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Lecture Notes on Introduction to Harmonic Analysis

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Lecture Notes on Introduction to Harmonic Analysis g Introduction to Harmonic Analysis LECTURE NOTES g Chengchun Hao Institute of Mathematics, CAS 2016 Contents 1 The Fourier Transform and Tempered Distributions .......................... 1 1.1 The L1 theory of the Fourier transform....................................1 1.2 The L2 theory and the Plancherel theorem.................................. 12 1.3 Schwartz spaces....................................................... 13 1.4 The class of tempered distributions....................................... 17 1.5 *Characterization of operators commuting with translations.................. 22 2 Interpolation of Operators ................................................. 27 2.1 Riesz-Thorin’s and Stein’s interpolation theorems........................... 27 2.2 The distribution function and weak Lp spaces............................... 33 2.3 The decreasing rearrangement and Lorentz spaces........................... 36 2.4 Marcinkiewicz’ interpolation theorem..................................... 41 3 The Maximal Function and Calderón-Zygmund Decomposition . 47 3.1 Two covering lemmas.................................................. 47 3.2 Hardy-Littlewood maximal function...................................... 49 3.3 Calderón-Zygmund decomposition....................................... 57 4 Singular Integrals ......................................................... 63 4.1 Harmonic functions and Poisson equation.................................. 63 4.2 Poisson kernel and Hilbert transform...................................... 67 4.3 The Calderón-Zygmund theorem......................................... 77 4.4 Truncated integrals..................................................... 79 4.5 Singular integral operators commuted with dilations......................... 82 4.6 The maximal singular integral operator.................................... 88 4.7 *Vector-valued analogues............................................... 92 5 Riesz Transforms and Spherical Harmonics.................................. 95 5.1 The Riesz transforms................................................... 95 5.2 Spherical harmonics and higher Riesz transforms........................... 99 5.3 Equivalence between two classes of transforms............................. 106 6 The Littlewood-Paley g-function and Multipliers .............................111 6.1 The Littlewood-Paley g-function......................................... 111 6.2 Fourier multipliers on Lp ................................................ 121 v - vi - Contents 6.3 The partial sums operators.............................................. 128 6.4 The dyadic decomposition............................................... 130 6.5 The Marcinkiewicz multiplier theorem.................................... 136 7 Sobolev and Hölder Spaces ................................................141 7.1 Riesz potentials and fractional integrals.................................... 141 7.2 Bessel potentials....................................................... 145 7.3 Sobolev spaces........................................................ 149 References................................................................ 153 Index .......................................................................155 Chapter 1 The Fourier Transform and Tempered Distributions In this chapter, we introduce the Fourier transform and study its more elementary properties, and extend the definition to the space of tempered distributions. We also give some characteri- zations of operators commuting with translations. 1.1 The L1 theory of the Fourier transform We begin by introducing some notation that will be used throughout this work. Rn de- notes n-dimensional real Euclidean space. We consistently write x = (x1;x2; ;xn), x = n n ··· (x1;x2; ;xn), for the elements of R . The inner product of x, x R is the number ···n ··· n 2 x x = ∑ x jx j, the norm of x R is the nonnegative number x = px x. Furthermore, · j=1 2 j j · dx = dx dx dx denotes the element of ordinary Lebesgue measure. 1 2 ··· n We will deal with various spaces of functions defined on Rn. The simplest of these p p n are the L = L (R ) spaces, 1 6 p < ¥, of all measurable functions f such that f p = R p 1=p p ¥ n k k ( Rn f (x) dx) < ¥. The number f p is called the L norm of f . The space L (R ) consists j j k k n ¥ n of all essentially bounded functions on R and, for f L (R ), we let f ¥ be the essential n n 2 k k supremum of f (x) , x R . Often, the space C0(R ) of all continuous functions vanishing at j j 2 infinity, with the L¥ norm just described, arises more naturally than L¥ = L¥(Rn). Unless other- wise specified, all functions are assumed to be complex valued; it will be assumed, throughout the note, that all functions are (Borel) measurable. In addition to the vector-space operations, L1(Rn) is endowed with a “multiplication” making this space a Banach algebra. This operation, called convolution, is defined in the following way: If both f and g belong to L1(Rn), then their convolution h = f g is the function whose value ∗ at x Rn is 2 Z h(x) = f (x y)g(y)dy: Rn − One can show by an elementary argument that f (x y)g(y) is a measurable function of the − two variables x and y. It then follows immediately from Fibini’s theorem on the interchange of 1 n the order of integration that h L (R ) and h 1 f 1 g 1. Furthermore, this operation is 2 k k 6 k k k k commutative and associative. More generally, we have, with the help of Minkowski’s integral R R inequality F(x;y)dy p 6 F(x;y) p dy, the following result: k kLx k kLx Theorem 1.1. If f Lp(Rn), p [1;¥], and g L1(Rn) then h = f g is well defined and 2 2 2 ∗ belongs to Lp(Rn). Moreover, h f g : k kp 6 k kpk k1 1 - 2 - 1. The Fourier Transform and Tempered Distributions Now, we first consider the Fourier1 transform of L1 functions. Definition 1.2. Let w R 0 be a constant. If f L1(Rn), then its Fourier transform is F f 2 n f g 2 or fˆ : Rn C defined by ! Z wix x F f (x) = e− · f (x)dx (1.1) Rn for all x Rn. 2 We now continue with some properties of the Fourier transform. Before doing this, we shall introduce some notations. For a measurable function f on Rn, x Rn and a = 0 we define the 2 6 translation and dilation of f by t f (x) = f (x y); (1.2) y − da f (x) = f (ax): (1.3) Proposition 1.3. Given f ;g L1(Rn), x;y;x Rn, a multiindex, a;b C, e R and e = 0, we 2 2 2 2 6 have (i) Linearity: F (a f + bg) = aF f + bF g. wiy x (ii) Translation: F ty f (x) = e− · fˆ(x). wix y (iii) Modulation: F (e · f (x))(x) = ty fˆ(x). n (iv) Scaling: F de f (x) = e − d 1 fˆ(x). j j e− (v) Differentiation: F ¶ a f (x) = (wix)a fˆ(x), ¶ a fˆ(x) = F (( wix)a f (x))(x). − (vi) Convolution: F ( f g)(x) = fˆ(x)gˆ(x). ∗ (vii) Transformation: F ( f A)(x) = fˆ(Ax), where A is an orthogonal matrix and x is a ◦ column vector. (viii) Conjugation: fd(x) = fˆ( x). − Proof. These results are easy to be verified. We only prove (vii). In fact, Z Z wix x wiA 1y x F ( f A)(x) = e− · f (Ax)dx = e− − · f (y)dy ◦ Rn Rn Z Z wiA y x wiy Ax = e− > · f (y)dy = e− · f (y)dy = fˆ(Ax); Rn Rn where we used the change of variables y = Ax and the fact that A 1 = A and detA = 1. − > j j ut Corollary 1.4. The Fourier transform of a radial function is radial. Proof. Let x;h Rn with x = h . Then there exists some orthogonal matrix A such that 2 j j j j Ax = h. Since f is radial, we have f = f A. Then, it holds ◦ F f (h) = F f (Ax) = F ( f A)(x) = F f (x); ◦ by (vii) in Proposition 1.3. ut It is easy to establish the following results: Theorem 1.5 (Uniform continuity). (i) The mapping F is a bounded linear transformation 1 n ¥ n from L (R ) into L (R ). In fact, F f ¥ f 1. k k 6 k k (ii) If f L1(Rn), then F f is uniformly continuous. 2 1 Jean Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier’s Law are also named in his honor. Fourier is also generally credited with the discovery of the greenhouse effect. 1.1. The L1 theory of the Fourier transform - 3 - Proof. (i) is obvious. We now prove (ii). By Z wix x wix h fˆ(x + h) fˆ(x) = e− · [e− · 1] f (x)dx; − Rn − we have Z wix h fˆ(x + h) fˆ(x) 6 e− · 1 f (x) dx j − j Rn j − jj j Z Z wix h 6 e− · 1 f (x) dx + 2 f (x) dx x 6r j − jj j x >r j j Zj j Z j j 6 w r h f (x) dx + 2 f (x) dx x r j j j jj j x >r j j j j6 j j =:I1 + I2; since for any q > 0 q eiq 1 = (cosq 1)2 + sin2 q = p2 2cosq = 2 sin(q=2) q : j − j − − j j 6 j j Given any e > 0, we can take r so large that I < e=2. Then, we fix this r and take h small 2 j j enough such that I1 < e=2. In other words, for given e > 0, there exists a sufficiently small d > 0 such that fˆ(x + h) fˆ(x) < e when h d, where e is independent of x. j − j j j 6 ut Example 1.6. Suppose that a signal consists of a single rectangular pulse of width 1 and height 1. Let’s say that it gets turned on at x = 1 and turned off at x = 1 . The standard name for this − 2 2 “normalized” rectangular pulse is 1 1; if 1 < x < 1 ; P(x) rect(x) := − 2 2 ≡ 0; otherwise: 1 1 x − 2 2 It is also called, variously, the normalized boxcar function, the top hat function, the indicator function, or the characteristic function for the interval ( 1=2;1=2).
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