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Lecture Notes: Harmonic Analysis

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Lecture Notes: Harmonic Analysis Lecture notes: harmonic analysis Russell Brown Department of mathematics University of Kentucky Lexington, KY 40506-0027 August 14, 2009 ii Contents Preface vii 1 The Fourier transform on L1 1 1.1 Definition and symmetry properties . 1 1.2 The Fourier inversion theorem . 9 2 Tempered distributions 11 2.1 Test functions . 11 2.2 Tempered distributions . 15 2.3 Operations on tempered distributions . 17 2.4 The Fourier transform . 20 2.5 More distributions . 22 3 The Fourier transform on L2. 25 3.1 Plancherel's theorem . 25 3.2 Multiplier operators . 27 3.3 Sobolev spaces . 28 4 Interpolation of operators 31 4.1 The Riesz-Thorin theorem . 31 4.2 Interpolation for analytic families of operators . 36 4.3 Real methods . 37 5 The Hardy-Littlewood maximal function 41 5.1 The Lp-inequalities . 41 5.2 Differentiation theorems . 45 iii iv CONTENTS 6 Singular integrals 49 6.1 Calder´on-Zygmund kernels . 49 6.2 Some multiplier operators . 55 7 Littlewood-Paley theory 61 7.1 A square function that characterizes Lp ................... 61 7.2 Variations . 63 8 Fractional integration 65 8.1 The Hardy-Littlewood-Sobolev theorem . 66 8.2 A Sobolev inequality . 72 9 Singular multipliers 77 9.1 Estimates for an operator with a singular symbol . 77 9.2 A trace theorem. 87 10 The Dirichlet problem for elliptic equations. 91 10.1 Domains in Rn ................................ 91 10.2 The weak Dirichlet problem . 99 11 Inverse Problems: Boundary identifiability 103 11.1 The Dirichlet to Neumann map . 103 11.2 Identifiability . 107 12 Inverse problem: Global uniqueness 117 12.1 A Schr¨odingerequation . 117 12.2 Exponentially growing solutions . 122 13 Bessel functions 127 14 Restriction to the sphere 129 15 The uniform sobolev inequality 131 16 Inverse problems: potentials in Ln=2 133 17 Scattering for a two-dimensional system 135 17.1 Jost solutions . 136 17.2 Estimates . 137 17.3 Notes . 142 CONTENTS v 18 Global existence of Jost solutions 143 18.1 Uniqueness . 143 18.2 Existence of solutions. 146 18.3 Behavior for large z .............................. 147 19 Differentiability of the Jost solutions 151 19.1 Differentiability of the Jost solution with respect to x. 151 19.2 Differentiability with respect to z . 153 19.3 Higher derivatives with respect to z . 156 20 Asymptotic expansion of the Jost solutions 159 20.1 Expansion with respect to x . 159 20.2 Expansion in z ................................ 161 20.3 The inverse of the scattering map. 163 21 The scattering map and evolution equations 167 21.1 A quadratic identity . 167 21.2 The tangent maps . 169 21.3 The evolution equations . 172 A Some functional analysis 175 A.1 Topologies . 175 A.2 Compact operators . 176 A.3 Derivatives . 176 vi CONTENTS Preface These notes are intended for a course in harmonic analysis on Rn which was offered to graduate students at the University of Kentucky in Spring of 2001. The background for this course is a course in real analysis which covers measure theory and the basic facts of life related to Lp spaces. The students who were subjected to this course had studied from Measure and integral by Wheeden and Zygmund and Real analysis: a modern introduction, by Folland. Much of the material in these notes is taken from the books of Stein Singular integrals and differentiability properties of functions, [19] and Harmonic analysis [20] and the book of Stein and Weiss, Fourier analysis on Euclidean spaces [21]. The monograph of Loukas Grafakos, Classical and modern Fourier analysis [8] provides an excellent treatment of the Fourier analysis in the first half of these notes. The exercises serve a number of purposes. They illustrate extensions of the main ideas. They provide a chance to state simple results that will be needed later. They occasionally give interesting problems. These notes are at an early stage and far from perfect. Please let me know of any errors. Participants in the 2008 version of the course include Jay Hineman, Joel Kilty, Justin Taylor, Zhongyi Nie, Jun Geng, Michael Shaw and Julie Miker. Their contributions to improving these notes are greatly appreciated. Russell Brown, [email protected] vii viii PREFACE This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. The latex source is available from the author. Chapter 1 The Fourier transform on L1 In this chapter, we define the Fourier transform and give the basic properties of the Fourier transform of an L1(Rn) function. Recall that L1(Rn) is the space of Lebesgue R measurable functions for which the norm kfk1 = Rn jf(x)j dx is finite. For 0 < p < 1, Lp(Rn) denotes the space of Lebesgue measurable functions for which the norm R p 1=p 1 n kfkp = ( Rn jf(x)j dx) is finite. When p = 1, the space L (R ) is the collection of measurable functions which are essentially bounded. For 1 ≤ p ≤ 1, the space Lp(Rn) is a Banach space. We recall that a vector space V over C with a function k · k is called a normed vector space if k · k : V ! [0; 1) and satisfies kf + gk ≤ kfk + kgk; f; g 2 V kλfk = jλjkfk; f 2 v; λ 2 C kfk = 0; if and only if f = 0: A function k · k which satisfies these properties is called a norm. If k · k is a norm, then kf − gk defines a metric. A normed vector space (V; k · k) is called a Banach space if V is complete in the metric defined using the norm. Throughout these notes, functions are assumed to be complex valued. 1.1 Definition and symmetry properties We define the Fourier transform. In this definition, x · ξ is the inner product of two n Pn elements of R , x · ξ = j=1 xjξj. Definition 1.1 If f 2 L1(Rn), then the Fourier transform of f, f^, is a function defined 1 2 CHAPTER 1. THE FOURIER TRANSFORM ON L1 on Rn and is given by Z f^(ξ) = f(x)e−ix·ξ dx: Rn The Fourier transform is a continuous map from L1 to the bounded continuous func- tions on Rn. Proposition 1.2 If f 2 L1(Rn), then f^ is continuous and ^ kfk1 ≤ kfk1: Proof. The estimate follows since e−ix·ξ is of modulus 1. Let fξjg be a sequence in Rn j −ix·ξj −ix·ξ −ix·ξj with limj!1 ξ = ξ, then we have limj!1 e f(x) = f(x)e and je f(x)j ≤ ^ j ^ jf(x)j. By the Lebesgue dominated convergence theorem, we have limj!1 f(ξ ) ! f(ξ). The inequality in the conclusion of Proposition 1.2 is equivalent to the continuity of the map f ! f^. This is an application of the conclusion of the following exercise. Exercise 1.3 A linear map T : V ! W between normed vector spaces is continuous if and only if there exists a constant C so that kT fkW ≤ CkfkV : In the following proposition, we use A−t = (A−1)t for the transpose of the inverse of an n × n matrix, A. Exercise 1.4 Show that if A is an n × n invertible matrix, then (A−1)t = (At)−1. Exercise 1.5 Show that A is an n × n matrix, then Ax · y = x · Aty. Proposition 1.6 If A is an n × n invertible matrix, then fd◦ A = j det Aj−1f^◦ A−t: Proof. If we make the change of variables, y = Ax in the integral defining fd◦ A, then we obtain Z fd◦ A(ξ) = f(Ax)e−ix·ξ dx Rn Z = j det Aj−1 f(y)e−iA−1y·ξ dy Rn Z = j det Aj−1 f(y)e−iy·A−tξ dy: Rn 1.1. DEFINITION AND SYMMETRY PROPERTIES 3 −n If we set f(x) = f(x/), then a simple application of Proposition 1.6 gives ^ ^ f(ξ) = f(ξ): (1.7) Recall that an orthogonal matrix is an n × n-matrix with real entries which satisfies t O O = In where In is the n×n identity matrix. Such matrices are clearly invertible since O−1 = Ot. The group of all such matrices is usually denoted by O(n). Corollary 1.8 If f 2 L1(Rn) and O is an orthogonal matrix, then f^◦ O = fd◦ O. Exercise 1.9 If x 2 Rn, show that there is an orthogonal matrix O so that Ox = (jxj; 0;:::; 0). Exercise 1.10 Let A be an n × n matrix with real entries. Show that A is orthogonal if and only if Ax · Ax = x · x for all x 2 Rn. We say that function f defined on Rn is radial if there is a function F on [0; 1) so that f(x) = F (jxj). Equivalently, a function is radial if and only if f(Ox) = f(x) for all orthogonal matrices O. Corollary 1.11 Suppose that f is in L1 and f is radial, then f^ is radial. Proof. We fix ξ in Rn and choose O so that Oξ = (jξj; 0;:::; 0): Since f ◦ O = f, we have that f^(ξ) = fd◦ O(ξ) = f^(Oξ) = f^(jξj; 0;:::; 0). We shall see that many operations that commute with translations can be expressed as multiplication operators using the Fourier transform.
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