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IMUS Lecture Notes on Harmonic Analysis, Metric Spaces and PDES, Sevilla 2011

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IMUS Lecture Notes on Harmonic Analysis, Metric Spaces and PDES, Sevilla 2011 IMUS Lecture Notes on Harmonic Analysis, Metric Spaces and PDES, Sevilla 2011 Edited By: Lucas Chaffee, Taylor Dupuy, Rafael Espinosa, Jarod Hart, Anna Kairema, Lyudmila Korobenko November 7, 2012 2 Contents 1 Shanmugalingam5 1.1 Lecture One: Sobolev Spaces.......................5 1.2 Lecture 2.................................9 1.3 Lecture 3................................. 13 1.4 Lecture 4................................. 18 1.5 Lecture 5................................. 25 2 Rafa 31 2.1 General Measure Theory. A review.................... 32 2.2 Integration................................. 37 2.3 Covering theorems............................ 41 2.4 Differentiation of Radon measures.................... 44 2.5 Lebesgue differentiation theorem..................... 46 2.6 Hausdorff Measure............................ 49 2.7 Extension of Lipschitz mappings..................... 52 2.8 Rademacher theorem........................... 55 2.9 Linear maps and Jacobians. The area formula.............. 56 2.10 Geodesic spaces of bounded curvature.................. 61 2.11 Gromov hyperbolicity........................... 69 3 Zhong 73 3.1 Lecture One. Poincar´einequalities.................... 73 3.2 Lecture Two................................ 75 3 4 CONTENTS 3.3 Lecture Three. Poncar´e=) Sobolev-Poncar´e.............. 79 3.4 Lecture Four................................ 82 3.5 Lecture Five................................ 85 4 Mateu 89 4.1 Lecture 1................................. 89 4.2 Lecture 2................................. 96 4.3 Appendix................................. 99 5 Jaramillo 101 5.1 Lecture One: The basic geometry of length in a metric space..... 101 5.2 Lecture Two: Defining line integral................... 104 5.3 Lecture Three: Spaces of pointwise-Lipschitz functions........ 107 5.4 Lecture Four: Newton{Sobolev Spaces of infinity type......... 111 5.5 Lecture Five: Poincar´einequalities................... 116 6 Magnani 119 6.1 Lecture 1: Preliminaries......................... 119 6.2 Lecture 2: Stratified Groups....................... 122 7 Valentino 129 7.1 Lecture One: Sobolev Spaces....................... 129 7.2 Lecture 2................................. 133 7.3 Lecture 3................................. 137 7.4 Lecture 4................................. 142 7.5 Lecture 5................................. 149 Bibliography 155 Chapter 1 Shanmugalingam 1.1 Lecture One: Sobolev Spaces 1.1.1 Differentiability and Smoothness in Rn n Let Ω ⊂ R be open, let x0 2 Ω define and let f :Ω ! R • Define what it means for f to be differentiable at x0. • Define what it means for f to be smooth at x0. We will denote n times differentiable functions by Cn(Ω) and smooth functions by C1(Ω). 1.1.2 Sobolev spaces in Rn by completing the smooth func- tions Let p 2 (1; 1) and f 2 C1(X), define the first Sobolev norm by kfkW 1;p(Ω) = kfkLp(Ω) + krfkLp(Ω): • Show that k · kW1;p is a norm. 1 The set of smooth functions C with the first Sobolev, k − kW1;p(Ω) is a normed vector space. • Show that the normed vector space (C1(Ω);W 1;p is not a Banach space. 5 6 CHAPTER 1. SHANMUGALINGAM The completion of the smooth functions on Ω with respect to the norm k−kW 1;p(Ω) is called the first p Sobolev space and is denoted by W 1;p(Ω). It consists of functions in Lp(Ω) with a gradient in Lp(Ω; Rn). 1.1.3 Integration by Parts and Weak Derivatives 1 1 n If f 2 C (Ω) and ' 2 Cc (Ω; R ) then (1.1.1) h'(x); rf(x)idx = − f(x)r · '(x)dx ˆΩ ˆΩ by applying integration by parts and the fact that ' has compact support. For 1 n f :Ω ! R measurable there exists an operator df : Cc (Ω; R ) ! R given by df(') = − f(x)r · '(x)dx: ˆΩ The idea is that when f is differentiable we have df = h−; rfi; by equation 7.1.1, so this operator is a replacement for derivatives for functions that don't have them. • What are the functions f such that there exists some V :Ω ! Rn such that 1 for all ' :2 Cc (Ω) we have df(') = hV (x);'(x)i? ˆΩ Functions which have these are called weakly differentiable. The following exer- cise shows that they don't always exist. • We claim that the function f : R2 ! R defined by ( 1; x > 0 f(x; y) = 0: x ≤ 0 does not have such a V . When a function does have a weak derivative we should observe we can change the definition of V on a set of measure zero and get another weak derivative. The weak derivitive is unique up to a set of measure zero. 1.1. LECTURE ONE: SOBOLEV SPACES 7 1.1.4 Sobolev spaces by existence of weak derivatives If f 2 Lp has a weak derivative rf then on the subset of functions which have a weak derivative we can define the first Sobolev norm as kfjW 1;p(Ω) = kfkLp(Ω) + krfkLp(Ω): The Sobolev Space could then be defined as the set of weak differentiable functions in Lp with bounded first Sobolev norm. 1.1.5 Equivalence of two ways of defining Sobolev spaces We have given two ways of defining the Sobolev Spaces. The first is as the completion 1 p of C (Ω) with respect to k−kW 1;p and the second as the subspace of L (Ω) of weakly differentiable functions with bounded Sobolev norm. Theorem 1.1.1 (Serrin). Both approaches to defining the Sobolev space Let : R ! R be a smooth non-negative function with compact support. Define η(x) = (jxj). For f 2 W 1;p(Ω) we can define for every " > 0 1 x − y (f ∗ η")(x) = n f(y)η( )dy: " ˆRn " By approximations of the identity this converges to f in Lp sense as " ! 0. Next we have @j(f ∗ η") = f ∗ @jη" = @jf ∗ η" which approaches @jf as " ! 0 for every j. This gives a sequence of approximations of f 2 W 1;p(Ω) by smooth functions. 1.1.6 absolute continuity A function f :Ω ! R is absolutely continuous along γ : [0; 1] ! Ω provided the following holds: For all " > 0 there exists δ > 0 such that for every collection of intervals [a1; b1]; [a2; b2];:::; [ak; bk] ⊂ [0; 1] we have X jbi − aij < δ =) f ◦ γ < ": ˆP i [ai;bi] 8 CHAPTER 1. SHANMUGALINGAM 1.1.7 convention: parametrizing in terms of arclength Let γ ⊂ Ω be a curve of finite length and let f 2 C1(Ω). We can view γ as γ : [0; 1] ! Ω. We have the inequality 1 0 jf(yγ) − f(xγ)j ≤ jrf(γ(t))jjγ (t)jdt; ˆ0 where y(γ) is the end point of the curve and xγ is the end point of the curve. We will parametrize curves in term of arclength so the the norm of the derivative will be equal to one reducing the above expression to 1 jf(yγ) − f(xγ)j ≤ jrf(γ(t))jdt: ˆ0 1.1.8 criteria for absolute continuity Proposition 1.1.2. Suppose that f :Ω ! R and γ is a rectifiable curve in Ω. Also suppose that there exists some g :Ω ! [0; 1] measurable such that gds < 1 ˆγ and such that for all subcurves β ⊂ γ we have jf(yβ) − f(xγ)j ≤ gds: ˆβ Then, f ◦ γ is absolutely continuous and j(f ◦ γ)0j ≤ g ◦ γ pointwise. • Using the definition of absolute continuity prove the above proposition. 1.1.9 the Poincar´einequality This number is about the Poincar´einequality. For a set A we use the following notation for integral average 1 − f = f: ˆA jAj ˆA 1 Proposition 1.1.3 (Poincar´eInequality). For f 2 C (Ω) and x0 2 Ω let r > 0 be such that B(x0; r)) ⊂ Ω jrf(z)j − jf(y) − f(x0)j ≤ cn n−1 : ˆB(x0;r) ˆB(x0;r) jz − x0j 1.2. LECTURE 2 9 Proof. The left hand side of the above we convert to polar coordinated r 1 1 n−1 n jf(y) − f(x)jdy = n ρ jf(ρ, θ) − f(x0)jdθdρ anr ˆB anr ˆ0 ˆSn−1 n where the volume of the n dimensional ball or radius r in anr we let y = (ρ, θ) where θ 2 Sn−1 and ρ 2 [0; r]. Since we have ρ ρ jf(ρ, θ) − f(x0)j = j rf(s; θ)dsj ≤ jrf(s; θ)jds ˆ0 ˆ0 we have r r ρ 1 n−1 1 jrf(s; θ)j n−1 n ρ jf(ρ, θ) − f(x0)jdθdρ ≤ n n−1 s dθdsdρ anr ˆ0 ˆSn−1 anr ˆ0 ˆ0 ˆSn−1 s 1 r jrf(z)j = n dzdρ anr ˆ0 ˆB(x0,ρ) jz − x0j 1 r jrf(z)j = n−1 dz: nan ˆ0 jz − x0j as an application of Fubini's Theorem. Remark 1.1.4. The expression jrf(z)j n−1 jz − x0j is called the Reisz Potential of jrfj 1.2 Lecture 2 1.2.1 Last time we showed the Poincare Inequality. This Inequality still holds if we enlarge the Ball we are taking the integral over. 1 Proposition 1.2.1. For x 2 B and f 2 Lloc(Ω) we have jrf(z)j (1.2.1) − jf(x) − f(y)j ≤ C n−1 dz ˆB ˆ4B jz − xj . n Consider the operator IB which takes in functions V : R ! R and is defined by jV (z)j IB(V )(x) := n−1 dz: ˆ4B jz − xj 10 CHAPTER 1. SHANMUGALINGAM p p Proposition 1.2.2. The operator IB is a bounded operator L (4B) ! L (4B). The strategy is to control this operator by the Hardy-Littlewood maximal function 1 Mf(x) := sup f(y)dy B3x jBj ˆB which is known to be a bounded function. The proof will allso use the Lebesgue differentiation theorem which states that 1 f(y)dy ! f(x) as jBj ! 0 jBj where the balls are contain x and the limit exists almost everywhere.
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