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Lectures in Harmonic Analysis Lectures in Harmonic Analysis Joseph Breen Lectures by: Monica Visan Last updated: June 29, 2017 Department of Mathematics University of California, Los Angeles Contents Preface 3 1 Preliminaries4 1.1 Notation and Conventions.............................4 1.2 Lp Spaces.......................................4 1.3 Complex Interpolation...............................5 1.4 Some General Principles..............................7 1.4.1 Integrating jxj−α ..............................8 1.4.2 Dyadic Sums................................8 2 The Fourier Transform 10 d 2.1 The Fourier Transform on R ........................... 10 d 2.2 The Fourier Transform on T ............................ 16 3 Lorentz Spaces and Real Interpolation 19 3.1 Lorentz Spaces.................................... 19 3.2 Duality in Lorentz Spaces............................. 23 3.3 The Marcinkiewicz Interpolation Theorem................... 26 4 Maximal Functions 29 4.1 The Hardy-Littlewood Maximal Function.................... 29 4.2 Ap Weights and the Weighted Maximal Inequality............... 30 4.3 The Vector-Valued Maximal Inequality...................... 36 5 Sobolev Inequalities 41 5.1 The Hardy-Littlewood-Sobolev Inequality.................... 41 5.2 The Sobolev Embedding Theorem........................ 44 5.3 The Gagliardo-Nirenberg Inequality....................... 47 6 Fourier Multipliers 49 6.1 Calderon-Zygmund Convolution Kernels.................... 49 6.2 The Hilbert Transform............................... 56 6.3 The Mikhlin Multiplier Theorem......................... 57 7 Littlewood-Paley Theory 61 7.1 Littlewood-Paley Projections........................... 61 7.2 The Littlewood-Paley Square Function...................... 68 7.3 Applications to Fractional Derivatives...................... 71 8 Oscillatory Integrals 79 8.1 Oscillatory Integrals of the First Kind, d = 1 ................... 80 8.2 The Morse Lemma................................. 87 8.3 Oscillatory Integrals of the First Kind, d > 1 ................... 89 1 8.4 The Fourier Transform of a Surface Measure.................. 92 9 Dispersive Partial Differential Equations 94 9.1 Dispersion...................................... 94 9.2 The Linear Schrodinger¨ Equation......................... 95 9.2.1 The Fundamental Solution........................ 95 9.2.2 Dispersive Estimates............................ 96 9.2.3 Strichartz Estimates............................ 96 10 Restriction Theory 97 10.1 The Restriction Conjecture............................. 97 10.2 The Tomas-Stein Inequality............................ 100 10.3 Restriction Theory and Strichartz Estimates................... 103 11 Rearrangement Theory 108 11.1 Definitions and Basic Estimates.......................... 108 11.2 The Riesz Rearrangement Inequality, d = 1 ................... 114 11.3 The Riesz Rearrangement Inequality, d > 1 ................... 117 11.4 The Polya-Szego inequality............................ 122 11.5 The Energy of the Hydrogen Atom........................ 125 12 Compactness in Lp Spaces 128 12.1 The Riesz Compactness Theorem......................... 128 12.2 The Rellich-Kondrachov Theorem........................ 131 12.3 The Strauss Lemma................................. 134 12.4 The Refined Fatou’s Lemma............................ 136 13 The Sharp Gagliardo-Nirenberg Inequality 137 13.1 The Focusing Cubic NLS.............................. 137 14 The Sharp Sobolev Embedding Theorem 138 15 Concentration Compactness in Partial Differential Equations 139 References 139 2 Preface **Eventually I’ll probably write a better introduction, this is a placeholder for now** The following text covers material from Monica Visan’s Math 247A and Math 247B classes, offered at UCLA in the winter and spring quarters of 2017. The content of each chapter is my own adaptation of lecture notes taken throughout the two quarters. Chapter1 contains a selection of background material which was not covered in lec- tures. The rest of the chapters roughly follow the chronological order of lectures given throughout the winter and spring, with some slight rearranging of my own choice. For example, basic results about the Fourier transform on the torus are included in Chapter 2, despite being the subject of lectures in the spring. A strictly chronological (and incom- plete) version of these notes can be found on my personal web page at http://www. math.ucla.edu/˜josephbreen/. Finally, any mistakes are likely of my own doing and not Professor Visan’s. Comments and corrections are welcome. 3 Chapter 1 Preliminaries The material covered in these lecture notes is presented at the level of an upper division graduate course in harmonic analysis. As such, we assume that the reader is comfortable with real and complex analysis at a sophisticated level. However, for convenience and completeness we recall some of the basic facts and inequalities that are used throughout the text. We also establish conventions with notation, and discuss a few select topics that the reader may be unfamiliar with (for example, the Riesz-Thorin interpolation theorem). Any standard textbook in real analysis or harmonic analysis is a suitable reference for this material, for example, [3], [6], and [8]. 1.1 Notation and Conventions If x ≤ Cy for some constant C, we say x . y. If x . y and y . x, then x ∼ y. If the implicit constant depends on additional data, this is manifested as a subscript in the inequality d d sign. For example, if A denotes the ball of radius r > 0 in R and j · j denotes R -Lebesgue d measure, we have jAj ∼d r . However, the dependencies of implicit constants are usually clear from context, or stated otherwise. FINISH 1.2 Lp Spaces ADD Proposition 1.1 (Young’s Convolution inequality). For 1 ≤ p; q; r ≤ 1, kf ∗ gk r d kfk p d kgk q d L (R ) . L (R ) L (R ) 1 1 1 whenever 1 + r = p + q . Young’s convolution inequality may be proved directly, or by using the Riesz-Thorin interpolation theorem (see Section 1.3). For details on both methods, refer to [3]. It is convenient to record a few special cases of Young’s inequality that arise frequently. Corollary 1.2 (Young’s Convolution inequality, special cases). The following convolution in- equalities hold: 1. For any 1 ≤ p ≤ 1, kf ∗ gkL1 . kfkLp kgkLp0 . In particular, kf ∗ gkL1 . kfkL2 kgkL2 and kf ∗ gkL1 . kfkL1 kgkL1 . 2. For any 1 ≤ p ≤ 1, kf ∗ gkLp . kfkL1 kgkLp . 4 1.3 Complex Interpolation An important tool in harmonic analysis is interpolation. Broadly speaking, interpolation considers the following question: given estimates of some kind on two different spaces, say Lp0 and Lp1 , what can be said about the corresponding estimate on Lp for values of p between p0 and p1? There are a number of theorems which answer this kind of question. In Chapter3, we will prove the Marcinkiewicz interpolation theorem using real analytic methods. Here, we prove a different interpolation theorem using complex analytic methods. Both theorems have their own advantages and disadvantages, and will be used frequently. Our treatment of the Riesz-Thorin interpolation theorem follows [8]. The main state- ment is as follows. Theorem 1.3 (Riesz-Thorin interpolation). Suppose that T is a bounded linear map from Lp0 + Lp1 ! Lq0 + Lq1 and that kT fkLq0 ≤ M0 kfkLp0 and kT fkLq1 ≤ M1 kfkLp1 for some 1 ≤ p0; p1; q0; q1 ≤ 1. Then 1−t t kT fkLqt ≤ M0 M1 kfkLpt where 1 = 1−t + t and 1 = 1−t + t , for any 0 ≤ t ≤ 1. pt p0 p1 qt q0 q1 As noted above, the proof of this theorem relies on complex analysis. Specifically, we need the following lemma. Lemma 1.4 (Three Lines lemma). Suppose that Φ(z) is holomorphic in S = f0 < Re z < 1g and continuous and bounded on S¯. Let M0 := sup jΦ(iy)j and M1 := sup jΦ(1 + iy)j: y2R y2R Then 1−t t sup jΦ(t + iy)j ≤ M0 M1 y2R for any 0 ≤ t ≤ 1. Proof. We can assume without loss of generality that Φ is nonconstant, otherwise the con- clusion is trivial. We first prove a special case of the lemma. Suppose that M0 = M1 = 1 and that ¯ sup0≤x≤1 jΦ(x + iy)j ! 0 as jyj ! 1. In this case, φ has a global (on S) supremum of M > 0. Let fzng be a sequence such that jΦ(zn)j ! M. Because sup0≤x≤1 jΦ(x + iy)j ! 0 as jyj ! 1, the sequence fzng is bounded, hence there is a point z1 2 S¯ such that a subsequence of fzng converges to z1. By continuity, jΦ(z1)j = M. Since Φ is nonconstant, by the maximum modulus principle, z1 is on the boundary of S¯. This means that jΦ(z)j is globally bounded by 1. Since M0 = M1 = 1, this proves the special case. Next, we remove the decay assumption and only suppose that M0 = M1 = 1. For " > 0, define "(z2−1) Φ"(z) := Φ(z)e : First observe that jΦ"(z)j ≤ 1 if Re z = 0 or Re z = 1. Indeed, note that for y 2 R we have 2 2 "((iy) −1) −"(y +1) jΦ"(iy)j = jΦ(iy)j e ≤ e ≤ 1 and 2 2 2 "((1+iy) −1) "(1+2iy−y −1) 2iy" −"y jΦ"(1 + iy)j = jΦ(1 + iy)j e ≤ e ≤ e e ≤ 1: 5 Next, we claim that Φ" satisfies the decay condition from the previous case. Indeed, note that 2 2 2 "((x+iy) −1) "(x −y −1+2xyi) jΦ"(x + iy)j = jΦ(x + iy)j e = jΦ(x + iy)j e 2 2 "(x −y −1) ≤ jΦ(x + iy)j e : Because 0 ≤ x ≤ 1 and because jΦj is bounded, as jyj ! 1, jΦ"(x + iy)j ! 0 uniformly in x. By the previous case, jΦ"(z)j ≤ 1 uniformly in S¯. Since this holds for all " > 0, letting " ! 0 gives the desired result for Φ. Finally we assume that M0 and M1 are arbitrary positive numbers. Define φ~(z) := z−1 −z M0 M1 φ(z). Note that if Re z = 0, then ~ iy−1 −iy iy −iy jΦ(z)j ≤ jM0 M1 jM0 ≤ jM0 j · jM1 j ≤ 1 and if Re z = 1, then ~ iy −1−iy iy −iy jΦ(z)j ≤ jM0 M1 jM1 ≤ jM0 j · jM1 j ≤ 1: The claim then follows by applying the previous case. Proof of Theorem 1.3. By scaling appropriately, we may assume without loss of generality that kfkLpt = 1.
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