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Harmonic and Real Analysis Herbert Koch Universit¨Atbonn Wintersemester 2014-2015

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Harmonic and Real Analysis Herbert Koch Universit¨Atbonn Wintersemester 2014-2015 Notes for Harmonic and real analysis Herbert Koch Universit¨atBonn Wintersemester 2014-2015 Recommended literature: [10, 7, 14, 13, 15] 1 Contents Chapter 1. Introduction 5 1. An example 5 Chapter 2. Fourier series 9 1. Definitions 9 2. The convolution and Young's inequality 10 3. Lp convergence of partial sums 15 4. Regularity and Fourier series 16 5. Complex Interpolation 17 6. The Hardy-Littlewood maximal function and real interpolation 21 7. Higher dimensions 30 Chapter 3. Harmonic functions and the Poisson kernel 31 1. Basic properties and the Poisson kernel 31 2. Boundary behaviour of harmonic functions 36 3. Almost everywhere convergence 37 4. Subharmonic functions 38 5. The theorems of Riesz 40 6. Conjugate functions 41 7. The Hilbert transform 43 n Chapter 4. The Fourier transform on R 47 1. Definition and first properties 47 2. The Fourier transform of Schwartz functions 48 3. The Fourier transform on tempered distributions 51 4. Oscillatory integrals and stationary phase 58 5. Quantization 64 6. Cotlar's Lemma and the Theorem of Calder´on-Vaillancourt 70 Chapter 5. Singular integrals of Calder´on-Zygmund type 75 1. The setting 75 2. The Calder´on-Zygmund theorem 77 3. Examples 81 Chapter 6. Hardy and BMO 87 1. More general maximal functions and the Hardy space Hp 87 2. The atomic decomposition 91 3. Duality and BMO 95 4. Relation to harmonic functions 98 5. Div-curl type results 100 6. Application to elliptic PDEs 101 7. Pointwise estimates and perturbations of elliptic equations 107 3 4 CONTENTS Chapter 7. Littlewood-Paley theory and square functions 111 1. The range 1 < p < 1 111 2. Square functions, tents and Carleson measures 114 Bibliography 117 CHAPTER 1 Introduction Harmonic analysis is concerned with describing, decomposing and ana- lyzing functions and operators with some 'structure' coming from the struc- ture of the Euclidean space. Its relevance comes from the insight that the same structures are relevant in different areas of mathematics like partial differential equations, signal processing, Fourier analysis and mathemati- cal physics. Key notions are Fourier transform, maximal functions, square function, BMO and Hardy spaces, Calder´on-Zygmund operators and their relation to partial differential equations. 1. An example The series 1 X 1 (1.1) zn n n=1 converges absolute for jzj < 1, and uniformly on every ball of radius < 1 since m m X 1 X 1 − jzjm+1 1 f(z) = jzjn ≤ jzjn = ≤ : n 1 − jzj 1 − jzj n=1 n=0 I claim that, given " > 0, it converges uniformly in fz : jzj ≤ 1; jz − 1j > "g: This is proven by an argument going back to Abel and mimics an integration by parts: m−1 m−1 m−1 j ! X zj 1 X X X 1 1 = zj + zk − j m j j + 1 j=n j=n j=n k=n m−1 m−1 1 X X zn − zj+1 1 1 = zj + − : m 1 − z j j + 1 j=n j=n To verify the formula, compare the coefficients of zj, n ≤ j ≤ m − 1: m−1 1 1 X 1 1 = + − : j m l l + 1 l=j Then m j X z 4 ≤ j n" j=n 5 6 1. INTRODUCTION and the partial sums are a Cauchy sequence. The limit is continuous, as a uniform limit of continuous functions. It is not hard to identify it: 1 f 0(x) = 1 − x which is the derivative of − ln(1 − x) and a check at x = 0 shows that f(x) = − ln(1 − x). Hence Im(1 − z) f(z) = − ln (1 − z) = − ln j1 − zj − i arctan C Re(1 − z) Im z = − ln j1 − zj + i arctan 1 − Re z for jzj ≤ 1, z 6= 1. Define 1 1 X (eix)n X einx 1 sin(x) g(x) := = = − ln j2 − 2 cos xj + i arctan n n 2 1 − cos(x) n=1 n=1 since p j1 − eixj = (1 − cos(x))2 + sin(x)2 = p2 − 2 cos(x): By elementary geometry of the inscribed angle or the addition theorem sin(x) 2 sin(x=2) cos(x=2) = 1 − cos(x) 1 − cos2(x=2) + sin2(x=2) π−x cos(x=2) sin( 2 ) π − x = = π−x = tan( ) sin(x=2) cos( 2 ) 2 and we obtain for the imaginary part 1 X sin(nx) π − x (1.2) h (x) := = : 0 n 2 n=1 for 0 < x < 2π. Define the absolute convergent series 1 X cos(nx) h (x) = : 1 n2 n=1 and for 0 < x < 2π n Z x Z x X sin(jx) h0(t)dt = lim dx n!1 j π π j=1 0 n n 1 X cos(jx) X cos(jπ) = − lim @ + A n!1 j2 j2 j=1 j=1 = − h1(x) + h1(π) and −h1 is a primitive of h0. Thus there exists a 2 R such that x2 πx h (x) = − + a 1 4 2 1. AN EXAMPLE 7 R 2π 1 and we want to determine a. Since 0 cos(nt)dt = n (sin(2πn)+sin(0)) = 0 we get Z 2π 3 2π 3 0 = h1(t)dt = − π + 2aπ 0 3 π2 and a = 6 . The evaluation at t = 0 gives the value of the Riemann ζ function at the value 2. Lemma 1.1. 1 X 1 π2 (1.3) ζ(2) = = : n2 6 n=1 Similarly 1 X 1 π4 (1.4) ζ(4) = = : n4 90 n=1 We have seen different notions of convergence. (1) Point-wise convergence (2) Uniform convergence (3) Absolute convergence (4) Convergence in L2 in connection with Fourier series. The notions are important since we want to differentiate resp. integrate term by term. CHAPTER 2 Fourier series 1. Definitions We denote the one dimensional torus by T = RnZ. Functions on the torus are identified with periodic functions on R with period 1. We denote the space of Radon measures ( complex Borel measures which are finite on compact sets) on a metric space X by M - these are objects which can be written as µ+ + µ− + iµi− − iµi− with non negative Radon measures µ+, µ−, µi+ and µi−. Definition 2.1. Let µ 2 T . We define the Fourier coefficients Z 1 −2inπx µ^(n) =µ ^n = e µ 0 and we write formally 1 X µ ∼ µ^(n)e2inπx n=−∞ for the relation between the series and the measure µ. If f is an integrable function we write Z 1 −2inπx f^(n) = f^n = e f(x)dx 0 and 1 X f ∼ f^(n)e2inπx: n=−∞ 2inπx The functions (e )n are orthonormal in the sense that Z 1 if n = m e\2iπnx(m) e2inπxe2imπxdx = 0 if n 6= m The convergence questions of the Fourier series are interesting, impor- tant, and a prototype for similar question in all areas of analysis. 9 10 2. FOURIER SERIES We define the Dirichlet kernel through N X ^ 2inπx SN f(x) = f(n)e n=−N N Z X 0 = e−2inπx f(x0)dx0e2inπx n=−N T Z X 0 = e2inπ(x−x )f(x0)dx0 T n=−N N Z 0 0 0 = DN (x − x )f(x )dx T =DN ∗ f(x) where N X sin[(2N + 1)πx] D (x) = einx = : N sin πx n=−N We may formulate the convergence question as: When and in which sense does DN ∗ f(x) converge to f(x)? 2. The convolution and Young's inequality Motivated by the occurrence of the convolution we take a closer look n at its properties. For X = T, R or R we define the convolution of two continuous functions with compact support by Z f ∗ g(x) = f(x − y)g(y)dy: We will need the convolution in much more general context. For that we consider Z If;g;h = f(x)g(x − y)h(y)dxdy Rn×Rn Lemma 2.2. Suppose that 1 ≤ p; q; r ≤ 1 and 1 1 1 + + = 2: p q r Then the integral defining I is integrable for all f 2 Lp, g 2 Lq and h 2 Lr and jIf;g;hj ≤ kfkLp kgkLq khkLr Proof. The case when one of the exponents is 1 and the others are 1 is simple. Hence we may assume that 1 ≤ p; q; r < 1. Let p=r0 q=r0 F1(x; y) =jf(x)j jg(x − y)j ; q=p0 r=p0 F2(x; y) =jg(x − y)j jh(y)j ; p=q0 r=q0 F3(x; y) =jf(x)j jh(y)j where 0 denotes the H¨olderdual exponent 1 1 1 1 1 1 + = + = + = 1; p p0 q q0 r r0 2. THE CONVOLUTION AND YOUNG'S INEQUALITY 11 with the obvious interpretation of the exponent 1. Then, applying H¨older's 1 1 1 inequality twice, since p0 + q0 + r0 = 1 Z Ifgh ≤ F1F2F3dxdy Z 1=q q ≤ (F1F2) dxdy kF3kLq0 ≤kF1kLr0 kF2kLp0 kF3kLq0 p q q q p r r0 r0 p0 p0 q0 q0 =kfkLp kgkLq kgkLq khkLr kfkLp khkLr =kfkLp kgkLq khkLr The lemma is the 'dual' statement to Young's inequality. Proposition 2.3 (Young's inequality). Suppose that 1 ≤ p; q; r ≤ 1 and 1 1 1 + = 1 + : p q r If f 2 Lp and g 2 Lq then for almost all x the integrand of Z f(x − y)g(y)dy r n is integrable and it defines a function in L (R ). Moreover p q kf ∗ gkLr(Rn) ≤ kfkL kgkL : If r = 1 then the integrand is integrable for all x and f ∗ g is a bounded continuous function.
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