Harmonic Analysis Short Lecture Notes

Table of contents :
Chapter 1. Fourier series on T
1.1. Fourier coefficients and series
1.2. Summability of Fourier series
1.3. Pointwise convergence of Fejér means
1.4. Fourier series in L2(T)
1.5. Convergence of Fourier series
1.6. Fourier multipliers
Chapter 2. The Fourier transform on Rd
2.1. The Fourier transform on L1(Rd)
2.2. Schwartz functions and the Fourier transform on L2(Rd)
2.3. The Fourier transform on Lp(Rd) for 1 p 2
2.4. Tempered distributions
2.5. Fourier multipliers
Chapter 3. Real harmonic analysis
3.1. Coverings and maximal functions
3.2. Real interpolation
3.3. Calderón-Zygmund operators
3.4. Mihlin's multiplier theorem
3.5. Littlewood-Paley theory
Chapter 4. Oscillatory integrals
4.1. The wave equation in Rd R
4.2. Oscillatory integrals
4.3. The Fourier transform of surface-carried measures
4.4. Application to the Schrödinger equation
Bibliography

Citation preview

Harmonic Analysis Short Lecture Notes Winter semester 2024/25

Dorothee Frey Fakultät für Mathematik Karlsruher Institut für Technologie 12.02.20251

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The lecture notes are continuously updated. It is intended as a supplement to the lecture and does not contain all the proofs, examples, sketches and explanations of the lecture.

Contents Chapter 1. Fourier series on T 1.1. Fourier coefficients and series 1.2. Summability of Fourier series 1.3. Pointwise convergence of Fejér means 1.4. Fourier series in L2 (T) 1.5. Convergence of Fourier series 1.6. Fourier multipliers

5 5 8 11 12 14 16

Chapter 2. The Fourier transform on Rd 2.1. The Fourier transform on L1 (Rd ) 2.2. Schwartz functions and the Fourier transform on L2 (Rd ) 2.3. The Fourier transform on Lp (Rd ) for 1 ≤ p ≤ 2 2.4. Tempered distributions 2.5. Fourier multipliers

19 19 23 26 27 33

Chapter 3. Real harmonic analysis 3.1. Coverings and maximal functions 3.2. Real interpolation 3.3. Calderón-Zygmund operators 3.4. Mihlin’s multiplier theorem 3.5. Littlewood-Paley theory

41 41 46 48 54 55

Chapter 4. Oscillatory integrals 4.1. The wave equation in Rd × R 4.2. Oscillatory integrals 4.3. The Fourier transform of surface-carried measures 4.4. Application to the Schrödinger equation

61 61 63 68 69

Bibliography

75

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CHAPTER 1

Fourier series on T In this first chapter, we give a brief introduction to Fourier series and the question of convergence and summability of Fourier series. For a more comprehensive study of Fourier series, we refer to the first three chapters of the textbook [10].

1.1. Fourier coefficients and series In the following, we consider 2π-periodic functions f : R → C, i.e. functions satisfying f (x + 2π) = f (x),

x ∈ R.

Then f is uniquely determined by f |[0,2π) (or any other interval of length 2π). Thus consider R/2πZ = {x + 2πZ : x ∈ R} = {x + 2πZ : x ∈ [0, 2π)}. R/2πZ is called the (one-dimensional) torus, denoted by T. We remark that the map (R/2πZ, +) → ({z ∈ C : |z| = 1}, · ),

[t] 7→ eit ,

is an isomorphism. We often take the interval [0, 2π) as a model for T. We use the Lebesgue measure on T, and write ˆ ˆ 2π f (t) dt := f (x) dx. 0

T

Note that the Lebesgue measure on T is translation invariant, that is, ˆ ˆ f (t − t0 ) dt = f (t) dt, t0 ∈ T. T

T

In this chapter, we use the following convention: For p ∈ [1, ∞), we equip Lp (T) with the norm 1/p  ˆ 1 p |f (t)| dt . ∥f ∥Lp (T) := 2π T The space (Lp (T), ∥ . ∥Lp (T) ) is a Banach space. We write C(T) := {f : R → C continuous : f 2π-periodic}, equipped with the supremum norm ∥ . ∥∞ . 5

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CHAPTER 1. FOURIER SERIES ON T

Definition 1.1. A trigonometric polynomial is an expression of the form (1.1)

P (t) :=

N X

ak eikt ,

t ∈ T,

k=−N

where N ∈ N and ak ∈ C for k ∈ {−N, . . . , N }. If aN ̸= 0 or a−N ̸= 0, then N is called the degree of P , ak are the coefficients of P , and the occurring k’s (with ak ̸= 0) are called frequencies. Remark. 1) We will frequently use the fact that for k ∈ Z, ( ˆ 2π 1 1, k = 0, eikt dt = 2π 0 0, k ̸= 0. 2) If a trigonometric polynomial P : T → C is given, one can compute the coefficients by ˆ ˆ N X 1 1 −int P (t)e dt = ak ei(k−n)t dt = an , |n| ≤ N. 2π T 2π T k=−N Thus, the function P : T → C uniquely determines (1.1). Definition 1.2. Let f ∈ L1 (T), let n ∈ Z. The n-th Fourier coefficient of f is defined by ˆ 1 fˆ(n) := f (t)e−int dt. 2π T The Fourier series of f is given by ∞ X (1.2) fˆ(n)eint , t ∈ T. S[f ] ∼ n=−∞

Note that the expression in (1.2) is only formal at first, and does not imply convergence in any sense. Lemma 1.3. Let f, g ∈ L1 (T), let n ∈ Z. Then c )(n) = αfˆ(n) for α ∈ C (linearity). (i) (f[ + g)(n) = fˆ(n) + gˆ(n), and (αf ´ 2π 1 (ii) |fˆ(n)| ≤ 2π |f (t)| dt = ∥f ∥L1 (T) . 0 I.e., the map (1.3) L1 (T) → ℓ∞ (Z), f 7→ (fˆ(n))n∈Z is linear and continuous. (iii) For τ ∈ T, denote by fτ := f ( . − τ ) the translation of f . Then fbτ (n) = e−inτ fˆ(n). The statement in (1.3) can be improved: The Fourier coefficients of an integrable function even form a null sequence, i.e. one has a map L1 (T) → c0 (Z).

1.1. FOURIER COEFFICIENTS AND SERIES

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Lemma 1.4 (Riemann-Lebesgue lemma). Let f ∈ L1 (T). Then lim fˆ(n) = 0. |n|→∞

Remark. 1) A function F : [a, b] → C is called absolutely continuous, if there exists f ∈ L1 ([a, b]) such that for all x ∈ [a, b] ˆ x f (t) dt. F (x) = F (a) + a ′

If F is absolutely continuous, then F = f almost everywhere. If we set F ′ (x) := 0 for F not differentiable in x, we have F ′ ∈ L1 ([a, b]). There exist functions that are differentiable almost everywhere, but are not absolutely continuous. Every Lipschitz continuous function is absolutely continuous. See [5, Section VII.4] for more details on absolutely continuous functions. 2) (Integration by parts for absolutely continuous functions) For F, G : [a, b] → C absolutely continuous with F ′ = f , G′ = g ∈ L1 ([a, b]), we have ˆ b ˆ b b F (t)g(t) dt = [F (t)G(t)]a − f (t)G(t) dt. a

a

Using the integration by parts formula above, we can show the following decay of Fourier coefficients of absolutely continuous functions with integral 0. ´t Lemma 1.5. Let f ∈ L1 (T), fˆ(0) = 0. Define F (t) := 0 f (s) ds, t ∈ T. Then F is continuous, 2π-periodic, and 1 Fˆ (n) = fˆ(n), n ∈ Z, n ̸= 0. in Remark. Iterating the statement of Lemma 1.5 directly implies the following: If f ∈ C k (T) = {f ∈ C(T) : f k-times continuously differentiable} for some k ∈ N, then |fˆ(n)| ≤

∥f ∥C k (T) , |n|k

n ∈ Z, n ̸= 0.

We introduce the convolution operator and show that the Fourier transform (1.3) turns convolutions of two functions into products. Definition and Lemma 1.6. (Convolution) Let f, g ∈ L1 (T). For a.e. t ∈ T, the function T → C, τ 7→ f (t − τ )g(τ ) is integrable. Define the function f ∗ g : T → C by ˆ 2π 1 f ∗ g(t) := f (t − τ )g(τ ) dτ, t ∈ T. 2π 0 Then f ∗ g ∈ L1 (T), and ∥f ∗ g∥L1 (T) ≤ ∥f ∥L1 (T) ∥g∥L1 (T) . Moreover, f[ ∗ g(n) = fˆ(n)ˆ g (n),

n ∈ Z.

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CHAPTER 1. FOURIER SERIES ON T

Remark. The map ∗ : L1 (T) × L1 (T) → L1 (T) is bilinear, continuous, associative, and distributive with respect to addition. The convolution of a function f with complex exponentials yields a trigonometric polynomial with coefficients (fˆ(n)). This representation will be the starting point for summability considerations in the next section. Lemma 1.7. Let f ∈ L1 (T), let n ∈ Z. Let φn (t) := eint , t ∈ T. Then φn ∗ f (t) = fˆ(n)eint , t ∈ T. PN More generally, if k(t) := n=−N an eint , t ∈ T, for some N ∈ N and an ∈ C, then k ∗ f (t) =

N X

an fˆ(n)eint .

n=−N

1.2. Summability of Fourier series We introduce the two convolution kernels (Dn )n∈N and (Fn )n∈N , the Dirichlet kernel and the Fejér kernel. We will see later on that there are large differences between the behaviour of the two kernels. Definition 1.8.

(i) The Dirichlet kernel (Dn )n∈N is defined by n X Dn (t) := eikt , t ∈ T, n ∈ N. k=−n

For f ∈ L1 (T), we denote by Sn (f ) the n-th partial sum of S[f ], i.e. n X fˆ(k)eikt , n ∈ N, t ∈ T. Sn (f )(t) := Sn (f, t) := k=−n

(ii) The Fejér kernel (Fn )n∈N is defined by  n  X |j| Fn (t) := 1− eijt , n + 1 j=−n

t ∈ T, n ∈ N.

For f ∈ L1 (T), we denote by σn (f )(t) := σn (f, t) := (Fn ∗ f )(t),

t ∈ T, n ∈ N,

the n-th Fejér means of f . Remark. 1) By Lemma 1.7, we have Sn (f ) = Dn ∗ f , n ∈ N. 2) We have sin(n + 21 )t , t ∈ T, n ∈ N, Dn (t) = sin 12 t  2 sin n+1 t 1 2 Fn (t) = , t ∈ T, n ∈ N. n+1 sin 12 t

1.2. SUMMABILITY OF FOURIER SERIES

9

3) The family (σn )n are the Cesàro means (arithmetic means) of (Sn )n , i.e. 1 σn (f )(t) = (S0 (f ) + S1 (f ) + . . . + Sn (f ))(t) n+1 n 1 X = (n + 1 − |j|)fˆ(j)eijt , t ∈ T. n + 1 j=−n We note that convergent series are Cesàro summable (i.e. have convergent Cesàro means), but not vice versa in general. We postpone the more difficult question of convergence of Fourier series (Sn (f ))n , to a later section, and focus first on the question of Cesàro summability σn (f ) → f , n → ∞, in Lp (T) or pointwise. We will see later, that, for example, there exist f ∈ L1 (T) with ∥Sn (f ) − f ∥L1 (T) ̸→ 0,

n → ∞.

We state important properties of the Fejér kernel. Lemma 1.9. For the Fejér kernel (Fn )n∈N , we have ˆ 1 (i) Fn (t) dt = 1, n ∈ N. 2π T (ii) (Fn )n is positive, i.e. Fn (t) ≥ 0 for all t ∈ T, n ∈ N. (iii) For all δ ∈ (0, π), 1 lim n→∞ 2π

ˆ

2π−δ

|Fn (t)| dt = 0. δ

Remark. There is a more general concept of Dirac kernels, also called summability kernels, i.e., families (kn )n∈N ⊆ C(T) satisfying (i), (iii) and (ii)’: supn∈N ∥kn ∥L1 (T) < ∞ of Lemma 1.9, for which the main statement below (Theorem 1.11) remains true for (kn )n instead of (Fn )n . See [10, Definition I.2.2 and Theorem I.2.3]. For the summability of Cesàro means, the most relevant properties of the underlying function spaces are the following. Lemma 1.10. Let p ∈ [1, ∞), let f ∈ Lp (T). (i) Let τ ∈ T. Then fτ ∈ Lp (T) with ∥fτ ∥Lp (T) = ∥f ∥Lp (T) . (ii) The map T → Lp (T), τ 7→ fτ is continuous. I.e., for every τ0 ∈ T, lim ∥fτ − fτ0 ∥Lp (T) = 0.

τ →τ0

Remark (Minkowski’s integral inequality). Let p ∈ [1, ∞), let (X, µ) and (T, ν) be two σfinite measure spaces. For F non-negative, measurable on the product space (X, µ)×(T, ν), we have ˆ ˆ p 1/p ˆ ˆ 1/p p F (x, t) dµ(x) dν(t) ≤ F (x, t) dν(t) dµ(x). T

X

X

T

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CHAPTER 1. FOURIER SERIES ON T

We state the main result of this section. Theorem 1.11 (Fejér’s theorem). Let p ∈ [1, ∞) and f ∈ Lp (T), or let p = ∞ and f ∈ C(T). Then ∥σn (f ) − f ∥Lp (T) → 0,

n → ∞.

Even if the statement does not (yet) give Lp convergence of Fourier series, we obtain the following two important consequences of Fejér’s theorem. Corollary 1.12 (Uniqueness theorem). Let f ∈ L1 (T). If fˆ(n) = 0 for all n ∈ Z, then f = 0 a.e. Equivalently, this states: If f, g ∈ L1 (T) with fˆ(n) = gˆ(n) for all n ∈ Z, then f = g a.e. Corollary 1.13. (i) Trigonometric polynomials are dense in Lp (T), p ∈ [1, ∞). (ii) (Weierstraß approximation theorem) Every continuous, 2π-periodic function can be approximated uniformly by trigonometric polynomials. We discuss another summation kernel, which considers Abel means instead of Cesàro means of Fourier series. It is closely linked to the theory of analytic functions and can be used to represent solutions to the Dirichlet problem on the disc. Example (Poisson kernel). For r ∈ (0, 1), we set ∞ X

P (r, t) :=

r|j| eijt = 1 + 2

j=−∞

∞ X

rj cos(jt),

t ∈ T.

j=1

Then P (r, t) = 1 + 2

∞ X

rj Re(eit )j = 1 + 2 Re

j=1

reit 1 + reit = Re 1 − reit 1 − reit

1 − r2 . 1 − 2r cos t + r2 The series converges absolutely with respect to ∥ . ∥L∞ (T) since r < 1, and (1.4)

=

P (r, t) ∗ f =

∞ X

fˆ(n)r|n| eint .

n=−∞

Pr ∗ f is the so-called Abel mean of S[f ], and convergence of Pr ∗ f for r → 1− is called Abel summability. We remark that (Pr )r∈(0,1) satsifies the following analogue properties of (Fn )n∈N shown in Lemma 1.9: (i) For every r ∈ (0, 1), we have because of uniform convergence of the series in t ∈ T ˆ ˆ 2π ˆ 2π ∞ X 1 1 1 P (r, t) dt = 1 dt + 2 rj cos(jt) dt = 1. 2π T 2π 0 2π 0 j=1

1.3. POINTWISE CONVERGENCE OF FEJÉR MEANS

11

(ii) P (r, t) ≥ 0 for all t ∈ T and r ∈ (0, 1). This follows from (1.4), since 1 − r2 > 0, and 1 − 2r cos t + r2 ≥ (1 − r)2 > 0. (iii) Let δ ∈ (0, π). Then ˆ 2π−δ ˆ 2π−δ 1 1 1 − r2 P (r, t) dt = dt 2π δ 2π δ 1 − 2r cos t + r2 1 ≤ (1 − r2 ) → 0, r →1−. 1 − 2r cos δ + r2 With analogous arguments as in Theorem 1.11, we obtain the following: For p ∈ [1, ∞) and f ∈ Lp (T), or for p = ∞ and f ∈ C(T), we have (1.5)

lim ∥Pr ∗ f − f ∥Lp (T) = 0.

r→1−

Dirchlet problem on the disc: Let p ∈ [1, ∞) and f ∈ Lp (T), or let p = ∞ and f ∈ C(T). We set u(reit ) := Pr ∗ f (t),

z = reit ∈ D = {z ∈ C : |z| < 1}.

Then u is harmonic on D. Thus u solves the Dirichlet problem ( ∆u = 0, |z| < 1, u = f, |z| = 1, where u = f on ∂D is interpreted in the sense of (1.5). 1.3. Pointwise convergence of Fejér means For f ∈ C(T), Theorem 1.11 yields pointwise and uniform convergence of Fejér means. We now discuss pointwise convergence for functions which might have jumps, but where the averaged left and right limit exists. Typical examples are step functions, or the sawtooth function f (t) = t, |t| < π. Theorem 1.14 (Fejér). Let f ∈ L1 (T), let t0 ∈ T. Assume that fˇ(t0 ) := lim (f (t0 + h) + f (t0 − h)) h→0

exists, with values ±∞ allowed. Then 1 σn (f, t0 ) → lim (f (t0 + h) + f (t0 − h)), 2 h→0 In particular, if t0 is a point of continuity, then σn (f, t0 ) → f (t0 ),

n → ∞.

n → ∞.

Remark. The proof uses the following two properties of the Fejér kernel: (1) Let θ ∈ (0, π), let t ∈ T. Then limn→∞ (supt∈(θ,2π−θ) Fn (t)) = 0. (2) Fn (t) = Fn (−t) (even kernel).

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CHAPTER 1. FOURIER SERIES ON T

Remark. Theorem 1.14 in particular implies: If t0 is a point of continuity of f and if the Fourier series S[f ] of f converges at t0 , then its sum is f (t0 ). 1.4. Fourier series in L2 (T) We now turn to convergence of Fourier series. Here, the simplest case is convergence in L2 (T), since in the Hilbert space setting, we can exploit orthogonality of the complex exponentials (ein( . ) )n∈Z . We recall some basic facts from Hilbert space theory: Let H be a Hilbert space over C, p equipped with an inner product ( . , . ) and the induced norm ∥ . ∥ = ( . , . ). Then (i) f is called orthogonal to g, if (f, g) = 0, where f, g ∈ H. (ii) S ⊆ H is an orthonormal system (ONS) if ∀f, g ∈ S, f ̸= g :

(f, g) = 0,

∀f ∈ S :

(f, f ) = 1.

(iii) An orthonormal system {φα }α∈I in H is complete, if for all f ∈ H: ∀α ∈ I : (f, φα ) = 0

⇒

f = 0.

Bessel’s inequality states that if one expands functions with respect to an orthonormal system, then the coefficients form a sequence in ℓ2 (I) (where I can be an uncountable index set, but for every f ∈ H, the set {α ∈ I : aα ̸= 0} will be countable; see, e.g., [14, Chapter V.4] for more details). Lemma 1.15 (Bessel’s inequality). Let {φα }α∈I be an orthonormal system in H, let f ∈ H. Set aα := (f, φα ). Then X |aα |2 ≤ ∥f ∥2 . α∈I

Proof. See Functional Analysis.

□

We recall the following characterization of completeness of orthonormal systems. We only consider countable orthonormal systems, which suffices for the application we have in mind. Theorem 1.16. Let {φn }∞ n=1 be an orthonormal system in H. Then the following are equivalent: (i) {φn } is complete. ∞ X 2 (ii) For all f ∈ H: ∥f ∥ = |(f, φn )|2 . n=1

(iii) For all f ∈ H:

∞ X f= (f, φn )φn . n=1

1.4. FOURIER SERIES IN L2 (T)

(iv) For all f, g ∈ H:

13

∞ X (f, g) = (f, φn )(g, φn ). n=1

□

Proof. See Functional Analysis.

We apply Theorem 1.16 to the Hilbert space H = L2 (T), equipped with the inner product ˆ 1 (f, g) = f (t)g(t) dt. 2π T Completeness of (ein( . ) )n∈Z can be shown by the uniqueness theorem. Proposition 1.17. The exponentials {ein( . ) }∞ n=−∞ form a complete orthonormal system 2 in L (T). Realizing that the Fourier series is the expansion of a function f ∈ L2 (T) with respect to this orthonormal system, we directly obtain L2 convergence of Fourier series from Theorem 1.16. Theorem 1.18. Let f, g ∈ L2 (T). Then ˆ 2π ∞ X 1 2 (i) |fˆ(n)|2 |f (t)| dt = 2π 0 n=−∞ (ii) f = lim

N →∞

1 (iii) 2π

ˆ

N X

fˆ(n)ein( . )

(Plancherel).

with respect to ∥ . ∥2 .

n=−N

2π

f (t)g(t) dt = 0

∞ X

fˆ(n)ˆ g (n)

(Parseval).

n=−∞

P∞ 2 2 (iv) Let (an )n∈Z ∈ ℓ (Z), i.e. n=−∞ |an | < ∞. Then there exists a unique f ∈ L (T) such that an = fˆ(n), n ∈ Z. 2

Remark. 1) (ii) states that ∥SN (f )−f ∥L2 (T) → 0 for N → ∞. Compare with the weaker statement ∥σN (f ) − f ∥L2 (T) → 0 for N → ∞ in Theorem 1.11. P int 2 2) In (iv), f is given by limN →∞ N n=−N an e , with the limit taken in L (T). 3) Theorem 1.18 in particular states: The map L2 (T) → ℓ2 (Z), f 7→ (fˆ(n))n∈Z , is an isometric isomorphism. We give an application to Plancherel’s theorem. Corollary 1.19 (Poincaré-Wirtinger). Let f ∈ C 1 (T) with infˆ(n), n ∈ Z, and ˆ 2π ˆ 2π 2 |f (t)| dt ≤ |f ′ (t)|2 dt. 0

0

´ T

f (t) dt = 0. Then fb′ (n) =

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CHAPTER 1. FOURIER SERIES ON T

Remark. In the proof of Corollary 1.19, we use that the sequence (m(n))n∈Z ∈ ℓ∞ (Z), 1 where m(n) = in , n ̸= 0, and m(0) = 0. From functional analysis, we know that the multiplication operator Mm , defined by Mm : ℓ2 (Z) → ℓ2 (Z),

(a(n))n∈Z 7→ (m(n)a(n))n∈Z ,

is linear and bounded if and only if (m(n))n ∈ ℓ∞ (Z). We will discuss such multipliers in more details in the next section.

1.5. Convergence of Fourier series If the underlying space is not a Hilbert space, the question of convergence of Fourier series is more subtle. We give a brief overview of some of the results in this section, but we will mostly refer to the literature for proofs. In the next chapter, similar questions for the Fourier transform on Rd will be discussed in more depth. Pointwise convergence of Fourier series We recall the definition of spaces of Hölder continuous functions: For α ∈ (0, 1), define C α (T) := {f ∈ C(T) : sup t∈T h̸=0

|f (t + h) − f (t)| < ∞}, |h|α

equipped with the norm ∥f ∥C α (T) = ∥f ∥∞ + sup t∈T h̸=0

|f (t + h) − f (t)| . |h|α

Theorem 1.20 (Bernstein). Let α > 12 , let f ∈ C α (T). Then the Fourier series S[f ] of f is absolutely convergent, and ∞ X

|fˆ(n)| ≤ cα ∥f ∥C α (T) ,

n=−∞

with a constant cα > 0 only depending on α. Proof. See [10, Theorem I.6.3].

□

P P∞ int ˆ ˆ Remark. 1) Note that ∞ converges n=−∞ |f (n)| < ∞ implies that t 7→ n=−∞ f (n)e uniformly on T. 2) The condition α > 21 is sharp for absolute convergence of Fourier series. There exists P ein log n int f ∈ C 1/2 (T) such that S[f ] does not converge absolutely. Consider e.g. ∞ e . n=1 n 3) There exists f ∈ C(T) such that the Fourier series S[f ] diverges at a point. See [10, Theorem II.2.1].

1.5. CONVERGENCE OF FOURIER SERIES

15

We refer to [10, Section II.2+II.3] for a more detailed discussion on pointwise convergence (and divergence) of Fourier series. Convergence in norm of Fourier series Notation: In the following, let B be one of the Banach spaces Lp (T) for p ∈ [1, ∞), or C(T). Using the uniform boundedness principle, we can rephrase the question of norm convergence of Fourier series as follows. Theorem 1.21. The following are equivalent: (i) For all f ∈ B, limn→∞ ∥Sn (f ) − f ∥B = 0. (ii) There exists a constant C > 0 such that for all n ∈ N and f ∈ B, ∥Sn (f )∥B ≤ C∥f ∥B . We recall Young’s inequality. See Analysis 3, or, e.g., [8, Theorem 1.2.12]. Remark (Young’s inequality). Let p, q, r ∈ [1, ∞] satisfy f ∈ Lp (T), g ∈ Lq (T), f ∗ g exists almost everywhere, and

1 r

+1 =

1 p

+ 1q . Then for all

∥f ∗ g∥Lr (T) ≤ ∥f ∥Lp (T) ∥g∥Lq (T) . Remark. 1) Note that (ii) in Theorem 1.21 is equivalent to supn∈N ∥Sn ∥L(B) < ∞. 2) Since Sn (f ) = Dn ∗ f , we have ∥Sn (f )∥B = ∥Dn ∗ f ∥B ≤ ∥Dn ∥L1 (T) ∥f ∥B by Young’s inequality, hence (1.6)

∥Sn ∥L(B) ≤ ∥Dn ∥L1 (T) .

However, ∥Dn ∥L1 (T) → ∞, n → ∞, as the next lemma shows. Lemma 1.22. For n ∈ N, the Lebesgue constants Ln := ∥Dn ∥L1 (T) satisfy 4 Ln = 2 log n + O(1). π For B = L1 (T) or B = C(T), the inequality in (1.6) is actually an equality, thus norm convergence of Fourier series in the two endpoint fails. Proposition 1.23. We have

(i) L1 (T) does not admit convergence in norm of Fourier series.

∥Sn ∥L(L1 (T)) = ∥Dn ∥L1 (T) , n ∈ N. (ii) C(T) does not admit convergence in norm. We have ∥Sn ∥L(C(T)) = ∥Dn ∥L1 (T) ,

n ∈ N.

For p ∈ (1, ∞), the Fourier series converges in norm. Compared to the case B = L2 (T), it is however much more difficult to prove. We omit the proof here, but we will prove a corresponding statement on Rd later.

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CHAPTER 1. FOURIER SERIES ON T

Theorem 1.24 (M. Riesz). Let p ∈ (1, ∞). For every f ∈ Lp (T), we have ∥Sn (f ) − f ∥Lp (T) → 0,

n → ∞. □

Proof. See [10, Theorem III.1.11]. 1.6. Fourier multipliers

In the proof of Corollary 1.19, we have seen that anti-derivatives can be represented as bounded multiplication operators on the sequence of Fourier coefficients, and are thus bounded on L2 (T). In this section, we discuss in more generality, under which conditions multiplication operators on Fourier coefficients give rise to bounded operators on Lp (T). Let B again be one of the spaces Lp (T) for p ∈ [1, ∞), or C(T). We recall the notation fτ = f ( . − τ ). Definition 1.25. An operator T ∈ L(B) is called translation invariant, if for all τ ∈ T and f ∈ B, we have T (fτ ) = (T f )τ = (T f )(· − τ ). Example (Convolution operator). Let f ∈ L1 (T). Define the convolution operator Tf : B → B,

g 7→ f ∗ g.

Then Tf is well-defined by Young’s inequality, with ∥Tf (g)∥B = ∥f ∗ g∥B ≤ ∥f ∥L1 (T) ∥g∥B , thus Tf ∈ L(B) with ∥Tf ∥L(B) ≤ ∥f ∥L1 (T) . We show that Tf is translation invariant: For τ ∈ T, g ∈ B, we have ˆ ˆ 1 1 Tf (gτ )(t) = (f ∗ gτ )(t) = f (s)gτ (t − s) ds = f (s)g((t − τ ) − s) ds 2π T 2π T = (f ∗ g)(t − τ ) = (Tf (g))τ (t), t ∈ T. We show that translation invariant operators can be represented as multiplication operators on the Fourier coefficients. Lemma 1.26. Let T ∈ L(B) be translation invariant. Then there exists a sequence (an )n∈Z with Tcf (n) = an fˆ(n), n ∈ Z, f ∈ B. Remark. 1) Lemma 1.26 states: Given T ∈ L(B) translation invariant, there exists a sequence (an )n∈Z with (1.7)

T en = an en ,

n ∈ Z,

where en = ein( . ) . 2) We in particular have ∥(an )∥ℓ∞ (Z) ≤ ∥T ∥L(B) , since |an | = ∥T en ∥B ≤ ∥T ∥L(B) ∥en ∥B = ∥T ∥L(B) .

1.6. FOURIER MULTIPLIERS

17

How about the reverse? Given a sequence (an ) ∈ ℓ∞ (Z), is there a corresponding translation invariant operator T ∈ L(B) satisfying (1.7)? Definition 1.27. A sequence m = (mn )n∈Z is called a Fourier multiplier for B if there exists an operator T ∈ L(B) such that T = Tm = F −1 Mm F. Remark. 1) If m is a Fourier multiplier, i.e. if there exists T ∈ L(B) such that T = Tm , then T en = mn en ,

(1.8)

n ∈ Z.

T is uniquely determined by (1.8), since (1.8) determines T for trigonometric polynomials, and these are dense in B. 2) If m = (mn )n∈Z is some sequence (not necessarily a Fourier multiplier), then necessarily Tm (p) =

N X

mn pˆ(n)ein( . ) .

n=−N

In order to check that T = Tm ∈ L(B), it suffices to check that there exists some C > 0 such that ∥Tm (p)∥B ≤ C∥p∥B for all trigonometric polynomials p. Tm is translation invariant by construction (exercise). Example. 1) Consider B = L2 (T). Then by Theorem 1.18, L2 (T) → ℓ2 (Z), f 7→ (fˆ(n))n∈Z , is an isometric isomorphism. We recall that a multiplication operator Mm : ℓ2 (Z) → ℓ2 (Z) is bounded if and only if m ∈ ℓ∞ (Z). Thus, a sequence m = (mn )n is a Fourier multiplier on L2 (T) if and only if m ∈ ℓ∞ (Z). 2) We denote by M (T) the space of finite (complex) Borel measures on T. By Riesz’ representation theorem (see, e..g, [14, Theorem II.2.5]), we have (C(T), ∥ . ∥∞ )´′ ∼ = M (T), equipped with the total variation norm ∥ . ∥M (T) . We denote ⟨f, µ⟩ := T f d¯ µ for f ∈ C(T), and define the Fourier coefficients of µ ∈ M (T) by ˆ int µ ˆ(n) := ⟨e , µ⟩ = e−int dµ, n ∈ Z. T

Then |ˆ µ(n)| = |µ(e−int )| ≤ ∥µ∥M (T) ∥e−in( . ) ∥∞ = ∥µ∥M (T) , thus (ˆ µ(n))n∈Z ∈ ℓ∞ (Z). We obtain the following result, stating that the sequence of Fourier coefficients of finite Borel measures are Fourier multipliers for all Banach spaces B under consideration. Theorem 1.28. Let µ ∈ M (T). Then µ ˆ = (ˆ µ(n))n∈Z is a Fourier multiplier on B with ∥Tµˆ ∥L(B) ≤ ∥µ∥M (T) .

18

CHAPTER 1. FOURIER SERIES ON T

□

Proof. See [10, Theorem I.7.9].

One can show that a sequence m is a Fourier multiplier on C(T) (or on L1 (T)), if and only if there exists some µ ∈ M (T) such that m = µ ˆ ([8, Theorem 4.3.4]). However, for p ∈ (1, ∞), p ̸= 2, a nontrivial characterization of the class of Fourier multipliers is not known. Additional comments Remark (Hausdorff-Young inequality). Let p ∈ [1, 2], and ′ have (fˆ(n))n∈Z ∈ ℓp (Z), with

1 p

+ p1′ = 1. For f ∈ Lp (T), we

∥(fˆ(n))∥ℓp′ (Z) ≤ ∥f ∥Lp (T) . This follows from the Riesz-Thorin interpolation theorem ([14, Theorem II.4.2]). Remark (Gibbs phenomenon). Consider f (t) = sgn(t), t ∈ (−π, π). Then one can show pointwise convergence of the Fourier series, i.e., for all t ∈ (−π, π), SN f (t) → f (t), N → ∞. But ˆ 2 π sin s ds ≈ 1, 17898 . . . , lim sup SN f (t) = N →∞ t∈(−π,π) π 0 s thus the amount by which SN f oversteps 1 does not tend to 0 for N → ∞ (overshoot at the jump discontinuity). Remark (Carleson-Hunt theorem). Let p ∈ (1, ∞), let f ∈ Lp (T). Then lim SN f (t) = f (t)

N →∞

for a.e. t ∈ T.

See [8, Theorem 4.3.14] and the reference there for the proof. The statement is false for p = 1 (Kolmogorov).

CHAPTER 2

The Fourier transform on Rd We study the Fourier transform on Rd . We will see that many results in L1 (Rd ) will be similar to the theory of Fourier series on the torus. However, extensions to Lp (Rd ) for p > 1 will be more involved as the Lebesgue measure of Rd is infinite, and we no longer have an inclusion of Lp spaces as it was the case for T. For this reason, we introduce the space of Schwartz functions and its dual space, the space of tempered distributions. 2.1. The Fourier transform on L1 (Rd ) Notation: We often write xξ = x · ξ for the inner product of x, ξ ∈ Rd . Definition 2.1. Let f ∈ L1 (Rd ). The Fourier transform fˆ of f is defined by ˆ ˆ (2.1) f (x)e−ixξ dx, ξ ∈ Rd . f (ξ) := Rd

We collect some basic properties of the Fourier transform on L1 (Rd ). Lemma 2.2. Let f ∈ L1 (Rd ). (i) Then fˆ ∈ L∞ (Rd ), and ∥fˆ∥∞ ≤ ∥f ∥1 . The map L1 (Rd ) → L∞ (Rd ),

f 7→ fˆ

is linear and bounded. (ii) fˆ is uniformly continuous on Rd . (iii) Denote fy := f ( . − y), y ∈ Rd . Then fby (ξ) = e−iξy fˆ(ξ),

ξ ∈ Rd .

(iv) Let λ > 0. Denote φ(x) := λd f (λx), x ∈ Rd . Then ξ φ(ξ) ˆ = fˆ( ), ξ ∈ Rd . λ Example. (a) Let f = χ[a,b] , for a < b. Then ˆ fˆ(0) = χ[a,b] (x) dx = b − a, R ˆ b e−ibξ − e−iaξ ˆ f (ξ) = e−ixξ dx = , −iξ a In particular, fˆ(ξ) → 0 for |ξ| → ∞, but fˆ ∈ / L1 (R). 19

ξ ̸= 0.

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

20

By Lemma 2.2, the Fourier transform maps L1 (Rd ) into BU C(Rd ). The next lemma shows that it even maps into C0 (Rd ). Proposition 2.3 (Riemann-Lebesgue lemma). Let f ∈ L1 (Rd ). Then lim fˆ(ξ) = 0.

|ξ|→∞

Remark. For f, g ∈ L1 (Rd ) and λ > 0, we have ˆ ˆ ˆ (2.2) f (ξ)g(λξ) dξ = f (λx)ˆ g (x) dx. Rd

Rd

Recall: A function f ∈ L1 (Rd ) is weakly differentiable in direction j ∈ {1, . . . , d}, if there exists some g ∈ L1 (Rd ) such that ˆ ˆ (2.3) f (x)∂j φ(x) dx = − g(x)φ(x) dx, for all φ ∈ Cc∞ (Rd ). Rd

Rd

g is called the weak derivative of f in direction j. Notation: g = ∂j f . One of the most important observations in Fourier analysis is the fact that und the Fourier transform, (weak) derivations are mapped into multiplications and vice versa. Lemma 2.4. (i) Let f ∈ L1 (Rd ) be weakly differentiable in direction j ∈ {1, . . . , d}, with ∂j f ∈ L1 (Rd ). Then d ˆ ∂ j f (ξ) = iξj f (ξ),

ξ ∈ Rd .

(ii) Let f ∈ L1 (Rd ) with x 7→ xj f (x) ∈ L1 (Rd ) for some j ∈ {1, . . . , d}. Then fˆ is differentiable in direction j, ∂j fˆ is continuous, and ∂j fˆ(ξ) = (x 7→ −ixj f (x))b(ξ), Example. (b) (Gauß kernel) Let G(x) = e−

|x|2 2

ξ ∈ Rd .

, x ∈ Rd . Then

2

−|ξ| b G(ξ) = (2π)d/2 e 2 , ξ ∈ Rd . Q 2 Proof: Since G(x) = dj=1 e−xj /2 , we can assume d = 1 without loss of generality. Now for x ∈ R, we have G′ (x) = −xG(x), hence by Lemma 2.4

b c′ (ξ) = (−xG(x))b(ξ) = −i(G) b ′ (ξ), iξ G(ξ) =G b ′ (ξ) = −ξ G(ξ). b b satisfy the same ODE, and i.e. (G) We see that G, G 2 √ ξ2 − ξ2 b b b G(ξ) = G(0)G(ξ) = G(0)e = 2πe− 2 , √ ´ x2 ˆ using the fact that G(0) = R e− 2 dx = 2π.

2.1. THE FOURIER TRANSFORM ON L1 (Rd )

21

1 −|ξ| ˆ (c) (Poisson kernel) Let P (x) := π1 1+x , ξ ∈ R. For the higher 2 , x ∈ R. Then P (ξ) = e dimensional analogue, the Poisson kernel is defined as

P (x) := cd

1 (1 +

|x|2 )

d+1 2

,

with

cd =

) Γ( d+1 2 π

d+1 2

,

x ∈ Rd .

Then Pˆ (ξ) = e−|ξ| , ξ ∈ Rd . For the proof, use the subordination formula: For t > 0, ˆ ∞ t2 dy 1 −2t e−y− y √ . e =√ y π 0 Theorem 2.5 (Fourier inversion formula). Let f ∈ L1 (Rd ) ∩ Cb (Rd ) with fˆ ∈ L1 (Rd ). Then ˆ 1 f (x) = fˆ(ξ)eixξ dξ, x ∈ Rd . (2π)d Rd Convolutions and Dirac kernels Similarly to the situation on the torus, we discuss approximations of Lp functions by convolution operators. The convolution with the Gauß kernel is, e.g., used as a representation of the solution to the heat equation with initial value in Lp , and the Poisson kernel for the Dirichlet problem on the upper half space with boundary data in Lp . Definition and Lemma 2.6 (Convolution). Let f, g ∈ L1 (Rd ). For almost every y ∈ Rd , the function Rd → C, y 7→ f (x − y)g(y), is integrable. We define the convolution f ∗ g : Rd → C by ˆ f ∗ g(y) :=

f (x − y)g(y) dy,

x ∈ Rd .

Rd 1

d

Then f ∗ g ∈ L (R ), and ∥f ∗ g∥1 ≤ ∥f ∥1 ∥g∥1 . Moreover, f[ ∗ g(ξ) = fˆ(ξ)ˆ g (ξ),

ξ ∈ Rd .

Note that Young’s inequality holds correspondingly. Definition 2.7. Let Λ = N or Λ = (0, ∞). A Dirac kernel (summability kernel) on Rd is a family (kλ )λ∈Λ of continuous functions in L1 (Rd ) such that ´ (i) for all λ ∈ Λ, Rd kλ (x) dx = 1, (ii) lim supλ→∞ ∥kλ ∥1 < ∞, ´ (iii) for all δ > 0, limλ→∞ |x|>δ |kλ (x)| dx = 0. A large class of Dirac kernels can be constructed by dilation of a normalized continuous L1 function with non-vanishing mean.

22

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

´ Example. Let φ ∈ L1 (Rd ) be continuous, with Rd φ(x) dx = 1. Write φλ (x) := λd φ(λx), λ > 0. Then ˆ ˆ ˆ d φλ (x) dx = φ(λx)λ dx = φ(y) dy = 1, Rd

Rd

Rd

and ∥φλ ∥1 = ∥φ∥1 . For δ > 0, we have ˆ ˆ |φλ (x)| dx = |x|>δ

|φ(y)| dy → 0,

λ → ∞,

|y|>λδ

by dominated convergence. Thus, (φλ )λ>0 is a Dirac kernel. Fejér kernel on R: Define (Fλ )λ>0 as above with  2 ˆ 1 1 sin(x/2) 1 F (x) = = (1 − |ξ|)eixξ dξ, 2π x/2 2π −1 Gauß kernel on Rd : Set H(x) :=

2

|x| 1 e− 2 (2π)d/2

, x ∈ Rd . Then for t > 0, define

√ ht (x) := t−d/2 H(x/ t) = Ht−1/2 (x) = Poisson kernel on Rd : Let P (x) :=

cd d+1 (1+|x|2 ) 2

x ∈ R.

|x|2 1 − 2t e , (2πt)d/2

, x ∈ Rd , with cd =

pt (x) := t−d P (x/t) = Pt−1 (x) = cd

t (t2

+

|x|2 )

d+1 2

,

x ∈ Rd . ) Γ( d+1 2 π

d+1 2

. For t > 0, set

x ∈ Rd .

Now check that for d ≥ 2, d

X d2 p (x) + ∂j2 pt (x) = 0, t dt2 j=1 thus pt is a harmonic function in the variables (x1 , . . . , xd , t). Setting u(t, x) := (pt ∗ f )(x), we see that u is a solution to the Dirichlet problem on Rd+1 + ( 2 d u + ∆x u = 0 on Rd+1 + , dt2 (2.4) u(0, . ) = f on Rd , where the boundary value f ∈ Lp (Rd ) is attained in the sense of Theorem 2.8 below. The question of almost everywhere convergence will be discussed in Chapter 3. We recall the notation BU C(Rd ) = {f : Rd → C : f bounded, uniformly continuous}. Theorem 2.8. Let p ∈ [1, ∞) and f ∈ Lp (Rd ), or let p = ∞ and f ∈ BU C(Rd ). Let (kλ )λ∈Λ be a Dirac kernel on Rd . Then ∥kλ ∗ f − f ∥p → 0,

λ → ∞.

Proof. See the proof of Theorem 1.11 on T. Remark. Poisson summation formula. See the exercises.

□

2.2. SCHWARTZ FUNCTIONS AND THE FOURIER TRANSFORM ON L2 (Rd )

23

2.2. Schwartz functions and the Fourier transform on L2 (Rd ) In order to extend the definition of the Fourier transform from L1 (Rd ) to L2 (Rd ), we first introduce the Schwartz space, which is a dense subspace of L2 (Rd ) and which is defined in such a way that the Fourier transform is a bijection on it. We recall some facts about multiindex notations. For α = (α1 , . . . , αd ) ∈ Nd0 , we set |α| := α1 + . . . + αd ,

∂ α f := ∂1α1 . . . ∂dαd f,

α! := α1 ! . . . αd !,

xα := xαa 1 . . . xαd d ,

x = (x1 , . . . , xd ) ∈ Rd .

Note that |xα | ≤ cd,α |x||α| ,

|x|k ≤ Cd,k

X

|xβ |,

x ∈ Rd ,

|β|=k

for some constants cd,α , Cd,k only depending on d, α, and k ∈ N. The Leibniz rule is the multidimensional analogue of the standard product rule: For f, g ∈ C |α| (Rd ), α ∈ Nd0 , we have X α1  αd  α ∂ (f g) = ... (∂ β f )(∂ α−β g), β β 1 d d β∈N0 β≤α

where β ≤ α means 0 ≤ βj ≤ αj for all j ∈ {1, . . . , d}. The class of Schwartz functions is defined as C ∞ functions such that the function and all its derivatives are decaying polynomially at any order at infinity. Definition 2.9. The Schwartz space S(Rd ) is defined by S(Rd ) = {f ∈ C ∞ (Rd ) : ∀α, β ∈ Nd0 : sup |xβ ∂ α f (x)| < ∞}. x∈Rd

The elements of S(Rd ) are called Schwartz functions. Remark. 1) If f, g ∈ S(Rd ), then also f g ∈ S(Rd ),

∂ α f ∈ S(Rd ),

x 7→ xα f (x) ∈ S(Rd ),

α ∈ Nd0 .

2) We have Cc∞ (Rd ) ⊆ S(Rd ) ⊆ C ∞ (Rd ). 2

3) For every a > 0, the function x 7→ e−a|x| is in S(Rd ) \ Cc∞ (Rd ). Note, however, that x 7→ e−|x| ∈ / S(Rd ), and x 7→ (1 + |x|4 )−a ∈ / S(Rd ). 4) Let p ∈ [1, ∞]. Then S(Rd ) ⊆ Lp (Rd ). Proof: Let f ∈ S(Rd ). Then for N ∈ N, N > dp , supx∈Rd |x|N |f (x)| =: M < ∞, and ˆ ˆ ˆ p p |f (x)| dx = |f (x)| dx + |f (x)|p dx d R |x|≤1 |x|≥1 ˆ 1 p p ≤ |B(0, 1)|∥f ∥∞ + M dx < ∞. Np |x|≥1 |x|

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

24

Definition 2.10. Let α, β ∈ Nd0 . The Schwartz seminorms ρα,β : S(Rd ) → [0, ∞) are defined by ρα,β (f ) := sup |xβ ∂ α f (x)|,

(2.5)

f ∈ S(Rd ).

x∈Rd

Let fk , f ∈ S(Rd ), k ∈ N. We say that fk → f in S(Rd ) for k → ∞, if ∀α, β ∈ Nd0 :

ρα,β (fk − f ) → 0,

k → ∞.

Remark. 1) For every α, β ∈ Nd0 , ρα,β is a seminorm on S(Rd ). 2) The notion of convergence is compatible with a topology in S(Rd ) induced by the countable family of seminorms ρα,β . A basis for open sets containing 0 in the topology is 1 {f ∈ S(Rd ) : ρα,β (f ) < }, N for all α, β ∈ Nd0 , N ∈ N. This topology is not derived from a norm, but S(Rd ) is metrisable, with e.g. a metric defined by X ρα,β (f − g) (2.6) . 2−|α|−|β| d(f, g) = 1 + ρ (f − g) α,β d α,β∈N0

3) S(Rd ) is complete with respect to the metric d in (2.6). S(Rd ) is a Fréchet space. (A Fréchet space is a complete, metrisable topological vector space, whose topology may be defined by a countable family of seminorms.) 4) S(Rd ) is sequentially complete with respect to ρα,β . I.e., if (fk )k∈N ⊆ S(Rd ) is a Cauchy sequence with respect to ρα,β for all α, β ∈ Nd0 , then there exists f ∈ S(Rd ) such that ρα,β (fk − f ) → 0,

k → ∞,

for all α, β ∈ Nd0 . Remark. The space C ∞ (Rd ), equipped with the family of seminorms pα,N (f ) := sup |∂ α f (x)|, |x|≤N

α ∈ Nd0 , N ∈ N,

is also a Fréchet space. We show that the Fourier transform of a Schwartz function is again a Schwartz function. The Fourier inversion formula directly follows from Theorem 2.5, since S(Rd ) ⊆ L1 (Rd ). Theorem 2.11 (Fourier inversion on S(Rd )). Let f ∈ S(Rd ). Then fˆ ∈ S(Rd ), and ˆ 1 f (x) = fˆ(ξ)eixξ dξ, x ∈ Rd . d (2π) Rd Notation: We denote the Fourier transform on S(Rd ) by F, and the reflected Fourier transform on S(Rd ) by FR , i.e. FR f (x) = Ff (−x), x ∈ Rd , f ∈ S(Rd ).

2.2. SCHWARTZ FUNCTIONS AND THE FOURIER TRANSFORM ON L2 (Rd )

25

Corollary 2.12. The Fourier transform F : S(Rd ) → S(Rd ),

f 7→ fˆ,

is linear and bijective, with FR F = FFR = (2π)d I

on S(Rd ).

From the Fourier inversion formula we obtain that the Fourier transform, defined on S(Rd ), is an isometry on L2 (Rd ) up to a normalising factor. Theorem 2.13 (Plancherel). Let f ∈ S(Rd ). Then 1 ∥f ∥22 = ∥fˆ∥22 . (2π)d An important corollary of Plancherel’s theorem is Heisenberg’s uncertainty principle. It shows that there is a limit to how well (Schwartz) functions can be localised simultaneously in time (space) and frequency domain. Proposition 2.14 (Heisenberg’s uncertainty principle). Let f ∈ S(Rd ), let x0 , ξ0 ∈ Rd . Then d ∥(x − x0 )f ∥2 ∥(ξ − ξ0 )fb∥2 ≥ (2π)d/2 ∥f ∥22 . 2 Explanation: Consider f ∈ S(R) with ∥f ∥2 = 1. Define the spread of f to be ˆ 2 σ (f ) = inf |x − x0 |2 |f (x)|2 dx. x0 ∈R

R

2

We interpret p(x) := |f (x)| as a probability density. By Plancherel’s theorem, also 1 ˆ q(ξ) := 2π |f (ξ)|2 is a probability density on R. We can thus recognize σ 2 (f ) as the variance of a random variable X with density p, and the minimising x0 ∈ R is given by ˆ xp(x) dx. x0 = EX = R

We use the analogous interpretation for fˆ, with a random variable Y and density q, and ξ0 given by ξ0 = EY . Heisenberg’s uncertainty principle then states that 1 σ(f )σ(fˆ) ≥ , 2 and equality holds if and only if f is a normalised Gaussian. I.e. f and fˆ cannot be both too well localised. Notation: Recall L1loc (Rd ) = {f : f ∈ L1 (K) for every K ⊆ Rd compact}. For a function g ∈ L1loc (Rd ), we define the support of g as supp g := Rd \ {x ∈ Rd : ∃ε > 0 : gχB(x,ε) = 0 a.e.}, that is, the smallest closed set S ⊆ Rd such that g(x) = 0 almost everywhere in S c . If g ̸ 0}. is continuous, then supp g = {x ∈ Rd : g(x) =

26

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

We use convolutions with Schwartz functions to approximate L1 functions by smooth functions. Lemma 2.15 (Mollifiers). Let φ ∈ S(Rd ), let f ∈ L1 (Rd ). Then φ ∗ f ∈ C ∞ (Rd ), and ∂ α (φ ∗ f ) = (∂ α φ) ∗ f , α ∈ Nd0 . Moreover, supp φ ∗ f ⊆ supp φ + supp f . Corollary 2.16. For p ∈ [1, ∞), Cc∞ (Rd ) is dense in Lp (Rd ). We can now extend the Fourier transform to L2 (Rd ). The theorem shows that it is an isometric isomorphism on L2 (Rd ) (up to a normalising factor). Theorem 2.17 (Fourier transform on L2 (Rd ); Plancherel). Let F : S(Rd ) → S(Rd ) be the Fourier transform on the Schwartz space. Then there exists a unique continuous extension F : L2 (Rd ) → L2 (Rd ), which is linear, bijective, and ∥f ∥22 =

1 ∥Ff ∥22 , d (2π)

f ∈ L2 (Rd ).

2.3. The Fourier transform on Lp (Rd ) for 1 ≤ p ≤ 2 In the last two sections, we have defined the Fourier transform on L1 (Rd ) explicitly by (2.1), and on L2 (Rd ) as a continuous extension in Theorem 2.17. We use complex interpolation for the definition on Lp (Rd ) for p ∈ (1, 2). Theorem 2.18 (Riesz-Thorin interpolation theorem). Let p0 , p1 , q0 , q1 ∈ [1, ∞], and let T be a linear operator defined on Lp0 (Rd ) and on Lp1 (Rd ). Assume that there exist M0 , M1 > 0 such that ∥T f ∥q0 ≤ M0 ∥f ∥p0 ,

f ∈ Lp0 (Rd ),

∥T f ∥q1 ≤ M1 ∥f ∥p1 ,

f ∈ Lp1 (Rd ).

Let θ ∈ [0, 1], and let p, q be defined by 1−θ θ 1 = + , p p0 p1 Then ∥T f ∥q ≤ M01−θ M1θ ∥f ∥q ,

1 1−θ θ = + . q q0 q1 f ∈ Lp0 (Rd ) ∩ Lp1 (Rd ).

Proof. See Functional Analysis or [8, Theorem 1.3.4].

□

We obtain the following bound for the Fourier transform on L1 (Rd ) ∩ L2 (Rd ). Proposition 2.19 (Hausdorff-Young inequality). Let p ∈ (1, 2], and let q be defined by 1 + 1q = 1. Let f ∈ L1 (Rd ) ∩ L2 (Rd ). Then p 1 ∥fˆ∥q ≤ ∥f ∥p . (2π)d/q

2.4. TEMPERED DISTRIBUTIONS

27

By density, we extend the Fourier transform to Lp (Rd ) for p ∈ [1, 2]. Corollary 2.20. Let p ∈ [1, 2], and p1 + 1q = 1. Then F : S(Rd ) → S(Rd ) has a unique continuous extension F : Lp (Rd ) → Lq (Rd ), with 1 ∥Ff ∥q ≤ ∥f ∥p , f ∈ Lp (Rd ). (2π)d/q Remark. For p ∈ (1, 2), one can show that F = Fp : Lp (Rd ) → Lq (Rd ) is not an isometry (up to a constant), and not surjective/onto. In particular, Fp is not invertible. See the exercises. 2.4. Tempered distributions An extension of the Fourier transform to Lp (Rd ) for p > 2 is not immediately possible by the above methods. For this purpose, we introduce the space of tempered distributions, which is the dual space of S(Rd ). Recall that the topology on S(Rd ) is given by the family of seminorms (ρα,β )α,β∈Nd0 defined in (2.5). We rephrase the continuity of a functional S(Rd ) → C. See [8, Proposition 2.3.4] or [14, Satz VIII.2.3, Korollar VIII.2.4] for a proof of 1). Remark. Let (Y, ∥ . ∥Y ) be a normed space. 1) Let T : (S(Rd ), (ρα,β )α,β∈Nd0 ) → Y be linear. Then T is continuous if there exist k, m ∈ N0 and a constant C > 0 such that X ∥T f ∥Y ≤ C ρα,β (f ), f ∈ S(Rd ). |α|≤k |β|≤m

2) Let S : Y → (S(Rd ), (ρα,β )α,β∈Nd0 ) → Y be linear. Then S is continuous if for all α, β ∈ Nd0 there exists Cα,β > 0 such that ρα,β (Sy) ≤ Cα,β ∥y∥Y ,

y ∈ Y.

3) Let T˜ : (S(Rd ), (ρα,β )α,β∈Nd0 ) → (S(Rd ), (ρα,β )α,β∈Nd0 ) be linear. Then T˜ is continuous if for every α, β ∈ Nd0 there exist k, m ∈ N0 , k = k(α, β), m = m(α, β) and Cα,β > 0 such that X ρα,β (T˜f ) ≤ Cα,β ργ,δ (f ), f ∈ S(Rd ). |γ|≤k |δ|≤m

Example. Let α, β ∈ Nd0 . The maps (a) S(Rd ) → S(Rd ), f 7→ ∂ α f , (b) S(Rd ) → S(Rd ), f 7→ xβ f ,

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

28

(c) S(Rd ) → S(Rd ), f 7→ fˆ are continuous. We introduce the dual space of S(Rd ). Definition 2.21. The space of tempered distributions is defined as S ′ (Rd ) := {T : S(Rd ) → C linear : T continuous} = {T : S(Rd ) → C linear : ∃k, m ∈ N, C > 0 ∀f ∈ S(Rd ) : |T f | ≤ C

X

ρα,β (f )}.

|α|≤k |β|≤m

We discuss some examples of tempered distributions. Important examples are Lp functions and finite Borel measures, but there are also examples which are neither a function or a measure. Example. (a) Dirac mass. Define δ0 by δ0 (f ) := f (0),

f ∈ S(Rd ).

Then δ0 ∈ S ′ (Rd ), since |δ0 (f )| = |f (0)| ≤ ∥f ∥∞ = ρ0,0 (f ),

f ∈ S(Rd ).

(b) Lp functions. Let p ∈ [1, ∞], let g ∈ Lp (Rd ). Define Tg by ˆ f (x)g(x) dx, f ∈ S(Rd ). Tg (f ) := Rd

Then Tg ∈ S ′ (Rd ): First note that by Hölder’s inequality for

1 p

+

1 q

=1

|Tg (f )| ≤ ∥g∥p ∥f ∥q . Moreover, Id : S(Rd ) → Lq (Rd ) is continuous. For q = ∞, this is clear. For q < ∞, let s ∈ R with s > d. Then ˆ ˆ q 1/q ∥f ∥q = ( |f (x)| dx) = ( (1 + |x|)−s (1 + |x|)s |f (x)|q dx)1/q Rd Rd ˆ X s/q ≤ sup (1 + |x|) |f (x)| · ( (1 + |x|)−s dx)1/q ≤ Cd,s,q ρ0,β (f ), x∈Rd

Rd

β∈Nd0 |β|≤l

where we choose l ∈ N with l ≥ qs . (c) Finite Borel measures. For µ a finite Borel measure, define Tµ by ˆ Tµ (f ) := f (x) dµ(x), f ∈ S(Rd ). Rd

Then Tµ ∈ S ′ (Rd ). (Exercise)

2.4. TEMPERED DISTRIBUTIONS

29

(d) Polynomials and polynomially growing functions. Let g : Rd → C be a function with |g(x)| ≤ C(1 + |x|)k ,

x ∈ Rd ,

for some k ∈ N, C > 0. Then Tg ∈ S ′ (Rd ), for Tg defined by ˆ (2.7) Tg (f ) := f (x)g(x) dx, f ∈ S(Rd ). Rd

(e) Principal value distribution on R. Define ˆ 1 dx p.v. (f ) := lim f (x) , ε→0+ x x |x|≥ε

f ∈ S(R).

Then p.v. x1 ∈ S ′ (R): First note that ˆ ˆ dx f (x) | ≤ |x|−2 dx · sup |xf (x)| ≤ Cρ0,1 (f ), f ∈ S(R). | x x∈R |x|≥1 |x|≥1 ´ On the other hand, since x 7→ x1 is odd, we have ε≤|x|≤1 dx = 0, and x ˆ ˆ dx dx f (x) | = | (f (x) − f (0)) | | x x ε≤|x|≤1 ε≤|x|≤1 ˆ dx ≤ ∥f ′ ∥∞ |x| ≤ 2∥f ′ ∥∞ = 2ρ1,0 (f ). |x| ε≤|x|≤1 (f) Derivations. Let x0 ∈ Rd , α ∈ Nd0 . Then T ∈ S ′ (Rd ), where T (f ) = ∂ α f (x0 ),

f ∈ S(Rd ). 2

Remark. Note that Tg ∈ / S ′ (Rd ) for g(x) = e|x| and Tg defined as in (2.7). If T = Tg ∈ S ′ (Rd ) for some function g, we often identify Tg with g. We will see in the following that operations on tempered distributions are defined in such a way that they are consistent with this identification. Definition and Lemma 2.22. Let T ∈ S ′ (Rd ). (i) (Derivative) Let α ∈ Nd0 . Define ∂ α T (f ) := (−1)|α| T (∂ α f ),

f ∈ S(Rd ).

Then ∂ α T ∈ S ′ (Rd ). (ii) (Translation.) Let y ∈ Rd . Define τy T by τy T (f ) := T (f−y ) = T (τ−y f ),

f ∈ S(Rd ).

Then τy T ∈ S ′ (Rd ). (iii) (Dilation.) Let λ > 0. Define δλ T by δλ T (f ) := T (λ−d δ1/λ (f )), Then δλ T ∈ S ′ (Rd ).

f ∈ S(Rd ).

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

30

Remark. If e.g. T = Tg for some g ∈ S(Rd ), then for f ∈ S(Rd ) we obtain from integration by parts ˆ ˆ α |α| α |α| α ∂ Tg (f ) = (−1) Tg (∂ f ) = (−1) g(x)∂ f (x) dx = ∂ α g(x)f (x) dx = T∂ α g (f ). Rd

Rd

α

Thus, ∂ Tg = T∂ α g . We introduce the Fourier transform of a tempered distribution. By the examples above, this in particular gives us a definition of the Fourier transform on Lp (Rd ) for p ∈ (2, ∞]. Note that the Fourier transform of a tempered distribution is again in S ′ (Rd ), because F is continuous on (S(Rd ), (ρα,β )α,β∈Nd0 ). Definition and Lemma 2.23 (Fourier transform of tempered distributions). Let T ∈ S ′ (Rd ). We define Tˆ by Tˆ(f ) := T (fˆ), f ∈ S(Rd ). Then Tˆ =: FT ∈ S ′ (Rd ). We calculate some examples. Part (c) states that the definition on S ′ (Rd ) is consistent with the definition of the Fourier transform on L1 (Rd ). Example. (a) ˆ1 = (2π)d δ0 , where 1(x) = 1 on Rd . This follows from ˆ ˆ1(f ) = Tˆ1 (f ) = T1 (fˆ) = fˆ(ξ) dξ = (2π)d f (0) = (2π)d δ0 (f ),

f ∈ S(Rd ).

Rd

(b) δˆ0 = 1. (Exercise) (c) Let g ∈ L1 (Rd ). Then Tˆg = Tgˆ, since (2.2) yields ˆ ˆ ˆ ˆ ˆ gˆ(ξ)f (ξ) dξ = Tgˆ(f ), Tg (f ) = Tg (f ) = g(x)f (x) dx = Rd

f ∈ S(Rd ).

Rd

Corollary 2.12 immediately implies that the Fourier transform is a bijection on S ′ (Rd ). Theorem 2.24. The Fourier transform F : S ′ (Rd ) → S ′ (Rd ) is a linear bijection with F −1 = and

1 FFR (2π)d

= IdS ′ ,

1 F4 (2π)2d

1 FR , (2π)d

= IdS ′ .

Here, the reflected Fourier transform on S ′ (Rd ) is defined by FR T (f ) := T (FR f ),

Definition 2.25. Let T ∈ S ′ (Rd ).

f ∈ S(Rd ).

2.4. TEMPERED DISTRIBUTIONS

31

(i) (Multiplication in S ′ (Rd )) Let h ∈ C ∞ (Rd ) be such that the map (2.8)

S(Rd ) → S(Rd ),

f 7→ hf

is continuous.

Then define hT by hT (f ) := T (hf ),

f ∈ S(Rd ).

(ii) (Convolution in S ′ (Rd )) Let h ∈ S(Rd ). Define h ∗ T by ˜ ∗ f ), h ∗ T (f ) := T (h

f ∈ S(Rd ),

˜ where h(x) := h(−x) is the reflection of h. Remark. 1) Note that (2.8) is e.g. true if for all α ∈ Nd0 there exists some kα > 0 such that |(∂ α h)(x)| ≤ C(1 + |x|)kα , x ∈ Rd . 2) If T = Tg for some g ∈ S(Rd ), and f, h ∈ S(Rd ), then (h ∗ Tg )(f ) = Th∗g (f ), i.e. h ∗ Tg = Th∗g . 3) If f ∈ S(Rd ) and T ∈ S ′ (Rd ), then f ∗ T is a C ∞ function. See [8, Theorem 2.3.20] for a proof. Example. Let x0 ∈ Rd , let h ∈ S(Rd ). Then for f ∈ S(Rd ), ˆ ˜ ˜ (h ∗ δx0 )(f ) = δx0 (h ∗ f ) = (h ∗ f )(x0 ) = h(x − x0 )f (x) dx = Th( . −x0 ) (f ), Rd

thus we can identify h ∗ δx0 with h( . − x0 ). For x0 = 0, we get that the operator S : S(Rd ) → S(Rd ), h 7→ h ∗ δ0 , is the identity operator on S(Rd ). We collect some properties of the Fourier transform on S ′ (Rd ), which are consequences of the corresponding statements on S(Rd ). Lemma 2.26. Let T ∈ S ′ (Rd ), f, h ∈ S(Rd ), α ∈ Nd0 , and y ∈ Rd . Then α T (f ) = (iξ)α T ˆ(f ); (i) ∂d (ii) ∂ α Tˆ(f ) = ((−ix)α T )b(f ); −iy( . ) ˆ (iii) τd T (f ); y T (f ) = e ˆ Tˆ(f ). (iv) h[ ∗ T (f ) = h For the proof of (iv), we use that for f, g ∈ S(Rd ), 1 ˆ f ∗ gˆ. fcg = (2π)d We calculate the Fourier transform of the function x 7→ |x|λ for λ ∈ (−d, 0). Note that T|x|λ ∈ S ′ (Rd ), since x 7→ |x|λ is locally integrable and has moderate growth at infinity.

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

32

Proposition 2.27. Let λ ∈ (−d, 0), let Hλ := T|x|λ . Then (Hλ )b = cλ H−d−λ ,

(2.9) where cλ =

Γ( d+λ ) 2 Γ(− λ ) 2

π −d/2−λ (2π)d+λ .

´∞ Recall that the Gamma function is defined as Γ(z) := 0 tz−1 e−t dt for Re z > 0. By substitution, we have for A > 0 and λ > 0, ˆ ∞ λ e−tA t−λ/2−1 dt = Aλ/2 Γ(− ). 2 0 We use this formula to deduce (2.9) from the known Fourier transform of the Gauß kernel by subordination. Remark (Fundamental solution). Let m ∈ N, let L be a differential operator of the form X L= aα ∂ α on Rd , α∈Nd0 |α|≤m

with constant coefficients aα ∈ C. Then a fundamental solution of L is a (tempered) distribution F such that L(F ) = δ0 .

(2.10)

We argue somewhat informally that F provides an inverse to L: Denote by T the convolution operator T : S(Rd ) → C ∞ (Rd ), f 7→ f ∗ F . By remark 3) above we know that f ∗ F ∈ C ∞ (Rd ), moreover, one can show that ∂ α (f ∗ F ) = (∂ α f ) ∗ F = f ∗ (∂ α F ). Because the convolution with δ0 is the identity on S(Rd ), we get from (2.10) X LT f = aα ∂ α (f ∗ F ) = f ∗ (LF ) = f ∗ ∂0 = f, |α|≤m

and, similarly, T Lf = Lf ∗ F = L(f ∗ F ) = f . We therefore see that LT = T L = I

on S(Rd ),

and in this sense, T is an inverse to L. Now let X aα (iξ)α P (ξ) := |α|≤m

be the symbol (characteristic polynomial) of L. For g ∈ S(Rd ), we then have (2.11)

(Lg)b(ξ) = P (ξ)ˆ g (ξ),

ξ ∈ Rd .

Because (f ∗ F ) b = fˆFˆ by Lemma 2.26, one is looking for a fundamental solution F formally defined by Fˆ (ξ) = P 1(ξ) , or ˆ 1 1 ixξ F (x) = e dξ. d (2π) Rd P (ξ) However, this can cause problems in general, as P can have zeros. For a more in-depth study of fundamental solutions, we refer to [13, Section 3.2.4]. We will have a look at some examples (see also the exercises).

2.5. FOURIER MULTIPLIERS

33

Example. Consider the Laplacian on Rd given by −∆ = −

d X

∂j2 .

j=1

Pd

The symbol of −∆ is P (ξ) = − j=1 (iξj )2 = |ξ|2 , ξ ∈ Rd , and P (ξ)−1 = |ξ|−2 . Note that |ξ|−2 is locally integrable for d ≥ 3. Proposition 2.28 (Fundamental solution for the Laplacian). (i) Let d ≥ 3. Then the function Γ( d2 − 1) −d+2 F : x 7→ Cd |x| , with Cd = , 4π d/2 is a fundamental solution for −∆ on Rd . 1 (ii) Let d = 2. Then the locally integrable function F : x 7→ 2π log |x|, is a fundamental 2 solution for −∆ on R . Proof. (i) follows from Proposition 2.27. For (ii), see [13, Theorem 3.2.9].

□

2.5. Fourier multipliers Motivated by (2.11), we aim to study operators that are given as multiplication operators on the Fourier side. We focus on bounded operators at first. As we will see, such operators are given as convolutions with a tempered distribution. Definition 2.29. Let X, Y be vector spaces of measurable functions on Rd . (i) X is called closed under translations, if f ∈ X implies τy f ∈ X for all y ∈ Rd . (ii) Let X and Y be closed under translations, let T : X → Y be an operator. Then T is translation invariant, if for all f ∈ X and all y ∈ Rd , T (τy f ) = τy (T f ). Remark. One can easily check that convolution operators (whenever defined) are translation invariant. The next theorem states a converse to it. Theorem 2.30. Let p, q ∈ [1, ∞], let T ∈ L(Lp (Rd ), Lq (Rd )) be translation invariant. Then there exists a unique S ∈ S ′ (Rd ) such that (2.12)

T (f ) = f ∗ S,

Proof. See [8, Theorem 2.5.2].

f ∈ S(Rd ). □

34

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

ˆ or, by taking the Fourier inverse on S ′ (Rd ), On the Fourier side, (2.12) reads (T f )b = fˆS, ˆ (2.13) T f = F −1 (fˆS). The question is now for which S ∈ S ′ (Rd ) the operator T in (2.13) extends to a bounded operator on Lp (Rd ). This leads to the notion of Lp Fourier multipliers. Definition 2.31. Let p ∈ [1, ∞). We denote by Mp (Rd ) the space of all m ∈ L∞ (Rd ) such that for all f ∈ S(Rd ) (2.14) Tm (f ) := F −1 (mfˆ) ∈ Lp (Rd ), and the operator Tm : S(Rd ) → Lp (Rd ) defined by (2.14) has a bounded extension Tm : Lp (Rd ) → Lp (Rd ). Elements of Mp (Rd ) are called Lp Fourier multipliers. Note that in (2.14), mfˆ is identified with the tempered distribution Tmfˆ, and F −1 denotes the inverse of the Fourier transform on S ′ (Rd ). Then F −1 (mfˆ) ∈ S ′ (Rd ) needs to be identified with a function in Lp (Rd ) to satisfy (2.14). For p = 2, Plancherel’s theorem allows to fully characterize the set of L2 Fourier multipliers. Theorem 2.32. Let T : L2 (Rd ) → L2 (Rd ) be linear and bounded. The following are equivalent: (i) T is translation invariant. (ii) There exists m ∈ L∞ (Rd ) such that T f = F −1 (mfˆ), f ∈ L2 (Rd ). Remark. Note that Theorem 2.32 in particular implies M2 (Rd ) = L∞ (Rd ). Remark. 1) Note that for f, g ∈ L1 (Rd ) ∩ L2 (Rd ), and T ∈ L(L2 (Rd )) translation invariant, we have T f ∗ g = f ∗ T g = T (f ∗ g). 1 d 2 d 2) For f ∈ L (R ), g ∈ L (R ), we have f ∗ g ∈ L2 (Rd ) with f[ ∗ g = fˆ · gˆ. Example. Let p ∈ [1, ∞). Let b ∈ Rd , let mb (ξ) = eiξb , ξ ∈ Rd . Then mb ∈ Mp (Rd ), with Tmb given by Tmb (f )(x) = f (x + b). p d Clearly, Tmb ∈ L(L (R )). Remark. 1) Let p ∈ [1, ∞), let M1 (Rd ), with

1 p

+

1 q

= 1. Then Mp (Rd ) = Mq (Rd ), and M∞ (Rd ) =

∥Tm ∥L(Lp ) = ∥Tm ∥L(Lq ) .

2.5. FOURIER MULTIPLIERS

35

2) We have M1 (Rd ) = {ˆ µ : µ ∈ (C0 (Rd ))′ }. That is, m ∈ M1 (Rd ) if and only if T is a convolution operator with a finite complexvalued Borel measure. See [8, Theorem 2.5.8] for a proof. 3) We can equip Mp (Rd ) with a norm: For m ∈ Mp (Rd ), set ∥m∥Mp := ∥Tm ∥L(Lp ) . From 1) we know that ∥m∥Mp = ∥m∥Mq for p1 + 1q = 1. Let p, r ∈ [1, 2] with 1 ≤ p ≤ r ≤ 2. Then M1 ⊆ Mp ⊆ Mr ⊆ M2 = L∞ (Rd ). This follows from the interpolation theorem of Riesz-Thorin (Theorem 2.18), since for m ∈ Mp ⊆ L∞ (Rd ), we know that Tm : Lp (Rd ) → Lp (Rd ) and Tm : L2 (Rd ) → L2 (Rd ) are bounded, thus also Tm : Lr (Rd ) → Lr (Rd ), with θ ∥Tm ∥L(Lr ) ≤ ∥Tm ∥1−θ L(Lp ) ∥Tm ∥L(L2 ) , + 2θ . where 1r = 1−θ p Exercise: One can even show that ∥m∥Mr ≤ ∥m∥Mp for 1 ≤ p ≤ r ≤ 2. 4) Let p ∈ [1, 2]. (Mp , ∥ . ∥Mp ) is a Banach space. Moreover, Mp is closed under pointwise multiplication, i.e., if m1 , m2 ∈ Mp , then m1 m2 ∈ Mp and ∥m1 m2 ∥Mp ≤ ∥m1 ∥Mp ∥m2 ∥Mp . 5) Let p, q ∈ [1, ∞] with q < p. Let T : Lp (Rd ) → Lq (Rd ) be translation invariant. Then T = 0. Sobolev spaces and Bessel potential spaces We use the theory of Fourier multipliers to identify Sobolev spaces on Rd with so-called Bessel potential spaces, where smoothness of functions is measured in terms of multiplication on the Fourier side. We first recall the definition of Sobolev spaces on open sets of Rd . Definition 2.33 (Sobolev spaces). Let Ω ⊆ Rd be open. Let p ∈ [1, ∞] and k ∈ N. We define the Sobolev space W k,p (Ω) as W k,p (Ω) := {f ∈ Lp (Ω) : ∀α ∈ Nd0 , |α| ≤ k, ∂ α f exists, and ∂ α f ∈ Lp (Ω)}. Here, ∂ α f denotes the α-th weak derivative of f , i.e. there exists gα = ∂ α f ∈ Lp (Ω) with ˆ ˆ |α| gα φ dx = (−1) f ∂ α φ dx, φ ∈ Cc∞ (Ω). Ω

Ω

Remark. For Ω = Rd , we can use tempered distributions to rewrite W k,p (Rd ) = {f ∈ Lp (Rd ) : ∀α ∈ Nd0 , |α| ≤ k ∃gα ∈ Lp (Rd ) : ∂ α (Tf ) = Tgα }.

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

36

α (T f ) = (iξ)α T ˆf in S ′ (Rd ) of the identity ∂ α (Tf ) = Tgα Taking the Fourier transform ∂\ motivates the following definition.

Definition 2.34 (Bessel potential space). Let p ∈ (1, ∞), let s ∈ R. We define the Bessel potential space H s,p (Rd ) := {f ∈ S ′ (Rd ) : F −1 (ξ 7→ (1 + |ξ|2 )s/2 fˆ(ξ)) ∈ Lp (Rd )}, equipped with the norm ∥f ∥H s,p (Rd ) := ∥F −1 (ξ 7→ (1 + |ξ|2 )s/2 fˆ(ξ))∥p . Note that H s,p (Rd ) is a space of tempered distribution, and that the expression in the definition is well-defined. Clearly, H 0,p (Rd ) = Lp (Rd ). We will show later that for s ≥ 0, elements of H s,p (Rd ) are Lp functions. For s < 0, however, one can show that this is not the case in general. Remark. For p ∈ (1, ∞) and s ∈ R, (H s,p (Rd ), ∥ . ∥H s,p ) is a Banach space. (H s,2 (Rd ), ( . , . )H s,2 ) is a Hilbert space, equipped with the inner product ˆ 1 (f, g)H s,2 = g (ξ) dξ (1 + |ξ|2 )s fˆ(ξ)ˆ (2π)d Rd ˆ F −1 ((1 + |ξ|2 )s/2 fˆ(ξ)) · F −1 ((1 + |ξ|2 )s/2 gˆ(ξ)) dξ. = Rd

Since F is an isometry on L2 (Rd ) (up to a normalising factor), we can identify Sobolev and Bessel potential spaces in L2 (Rd ) for k = s ∈ N. Proposition 2.35. Let k ∈ N. We have W k,2 (Rd ) = H k,2 (Rd ). We have an analogue result for p ∈ (1, ∞). Theorem 2.36. Let p ∈ (1, ∞), let k ∈ N. Then W k,p (Rd ) = H k,p (Rd ). Proof. Exercise.

□

For the proof of Theorem 2.36, we will need to check that for α ∈ Nd0 with |α| ≤ k, the function m defined by ξα m(ξ) := , ξ ∈ Rd , (1 + |ξ|2 )k/2 is an Lp Fourier multiplier, i.e. m ∈ Mp (Rd ). We make use of the following sufficient condition for Lp Fourier multipliers. We will give a proof of Theorem 2.37 at the end of Chapter 3.

2.5. FOURIER MULTIPLIERS

37

Theorem 2.37 (Mihlin multiplier theorem). Let p ∈ (1, ∞). Let m ∈ C L (Rd \ {0}) for some L ∈ N with L > d2 . Assume that there exists A > 0 such that for all α ∈ Nd0 with |α| ≤ L, |∂ α m(ξ)| ≤ A|ξ|−|α| ,

ξ ∈ Rd \ {0}.

Then m ∈ Mp (Rd ), with ∥Tm ∥L(Lp (Rd )) ≤ Cd max{p,

1 }(A + ∥m∥∞ ). p−1

If a linear operator defined on S(Rd ) is given by multiplication with some function m on the Fourier side, we call m the symbol of the operator. Note that m does not need to be bounded for this definition. It is even possible to define the operator only on a subspace of S(Rd ), as we will see in examples later on. Definition 2.38. Let T : S(Rd ) → S ′ (Rd ) be a linear operator. Let m : Rd → C be a function such that Tcf = mfˆ, f ∈ S(Rd ). Then m is called the symbol of the operator T . Example. (a) T = ∂j , j = 1, . . . , d, with the symbol m(ξ) = iξj . (b) T = −∆, with the symbol m(ξ) = |ξ|2 . P P (c) T = |α|≤k aα ∂ α , for coefficients aα ∈ C, with the symbol P (ξ) = |α|≤k aα (iξ)α . (d) Tt = et∆ , t ≥ 0, the heat semigroup, with symbol mt (ξ) = e−t|ξ| (see the example below).

2

We study some applications of Mihlin’s theorem. Example (The spectrum of the Laplace operator on Lp (Rd )). Let p ∈ (1, ∞), and consider the Laplace operator on Lp (Rd ) defined by D(−∆) = W 2,p (Rd ), −∆u = −

d X

∂j2 u,

u ∈ D(−∆).

j=1

We would like to identify the spectrum σ(−∆). Vice versa, given f ∈ Lp (Rd ), we need to identify for which λ ∈ C is there a unique solution u ∈ W 2,p (Rd ) of λu + ∆u = f. Taking the Fourier transform (in S ′ (Rd )), this leads to the equation (λ − |ξ|2 )ˆ u(ξ) = fˆ(ξ), ξ ∈ Rd , and uˆ(ξ) = (λ − |ξ|2 )−1 fˆ(x) = mλ (ξ)fˆ(ξ), ξ ∈ Rd , with mλ (ξ) := (λ − |ξ|2 )−1 , if λ ∈ C \ [0, ∞). One can show that for λ ∈ [0, ∞), the multiplication operator M defined by M g(ξ) = (λ − |ξ|2 )g(ξ) is not invertible on Lp (Rd ).

CHAPTER 2. THE FOURIER TRANSFORM ON Rd

38

It remains to check that mλ ∈ Mp (Rd ) for λ ∈ C \ [0, ∞). Thus, by Mihlin’s multiplier theorem, σ(−∆) = [0, ∞), and ρ(−∆) = C \ [0, ∞). (Exercise). Example. Let p ∈ (1, ∞) in the following. (a) (Bounded imaginary powers) Let τ ∈ R. Then mτ : ξ 7→ |ξ|iτ ∈ Mp (Rd ). This follows from Mihlin’s multiplier theorem, since m ∈ L∞ (Rd ), and |∂j m(ξ)| ≲ |ξ|−1 for ξ ∈ Rd \{0}, and corresponding estimates for higher derivatives follow by induction. Thus, defining (−∆)iτ f := Tmτ /2 f = F −1 (mτ /2 (| . |2 )fˆ),

f ∈ S(Rd ),

we obtain (−∆)iτ ∈ L(Lp (Rd )) for all τ ∈ R, i.e., −∆ has bounded imaginary powers in Lp (Rd ), for p ∈ (1, ∞). (b) (Holomorphic functional calculus) Denote by Σω := {λ ∈ C \ {0} : | arg λ| < ω} the open sector with angle ω ∈ (0, π), and by H ∞ (Σω ) = {f ∈ H(Σω ) : f bounded} the set of holomorphic, bounded functions on Σω . For ψ ∈ H ∞ (Σω ) for some ω > 0, we then define the operator ψ(−∆) by ψ(−∆)g := Φ−∆ (ψ)(g) := F −1 (ξ 7→ ψ(|ξ|2 )ˆ g (ξ)),

g ∈ S(Rd ).

Then by Mihlin’s multiplier theorem, ψ(−∆) ∈ L(Lp (Rd )), since ψ(| . |2 ) ∈ L∞ (Rd ), and |α| α

2

|α| |α| α

|ξ| ∂ ψ(|ξ| ) = |ξ| 2 ξ ψ

(|α|)

2

(|ξ| ) =



2ξ |ξ|

α

(|ξ|2 )|α| ψ |α| (|ξ|2 ),

ξ ∈ Rd .

Now for n ∈ N, t 7→ tn ψ (n) (t) is bounded, t > 0, as a consequence of Cauchy’s integral formula, writing for 0 < ν < ω, ˆ tn n! n n f (ζ) dζ, t ψ (t) = 2πi ∂Σν (ζ − t)n+1 and using that ζ 7→

tn (ζ−t)n+1

∈ L1 (∂Σν ) uniformly in t > 0.

2.5. FOURIER MULTIPLIERS

39

(c) (Heat semigroup) For t > 0, set ψt (z) := e−tz , z ∈ Σω for some ω ∈ (0, π2 ). Then 2

ψt (−∆)g = F −1 (ξ 7→ e−t|ξ| gˆ(ξ)),

g ∈ S(Rd ),

defines the heat semigroup et∆ := ψt (−∆) on Lp (Rd ), with et∆ ∈ L(Lp (Rd )). This definition is consistent with the definition of et∆ as a convolution operator with 2 the heat kernel, as F −1 (ξ 7→ e−t|ξ| gˆ(ξ)) can be represented as a convolution with the 2 function F −1 (e−t| . | ). (d) Consider A = dtd on Lp (R), with D(A) = W 1,p (R). For ψ ∈ H ∞ (Σω ) for some ω > π2 , define ψ(A)g := ΦA (ψ)g := F −1 (ξ 7→ ψ(iξ)ˆ g (ξ)), g ∈ S(R). Again by Mihlin’s multiplier theorem, we obtain ψ(A) ∈ L(Lp (R)). Note that the operator A has symbol iξ, and that the spectrum of A in Lp (R) is σ(A) = iR. We finish the section with a useful criterion to check the assumptions of Mihlin’s multiplier theorem. Definition 2.39. A function m : Rd \ {0} → C is called positive homogeneous of degree a ∈ R, if for all λ > 0 m(λξ) = λa m(ξ), ξ ∈ Rd \ {0}. Lemma 2.40. Let m ∈ C L (Rd \ {0}) for some L ∈ N, L > d2 , be positive homogeneous of degree 0. Then m ∈ Mp (Rd ) for all p ∈ (1, ∞).

CHAPTER 3

Real harmonic analysis One of the goals of this chapter is to give a proof of Mihlin’s multiplier theorem (Theorem 2.37). The arguments here will not rely on the Fourier transform directly, but we will consider translation invariant operators Tm f = F −1 (mfˆ) as convolution operators (F −1 m) ∗ f instead and use more geometric arguments based on set and function decompositions. Parts of this chapter follow the presentation in [2]. See also [11] and [7].

3.1. Coverings and maximal functions We use the following notation: For x ∈ Rd , r > 0 and β > 0, we write B(x, r) := {y ∈ Rd : |y − x| < r}, βB(x, r) := B(x, βr), rB or rad B

radius of a ball B ⊆ Rd ,

|B|

Lebesgue measure of B.

Given a covering of a set in Rd by balls, we would like to choose a sub-covering such that we can control the Lebesgue measure of the original set by (a multiple of) the sum of the Lebesgue measure of the balls of the sub-covering. One possibility to do so is as follows. Lemma 3.1 (Vitali covering lemma). Let {Bα }α∈I be a family of balls in Rd , and suppose that sup rad Bα < ∞. α∈I

Then there exists a countable subset I0 ⊆ I such that (i) {Bβ }β∈I0 are mutually disjoint; S S (ii) α∈I Bα ⊆ β∈I0 5Bβ . We introduce the Hardy-Littlewood maximal function as a central object of real harmonic analysis. Maximal functions are e.g. used to show almost everywhere convergence of families of integral operators. Definition 3.2 (Hardy-Littlewood maximal function). Let f ∈ L1loc (Rd ). 41

42

CHAPTER 3. REAL HARMONIC ANALYSIS

(i) We define the uncentred Hardy-Littlewood maximal function M (f ) by ˆ 1 M (f )(x) = sup |f (y)| dy, x ∈ Rd , B∋x |B| B where the supremum is taken over all balls B in Rd . (ii) We define the centred Hardy-Littlewood maximal function Mc (f ) by ˆ 1 Mc (f )(x) = sup |f (y)| dy, x ∈ Rd . r>0 |B(x, r)| B(x,r) Remark. There exists C = C(d) > 0 such that (exercise) (3.1)

Mc (f ) ≤ M (f ) ≤ C(d)Mc (f ).

We first show measurability of M (f ). We recall that a real-valued function g is called lower semi-continuous, if for all λ ∈ R, the set Oλ = {x ∈ Rd : g(x) > λ} is open. Lemma 3.3. Let f ∈ L1loc (Rd ). Then M (f ) is a lower semi-continuous function, and hence a Borel function. One can similarly show that Mc (f ) is lower semi-continuous. Remark. 1) If f ∈ L1loc (Rd ), then 1 Ar f (x) := |B(x, r)|

ˆ f (y) dy, B(x,r)

is continuous in r > 0 and x ∈ Rd . S 2) For every λ ∈ R, the set (Mc (f ))−1 ((λ, ∞)) = r>0 (Ar |f |)−1 ((λ, ∞)) is open. Example. (d = 1) Consider f = χ[a,b] for some a < b. Then for x ∈ R, Mc (f )(x) = sup r>0

|B(x, r) ∩ [a, b]| ≤ 1. |B(x, r)|

If x ∈ (a, b), we obtain Mc (f )(x) = 1 by choosing r = dist(x, ∂[a, b]). If x ≤ a, then |a−b| Mc (f )(x) = 2|x−b| with the choice r = |b − x|. We obtain  b−a   2|x−b| , x ≤ a, Mc (f )(x) = 1, x ∈ (a, b),   b−a , x ≥ b. 2|x−a| Note that Mc (f )(a) = Mc (f )(b) = 12 , thus Mc (f ) has a jump at a and b. Exercise: For the uncentred maximal function, we have  b−a   |x−b| , x ≤ a, M f (x) = 1, x ∈ (a, b),   b−a , x ≥ b, |x−a|

3.1. COVERINGS AND MAXIMAL FUNCTIONS



which shows that M f (x) = 1 +

dist(x,I) |I|

−1

43

, with I = [a, b].

Remark. 1) If f ∈ L1loc (Rd ), then for R > 0 and x ∈ Rd , ˆ 1 Mc (f )(x) ≥ |f (y)| dy. B(x, |x| + R) B(0,R) 2) If f ∈ L1 (Rd ) with M f ∈ L1 (Rd ), then f = 0 almost everywhere (exercise). The main result of this section is the following. Theorem 3.4 (Maximal theorem). There exists C = C(d) > 0 such that for all f ∈ L1 (Rd ) and all λ > 0, ˆ C d |{x ∈ R : M (f )(x) > λ}| ≤ |f (y)| dy. λ Rd By comparability of M (f ) and Mc (f ), the result holds correspondingly for Mc (f ). As an important consequence of the maximal theorem, we show the Lebesgue differentiation theorem. It provides a generalisation of the fundamental theorem of calculus for Lebesgue measurable functions. Theorem 3.5 (Lebesgue differentiation theorem). Let f ∈ L1loc (Rd ). Then there exists a measurable set Lf ⊆ Rd such that |Lcf | = 0 and ˆ 1 lim |f (x) − f (y)| dy = 0, x ∈ Lf . r→0 |B(x, r)| B(x,r) In particular, 1 f (x) = lim r→0 |B(x, r)|

ˆ f (y) dy,

x ∈ Lf .

B(x,r)

Lf is called the set of Lebesgue points. Remark. If f is continuous, one can choose Lcf = ∅. The result in particular shows that the maximal function Mc (f ) pointwise controls f . Corollary 3.6. Let f ∈ L1loc (Rd ). Then |f (x)| ≤ Mc f (x) for almost every x ∈ Rd . As we have seen, the maximal function cannot map into L1 (Rd ). We do however obtain Lp boundedness of the maximal operator for p > 1. For p = ∞, this is a simple observation. Remark. Let f ∈ L∞ (Rd ). Then Mc f (x) ≤ M f (x) ≤ ∥f ∥∞ , x ∈ Rd . For p ∈ (1, ∞), it is possible to interpolate between the result in Theorem 3.4 and the L∞ boundedness of M . See the next section for the interpolation theorem. We give a direct proof here instead.

44

CHAPTER 3. REAL HARMONIC ANALYSIS

Theorem 3.7 (Lp boundedness of maximal function). Let p ∈ (1, ∞]. Then there exists C = C(d, p) > 0 such that for every f ∈ Lp (Rd ), we have M f ∈ Lp (Rd ), and ∥M f ∥p ≤ C∥f ∥p . To be able to apply the maximal theorem in the proof, we use Cavalieri’s principle to rewrite the Lp norm in terms of level sets. Lemma 3.8 (Cavalieri’s principle). Let p ∈ (0, ∞), and let g ∈ Lp (Rd ). Then ˆ ∞ p |{x ∈ Rd : |g(x)| > λ}|λp−1 dλ. ∥g∥p = p 0

As an application of the maximal function, we show the Hardy-Littlewood-Sobolev inequality and, as a consequence thereof, Sobolev embeddings. We first define Riesz potentials as negative fractional powers of the Laplacian. At first formally, we can define ξ ∈ Rd \ {0}, ((−∆)−α/2 f )b(ξ) = |ξ|−α fˆ(ξ), or (−∆)−α/2 f := F −1 (|ξ|−α fˆ(ξ)) = F −1 (|ξ|−α ) ∗ f. From Proposition 2.27, we know that as elements in S ′ (Rd ), we have for λ ∈ (−d, 0), Tˆ|x|λ = cλ T|x|−d−λ , with cλ given in Proposition 2.27. Applying this identity with λ = −d + α for α ∈ (0, d), we have for alle f ∈ S(Rd ), 1 1 ξ ∈ Rd \ {0}. ( d−α ∗ f )(ξ) = |ξ|−α fˆ(ξ), c−d+α |x| This motivates the following definition. Definition 3.9 (Riesz potential). Let α ∈ (0, d), let f ∈ L1loc (Rd ). The Riesz potential Iα f of order α is defined by ˆ 1 f (y) Iα f (x) = dy, x ∈ Rd , cα Rd |x − y|d−α Γ( α )

2 with cα = π d/2 2α Γ( d−α . ) 2

Remark. We denote (−∆)−α/2 f := Iα f for f ∈ S(Rd ) and α ∈ (0, d). Theorem 3.10 (Hardy-Littlewood-Sobolev inequality). Let α ∈ (0, d), let p ∈ [1, αd ), and let q satisfy 1 1 α − = . p q d (i) Let p = 1. There exists a constant C = C(d, α) > 0 such that for all f ∈ S(Rd ), d C |{x ∈ Rd : |Iα f (x)| > λ}| ≤ d ∥f ∥1d−α . λ d−α

3.1. COVERINGS AND MAXIMAL FUNCTIONS

45

(ii) Let p > 1. There exists a constant C = C(d, α, p) > 0 such that for all f ∈ S(Rd ), ∥Iα f ∥q ≤ C∥f ∥p . Consequently, Iα has a unique extension on Lp (Rd ) for all p ∈ [1, αd ), which satisfies the above estimates. Remark. (Homogeneity properties of Iα ). Let r > 0, and consider the dilation δr f (x) := f (rx). Then Iα (δr f ) = r−α δr (Iα f ). Using this homogeneity property, one can show the following: If the inequality ∥Iα f ∥q ≤ C∥f ∥p holds for all f ∈ S(Rd ) and some C > 0 finite, then necessarily p1 − 1q = αd . We use the Hardy-Littlewood-Sobolev inequality to show Sobolev embeddings. We only give the statement for derivatives of order 1, using the following representation of f by ∇f . We refer to [11, Theorem V.2.2] and [9, Theorem 1.3.5] for more general statements on Sobolev embeddings. Lemma 3.11. Let f ∈ Cc∞ (Rd ). Then ˆ ∇f (y) · (x − y) 1 dy, f (x) = ωd−1 Rd |x − y|d

x ∈ Rd ,

where we denote by ωd−1 the surface measure of the unit sphere in Rd . Theorem 3.12 (Sobolev inequality). Let d ≥ 2, let p ∈ (1, d) and q with there eixsts C = C(d, p) such that for all f ∈ Cc∞ (Rd ),

1 p

− 1q = d1 . Then

∥f ∥q ≤ C∥∇f ∥p . Application: Almost everywhere convergence of solutions to the Dirichlet problem See [11, Section III.2]. Let d ≥ 2. Let f ∈ Lp (Rd ) be given for p ∈ [1, ∞), or f ∈ BU C(Rd ) for p = ∞. We denote by t pt (x) = cd x ∈ Rd , t > 0, d+1 , (t2 + |x|2 ) 2 the Poisson kernel, where cd =

) Γ( d+1 2 π

d+1 2

, and set u(t, x) := (pt ∗ f )(x). Recall that then u is

a solution to the Dirichlet problem (2.4) on Rd+1 + . From Theorem 2.8, we know that the boundary value u(0, . ) = f is attained in the sense that ∥u(t, . ) − f ∥p → 0, t → 0. For p < ∞, this only implies almost everywhere convergence for some subsequence (ptn ∗ f )n∈N . Using the maximal theorem, we can now show almost everywhere convergence for (pt ∗ f )t>0 . Theorem 3.13. Let p ∈ [1, ∞) and f ∈ Lp (Rd ), or let p = ∞ and f ∈ BU C(Rd ). Denote u(t, x) := pt ∗ f (x), (t, x) ∈ Rd+1 + . Then

46

CHAPTER 3. REAL HARMONIC ANALYSIS

x ∈ Rd ,

(i) sup |u(t, x)| ≤ Mc f (x), t>0

for almost every x ∈ Rd ,

(ii) lim u(t, x) = f (x), t→0

(iii) lim ∥u(t, . ) − f ∥p = 0. t→0

3.2. Real interpolation Definition 3.14 (Distribution function). Let f be a measurable function on Rd . The distribution function df of f is defined on [0, ∞) by df (λ) := |{x ∈ Rd : |f (x)| > λ}|,

λ ≥ 0.

df gives information about the size of f , but not about its behaviour near a given point. E.g. translations do not change the distribution function. P Example. Consider the simple function f = 3k=1 ak χEk , with a1 > a2 > a3 > 0, and Ek P P disjoint Borel sets in Rd . Set Bj := jk=1 |Ek |. Then df (λ) = 3j=1 Bj χ[aj+1 ,aj ) (λ), with a4 := 0. We collect some simple properties of the distribution function. Lemma 3.15. Let f, g : Rd → C be measurable. For λ, µ > 0, we have (i) |g| ≤ |f | for a.e. x ∈ Rd implies dg ≤ df , λ (ii) dcf (λ) = df ( |c| ), c ∈ C \ {0},

(iii) df +g (λ + µ) ≤ df (λ) + dg (µ), (iv) df g (λµ) ≤ df (λ) + dg (µ). In the previous section, we have seen that the maximal operator M does not map L1 functions into L1 function, since x 7→ |x|1 d is not integrable. We now define function spaces, which are larger than Lp spaces, and which are tailored to this situation. Definition 3.16 (Weak Lp spaces). Let p ∈ (0, ∞). The space Lp,∞ (Rd ) is defined as the set of all measurable functions f on Rd with sup λp |{x ∈ Rd : |f (x)| > λ}| = sup λp df (λ) < ∞, λ>0

λ>0

equipped with the quasi-norm ∥f ∥p,∞ := sup(λp df (λ))1/p . λ>0 ∞,∞

When p = ∞, we define L

d

∞

(R ) := L (Rd ).

One of the most important examples of functions in Lp,∞ (Rd ) is the following.

3.2. REAL INTERPOLATION

47

Example. Consider f (x) := |x|1d/p , for x ∈ Rd \ {0}. Then f ∈ / Lp (Rd ), but f ∈ Lp,∞ (Rd ), since 1 df (λ) = |{x : > λ}| = |{x : |x| < λ−p/d }| = |B(0, λ−p/d )| = νd λ−p , |x|d/p 1/p

and ∥f ∥p,∞ = supλ>0 (λp df (λ))1/p = νd . The inclusion Lp (Rd ) ⊆ Lp,∞ (Rd ) holds by Tchebytchev-Markov’s inequality. Lemma 3.17 (Tchebytchev-Markov inequality). Let p ∈ (0, ∞), let g be a positive, measurable function on Rd . Then ˆ 1 1 d |{x ∈ R : g(x) > λ}| ≤ p g(x)p dx ≤ p ∥g∥pp . λ {x : g(x)>λ} λ In particular, Lp (Rd ) ⫋ Lp,∞ (Rd ). Remark. 1) Lp,∞ (Rd ) is called weak Lp space. 2) For p ∈ (0, ∞), ∥f ∥p,∞ is not a norm, but a quasi-norm with ∥f + g∥p,∞ ≤ 2(∥f ∥p,∞ + ∥g∥p,∞ ),

f, g ∈ Lp,∞ (Rd ).

Note that ∥f ∥p,∞ = 0 implies f = 0 a.e. 3) Lp,∞ (Rd ) is metrisable for all p ∈ (0, ∞), and normable for all p ∈ (1, ∞). It is not normable for p = 1. Proposition 3.18. Let p ∈ (0, ∞). The space Lp,∞ (Rd ) is complete with respect to the quasi-norm ∥ . ∥p,∞ . □

Proof. See [8, Theorem 1.4.11].

We want to state the next interpolation theorem for sublinear operators and not only for linear operators, so that e.g. applications to the maximal operator are possible. For the maximal operator M : f 7→ M f , we can choose the set D below as L1loc (Rd ). Definition 3.19 (Sublinear operator). Let K ∈ {R, C}, let F := {f : Rd → K : f measurable}, and let D be a subspace of F. Let T : D → F be an operator. T is called sublinear, if for all f, g ∈ D and all α ∈ K, |T (f + g)(x)| ≤ |T f (x)| + |T g(x)|, |T (αf )(x)| = |α||T f (x)|,

for a.e. x ∈ Rd , for a.e. x ∈ Rd .

Definition 3.20. Let p, q ∈ [1, ∞], and let T : D → F be a sublinear operator. We say that

48

CHAPTER 3. REAL HARMONIC ANALYSIS

(i) T is of strong type (p, q) if T : D ∩ Lp (Rd ) → Lq (Rd ) is a bounded map. I.e. there exists a constant C > 0 such that whenever f ∈ D ∩ Lp (Rd ), then T f ∈ Lq (Rd ), and ∥T f ∥q ≤ C∥f ∥p . (ii) Let q < ∞. T is of weak type (p, q), if T : D ∩ Lp (Rd ) → Lq,∞ (Rd ) is a bounded map. (iii) T is of weak type (p, ∞) if it is of strong type (p, ∞). Example. (a) The maximal operator M : f 7→ M f is of weak type (1, 1), and of strong type (p, p), for p ∈ (1, ∞]. d (b) The Riesz potential Iα is of weak type (1, d−α ), 0 < α < d, and of strong type (p, q), 1 1 α d − q = d , p ∈ [1, α ). p Theorem 3.21 (Marcinkiewicz interpolation theorem). Let D be a subspace of F that is stable under multiplication by indicator functions, i.e., f ∈D

⇒

χE f ∈ D

for all E measurable.

Let p1 , p2 , q1 , q2 ∈ [1, ∞] with p1 < p2 , q1 < q2 , and p1 ≤ q1 , p2 ≤ q2 . Let T : D → F be a sublinear operator such that T : D ∩ Lp1 (Rd ) → Lq1 ,∞ (Rd ), T : D ∩ Lp2 (Rd ) → Lq2 ,∞ (Rd ) is bounded (i.e., T is of weak type (p1 , q1 ), and of weak type (p2 , q2 )). Then T : D ∩ Lp (Rd ) → Lq (Rd ) is bounded (i.e., T is of strong type (p, q)), with p, q satisfying 1−θ θ 1 = + , p p1 p2

1 1−θ θ = + , q q1 q2

for some θ ∈ (0, 1).

3.3. Calderón-Zygmund operators The two most important examples of Calderón-Zygmund operators are the Hilbert transform in one dimension, and its higher-dimensional analogues, the Riesz transforms. The Hilbert transform

3.3. CALDERÓN-ZYGMUND OPERATORS

49

Definition 3.22 (Hilbert transform). We define the map   1 ′ H : S(R) → S (R), f 7→ p.v. ∗ f. πx That is,  (3.2)

(p.v.

1 πx

ˆ

 ∗ f )(y) = lim

ε→0

{x∈R : |x|≥ε}

1 f (y − x) dx, πx

for a.e. y ∈ R.

H is called the Hilbert transform. 1 Remark. 1) Recall that by Example (e) in Section 2.4, p.v. πx ∈ S ′ (R). By definition of the convolution on S ′ (R), we have   1 ∗ f ∈ S ′ (R), f ∈ S(R), p.v. πx

with for g ∈ S(R),   ˆ ˆ 1 dx 1 ˜ (p.v. f (y − x)g(y) dy ∗ f )(g) = p.v. (f ∗ g) = lim ε→0 |x|≥ε R πx πx πx   ˆ ˆ f (y − x) dx dy. = g(y) lim ε→0 |x|≥ε πx R This shows (3.2). 2) As shown in the exercises (cf. [8, (5.1.12)]), we have   1 (3.3) F p.v. = −i sgn( . ). πx Using the Fourier symbol (3.3) of the Hilbert transform, we can extend the operator to L2 (R). Proposition 3.23. The Hilbert transform H : S(R) → S ′ (R) extends to a bounded operator on L2 (R), with (Hf )b(ξ) = −i sgn(ξ)fˆ(ξ),

f ∈ L2 (R).

Note that the explicit expression of Hf in (3.2) is only valid for f ∈ S(R), not for f ∈ L2 (R) in general. If f is compactly supported, we get the following representation away from the support. Lemma 3.24. Let f ∈ L2 (R) with supp f compact. Then ˆ 1 f (y) dy, for a.e. x ∈ / supp f. Hf (x) = π R x−y

50

CHAPTER 3. REAL HARMONIC ANALYSIS

Motivation: Connection with holomorphic extensions. Let f ∈ Cc∞ (R). The Cauchy extension of f to C \ R is given by ˆ 1 f (t) F (z) = dt 2πi R t − z for z = x + iy, y ̸= 0. F is holomorphic on C \ R, but note that C \ R is not simply connected. Question: What is limy→0+ F (x ± iy)? We decompose F into real and imaginary part and consider the limits separately. Recall that the Poisson kernel on R is given by y 1 Py (x) = , x ∈ R, y > 0. π y 2 + x2 Then for z = x + iy with y > 0, we have  ˆ   ˆ  i f (t) i f (t) 2 Re F (z) = Re dt = Re dt π R z−t π R x − t + iy ˆ y f (t) = dt = Py ∗ f (y), 2 π R y + (x − t)2 which is harmonic on the upper half-plane. Its harmonic conjugate is given by ˆ 1 f (t)(x − t) 2 Im F (z) = dt = Qy ∗ f (x), π R (x − t)2 + y 2 with Qy (x) = p ∈ [1, ∞),

1 x π y 2 +x2

the conjugate Poisson kernel. By Fejér’s theorem we know that for Py ∗ f → f,

y → 0+,

in Lp (R).

Exercise: For f ∈ S(R), Qy ∗ f → Hf,

y → 0+,

in Lp (R).

By Theorem 3.13 and an analogue result for the conjugate Poisson kernel, one also obtains pointwise convergence almost everywhere. We thus get 1 F (x ± iy) → (f (x) ± iHf (x)), y → 0+, 2 as well as F (x + iy) − F (x − iy) Hf (x) = lim , y→0+ i (3.4) f (x) = lim (F (x + iy) + F (x − iy)). y→0+

Remark. 1) Set F± (x) := limy→0+ F (x ± iy). Since H is bounded in L2 (R), (3.4) implies that the decomposition f = F+ + F− 2 is topological in L (R), i.e. ∥f ∥2 ≃ ∥F+ ∥2 + ∥F− ∥2 .

3.3. CALDERÓN-ZYGMUND OPERATORS

51

2) For f ∈ S(R), one can show the identity (Hf )2 = f 2 + 2H(f · Hf ), which can be used to show the Lp boundedness of H. See [8, Theorem 5.1.7] for the details. The Riesz transforms The higher-dimensional analogues of the Hilbert transform are defined as follows. We will discuss applications of Riesz transforms at the end of the section. Definition 3.25. Let j ∈ {1, . . . , d}. We define Rj : S(Rd ) → S ′ (Rd ), that is,

ˆ Rj f (y) = lim cd ε→0

where cd =

Γ( d+1 ) 2 d+1 π 2

{x∈Rd : |x|≥ε}

f 7→ p.v. cd

xj ∗ f, |x|d+1

xj f (y − x) dx, |x|d+1

for a.e. y ∈ Rd ,

.

′ d Remark. Similarly to ´ the casexj of the Hilbert transform, one can show that Rj f ∈ S (R ) d for f ∈ S(R ), since ε≤|x|≤1 |x|d+1 dx = 0 by symmetry.

Proposition 3.26. Let j ∈ {1, . . . , d}. Rj extends to a bounded operator on L2 (Rd ), with (Rj f )b(ξ) = −i

ξj ˆ f (ξ), |ξ|

f ∈ L2 (Rd ), ξ ̸= 0.

Our aim is to show Lp boundedness of e.g. the Hilbert transform or the Riesz transforms. xj 1 Since the kernels p.v. πx and p.v. cd |x|d+1 are not integrable at 0, this does not immediately follow from Young’s inequality. We start with the Calderón-Zygmund decomposition, which yields a decomposition of L1 functions into a “good” and a “bad” part. Lemma 3.27 (Calderón-Zygmund decomposition). Let f ∈ L1 (Rd ), let λ > 0. Then there exists a sequence (Qj )j of cubes with sides parallel to the axes and pairwise disjoint interior such that ˆ 1 λ< |f (y)| dy ≤ 2d λ, j ∈ N, |Qj | Qj and |f (x)| ≤ λ

for a.e. x ∈ Rd \

[ j∈N

This immediately implies the following decomposition.

Qj .

52

CHAPTER 3. REAL HARMONIC ANALYSIS

Corollary 3.28. Let f ∈ L1 (Rd ), let λ > 0. Let (Qj )j be a sequence of cubes as in Lemma 3.27. Then there exists a decomposition f =g+b

a.e. on Rd ,

where (i) g ∈ L∞ (Rd ), and ∥g∥∞ ≤ 2d λ, ∞ X (ii) b = bj , with supp bj ⊆ Qj , ˆ

j=1

bj (y) dy = 0,

(iii) Qj ∞ X

ˆ ∞ X 1 ∥f ∥1 (iv) |Qj | ≤ , |f (y)| dy ≤ λ Qj λ j=1 j=1 ˆ ˆ ∞ ∞ X X (v) |bj (y)| dy ≤ 2 |f (y)| dy, thus ∥b∥1 = ∥ bj ∥ 1 = ∥bj ∥1 ≤ 2∥f ∥1 . Qj

Qj 1

d

(vi) g ∈ L (R ), and ∥g∥1 ≤ ∥f ∥1 ,

j=1

∥g∥22

j=1 d

≤ ∥g∥1 ∥g∥∞ ≤ 2 λ∥f ∥1 .

Remark. For f ∈ L1 (Rd ) with f = g + b as above, g is called “good” function, because 1 d g ∈ L2 (Rd ), and ´ b is called “bad” function, because b ∈ L (R ) only, but with the additional cancellation bj = 0. Notation: ∆ := {(x, x) : x ∈ Rd } the diagonal of Rd × Rd . Definition 3.29 (Calderón-Zygmund kernel). Let α ∈ (0, 1]. A Calderón-Zygmund kernel of order α is a continuous function K : Rd × Rd \ ∆ → K such that there exists a C > 0 with C (a) |K(x, y)| ≤ , (x, y) ∈ ∆c , d |x − y|  α |y − y ′ | 1 ′ ′ (b) |K(x, y) − K(x, y )| + |K(y, x) − K(y , x)| ≤ C , |x − y| |x − y|d for all x, y, y ′ ∈ Rd with |y − y ′ | ≤ 12 |x − y|, x ̸= y. Remark. When α = 1, then ∇y K(x, y) exists a.e. with |∇y K(x, y)| ≤

C˜ , |x − y|d+1

(x, y) ∈ ∆c ,

and analogously for ∇x K(x, y). Definition 3.30 (Calderón-Zygmund operator). Let T ∈ L(L2 (Rd )).

3.3. CALDERÓN-ZYGMUND OPERATORS

53

(i) K : ∆c → K is said to be associated with T if for all f ∈ L2 (Rd ) with supp f compact, ˆ T f (x) = K(x, y)f (y) dy Rd c

for a.e. x ∈ (supp f ) . (ii) Let α ∈ (0, 1]. T is called Calderón-Zygmund operator of order α if it is associated with a Calderón-Zygmund kernel K of order α. Notation: CZOα := {T ∈ L(L2 (Rd )) : T Calderón-Zygmund operator of order α} Remark. 1) The map T 7→ K, where T ∈ CZOα , and K is the associated CalderónZygmund kernel, is not injective. Thus, T is not uniquely determined by the associated kernel. Example: Let m ∈ L∞ (Rd ), and let Tm f = mf be the multiplication operator on L2 (Rd ). Then Tm ∈ L(L2 (Rd )). Let K be a kernel with K = 0 on ∆c . For f ∈ L2 (Rd ) with supp f compact, we then have Tm f (x) = m(x)f (x) = 0, and

x∈ / supp f,

ˆ K(x, y)f (y) dy = 0,

x∈ / supp f,

Rd

thus K = 0 is the associated kernel to any of the multiplication operators Tm . 2) Let T ∈ L(L2 (Rd )), and denote by T ∗ its Hilbert space adjoint, by T ′ its real dual operator. Then T ∈ CZOα

⇔

T ∗ ∈ CZOα

⇔

T ′ ∈ CZOα .

If K is the associated kernel to T , then K ∗ and K ′ are the associated kernels to T ∗ and T ′ , respectively, where K ∗ (x, y) = K(y, x) and K ′ (x, y) = K(y, x). Weak type (1, 1) boundedness of Calderón-Zygmund operators We first introduce a weaker kernel condition compared to Definition 3.29 (b), which will be sufficient for showing weak type (1, 1) boundedness. Definition 3.31 (Hörmander condition). Let K ∈ L1loc (Rd \ ∆), and suppose that there exists CH > 0 such that ˆ (3.5) |K(x, y) − K(x, y ′ )| dx ≤ CH for a.e. y, y ′ ∈ Rd . |x−y|≥2|y−y ′ |

Then K is said to satisfy the Hörmander condition. Remark. Every Calderón-Zygmund kernel K of order α, α ∈ (0, 1], satisfies the Hörmander condition (exercise). Theorem 3.32. Let α ∈ (0, 1], let T be a CZO of order α. Then T is of weak type (1, 1).

54

CHAPTER 3. REAL HARMONIC ANALYSIS

Remark. Theorem 3.32 remains true if T satisfies all conditions of a Calderón-Zygmund operator except for Definition 3.29 (b) replaced by the Hörmander condition (3.5). Using the interpolation theorem of Marcinkiewicz, we can now deduce Lp boundedness of Calderón-Zygmund operators for p ∈ (1, 2). Since the dual operator is again a CalderónZygmund operator, we obtain Lp boundedness for all p ∈ (1, ∞). Corollary 3.33. Let α ∈ (0, 1], let T be a CZO of order α. Then for every p ∈ (1, ∞), T is of strong type (p, p).

Remark. Similarly to above, the result remains true if Definition 3.29 (b) is replaced by the Hörmander condition (3.5) for both K and the dual kernel K ′ .

3.4. Mihlin’s multiplier theorem We are now in the position to give a proof of Mihlin’s multiplier theorem, Theorem 2.37. Recall: Given m ∈ L∞ (Rd ), we define the operator Tm by Tm (f ) := F −1 (mfˆ),

f ∈ S(Rd ).

We call Mp (Rd ) the space of all Fourier multipliers in Lp (Rd ), i.e. the space of all m ∈ L∞ (Rd ) such that Tm extends to a bounded operator on Lp (Rd ). Idea of proof: We show that Tm , defined by Tm f = F −1 (mfˆ), satisfies the Hörmander condition. Since Tm is a convolution operator, the Hörmander condition for K and K ′ is the same. Formally, we have Tm f = K ∗ f with K = F −1 m, but, in general, K ∈ / L1 (Rd ). We therefore decompose m as a sum of functions mj with compact support such that Kj = F −1 mj ∈ L1 (Rd ). Lemma 3.34 (Partition of unity). There exists φ ∈ C ∞ (Rd \ {0}), radial, bounded, with supp φ ⊆ {ξ ∈ Rd :

1 ≤ |ξ| ≤ 4}, 2

and for all α ∈ Nd0 , ∥∂ α φ∥∞ ≤ C(d, α), such that (φj )j∈Z with φj (ξ) := φ(2−j ξ) form a partition of unity, i.e., ∞ X

φj (ξ) = 1,

ξ ∈ Rd \ {0}.

j=−∞

For given ξ ̸= 0, at most 4 terms in the sum are nonzero.

3.5. LITTLEWOOD-PALEY THEORY

55

3.5. Littlewood-Paley theory 1/2

In the following, let H be a separable complex Hilbert space, with norm | . |H = ( . , . )H . We want to define Lp spaces for H-valued functions. Let f : Rd → H. Then f is strongly measurable if given an orthonormal basis (ej )j of H, all coefficients (f, ej ) : Rd → C are measurable. We refer to [1, Chapter X] for the integration theory of Banach space valued functions. See also [15, Section V.4]. (i) Let p ∈ [1, ∞). Define the Hilbert-space valued Lp space as ˆ p d d L (R ; H) := {f : R → H : f strongly measurable, |f |pH dx < ∞},

Definition 3.35.

Rd

equipped with the norm ∥f ∥Lp (H) := ∥|f |H ∥p . (ii) For p = ∞, define L∞ (Rd ; H) := {f : Rd → H : f strongly measurable, |f |H ∈ L∞ (Rd )}, equipped with the norm ∥f ∥L∞ (H) := ∥|f |H ∥∞ . In this section, we will be interested in the case H = ℓ2 (Z), for which we get Lp (Rd ; ℓ2 (Z)) = {f = (fj )j∈Z : Rd → C : fj measurable for all j, ∥f ∥p < ∞}, equipped with ∥f ∥p = ∥f ∥Lp (ℓ2 (Z)) = ∥

X

|fj |2


1/2

∥Lp (Rd ) .

j∈Z

Remark. Replacing the norm on C by the norm | . |H on H in Section 3.3, it is not hard to see that one can extend the Calderón-Zygmund theory to operators T : L2 (Rd ; C) → L2 (Rd ; H), where the conditions (a), (b) in Definition 3.29 are replaced by c (a)H |K(x, y)|H ≤ |x−y| (x, y) ∈ ∆c , d,   ′| α (b)H |K(x, y) − K(x, y ′ )|H + |K(y, x) − K(y ′ , x)|H ≤ C |y−y |x−y|

1 , |x−y|d

for all x, y, y ′ ∈ Rd with |y − y ′ | ≤ 12 |x − y|, x ̸= y. As a consequence, we obtain the following generalisation of Corollary 3.33. Corollary 3.36. Let p ∈ (1, ∞), let T ∈ L(L2 (Rd ; C), L2 (Rd ; H)) be a Calderón-Zygmund operator. Then T extends to a bounded operator T : Lp (Rd ; C) → Lp (Rd ; H).

56

CHAPTER 3. REAL HARMONIC ANALYSIS

We use Corollary 3.36 to show the Littlewood-Paley theorem. For p ∈ (1, ∞), it gives an alternative characterisation of the Lp norm of a function f by the Lp norm of the square P 1/2 2 function f 7→ of frequency-localised Littlewood-Paley blocks ∆j f . j∈Z |∆j f | Theorem 3.37. Let p ∈ (1, ∞). Let ψ ∈ S(Rd ) with 1 ˆ = 1, |ξ| ∈ [1, 2]. and ψ(ξ) supp ψˆ ⊆ {ξ ∈ Rd : ≤ |ξ| ≤ 4}, 2 Then there exists C1 , C2 > 0 depending only on ψ, d, p such that for all f ∈ Lp (Rd ) !1/2 X C1 ∥f ∥p ≤ ∥ |∆j f |2 ∥p ≤ C2 ∥f ∥p , j∈Z

where ∆j f = ψj ∗ f , with ψj (x) = 2jd ψ(2j x). d ˆ ˆ ˆ Remark. Note that the ∆j ’s are almost orthogonal : We have ∆ j f = ψj f , with ψj (ξ) = ˆ −j ξ), and supp ψˆj ⊆ {ξ ∈ Rd : 2j−1 ≤ |ξ| ≤ 2j+2 }. Thus, ψ(2 1 d d (∆ whenever |j − k| ≥ 3. (∆j f, ∆k g) = j f , ∆k g) = 0, (2π)d Lack of orthogonality in Lp Remark. 1) Consider (fj )j∈Z ⊆ Lp (Rd ) with supp fˆj mutually disjoint. In L2 (Rd ): By Plancherel, the disjointness of the Fourier support implies that the sequence (fj )j is orthogonal, i.e. X X X X 1 ˆj ∥2 = 1 ˆj ∥2 = f ∥ fj ∥22 = ∥ ∥ f ∥fj ∥22 . 2 2 d d (2π) (2π) j j j j In Lp (Rd ): In general, the inequalities X X (3.6) ∥ fj ∥pp ≤ Cp ∥fj ∥pp , j

X

p > 2,

j

∥fj ∥pp

≤ Cp ∥

j

X

fj ∥pp ,

p < 2,

j

are false. See the example below and [8, Section 6.1]. 2) Let p ∈ (1, ∞), let f ∈ S(Rd ). Then, in general, the inequalities X
1/q (3.7) ∥ |∆j f |q ∥p ≤ Cp,q ∥f ∥p , q < 2, j∈Z

∥f ∥p ≤ Cp,q ∥

X

|∆j f |q


1/q

∥p ,

q > 2,

j∈Z

are false. Thus, Theorem 3.37 is false for ℓ2 (Z) replaced by ℓq (Z), q ̸= 2.

3.5. LITTLEWOOD-PALEY THEORY

57

Example. (a) Counterexample for (3.6) with p > 2. Let φ ∈ S(R), φ ̸= 0, with φˆ ≥ 0, and supp φˆ ⊆ {ξ : |ξ| ≤ 41 }. For j ∈ N0 , define fj (x) := eijx φ(x). Then fˆj (ξ) = φ(ξ ˆ − j), supp fˆj ⊆ {ξ : |ξ − j| ≤ 41 }, and for N ∈ N, N X

∥fj ∥pp = (N + 1)∥φ∥pp .

j=0

On the other hand, ∥

N X

ˆ fj ∥pp

|

= R

j=0

ˆ

N X

ˆ e

ijx p

p

| |φ(x)| dx =

j=0

≥c |x|< 10(N1 +1)

ei(N +1)x − 1 p |φ(x)|p dx ix − 1 e R

(N + 1)x p |φ(x)|p dx = Cp (N + 1)p−1 , x

´ 1 since by assumption, φ(0) = 2π φ(ξ) ˆ dξ > 0. If p > 2, this shows that (3.6) cannot R hold. (b) (Sketch) Counterexample for (3.7) with q < 2. Let φ ∈ S(R). One can construct a sequence of functions (fj )j with fj (x) = eicj x φ(x),

∆j (fj ) = fj , j ≥ 3, PN for suitably chosen coefficients cj . Then for f = j=3 fj , we have and

N X ∥( |∆j (fj )|q )1/q ∥p = (N − 2)1/q ∥φ∥p , j=3

and by Theorem 3.37, ∥f ∥p = ∥

N X

f j ∥ p ≤ cp ∥

j=3

N X

|∆j (fj )|2


1/2

∥p = cp (N − 2)1/2 ∥φ∥p .

j=3

This shows that (3.7) is false for q < 2. Application of the Littlewood-Paley decomposition We use the Littlewood-Paley decomposition of Theorem 3.37 to establish a Sobolev embedding for homogeneous Sobolev spaces. For f ∈ S(Rd ) and s ≥ 0, set ∥f ∥ ˙ s := ∥ξ 7→ |ξ|s fˆ(ξ)∥2 . H

Proposition 3.38. Let s ∈ [0, d2 ), and 12 − p1 = ds . There exists a constant Cs,d > 0 such that for all f ∈ S(Rd ), ∥f ∥p ≤ Cs,d ∥f ∥H˙ s . Compare with Theorem 3.12 for s = 1.

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CHAPTER 3. REAL HARMONIC ANALYSIS

Remark. In the proof, we use the generalised Minkowski inequality: For p, q ∈ [1, ∞], p ≥ q, and fj ∈ Lp (Rd ) for j ∈ Z, we have X X
1/q
1/q ∥ ∥p ≤ . |fj |q ∥fj ∥qp j∈Z

j∈Z

The proof of Proposition 3.38 relies on the following estimate for frequency-localised functions. Lemma 3.39. Let f ∈ S(Rd ) with supp fˆ ⊆ {ξ : |ξ| ≤ R} for some R > 0. Then for p, q ∈ [1, ∞], p ≤ q, we have 1

1

∥f ∥q ≤ Cd Rd( p − q ) ∥f ∥p . Almost everywhere convergence of Calderón-Zygmund operators Recall: The Hilbert transform is defined as a principal value integral, i.e.   ˆ 1 1 f (x − y) dy ∗ f = lim Hf (x) = p.v. ε→0 |y|>ε πy πy f ∈ S(R).

=: lim Hε f (x), ε→0

From Corollary 3.33, we know that H extends to a bounded operator on Lp (R), p ∈ (1, ∞), i.e. there exists Cp > 0 such that ∥Hf ∥p ≤ Cp ∥f ∥p ,

f ∈ Lp (R),

Hε f → Hf, ε → 0

w.r.t. ∥ . ∥p .

and

Question: Almost everywhere convergence: For f ∈ Lp (R), do we have (3.8)

Hε f (x) → Hf (x), ε → 0,

for a.e. x ∈ R?

To establish (3.8), we define the associated maximal operator H ∗ by H ∗ f (x) := sup |Hε f (x)|. ε>0

We then get the following result. See [4, Theorem 3.4 and Lemma 3.5] for a proof. Theorem.

(i) (Cotlar’s lemma). For f ∈ S(R), H ∗ f (x) ≤ M (Hf )(x) + cM f (x),

x ∈ R.

(ii) The maximal operator H ∗ is of strong type (p, p) for p ∈ (1, ∞), and of weak type (1, 1). (iii) Let p ∈ [1, ∞), let f ∈ Lp (R). Then Hf (x) = lim Hε f (x) ε→0

for a.e. x ∈ R.

3.5. LITTLEWOOD-PALEY THEORY

59

Remark. More generally, for a Calderón-Zygmund operator T with associated kernel K, define ˆ Tε f (x) := K(x, y)f (y) dy, f ∈ S(Rd ). |x−y|>ε

Then one can show that lim Tε f (x)

(3.9)

ε→0

for f ∈ Cc∞ (Rd ) if and only if ˆ lim ε→0

exists a.e.

K(x, y) dy

exists a.e.

εε} γ(t) − γ(s) ˆ 1 f (w) = lim dw. ε→0 πi {w∈Γ : | Re z−Re w|>ε} z − w Note that for A = 0, this gives us the formula for the Hilbert transform. For g(s) = f (γ(s))γ ′ (s), we arrive at the operator T defined by ˆ g(y) (3.10) T g(x) := lim dy, ε→0 |x−y|>ε x − y + i(A(x) − A(y))

60

CHAPTER 3. REAL HARMONIC ANALYSIS

with associated kernel 1 , (x, y) ∈ R2 \ ∆. x − y + i(A(x) − A(y)) It is straightforward to check that K is a Calderón-Zygmund kernel of order 1. In order to show that T is a Calderón-Zygmund operator, one needs to check that T ∈ L(L2 (R)) first. This is difficult for the Cauchy integral operator. K(x, y) :=

Theorem (Coifman, McIntosh, Meyer, 1982). The operator T as defined in (3.10) extends to a bounded operator in L2 (R).

CHAPTER 4

Oscillatory integrals 4.1. The wave equation in Rd × R As a motivation for the study of oscillatory integrals, we shortly have a look at solutions to the wave equation in Rd × R. For more details, see e.g. [12, Section 6.3] or [6, Section 2.4]. Given initial conditions f, g ∈ S(Rd ), we consider the d-dimensional wave equation in L2 (Rd ) given by  d 2  ∂t u − ∆u = 0 on R × R, (W) u( . , 0) = f in Rd ,  ∂ u( . , 0) = g in Rd . t In order to get a candidate for a solution to (W), we take the Fourier transform with respect to x ∈ Rd , which transforms (W) into  2 2  ∂t uˆ(ξ, t) + |ξ| uˆ(ξ, t) = 0, uˆ( . , 0) = fˆ,  uˆ ( . , 0) = gˆ. t For fixed ξ ∈ Rd , we now have to solve an ordinary differential equation, with solution uˆ(ξ, t) = A(ξ) cos(|ξ|t) + B(ξ) sin(|ξ|t), where A and B are chosen to satisfy the initial conditions uˆ(ξ, 0) = A(ξ) = fˆ(ξ),

uˆt (ξ, 0) = B(ξ)|ξ| = gˆ(ξ).

Hence, we get sin(|ξ|t) , uˆ(ξ, t) = fˆ(ξ) cos(|ξ|t) + gˆ(ξ) |ξ|

(ξ, t) ∈ Rd × R,

which is indeed a solution to (W). Proposition 4.1. Given f, g ∈ S(Rd ), a solution to the wave equation (W) in L2 (Rd ×R) is given by ˆ  1 sin(|ξ|t)  ixξ (4.1) u(x, t) = fˆ(ξ) cos(|ξ|t) + gˆ(ξ) e dξ. d (2π) Rd |ξ| □

Proof. See [12, Section 6.3]. 61

62

CHAPTER 4. OSCILLATORY INTEGRALS

Proposition 4.1 provides us with an explicit formula for the solution to the wave equation. It can however be somewhat difficult to calculate the Fourier inverse. For dimensions d = 1 and d = 3, we give more direct expressions. Case d = 1: We look at the wave equation on some interval [0, L]. Then the solution of the wave equation is given by d’Alemnbert’s formula ˆ 1 1 x+t u(x, t) = [f (x + t) − f (x − t)] + g(y) dy, 2 2 x−t with f, g extended outside [0, L] such that f, g are odd on [−L, L] and f, g are 2L-periodic on R. Case d = 3: We define the spherical mean of f ∈ S(R3 ) over the sphere with centre x and radius t by ˆ 1 At f (x) := f (x − tγ) dσ(γ), (x, t) ∈ R3 × R. 4π S 2 Recall that σ(S 2 ) = 4π. We first note that the averaging operator At preserves smoothness and decay properties of functions, and is smooth with respect to t. Lemma 4.2. Let f ∈ S(R3 ), let t ∈ R be fixed. Then At f ∈ S(R3 ), and At f ∈ C ∞ (R) is k indefinitely differentiable with respect to t, with dtd k At f ∈ S(R3 ). Since At f ∈ S(R3 ), we can calculate the Fourier transform as   ˆ ˆ 1 −ix·ξ d f (x − γt) dσ(γ) dx e A t f (ξ) = 4π S 2 R3 ˆ ˆ 1 = ( e−i(x−γt)·ξ f (x − γt) dx)e−iγt·ξ dσ(γ) 4π S 2 R3 ˆ 1 = fˆ(ξ) · e−iγt·ξ dσ(γ), ξ ∈ R3 . 4π S 2 Changing into spherical coordinates, we can calculate the latter integral. Lemma 4.3. For ξ ∈ R3 , we have ˆ 1 sin(|ξ|) e−iξ·γ dσ(γ) = . 4π S 2 |ξ| We thus get for t ∈ R sin(|ξ|t) ˆ d A f (ξ), t f (ξ) = |ξ|t

ξ ∈ R3 ,

which we can now use to rewrite (4.1) for the case d = 3. Proposition 4.4. Given f, g ∈ S(R3 ), a solution to (W) is given by u(x, t) = ∂t (tAt f (x)) + tAt g(x),

(x, t) ∈ R3 × R.

4.2. OSCILLATORY INTEGRALS

63

Smoothing properties of At We define the spherical means for any dimension d > 1, but fix t = 1. For f ∈ S(Rd ), define the averaging operator ˆ 1 f (x − y) dσ(y), x ∈ Rd . Af (x) := σ(S d−1 ) S d−1 We can show that A not only extends to a bounded operator on L2 (Rd ), but has smoothing properties, with the order of smoothing depending on the dimension. Proposition 4.5. Let s =

d−1 . 2

The map

A : L2 (Rd ) → H s (Rd ),

f 7→ Af

is bounded. Remark. In the proof, we implicitly use that for f ∈ L2 (Rd ), Af = σ(S1d−1 f ∗ dσ. In the sense of tempered distributions with dσ a finite Borel measure, we then get 1 c = c Af fˆdσ. σ(S d−1 For d = 3, Lemma 4.3 shows that

1 c dσ(ξ) 4π

=

sin(|ξ| , |ξ|

with

1 c ≤ c(1 + |ξ|)−1 . dσ 4π 4.2. Oscillatory integrals We study so-called oscillatory integrals, which are of the following form and which play an important role in the study of dispersive equations. Definition 4.6. Let λ ∈ R, let ϕ : Rd → R, and ψ : Rd → C be smooth and integrable. Then ˆ I(λ) := eiλϕ(x) ψ(x) dx Rd

is called oscillatory integral, ϕ is the phase function, and ψ the amplitude. We would like to understand the decay of I(λ) for λ → ±∞, depending on the precise assumptions on the phase and the amplitude. Remark. 1) It suffices to consider λ > 0, since the case λ < 0 follows from taking the ´ complex conjugate, since I(−λ) = eiλϕ(x) ψ(x) dx. 2) We have the trivial bound |I(λ)| ≤ ∥ψ∥1 , which is sharp for ϕ = const. For ϕ ̸= const, one expects decay for I(λ), λ → ±∞.

64

CHAPTER 4. OSCILLATORY INTEGRALS

Non-stationary phase The most simple situation in dimension 1 is as follows. The crucial assumption here is that ϕ′ is non-vanishing, which in particular excludes the case ϕ = const . Lemma 4.7. Let d = 1, and let ϕ ∈ C ∞ (R, R), ψ ∈ C ∞ (R, C) with supp ψ compact. Assume that ϕ′ (x) ̸= 0, x ∈ supp ψ. Then for all N ∈ N, there exists CN > 0 such that |I(λ)| ≤ CN λ−N ,

λ > 0.

In the proof, the key idea is to use integration by parts to write eiλu =

1 dn iλu (e ). (iλ)N dλN

When we choose the amplitude function ψ = χ[a,b] for some interval [a, b], we get boundary terms in the integration by parts argument used in the proof of Lemma 4.7, which prevent higher order decay. Note however, that we can bound the size of I1 (λ) independently of the length of the interval [a, b]. Lemma 4.8. Let a, b ∈ R, a < b, let ϕ ∈ C 2 (R, R), and let ϕ′ be monotonic, with |ϕ′ (x)| ≥ 1 on [a, b]. Then for ˆ b I1 (λ) := eiλϕ(x) dx, a

we have |I1 (λ)| ≤

3 , λ

λ > 0.

Remark. The decrease of order λ1 is optimal. Example: Consider ϕ(x) = x, then ϕ′ (x) = 1, and I1 (λ) =

1 (eiλb iλ

− eiλa ).

For the higher dimensional analogue, we also restrict to the case ∇ϕ non-vanishing at first. Definition 4.9. Let x0 ∈ Rd . Then x0 is called a critical point of a phase function ϕ, if ∇ϕ(x0 ) = 0. Proposition 4.10 (Principle of non-stationary phase). Let ϕ ∈ C ∞ (Rd ; R) and ψ ∈ C ∞ (Rd ; C) with supp ψ compact. Suppose that ϕ has no critical points on supp ψ. Then for all N ∈ N, there exists CN > 0 such that |I(λ)| ≤ CN λ−N ,

λ > 0.

4.2. OSCILLATORY INTEGRALS

65

Stationary phase We now investigate what happens if the phase ϕ has some critical point, i.e., ϕ′ (x0 ) = 0 for some x0 ∈ Rd . We will assume in the following that the critical point x0 is non-degenerate, that is, ϕ′′ (x0 ) ̸= 0. We start with the example of a quadratic phase ϕ(x) = x2 . Example. (d = 1) Consider the phase function ϕ(x) = x2 , with critical point x0 = 0, which is non-degenerate, since ϕ′′ (0) ̸= 0. Claim: We have the expansion ˆ 2 eiλx ψ(x) dx = c0 λ−1/2 + O(|λ|−3/2 ), λ → ∞, R

or, more precisely, for N ∈ N0 , ˆ N X 2 (4.2) eiλx ψ(x) dx = ck λ−1/2−k + O(|λ|−3/2−N ), R

k=0

Thus, the best possible decay is of order λ−1/2 . Proof of (4.2): Using the Fourier transform of the Gauß kernel, we know that for s > 0, ˆ ˆ x2 1 1 −1/2 −sx2 ˆ dx. e ψ(x) dx = √ s e− 4s ψ(x) 2 π R R We can extend this analytically to Re s > 0. It is then possible to take the limit to s = −iλ (see the exercises), which yields  1/2 ˆ ˆ 1 1 i 2 iλx2 ˆ dx. e ψ(x) dx = √ e−ix /4λ ψ(x) 2 π λ R R P 2 (iu2 )k 2N +2 Using the expansion eiu = N ), this gives k=0 k! + O(|u|  1/2 1 1 i I(λ) = √ 2πψ(0) + . . . 2 π λ = λ−1/2 (iπ)1/2 ψ(0) + . . . . It is then straightforward to check that ck = (iπ)1/2

ik (2k) ψ (0), 22k k!

k ∈ N0 .

In the one-dimensional situation with constant amplitude, we obtain the following. Proposition 4.11. Let a, b ∈ R, a < b, and let ϕ ∈ C 2 ([a, b]; R) with |ϕ′′ (x)| ≥ 1 on [a, b]. Then for ˆ b I1 (λ) = eiλϕ(x) dx, a

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CHAPTER 4. OSCILLATORY INTEGRALS

we have |I1 (λ)| ≤

8 λ1/2

,

λ > 0.

With an integration by parts argument, we can extend this to oscillatory integrals with smooth amplitude and obtain the estimate of van der Corput. Corollary 4.12. Let a, b ∈ R, a < b. Let ϕ be as in Proposition 4.11, and let ψ ∈ C 1 ([a, b]; C). Then ˆ b eiλϕ(x) ψ(x) dx| ≤ cψ λ−1/2 , λ > 0, | a ´b with cψ = 8( a |ψ ′ (x)| dx + |ψ(b)|). Example. (a) (Bessel functions) For m ∈ Z, define the Bessel function Jm by ˆ 2π 1 Jm (r) := (4.3) eir sin x e−imx dx, r ∈ R. 2π 0 Jm (r) is the m-th Fourier coefficient of x 7→ eir sin x , hence, we have a representation X eir sin x = Jm (r)eimx , x ∈ T. m∈Z

One can extend the definition of Bessel functions to m ∈ R with m > − 21 by the formula
ˆ 1 r m 1 2 Jm (r) := r ∈ R. eirt (1 − t2 )m− 2 dt, 1 1 Γ(m + 2 )Γ( 2 ) −1 For m ∈ Z, this definition is consistent with (4.3), see e.g. [8, Appendix B.2]. Claim: We have the asymptotic behaviour (4.4)

Jm (r) = O(r−1/2 ),

r → ∞.

Proof of (4.4): Jm (r) is given as oscillatory integral in (4.3) with λ = r, ϕ(x) = sin x, 1 −imx e , [a, b] = [0, 2π]. We decompose the interval and consider first A1 := ψ(x) = 2π π 3 [ 4 , 4 π] ∪ [ 45 π, 74 π]. In this region, ϕ has critical points, but we have 1 |ϕ′′ (x)| = | sin x| ≥ √ , 2 We can thus apply Corollary 4.12, which yields A1 Jm (r) = O(r−1/2 ), √1 2

x ∈ A1 .

r → ∞.

Note that the lower bound instead of 1 changes the constant, but not the asymptotics in the estimate. On the other hand, on A2 := [0, 2π] \ A1 , we are in the non-stationary phase regime, since 1 |ϕ′ (x)| = | cos x| ≥ √ , x ∈ A2 . 2

4.2. OSCILLATORY INTEGRALS

67

Thus, Lemma 4.8 applies, where, as in previous proofs, we can include an amplitude ψ using integration by parts. We obtain A2 (r) = O(r−1 ), Jm

r → ∞.

Combining the two estimates concludes the proof. (b) (Radial functions) If f ∈ S(Rd ) is radial, i.e. f (x) = f0 (|x|) for some function f0 defined on R, then fˆ is radial, i.e. fˆ(ξ) = F0 (|ξ|) for some function F0 . (See [12, Section 6.4].) Using the Bessel functions, one can express F0 in terms of f0 in the following way. We set ρ = |ξ|. For d = 1, radial functions are even functions, thus ˆ ∞ ˆ cos(rρ)f0 (r) dr. F0 (ρ) = f (|ξ|) = 2 0

For d = 3, one can use Lemma 4.3 to show that ˆ 1 ∞ F0 (ρ) = 4π sin(ρr)f0 (r)r dr. ρ 0 In general, one arrives at ([8, Appendix B.5]) ˆ ∞ d/2 −d/2+1 (4.5) F0 (ρ) = (2π) ρ Jd/2−1 (ρr)rd/2−1 f0 (r)r dr. 0 q Since J1/2 (r) = π2 r−1/2 sin(r), this is consistent with the above formula for d = 3. (c) Because the surface measure dσ on S d−1 is radial, one can use (4.5) to show that c dσ(ξ) = (2π)d/2 |ξ|−d/2+1 Jd/2−1 (ξ), which, together with Example (a), implies (4.6)

c |dσ(ξ)| = O(|ξ|−

d−1 2

),

|ξ| → ∞.

The latter is the key estimate for the proof of Proposition 4.5. We generalise Corollary 4.12 to higher dimensions. We denote by Hϕ the Hessian of ϕ. Proposition 4.13 (Principle of stationary phase). Let ϕ ∈ C ∞ (Rd ; R) and ψ ∈ C ∞ (Rd ; C) with supp ψ compact. Suppose that det Hϕ (x) ̸= 0, Then for

x ∈ supp ψ.

ˆ eiλϕ(x) ψ(x) dx,

I(λ) = Rd

we have (4.7)

|I(λ)| = O(λ−d/2 ),

λ → ∞.

Remark. Checking carefully the proof, we see that in (4.7), we get an estimate of the form |I(λ)| ≤ Aλ−d/2 , λ > 0, where A only depends on ∥ϕ∥C d+2 , ∥ψ∥C d+1 , the lower bound for | det Hϕ (x)|, and diam supp ψ.

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CHAPTER 4. OSCILLATORY INTEGRALS

4.3. The Fourier transform of surface-carried measures Our aim is to apply Proposition 4.13 to the Fourier transform of finite measures on hypersurfaces. These naturally appear as solution formulas in the study of dispersive differential equations, and the decay behaviour of the surface measure is closely linked to smoothing properties of the solution operators. See the forthcoming section for an application to the Schrödinger equation. We quickly recall the local parametrisation of a hypersurface as a graph of a function. Let M ⊆ Rd be a C ∞ hypersurface. Then for every a = (a′ , ad ) ∈ M , after possibly rotating the initial coordiante system, there exist U1 ⊆ Rd−1 open, a′ ∈ U1 , U2 ⊆ R open, ad ∈ U2 , and φ ∈ C ∞ (U1 , R), with φ(U1 ) ⊆ U2 , such that (U1 × U2 ) ∩ M = {(x′ , xd ) ∈ U1 × U2 : xd = φ(x′ )}. By translation and rotation of the coordinate system, one can further assume that φ(0) = 0,

∇x′ φ(x′ )|x′ =0 = 0.

Then a ∈ M corresponds to (0, 0), and the tangent plane to M at 0 is the hyperplane xd = 0. With this choice, Taylor’s theorem yields the representation 1 X φ(x′ ) = ajk xj xk + O(|x′ |3 ), 2 1≤j,k≤d−1  2  ∂ φ(0) with the coefficient matrix (ajk )jk = Hφ (0) = ∂j ∂k . Since Hφ is symmetric, we can jk

use another rotation in Rd−1 to write d−1

φ(x′ ) =

1X λj x2j + O(|x′ |3 ). 2 j=1

The occurring λj ’s are called the principal curvatures of M at a, and det Hφ (0) = is the Gauß curvature of M at a.

Qd−1 j=1

λj

Recall that if M is covered by only one chart, with the local parametrisation M = {(x′ , xd ) : xd = φ(x′ )}, then for f ∈ C(M ) with supp f compact, we can calculate the surface integral as ˆ ˆ p f dσ = f (x′ , φ(x′ )) 1 + |∇x′ φ(x′ )|2 dx′ . M

Rd−1

We call a measure dµ a surface-carried measure on M with smooth density, if dµ = ψdσ for some ψ ∈ C ∞ (M ) with compact support.

4.4. APPLICATION TO THE SCHRÖDINGER EQUATION

69

c where As a generalisation of (4.6), we are interested in the decay behaviour of dµ, ˆ c= dµ e−ixξ dµ(x). M

We can show that (4.6) remains valid whenever the hypersurface M has non-vanishing curvature. This in particular applies to the sphere in Rd or the paraboloid, but not to the circular cone {x2d = |x′ |2 , xd ̸= 0}. Theorem 4.14. Let M ⊆ Rd be a C ∞ hypersurface, let dµ be a surface-carried measure with smooth density on M . Suppose that M has non-vanishing Gaussian curvature at each point of supp dµ. Then d−1 c |dµ(ξ)| = O(|ξ|− 2 ), |ξ| → ∞. 4.4. Application to the Schrödinger equation In this last section, we want to apply the results on oscillatory integrals to the solution of the Schrödinger equation. The Schrödinger equation is an example of a dispersive equation, which are, roughly speaking, equations that obey some conservation properties, such as conservation of mass or energy (the L2 norm of the solution or the gradient of the solution), yet the solutions disperse in the sense that the L∞ norm decays over time. Given f ∈ S(Rd ), consider the linear Schrödinger equation ( i∂t u + ∆u = 0, (x, t) ∈ Rd × R, (S) u( . , 0) = f. To find a candidate for a solution to (S), we first argue formally and take the Fourier transform with respect to x, which transforms (S) into the equation ( i∂t uˆ(ξ, t) − |ξ|2 uˆ(ξ, t) = 0, uˆ( . , 0) = fˆ. This reduces the problem to the ordinary differential equation i∂t uˆ = |ξ|2 uˆ, which we can directly solve. Taking the initial condition into account, we arrive at 2 uˆ(ξ, t) = e−it|ξ| fˆ(ξ), (ξ, t) ∈ Rd × R. We thus obtain (4.8)

1 u(x, t) = (2π)d =F

−1

(e

ˆ 2 eixξ e−it|ξ| fˆ(ξ) dξ

Rd −it|ξ|2

fˆ(ξ))(x) =: eit∆ (f )(x),

(x, t) ∈ Rd × R.

One can check that u defined in (4.8) is actually a solution in the classical sense if the initial datum is in S(Rd ). Proposition 4.15. Let t ∈ R, let f ∈ S(Rd ). Then

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CHAPTER 4. OSCILLATORY INTEGRALS

(i) eit∆ : S(Rd ) → S(Rd ). (ii) For u defined by (4.8), we have u ∈ C ∞ (Rd × R) is a solution to (S). (iii) We have u(x, t) = Kt ∗ f (x), t ̸= 0, where Kt is the Schrödinger kernel given by 1 2 Kt (x) = e−|x| /4it , x ∈ Rd , t ̸= 0. d/2 (4πit) (iv) ∥u( . , t)∥∞ ≤

1 ∥f ∥1 , (4π|t|)d/2

t ̸= 0. □

Proof. See exercise sheet 6 or [13, Proposition 8.6.1].

We can extend the operator eit∆ defined in (4.8) to L2 (Rd ) and obtain the following result for initial data f ∈ L2 (Rd ). Proposition 4.16. Let t ∈ R, let f ∈ L2 (Rd ). Then (i) eit∆ : L2 (Rd ) → L2 (Rd ) is unitary. (ii) The map R → L2 (Rd ), t 7→ eit∆ f , is continuous. (iii) u defined by (4.8) satisfies (S) in the weak sense. That is, for all φ ∈ Cc∞ (Rd ), ˆ (−i∂t + ∆)(φ)(x, t) · u(x, t) dxdt = 0. Rd ×R

□

Proof. See [13, Proposition 8.6.2].

If the initial datum f is in L1 (Rd ), then Proposition 4.15 shows that the solution u decays in the supremum norm for |t| → ∞. Since eit∆ is a unitary operator in L2 (Rd ), we cannot hope for the same behaviour for f ∈ L2 (Rd ). We are thus looking for averaged decay in x and t. One result of this kind are the following Strichartz estimates. See for example [3, Chapter 8] for more details on Strichartz estimates. Theorem 4.17 (Strichartz estimates). Let f ∈ L2 (Rd ), let q = there exists some c > 0 independent of f such that (4.9)

2d+4 . d

For u as in (4.8),

∥u∥Lq (Rd ×R) ≤ c∥f ∥L2 (Rd ) .

Proof. See e.g. [13, Theorem 8.6.3] for a proof using the theory of oscillatory integrals and complex interpolation, or see [3, Theorem 8.18 and Proposition 8.6] for a more direct proof based on the Hardy-Littlewood-Sobolev inequality (Theorem 3.10). □ Remark. If u is a solution to the Schrödinger equation (S), (4.9) can only be true for q = 2d+4 , as can be seen from a scaling argument: For δ > 0, set d fδ (x) := f (δx),

uδ (x, t) := u(δx, δ 2 t).

Then uδ solves (S) with initial data fδ . We have ∥fδ ∥2 = δ −d/2 ∥f ∥2 ,

∥uδ ∥Lqt,x = δ −

d+2 q

∥u∥Lqt,x ,

4.4. APPLICATION TO THE SCHRÖDINGER EQUATION

71

hence in order to satisfy (4.9), one requires δ−

d+2 q

∥u∥q ≤ cδ −d/2 ∥f ∥2 ,

δ > 0.

If we distinguish the cases δ < 1 and δ > 1, it becomes clear that this is only possible for d+2 = d2 . q We do not give a proof of Theorem 4.17 here, but instead prove the following weaker result. Proposition 4.18. Let f ∈ L2 (Rd ) with supp fˆ ⊆ B(0, 1). Let q > exists some c > 0 independent of f such that (4.10)

2d+4 . d

Then there

∥u∥Lq (Rd ×R) ≤ c∥f ∥L2 (Rd ) .

Idea of proof: We want to write the solution u to the linear Schrödinger equation (S) in the form u(x, t) = eit∆ f (x) ≈ Fd dµ(x) for a suitably defined measure dµ on a manifold M , where M is determined by the solution formula in (4.8), and then apply Theorem 4.14. To see this, we consider the Fourier transform on Rd+1 , with x¯ = (x, xd+1 ) ∈ Rd+1 , xd+1 = t, and the dual variable ξ¯ = (ξ, ξd+1 ) ∈ Rd+1 . We define the paraboloid M := {(ξ, ξd+1 ) ∈ Rd+1 : ξd+1 = −|ξ|2 }, with the parametrisation ξd+1 = φ(ξ) = −|ξ|2 , and ∇ξ φ(ξ) = −2ξ. Note that the paraboloid has non-vanishing Gaussian curvature at every point. We can then write

ˆ 1 2 u(x, t) = u(x, xd+1 ) = eixξ e−it|ξ| fˆ(ξ) dξ d (2π) Rd ˆ p −1 p 1 ixξ ixd+1 φ(ξ) 2 = e e ψ (ξ, φ(ξ)) 1 + 4|ξ| 1 + 4|ξ|2 fˆ(ξ) dξ, 0 (2π)d Rd

where we choose some ψ0 ∈ Cc∞ (Rd+1 ) with ψ0 = 1 for (ξ, ξd+1 ) ∈ M and |ξ| ≤ 1, and use p ¯ := ψ0 (ξ) ¯ 1 + |∇ξ φ(ξ)|2 −1 , and that supp fˆ ⊆ B(0, 1). We now set ψ(ξ) (4.11)

dµ = ψdσ.

Moreover, denote F (ξ, ξd+1 ) := fˆ(ξ) for (ξ, ξd+1 ) ∈ Rd+1 . This yields the representation ˆ ˆ 1 1 ¯ i¯ xξ¯ ¯ (ξ)dσ( ¯ ¯ = ¯ ¯ u(x, t) = e ψ(ξ)F ξ) ei¯xξ F (ξ)dµ( ξ) d d (2π) M (2π) M 1 d = F dµ(−¯ x). (2π)d

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CHAPTER 4. OSCILLATORY INTEGRALS

By definition of ψ0 , we have ∥F ∥L2 (M,dµ) = ∥fˆ∥L2 (Rd ) . Thus, in order to prove (4.10), we need to show for F (ξ, ξd+1 ) = fˆ(ξ), 1 (4.12) ∥u∥Lqt,x = ∥Fd dµ∥Lqt,x ≤ c∥F ∥L2 (M,dµ) = c∥f ∥L2 (Rd ) , (2π)d with dµ defined in (4.11) a surface-carried measure with smooth density ψ with compact support on the paraboloid M in Rd+1 . Instead of directly proving (4.12), it is simpler to restate the problem as a convolution c and then use Theorem 4.14. This relies on the following equivalence (which can with dµ, ′ abstractly be understood as an equivalence of T : L2 → Lq and T T ∗ : Lq → Lq , and is also called a T T ∗ argument). Lemma 4.19. Let dµ be a finite positive measure. Let q ≥ 1, with c > 0. The following are equivalent: (i) ∥Fd dµ∥q ≤ c∥F ∥L2 (M,dµ) , for all F ∈ L2 (M, dµ). c ∗ f ∥q ≤ c2 ∥f ∥q′ , (ii) ∥dµ

1 q

+

1 q′

= 1, and let

for all f ∈ S(Rd ). □

Proof. See [13, Proposition 8.5.3]. The inhomogeneous Schrödinger equation

Given F ∈ Cc∞ (Rd × R), consider the inhomogeneous Schrödinger equation with initial condition zero ( i∂t u + ∆u = F, (x, t) ∈ Rd × R, (Sinh ) u( . , 0) = 0. Formally, we can then use Duhamel’s formula to write the solution operator as ˆ t ei(t−s)∆ F (s, . ) ds, SF ( . , t) := −i 0

which is well-defined by Proposition 4.15. One can show that SF indeed solves the inhomogeneous problem (Sinh ). Proposition 4.20. Let F ∈ Cc∞ (Rd × R), let q =

2d+4 d

and

1 q

+

1 q′

= 1. Then

(i) SF ∈ C ∞ (Rd × R) and solves (Sinh ). (ii) There exists c > 0 independent of F such that ∥SF ∥Lq (Rd ×R) ≤ c∥F ∥Lq′ (Rd ×R) .

(4.13)

Proof. See [13, Proposition 8.6.6, Theorem 8.6.7]. ′

□

Remark. For F ∈ Lq (Rd × R), SF can be redefined on a set of measure zero such that for all t, SF ( . , t) ∈ L2 (Rd ), and the map R → L2 (Rd ), t 7→ SF ( . , t) is continuous.

4.4. APPLICATION TO THE SCHRÖDINGER EQUATION

73

The critical nonlinear Schrödinger equation In this last paragraph, we give a brief sketch for the solvability of the nonlinear Schrödinger equation in the scale-invariant case. In order to run a fixed-point argument, we rely on the Strichartz estimates from Theorem 4.17. Let λ > 1 and σ ∈ R, σ ̸= 0. Consider for f ∈ L2 (Rd ) the nonlinear Schrödinger equation ( −i∂t u − ∆u = σ|u|λ−1 u, (x, t) ∈ Rd × R, (NLS) u( . , 0) = f.

Remark. 1) Solutions to (NLS) the two conservation properties ´ have 2 (a) conservation of “mass” R´d |u| dx, and σ |u|λ ) dx, (b) conservation of “energy” Rd ( 21 |∇u|2 − λ+1 which are independent of t. 2) Scale-invariant case: We are looking for the choice of λ, for which the problem is scaleinvariant. If u solves (NLS) with initial data f , then δ a u(δx, δ 2 t) solves (NLS) with initial data δ a f (δx), for every δ > 0, for the choice a + 2 = λa. We moreover want that ∥f ∥2 = ∥fδ ∥2 , which is the case for a = d2 . We thus get the condition 4 λ=1+ . d 2d+4 2d+4 ′ Note that for q = d , we have q = d+4 and q ′ λ = q. 3) By rescaling the equation (x, t) 7→ (|σ|1/2 x, |σ|t), we can restrict to the cases σ = ±1. The main result of this section now states the existence of a solution to (NLS) for all times if the initial data is sufficiently small. Let f ∈ L2 (Rd ) be given. We call u ∈ Lq (Rd × R) a strong solution to (NLS) if (i) u satisfies the differential equation in the weak sense; (ii) for all t ∈ R, u( . , t) ∈ L2 (Rd ), and the map R 7→ L2 (Rd ), t 7→ u( . , t) is continuous with u( . , 0) = f . Theorem 4.21. Let q = 2d+4 , 1q + q1′ = 1 and λ = d+4 . Let f ∈ L2 (Rd ). Then there exists d d ε > 0 such that whenever ∥f ∥2 < ε, there exists a strong solution to (NLS) in Lq (Rd × R). Proof. See [13, Theorem 8.6.9].

□

Idea of proof: Let u0 := eit∆ f be the solution to the linear equation. Then, using the solution operator (4.13) to the inhomogeneous problem, we need to find u such that u = σS(|u|λ−1 u) + u0 .

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This can be solved by an iteration argument, provided ∥f ∥2 is sufficiently small. In order to apply the Banach fixed point theorem, we define a metric space B and a contraction mapping M. From the previous results, we know that (1) the map L2 (Rd ) → Lq (Rd × R), f 7→ u0 = eit∆ f is bounded (Strichartz estimates, Theorem 4.17); ′ (2) the map S : Lq (Rd × R) → Lq (Rd × R) is bounded (Proposition 4.20); ′ (3) the nonlinear map N : Lq (Rd × R) → Lq (Rd × R), u 7→ |u|λ−1 u is bounded with ∥N u∥q′ = ∥u∥λq . We therefore define the metric space B := {u ∈ Lq (Rd × R) : ∥u∥q ≤ δ} for some δ > 0, and the map M on B by M(u) := σS(|u|λ−1 u) + u0 . For δ > 0 and ε > 0 sufficiently small, one can then use (1)-(3) to show that for ∥f ∥2 ≤ ε, (i) M : B → B, (ii) ∥M(u) − M(v)∥q ≤ 21 ∥u − v∥q , u, v ∈ B. This allows to apply the Banach fixed point theorem, and it remains to show that the obtained fixed point is indeed a strong solution to (NLS).

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