓV (ln PT −t f )(x), which are defined on M × [0,T ). We have ∆H φ1 +

∂φ1 = 2(PT −t f )Γ2 (ln PT −t f ) ∂t

∆H φ2 +

∂φ2 = 2(PT −t f )Γ2V (ln PT −t f ). ∂t

and

Proof. This is direct computation without a trick. Let us just point out that the formula ∂φ2 ∆H φ2 + = 2(PT −t f )Γ2V (ln PT −t f ) ∂t uses the fact that Γ(g, ΓV (g)) = ΓV (g, Γ(g)) and thus that the foliation is totally geodesic.  We now show how to prove the Li–Yau estimates for the horizontal semigroup. The method we use is adapted from [7]. Theorem 5.9. Let α > 2. For f ∈ C0∞ (M), f ≥ 0, f , 0, the following inequality holds for t > 0: ! ακ 2%1 ∆H Pt f n%21 2%2 V tΓ (ln Pt f ) ≤ 1 + − t + t Γ(ln Pt f ) + α (α − 1) %2 α Pt f 2α ! n(α − 1) 2 1 + ακ  2 %1 n ακ (α−1) % 2 − 1+ + . 2 (α − 1) %2 8(α − 2)t Proof. We fix T > 0 and consider two functions a, b : [0,T] → R ≥0 to be chosen later. Let f ∈ C ∞ (M), f ≥ 0. Consider the function φ(x,t) = a(t)(PT −t f )(x)Γ(ln PT −t f )(x) + b(t)(PT −t f )(x)ΓV (ln PT −t f )(x).

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Applying Lemma 5.8 and the curvature-dimension inequality in Theorem 4.8, we obtain ∂φ ∆H φ + = a 0 (PT −t f )Γ(ln PT −t f ) + b0 (PT −t f )ΓV (ln PT −t f ) ∂t + 2a(PT −t f )Γ2 (ln PT −t f ) + 2b(PT −t f )Γ2V (ln PT −t f ) ! a2 0 ≥ a + 2%1 a − 2κ (PT −t f )Γ(ln PT −t f ) b + (b0 + 2%2 a)(PT −t f )ΓV (ln PT −t f ) 2a (PT −t f )(∆H (ln PT −t f )) 2 . + n But, for any function γ : [0,T] → R, (∆H (ln PT −t f )) 2 ≥ 2γ∆H (ln PT −t f ) − γ 2 , and from the chain rule, ∆H (ln PT −t f ) =

∆H PT −t f − Γ(ln PT −t f ). PT −t f

Therefore, we obtain ! ∂φ a2 4aγ 0 ∆H φ + ≥ a + 2%1 a − 2κ − (PT −t f )Γ(ln PT −t f ) ∂t b n + (b0 + 2%2 a)(PT −t f )ΓV (ln PT −t f ) +

4aγ 2aγ 2 ∆H PT −t f − PT −t f . n n

The idea is now to choose a, b, γ such that a2 4aγ   0   a + 2%1 a − 2κ b − n = 0,   b0 + 2% a = 0. 2  With this choice we get ∂φ 4aγ 2aγ 2 ≥ ∆H PT −t f − PT −t f . (5.4) ∂t n n We wish to apply Proposition 5.7. So, we take f ∈ C0∞ (M) and apply the previous inequality with f ε = f + ε instead of f , where ε > 0. If moreover a(T ) = b(T ) = 0, we end up with the inequality ∆H φ +

a(0)(PT f ε )(x)Γ(ln PT f ε )(x) + b(0)(PT f )(x)ΓV (ln PT f ε )(x) Z T Z T 4aγ 2aγ 2 ≤− dt∆H PT f ε (x) + dtPT f ε (x). n n 0 0

(5.5)

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 301

If we now choose b(t) = (T − t) α and b, γ such that 2   0 + 2% a − 2κ a − 4aγ = 0,  a 1  b n   b0 + 2% a = 0, 2 

the result follows by a simple computation and then sending ε → 0.



Observe that if Ric H ≥ 0, then we can take %1 = 0 and the estimate simplifies to Γ(ln Pt f ) +

2%2 V tΓ (ln Pt f ) α

 2 ! ακ 2 ∆H Pt f n(α − 1) 1 + (α−1) %2 ακ ≤ 1+ + . (α − 1) %2 Pt f 8(α − 2)t By adapting a classical method of Li and Yau [43] and integrating this last inequality on sub-Riemannian geodesics leads to a parabolic Harnack inequality (details are in [14]). For α > 2, we denote Dα =

 n(α − 1) 2 1 +

ακ (α−1) % 2

4(α − 2)

 .

(5.6)

The minimal value of Dα is difficult to compute, depends on κ, %2 , and does not seem relevant because the constants we get are anyhow not optimal. We just point out that the choice α = 3 turns out to simplify many computations and is actually optimal when κ = 4%2 . Corollary 5.10. Let us assume that Ric H ≥ 0. Let f ∈ L ∞ (M), f ≥ 0, and consider u(x,t) = Pt f (x). For every (x, s), (y,t) ∈ M × (0, ∞) with s < t, one has, with Dα as in (5.6), !  t  D α /2 Dα d(x, y) 2 u(x, s) ≤ u(y,t) exp . s n 4(t − s) Here d(x, y) is the sub-Riemannian distance between x and y. Since the work by Li and Yau (see [43]), it is classical and not difficult to prove that a parabolic Harnack inequality implies a Gaussian upper bound on the heat kernel. With the curvature-dimension inequality in hand, it is actually also possible, but much more difficult, to prove a lower bound. The final result proved in [11] is as follows.

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Theorem 5.11. Let us assume that Ric H ≥ 0; then for any 0 < ε < 1 there exists a constant C(ε) = C(n, κ, %2 , ε) > 0, which tends to ∞ as ε → 0+ , such that for every x, y ∈ M and t > 0 one has ! ! C(ε) −1 Dα d(x, y) 2 C(ε) d(x, y) 2 ≤ pt (x, y) ≤ , √ exp − √ exp − n(4 − ε)t (4 + ε)t µ(B(x, t)) µ(B(x, t)) where pt (x, y) is the heat kernel of ∆H . We mention that those results are not restricted to the case %1 = 0 but that similar results may also be obtained when %1 ≤ 0. We refer to [12].

6 The horizontal Bonnet–Myers theorem Let M be a smooth, connected manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation F with bundle-like complete metric g and totally geodesic m-dimensional leaves. We also assume that the horizontal distribution is of Yang–Mills type. In this section, we prove the following result: Theorem 6.1. Assume that for any smooth horizontal one-form η ∈ Γ∞ (H∗ ), D E 2 2 hRic H (η), ηiH ≥ %1 kη kH , −J2 (η), η ≤ κkη kH , H

and that for any vertical one-form η ∈

Γ∞ (V ∗ ),

1 Tr(Jη∗ Jη ) ≥ %2 kη kV2 , 4 with %1 , %2 > 0 and κ ≥ 0. Then the manifold M is compact and we have s ! √ %2 + κ 3κ 1+ n, diam M ≤ 2 3π %1 %2 2%2 where diam M is the diameter of M for the sub-Riemannian distance. We mention that the bound √ diam M ≤ 2 3π

s

! κ + %2 3κ 1+ n %1 %2 2%2

is not sharp. This is because the method we use, that comes from joint work with Garofalo [14], is an adaptation of the energy-entropy inequality methods developed

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 303

by Bakry in [3]. Even in the Riemannian case, Bakry’s methods are known to lead to non-sharp constants. An analytical method that leads to sharp diameter constants is based on sharp Sobolev inequalities (see [6]), however as of today, it is still an open question to prove those sharp Sobolev inequalities.

6.1 Ultracontractivity bounds and diameter estimates. In this section, we show how ultracontractivity bounds for the heat semigroup can be used to get diameter bounds on a space. The result we give below is a variation on results due to Bakry [3] and Davies [27]. The result holds true in great generality in the context of Dirichlet spaces. Let µ be a probability measure on a locally compact topological space Ω. We assume that there is on Ω a regular Dirichlet form E which is symmetric in L 2 (Ω, µ) (see Fukushima [32]) . Let (Pt )t ≥0 be the symmetric Markov semigroup associated to E . It is well known that we can associated to E a distance which is defined as follows. Let D be the domain in L 2 (Ω, µ) of the Dirichlet form E . We denote by D∞ the set of bounded functions in D. For f ∈ D∞ , we define I f (h) =

1 (2E ( f g, g) − E ( f 2 , g)), 2

g ∈ D∞ .

We then say that f ∈ Lip if for every g ∈ D∞ , |I f (g)| ≤ kgk1 . For x, y ∈ Ω, we define d(x, y) = sup{ f (x) − f (y), f ∈ Lip} and assume that d is a distance everywhere finite that induces the topology of Ω. Theorem 6.2. Assume that for every f ∈ L 2 (Ω, µ) and t ≥ 0, kPt f k∞ ≤

1 (1 −

e−αt ) D/2

k f k2 ,

with α, D > 0. Then Ω is compact and its diameter for the distance d satisfies r 2D diam(Ω) ≤ 2π . α Proof. Since kPt f k∞ ≤

1 (1 −

e−αt ) D/2

k f k2 ,

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Fabrice Baudoin

R from Davies’ theorem ([27, Theorem 2.2.3]), for f ∈ L 2 (Ω) such that Ω f 2 dµ = 1, we obtain Z Z   f 2 ln f 2 dµ ≤ 2t Γ( f )dµ − D ln 1 − e−αt , t > 0. Ω



By minimizing the right-hand side of the above inequality over t, we get that for R f ∈ L 2 (M) such that M f 2 dµ = 1, ! Z Z 2 2 f ln f dµ ≤ Φ Γ( f )dµ , M

M

where

! ! !# 2 2 2 2 x ln 1 + x − x ln x . αD αD αD αD The function Φ enjoys the following properties: "

Φ(x) = D

1+

• Φ0 (x)/x 1/2 and Φ(x)/x 3/2 are integrable on (0, ∞); • Φ is concave; R +∞ R +∞ dx = 0 • 21 0 Φ(x) x 3/2

Φ0 (x) √ dx x

= −2

R

+∞ 0



xΦ00 (x)dx < +∞.

We can therefore apply [3, Theorem 5.4] to deduce that the diameter of M is finite and Z +∞ √ 00 xΦ (x)dx. diam(Ω) ≤ −2 0 2D Since Φ00 (x) = − x (2x+α D) , a routine calculation shows r Z +∞ √ 00 2D −2 xΦ (x)dx = 2π . α 0



6.2 Proof of the compactness theorem. We now turn to the proof of Theorem 6.1. Let M be a smooth, connected manifold with dimension n + m. As usual, we assume that M is equipped with a Riemannian foliation with bundle-like complete metric g and totally geodesic m-dimensional leaves. We also assume that the horizontal distribution is of Yang–Mills type and that for any smooth horizontal oneform η ∈ Γ∞ (H∗ ), D E 2 2 hRic H (η), ηiH ≥ %1 kη kH , −J2 (η), η ≤ κkη kH , H

and that for any vertical η ∈ Γ∞ (V ∗ ), 1 Tr(Jη∗ Jη ) ≥ %2 kη kV2 , 4 with %1 , %2 > 0 and κ ≥ 0.

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 305

The first step is to prove that the volume of M is finite. Lemma 6.3. The measure µ is finite, i.e., µ(M) < +∞, and for every x ∈ M, f ∈ L 2 (M), Z 1 Pt f (x) →t→+∞ f dµ. µ(M) M Proof. We first prove a gradient bound for the semigroup following an argument close to the one in the proof of Theorem 5.9. We fix T > 0 and consider two functions a, b : [0,T] → R ≥0 to be chosen later. Let f ∈ C0∞ (M). Consider the function φ(x,t) = a(t)Γ(PT −t f )(x) + b(t)ΓV (PT −t f )(x). Computing derivatives and applying the curvature-dimension inequality in Theorem 4.8, we obtain ∆H φ +

∂φ = a 0 Γ(PT −t f ) + b0 ΓV (PT −t f ) + 2aΓ2 (PT −t f ) + 2bΓ2V (PT −t f ) ∂t ! a2 0 ≥ a + 2%1 a − 2κ Γ(PT −t f ) + (b0 + 2%2 a)ΓV (PT −t f ). b

Let us now choose

b(t) = e−2%1 %2 t /(κ+%2 )

and a(t) = −

b0 (t) , 2%2

so that b0 + 2%2 a = 0 and a 0 + 2%1 a − 2κ

a2 = 0. b

With this choice, we get

∂φ ≥ 0. ∂t From the parabolic comparison theorem, Theorem 5.7, we deduce ∆H φ +

! κ + %2 V κ + %2 −2% 1 % 2 t /(κ+% 2 ) V Γ(Pt f ) + Γ (Pt f ) ≤ e Pt (Γ( f )) + Pt (Γ ( f )) . %1 %1

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Let f , g ∈ C0∞ (M); we have ! Z tZ Z ∂ (Pt f − f )g dµ = Ps f g dµ ds M 0 M ∂s Z tZ = (∆H Ps f ) g dµ ds 0 M Z tZ =− Γ(Ps f , g)dµ ds. 0

M

By means of the previous bound and the Cauchy–Schwarz inequality we find that Z (Pt f − f )g dµ (6.1) M ! r Z t Z κ + %2 V −% 1 % 2 s/(κ+% 2 ) kΓ( f )k∞ + e ds kΓ ( f )k∞ Γ(g) 1/2 dµ. ≤ %1 M 0 It is seen from the spectral theorem that in L 2 (M, µ) we have a convergence Pt f → P∞ f , where P∞ f belongs to the domain of ∆H . Moreover ∆H P∞ f = 0. By hypoellipticity of ∆H we deduce that P∞ f is a smooth function. Since ∆H P∞ f = 0, we have Γ(P∞ f ) = 0 and therefore P∞ f is constant. Let us now assume that µ(M) = +∞. This implies in particular that P∞ f = 0 because no constant besides 0 is in L 2 (M, µ). Then using (6.1) and letting t → +∞, we infer Z f g dµ M ! r Z Z +∞ κ + %2 V −% 1 % 2 s/(κ+% 2 ) kΓ( f )k∞ + kΓ ( f )k∞ Γ(g) 1/2 dµ. ≤ e ds %1 M 0 Let us assume g ≥ 0, g , 0 and take for f a sequence hn increasing in C0∞ (M), 0 ≤ hn ≤ 1, such that hn % 1 on M, and ||Γ(hn )||∞ → 0. Letting n → ∞, we deduce Z g dµ ≤ 0, M

which is clearly absurd. As a consequence, µ(M) < +∞. The invariance of µ by the semigroup implies Z Z P∞ f dµ = f dµ, M

and thus

1 P∞ f = µ(M)

M

Z f dµ. M

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 307

Finally, using the Cauchy–Schwarz inequality, we find that for x ∈ M, f ∈ L 2 (M, µ), s,t, τ ≥ 0, |Pt+τ f (x) − Ps+τ f (x)| = |Pτ (Pt f − Ps f )(x)| Z = p(τ, x, y)(Pt f − Ps f )(y) µ(dy) Z M ≤ p(τ, x, y) 2 µ(dy)kPt f − Ps f k22 M

≤ p(2τ, x, x)kPt f − Ps f k22 . Thus, we have Pt f (x) →t→+∞

1 µ(M)

Z f dµ. M

 Since µ(M) < +∞, we can assume µ(M) = 1. The second lemma we need is a uniform bound on the heat kernel of ∆H , from which we will immediately deduce Theorem 6.1 by using Theorem 6.2. Lemma 6.4. Let β > 2. For f ∈ C0∞ (M), f ≥ 0, and t ≥ 0, Pt f ≤ where

Z

1 1 − e−2%1 %2 t /(β (%2 +κ))

n β−1 Dβ = 4 β−2

 D β /2

f dµ, M

! ! κ 1+ β−1 . %2

Proof. We fix T > 0 and consider functions a, b : [0,T] → R ≥0 and γ : [0,T] → R such that a2 4aγ  0   − = 0, a + 2% a − 2κ  1   b n  0 b + 2%2 a = 0,      a(T ) = b(T ) = 0.  Let f ∈ C0∞ (M), f ≥ 0. Recall that from inequality (5.5), a(0)(PT f )(x)Γ(ln PT f )(x) + b(0)(PT f )(x)ΓV (ln PT f )(x) Z T Z T 4aγ 2aγ 2 ≤− dt∆H PT f (x) + dtPT f (x). n n 0 0

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Since the left-hand side is non-negative we deduce Z T Z T −2 aγ dt ∆H PT f (x) + aγ 2 dt PT f (x) ≥ 0. 0

Let us choose with β > 2, α =

0

b(t) = (e−αt − e−αT ) β , 2% 1 % 2 β (% 2 +κ) ,

0 ≤ t ≤ T,

and a, γ such that

a2 4aγ  0   a + 2% a − 2κ − = 0, 1  b n   b0 + 2% a = 0. 2  We obtain, after some computations, 0≤

(2%1 − α) −αT ∆H PT f n(2%1 − α) 2 e−2αT  e   . + −αT PT f 2%2 1 − β1 16%2 1 − β2 1 − e

Since T is arbitrary, this implies that for every t > 0, ∆H Pt f n β−1 e−αt ≥− (2%1 − α) . Pt f 8 β−2 1 − e−αt R Taking into account that Pt f (x) →t→+∞ M f dµ and integrating from 0 to +∞ yields the claim.  As a consequence of Theorem 6.2, we deduce that M is compact and that for every β > 2, ! r s ! %2 κ n β( β − 1) β− . diam(M) ≤ π 1 + %2 %1 β−2 %2 + κ The optimal β does not lead to a nice formula. The value β = 3 yields the bound s ! √ %2 + κ 3κ diam M ≤ 2 3π n. 1+ %1 %2 2%2

7 Riemannian foliations and hypocoercivity We now show how the geometry of foliations can be used to study some hypoelliptic diffusion operators that we call Kolmogorov-type operators. We shall mainly be

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 309

interested in the problem of convergence to equilibrium for the parabolic equation associated with those operators. The methods we develop to prove convergence to equilibrium come with estimates that Villani calls hypocoercive (see [50]). As an illustration, we study the kinetic Fokker–Planck equation.

7.1 Kolmogorov-type operators. Let M be a smooth, connected manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation F with m-dimensional leaves. As before, we indicate by ∆H the horizontal Laplacian and by ∆V the vertical Laplacian. Definition 7.1. We define a Kolmogorov-type operator as a hypoelliptic diffusion operator L on M that can be written as L = ∆V + Y, where Y is a smooth vector field on M. The simplest example of such an operator was studied by Kolmogorov himself. Let us consider the operator ∂2 ∂ L = 2 +v . ∂x ∂v Then, by considering the trivial foliation on R2 that comes from the submersion ∂2 ∂ (v, x) → x, we can write L = ∆V + Y with ∆V = ∂v 2 and Y = v ∂x . More interesting is the operator on R2n = {(v, x), v ∈ Rn , x ∈ Rn }, L = ∆v − v · ∇v + ∇ x V · ∇v − v · ∇ x , where V : Rn → R is a smooth potential. The parabolic equation ∂h = ∆v h − v · ∇v h + ∇V · ∇v h − v · ∇ x h, ∂t

(x, v) ∈ R2n

(7.1)

is then known as the kinetic Fokker–Planck equation with confinement potential V . It has been extensively studied due its importance in mathematical physics. We refer for instance to [29, 37, 50, 56]. This equation is the Kolmogorov–Fokker–Planck equation associated to the stochastic differential system    dx t = vt dt,  dvt = −vt dt − ∇V (x t )dt + dBt ,  where (Bt )t ≥0 is a Brownian motion in Rn .

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We can obviously write L = ∆V + Y, where ∆V = ∆v and Y = −v · ∇v + ∇ x V · ∇v − v · ∇ x , and consider the trivial foliation on R2n that comes from the submersion (v, x) → x. However we will see that the metric on R2n to choose is not the standard Euclidean metric but rather the metric that makes ( ) ∂ ∂ ∂ + , ,1 ≤ i ≤ n 2 ∂x i ∂vi ∂vi an orthonormal basis at any point.

7.2 Convergence to equilibrium and hypocoercive estimates. We consider a Kolmogorov-type operator L = ∆V + Y, and assume in this section that the Riemannian foliation is totally geodesic with a bundle-like metric. Our first task will be to prove a Bochner-type inequality for L. If f ∈ C ∞ (M), we denote

T2 ( f ) =

 1 L(k∇ f k 2 ) − 2h∇ f , ∇L f i , 2

where ∇ is the whole Riemannian gradient. We denote by RicV the Ricci curvature of the leaves and we denote by DY the tensor defined by DY (U,V ) = hDU Y,V i where D is the Levi-Civita connection. We have then the following Bochner inequality. Theorem 7.2. For every f ∈ C ∞ (M),

T2 ( f ) ≥ (RicV − DY )(∇ f , ∇ f ). Proof. We can split T2 into three parts:

T2 ( f ) = Γ2H ( f ) + Γ2V ( f ) + Γ2Y ( f ), where

 1 ∆V (k∇H f k 2 ) − 2h∇H f , ∇H ∆V f i , 2  1 Γ2V ( f ) = ∆V (k∇V f k 2 ) − 2h∇V f , ∇V ∆V f i , 2

Γ2H ( f ) =

and Γ2Y ( f ) =

 1 Y (k∇ f k 2 ) − 2h∇ f , ∇Y f i . 2

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 311

We now compute these three terms separately. Since ∇H and ∆V commute, we find Γ2H ( f ) = k∇V ∇H f k 2 . So we have Γ2H ( f ) ≥ 0. Since ∆V is the Laplace–Beltrami operator on the leaves, from the usual Bochner formula we have Γ2V ( f ) = k∇2V f k 2 + RicV (∇ f , ∇ f ). Thus we have Γ2V ( f ) ≥ RicV (∇ f , ∇ f ). Finally, we see that 1 1 Y k∇ f k 2 = DY k∇ f k 2 = h∇ f , DY ∇ f i 2 2 and h∇ f , ∇Y f i = h∇ f , ∇hY, ∇ f ii = DY (∇ f , ∇ f ) + h∇ f , DY ∇ f i.  A difficulty that arises when studying Kolmogorov-type operators is that, in general, they are not symmetric with respect to any measure. As a consequence, we cannot use functional analysis and the spectral theory of self-adjoint operators to define the semigroup generated by L. A typical assumption to ensure that L generates a well-behaved semigroup is the existence of a nice Lyapounov function. So, in the sequel, we will assume that there exists a function W such that W ≥ 1, k∇W k ≤ CW , LW ≤ CW for some constant C > 0 and {W ≤ m} is compact for every m. This condition is actually not too restrictive and may be checked in concrete situations. If M is compact, it is obviously satisfied. A non-compact situation where it is satisfied is the following: Assume that M is non-compact and that any two points of M can be joined by a unique geodesic. Also assume that the Riemannian foliation comes from a Riemannian submersion π : M → B and that the Ricci curvature of the leaves RicV is bounded from below by a negative constant −K. If x ∈ B, we denote L x = π −1 ({x}). Any geodesic γ : [0, L] → B in the base space can be lifted into a geodesic in M. For x ∈ Lγ(0) , denote by τγ (x) the endpoint of the unique horizontal lift of γ starting from x. Since the leaves are assumed to be totally geodesic, the map τγ induces an isometry between Lγ(0) and Lγ(L) . Now fix a base point x 0 ∈ M and for x ∈ Lπ(x0 ) define %V (x) = d(x 0 , x). If x < Lπ(x0 ) , then consider γ : [0, L] → B to be the unique geodesic between π(x 0 ) and π(x) and define %V (x) = d(τγ (x 0 ), x). Consider also the function %H (x) = d(π(x 0 ), π(x)) and finally define W (x) = 1 + %V (x) 2 + %H (x) 2 .

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Obviously W ≥ 1 is smooth, and such that {W ≤ m} is compact for every m. We have k∇W k = 2%V k∇%V k + 2%H k∇%H k ≤ 2%V + 2%H ≤ CW and LW = 2%V ∆V %V + 2%H ∆V %H + 2k∇V %V k 2 + 2k∇V %H k 2 + Y W. On the other hand, from the Laplacian comparison theorem on the leaves, r r K K * ∆V %V ≤ (m − 1) coth %V + , m−1 m − 1 , so we have for some constant C, LW ≤ CW + Y W. So, if we additionally assume that Y is a Lipschitz vector field, that is, kDY k ≤ C, then W satisfies all the requirements. The assumption about the existence of the function W such that LW ≤ CW easily implies that L is the generator of a Markov semigroup (Pt )t ≥0 that uniquely solves the heat equation in L ∞ . Moreover, consider a smooth and decreasing function h : R ≥0 → R such that h = 1 on [0, 1] and h = 0 on [2, +∞). Then define hn = h W n and consider the compactly supported diffusion operator L n = h2n L. Since L n is compactly supported, a Markov semigroup Ptn with generator L n is ∂P n f easily constructed as the unique bounded solution of ∂tt = L n Ptn f , f ∈ L ∞ . Then, for every bounded f , Ptn f → Pt f ,

n → ∞.

We now prove our first gradient bound for the Kolmogorov-type operator. Theorem 7.3. Let us assume that for some K ∈ R, RicV − DY ≥ −K; then for every bounded and Lipschitz function f ∈ C ∞ (M), we have, for t ≥ 0, k∇Pt f k 2 ≤ e2K t Pt (k∇ f k 2 ). Proof. We follow an approach by Wang [52]. We fix t > 0, n ≥ 1, and f ∈ C ∞ (M) compactly supported inside the set {W ≤ n}. Consider the functional defined for s ∈ [0,t] and evaluated at a fixed point x 0 in the set {W ≤ n}: n Φn (s) = Psn (k∇Pt−s f k 2 ).

3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 313

We have n n n Φn0 (s) = Psn (L n k∇Pt−s f k 2 − 2h∇L n Pt−s f , ∇Pt−s f i).

Now, observe that by assumption, and denoting by K − the negative part of K, n n n L n k∇Pt−s f k 2 − 2h∇L n Pt−s f , ∇Pt−s fi n n n n = h2n T2 (Pt−s f , Pt−s f ) − 4hn LPt−s f h∇hn , ∇Pt−s fi n n n ≥ −2K h2n k∇Pt−s f k 2 − 4hn LPt−s f h∇hn , ∇Pt−s fi n n n ≥ −2K h2n k∇Pt−s f k 2 − 4Pt−s L n f h∇ ln hn , ∇Pt−s fi n n ≥ −2K h2n k∇Pt−s f k 2 − 4kL f k∞ k∇ ln hn kk∇Pt−s fk n 2 ≥ −(2K − + 2)k∇Pt−s f k 2 − 2kL f k∞ k∇ ln hn k 2 .

The term k∇ ln hn k can be estimated as follows inside the set {W ≤ 2n}: ! C 1 0 W h k∇W k ≤ , k∇ ln hn k = − nhn n hn where C is a constant independent of n. On the other hand a direct computation and the assumptions on W show that ! 1 C Ln 2 ≤ 2 , hn hn where, again, C is a constant independent from n. This last estimate classically implies ! 1 eC s n Ps ≤ . h2n h2n Putting the pieces together we end up with a differential inequality Φn0 (s) ≥ −(2K − + 2)Φn (s) − C, where C now depends on f and t, but still does not depend on n. Integrating this inequality from 0 to t yields a bound of the type k∇Ptn f k ≤ C, where C depends on f and t. This bounds holds uniformly on the set {W ≤ n}. We now pick any x, y ∈ M, f ∈ C0∞ (M), and n big enough so that x, y ∈ {W ≤ n} and Supp( f ) ⊂ {W ≤ n}. We have from the previous inequality |Ptn f (x) − Ptn f (y)| ≤ C d(x, y),

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Fabrice Baudoin

and thus, by taking the limit when n → ∞, |Pt f (x) − Pt f (y)| ≤ C d(x, y). We therefore reach the important conclusion that Pt transforms C0∞ (M) into a subset of the set of smooth and Lipschitz functions. With this conclusion in hand, we can now run the usual Bakry–Émery machinery. Let f ∈ C0∞ (M), and T > 0, and consider the function φ(x,t) = k∇PT −t f k 2 (x). We have

∂φ = 2T2 (PT −t f , PT −t f ) ≥ −2K φ. ∂t Since we know that φ is bounded, we can use a parabolic comparison principle similar to the one in Proposition 5.7 to conclude, thanks to Gronwall’s inequality, that Lφ +

k∇Pt f k 2 ≤ e2K t Pt (k∇ f k 2 ). This inequality is then easily extended to any bounded and Lipschitz function f .  Under the same assumptions, we can actually get slightly stronger bounds: Theorem 7.4. Let us assume that for some K ∈ R, RicV − DY ≥ −K; then for every non-negative function f ∈ C ∞ (M) such that Lipschitz, we have, for t ≥ 0,

p

f is bounded and

(Pt f )k∇ ln Pt f k 2 ≤ e2K t Pt ( f k∇ ln f k 2 ). Proof. Notice that if φ(x,t) = (PT −t f )k∇ ln PT −t f k 2 (x), we have

∂φ = 2(PT −t f ) T2 (ln PT −t f , ln PT −t f ), ∂t where we use the fact that since the foliation is totally geodesic we have for every smooth g, h∇H g, ∇H k∇V gk 2 i = h∇V g, ∇V k∇H gk 2 i. Lφ +

The proof then follows the same lines as the proof of Theorem 7.3.



3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 315

We now turn to the problem of convergence to an equilibrium for the semigroup Pt and connect this problem to functional inequalities satisfied by the equilibrium measure. Our first result is the counterpart to Kolmogorov-type operators of the famous Bakry–Émery criterion [5]. Theorem 7.5. Assume that for some % > 0, RicV − DY ≥ %, and that there exists a probability measure µ on M such that for every x ∈ M and bounded f , Z lim Pt f (x) = f dµ. t→+∞

M

Then, µ satisfies the log-Sobolev inequality "Z ! !# Z Z Z 1 2 f k∇ ln f k dµ ≥ f ln f dµ − f dµ ln f dµ . 2% M M M M Proof. Let g ∈ C0∞ (M), g ≥ 0 and define f = g + ε where ε > 0. Since µ needs to be an invariant measure for L, we have Z Z Z Z +∞ Z ∂ f ln f dµ − f dµ ln f dµ = − (Pt f )(ln Pt f )dµ dt ∂t M M M Z0 +∞ Z M =− (LPt f )(ln Pt f )dµ dt 0 M Z +∞ Z k∇V Pt f k 2 = dµ dt Pt f 0 M Z +∞ Z = Pt f k∇V ln Pt f k 2 dµ dt Z0 +∞ M Z −2%t ≤ e dt k f ∇ ln f k 2 dµ 0 M Z 1 ≤ f k∇ ln f k 2 dµ. 2% M p We then extend the inequality to any non-negative f such that f is bounded and Lipschitz.  We now study the converse question which is to understand how a functional inequality satisfied by an invariant measure implies the convergence to equilibrium of the semigroup. The easiest convergence to deal with is L 2 convergence and, as is well known, it is connected to the Poincaré inequality.

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Theorem 7.6. Assume that there exist two constants %1 ≥ 0, %2 > 0 such that for every X ∈ Γ∞ (TM), 2 h(RicV − DY )(X ), Xi ≥ −%1 k X kV2 + %2 k X kH .

Assume moreover that the operator L admits an invariant probability measure µ that satisfies the Poincaré inequality ! 2 Z Z  k∇ f k 2 dµ ≥ κ  f 2 dµ − f dµ  .  M  M M R Then, for every bounded and Lipschitz function f such that M f dµ = 0, Z Z 2 ( %1 + %2 ) (Pt f ) dµ + k∇Pt f k 2 dµ M M ! Z Z ≤ e−λt ( %1 + %2 ) f 2 dµ + k∇ f k 2 dµ , Z

M

where λ =

M

2% 2 κ κ+% 1 +% 2 .

Proof. We fix t > 0 and consider the functional Ψ(s) = ( %1 + %2 )Ps ((Pt−s f ) 2 ) + Ps (k∇Pt−s f k 2 ). By repeating the arguments of the proof of the previous theorem, we get the differential inequality Z s Ψ(s) − Ψ(0) ≥ 2%2 Pu (k∇Pt−u f k 2 )du. 0

Now define ε =

% 1 +% 2 κ+% 1 +% 2 .

ε

We have, from the assumed Poincaré inequality,

Z

k∇Pt−u f k 2 dµ ≥ εκ M

Therefore, defining Θ(s) =

Z (Pt−u f ) 2 dµ. M

R R2n

Ψ(s)dµ, we obtain

Z sZ Z sZ Θ(s) − Θ(0) ≥ 2η(1 − ε) k∇Pt−u f k 2 dµ du + 2εκ (Pt−u f ) 2 dµ du 0 M 0 M Z s Θ(u)du. ≥λ 0

We then conclude with Gronwall’s differential inequality.



3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations 317

We can similarly prove convergence to equilibrium in the entropic distance provided the assumption that the invariant measure satisfies a log-Sobolev inequality. Theorem 7.7. Assume that there exist two constants %1 ≥ 0, %2 > 0 such that for every X ∈ Γ∞ (TM), 2 h(RicV − DY )(X ), Xi ≥ −%1 k X kV2 + %2 k X kH .

Assume moreover that the operator L admits an invariant probability measure µ that satisfies the log-Sobolev inequality "Z ! !# Z Z Z 2 f k∇ ln f k dµ ≥ κ f ln f dµ − f dµ ln f dµ . M

M

M

M

p

∞ RThen for every positive and bounded f ∈ C (M), such that k∇ f k is bounded and f dµ = 1, M Z Z 2( %1 + %2 ) Pt f ln Pt f dµ + Pt f k∇ ln Pt f k 2 dµ M M ! Z Z −λt 2 ≤e 2( %1 + %2 ) f ln f dµ + f k∇ ln f k dµ , M

where λ =

M

2% 2 κ κ+2(% 1 +% 2 ) .

7.3 The kinetic Fokker–Planck equation. In this section we study the kinetic Fokker–Planck equation which is an important example of an equation to which our methods apply. Let V : Rn → R be a smooth function. The kinetic Fokker–Planck equation with confinement potential V is the parabolic partial differential equation ∂h = ∆v h − v · ∇v h + ∇ x V · ∇v h − v · ∇ x h, ∂t

(x, v) ∈ R2n .

(7.2)

The operator L = ∆v − v · ∇v + ∇ x V · ∇v − v · ∇ x is a Kolmogorov-type operator. The foliation on R2n which is relevant here is not the trivial one. We endow R2n with the translation-invariant metric that makes ( ) ∂ ∂ ∂ 2 + , ,1 ≤ i ≤ n ∂x i ∂vi ∂vi an orthonormal basis at any point. We then consider the foliations with leaves {(x, v), v ∈ Rn }. It is obviously totally geodesic.

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The operator L admits as invariant measure the measure dµ = e−V (x)−( kv k

2 /2)

dx dv.

It is readily checked that L is not symmetric with respect to µ. The operator L is the generator of a strongly continuous sub-Markov semigroup (Pt )t ≥0 . If we assume that the Hessian ∇2V is bounded, which we do in the sequel, then Pt is Markovian. Observe that since ∇V is Lipschitz, the function W (x, v) = 1 + k xk 2 + kvk 2 is such that, for some constant C > 0, LW ≤ CW and k∇W k ≤ CW . The quadratic form T2 is easy to compute in this case and we then obtain the following result that was first obtained in [9]: Proposition 7.8. For every 0 < η < 21 , there exists K (η) ≥ − 12 such that for every f ∈ C ∞ (R2n ), T2 ( f , f ) ≥ −K (η)k∇V f k 2 + η k∇H f k 2 . The previous proposition shows that Theorems 7.6 and 7.7 thus apply to the kinetic Fokker–Planck operator. We mention that the entropic of the semigroup under the assumption that the invariant measure satisfies a log-Sobolev inequality was first established by Villani (see [50, Theorem 35]) but the rate of convergence given by Theorems 7.6 and 7.7 is more explicit.

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Index area formula, 26, 46–52, 56, 79 Baker–Campbell–Hausdorff formula, 133, 138 ball-box theorem, 205 basic vector field, 262∗ , 263 Bishop–Gromov theorem, 234 blow-up theorem, 90 Bochner’s identity, 219 Bott connection, 282∗ bounded variation, 179∗ BV -function, 32∗ Caccioppoli set (or G-Caccioppoli set), 33∗ , 34, 92 Carnot group, 3∗ , 131∗ , 136 Carnot–Caratheódory distance, 8∗ , 162∗ carré du champ, 240 characteristic point, 44 of a function, 97∗ of a regular surface, 95 of a set with finite Euclidean perimeter, 87∗ Chow–Rashevsky theorem, 164 Cohn-Vossen theorem, 171 complementary subgroup, 35∗ , 38–41 cone, 57∗ connection, 218∗ contact manifold, 267∗ , 283 corkscrew condition, 163, 191 CR Yamabe problem, 154∗ curvature-dimension inequality, 224∗ , 289∗ cut locus, 220 cut point, 220 dilation, 4∗ , 131∗ Dirichlet form, 303∗

distance d ∞ , 9∗ divergence theorem, 93 doubling condition, 189, 233–234 Engel group, 37∗ , 91, 107, 139∗ Euclidean group, 4∗ , 105 exponential coordinates, 3∗ , 135∗ finite rank condition, 125∗ Folland–Stein embedding theorem, 156 fractional integration operator, 198∗ G-perimeter, 33∗ , 34, 46, 83–107, 181∗ G-regular hypersurface, 43∗ , 45 gauge, 132, 144, 175 generalized curvature-dimension inequality, 244∗ generating system of vector fields, 3∗ group constants, 136 group law structure, 5 group of Heisenberg type, 141–143, 146–147, 149–150 H-regular surface, 54∗ , 55, 72, 75, 76 harmonic distribution, 203∗ harmonic function, 134∗ Harnack inequality, 230 Hausdorff dimension, 13∗ , 14, 17, 49, 50 Hausdorff measure, 12∗ , 16–19, 48–51 heat kernel, 294∗ , 301 heat semigroup, 292 Heisenberg group, 4∗ , 52, 72, 96, 106, 137∗ , 268∗ H-linear map, 21∗ , 23, 26 homogeneous dimension, 10∗ , 17, 26, 31, 34, 45, 49, 50, 128, 132∗ , 190 homogeneous norm, 7∗ Hopf fibration, 271∗ , 290

324

Index

horizontal curve, 168∗ horizontal gradient, 134∗ horizontal Laplacian, 263∗ , 290, 293 horizontal layer, 131 horizontal p-Laplacian, 158∗ hypocoercivity, 310∗ hypoelliptic operator, 125∗ implicit function theorem, 45 intrinsic differential, 67∗ –71 intrinsic graph, 41∗ , 45, 55–72 intrinsic linear map, 65∗ –67 intrinsic Lipschitz function, 57∗ , 58, 60–64, 76 intrinsic Lipschitz graph, 58∗ , 64–65 invariant distance, 7∗ isoperimetric inequality, 213 kinetic Fokker–Planck equation, 317∗ Kolmogorov-type operator, 309∗ Laplacian comparison theorem, 220 left translation, 4∗ length-space, 170∗ Li–Yau inequality, 217, 228–230, 299 local doubling condition, 204 locally subelliptic operator, 239 log-Sobolev inequality, 315∗ measure-theoretic boundary, 47∗ , 93 Metivier group, 141∗ metric measure, 12∗ Morrey space, 193∗ Nagel–Stein–Wainger polynomial, 203∗ P-differential, 23∗ , 26 Pansu differential, 23∗ , 26 Pansu–Rademacher theorem, 23 perimeter, 33∗ , 34, 46, 83–107, 181∗ perimeter minimizer, 94∗ , 105 Poincaré inequality, 31 quaternionic Hopf fibration,

275∗ ,

Rademacher-type theorem, 73

291

rectifiability in Carnot groups, 80∗ in Heisenberg groups, 81∗ –83 in metric spaces, 79∗ reduced boundary, 85∗ , 92 regular hypersurface, 43∗ , 45 regular surface, 54∗ , 55, 72, 75, 76 Ricci tensor, 218∗ Riemann curvature tensor, 218∗ Riemannian foliation, 266∗ , 309 Riemannian submersion, 261∗ segment property, 171∗ Sobolev embedding, 212, 215 Sobolev space, 28∗ –29 strong, 176∗ weak, 176∗ space of homogeneous type, 189 step, 3∗ stochastic completeness, 297∗ stratification, 131 strong Sobolev space, 176∗ sub-Laplacian, 134∗ , 137 subelliptic mollifier, 209∗ t-graph, 41∗ , 81, 96–101 tangent group to a regular hypersurface, 43∗ to a regular surface, 55 torsion tensor, 218∗ total variation, 179∗ totally geodesic, 262∗ , 265, 266, 310 variation bounded, 179∗ total, 179∗ vertical Laplacian, 263∗ weak Sobolev space, 176∗ Whitney extension theorem, 27 X1 -graph, 101∗ , 102–105 Yamabe equation, 153 Yang–Mills connection, 284∗

Series of Lectures in Mathematics

Volume I Davide Barilari, Ugo Boscain and Mario Sigalotti Editors Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds.

The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.

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Davide Barilari, Ugo Boscain and Mario Sigalotti, Editors

In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology.

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds, Volume I

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds

Geometry, Analysis and Dynamics on sub-Riemannian Manifolds Volume I Davide Barilari Ugo Boscain Mario Sigalotti Editors