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[Math.CA] 18 Sep 2001 Lmnso Ieradra Analysis Real and Linear of Elements Tpe Semmes Stephen Oso,Texas Houston, Ieuniversity Rice Preface

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[Math.CA] 18 Sep 2001 Lmnso Ieradra Analysis Real and Linear of Elements Tpe Semmes Stephen Oso,Texas Houston, Ieuniversity Rice Preface Elements of Linear and Real Analysis Stephen Semmes Rice University Houston, Texas arXiv:math/0108030v5 [math.CA] 18 Sep 2001 Preface This book deals with some basic themes in mathematical analysis along the lines of classical norms on functions and sequences, general normed vector spaces, inner product spaces, linear operators, some maximal and square- function operators, interpolation of operators, and quasisymmetric mappings between metric spaces. Aspects of the broad area of harmonic analysis are entailed in particular, involving famous work of M. Riesz, Hardy, Littlewood, Paley, Calder´on, and Zygmund. However, instead of working with arbitrary continuous or integrable func- tions, we shall often be ready to use only step functions on an interval, i.e., functions which are piecewise-constant. Similarly, instead of infinite- dimensional Hilbert or Banach spaces, we shall frequently restrict our atten- tion to finite-dimensional inner product or normed vector spaces. We shall, however, be interested in quantitative matters. We do not attempt to be exhaustive in any way, and there are many re- lated and very interesting subjects that are not addressed. The bibliography lists a number of books and articles with further information. The formal prerequisites for this book are quite limited. Much of what we do is connected to the notion of integration, but for step functions ordinary integrals reduce to finite sums. A sufficient background should be provided by standard linear algebra of real and complex finite-dimensional vector spaces and some knowledge of beginning analysis, as in the first few chapters of Rudin’s celebrated Principles of Mathematical Analysis [Rud1]. This is not to say that the present monograph would necessarily be easy to read with this background, as the types of issues considered may be unfamiliar. On the other hand, it is hoped that this monograph can be helpful to readers with a variety of perspectives. ii Contents Preface ii 1 Notation and conventions 1 2 Dyadic intervals 3 2.1 The unit interval and dyadic subintervals . 3 2.2 Functionsontheunitinterval . 5 2.3 Haarfunctions .......................... 6 2.4 Binarysequences ......................... 8 3 Convexity and some basic inequalities 9 3.1 Convexfunctions ......................... 9 3.2 Jensen’s inequality . 12 3.3 H¨older’s inequality . 13 3.4 Minkowski’s inequality . 14 3.5 p< 1................................ 17 4 Normed vector spaces 18 4.1 Definitions and basic properties . 18 4.2 Dualspacesandnorms . .. 20 4.3 Secondduals ........................... 21 4.4 Lineartransformationsandnorms . 25 4.5 Lineartransformationsandduals . 25 4.6 Innerproductspaces .. .. 27 4.7 Inner product spaces, continued . 29 4.8 Separationofconvexsets. 30 4.9 Somevariations .......................... 32 iii iv CONTENTS 5 Strict convexity 37 5.1 Functionsofonerealvariable . 37 5.2 Theunitballinanormedvectorspace . 38 5.3 Linearfunctionals......................... 39 5.4 Uniqueness of points of minimal distance . 42 5.5 Clarkson’s inequalities . 43 6 Spectral theory 44 6.1 Thespectrumandspectralradius . 44 6.2 Spectralradiusandnorms . 45 6.3 Spectralradiusandnorms,2. 46 6.4 Innerproductspaces . .. .. 50 6.5 The C∗-identity.......................... 53 6.6 Projections ............................ 55 6.7 Remarksaboutdiagonalizableoperators . 58 6.8 Commuting families of operators . 59 7 Linear operators between inner product spaces 61 7.1 Preliminaryremarks ....................... 61 7.2 Schmidtdecompositions . 62 7.3 TheHilbert-Schmidtnorm . 63 7.4 Anumericalfeature........................ 64 7.5 Numericalrange.......................... 65 8 Subspaces and quotient spaces 67 8.1 Linearalgebra........................... 67 8.2 Quotientspacesandnorms. 68 8.3 Mappings between vector spaces . 70 9 Variation seminorms 73 9.1 Basicdefinitions.......................... 73 9.2 The p = 2 and n = 1, p =1cases ................ 75 9.3 Minimization . 76 9.4 Truncations ............................ 78 10 Groups 81 10.1Generalnotions .......................... 81 10.2 Some operators on F(G)..................... 83 CONTENTS v 10.3 Commutativegroups . 83 10.4Specialcases............................ 85 10.5 Groupsofmatrices ........................ 86 11 Some special families of functions 89 11.1 Rademacherfunctions . 89 11.2 Linearfunctionsonspheres . 92 11.3 Linearfunctions,continued. 94 11.4 Lacunary sums, p =4....................... 95 12 Maximal functions 97 12.1 Definitionsandbasicproperties . 97 12.2 Thesizeofthemaximalfunction . 99 12.3 Somevariations ..........................101 12.4 Moreonthesizeofthemaximalfunction . 102 13 Square functions 106 13.1 S-functions ............................106 13.2Estimates,1............................108 13.3Estimates,2............................113 13.4Duality...............................117 13.5 Duality,continued. .120 13.6 Some inequalities . 122 13.7 Another inequality for p =1...................125 13.8Variants ..............................126 13.9 Some remarks concerning p =4 .................127 14 Interpolation of operators 130 14.1 Thebasicresult..........................130 14.2 Adigressionaboutconvexfunctions. 133 14.3 A place where the maximum is attained . 135 14.4 Therestoftheargument. .136 14.5 Areformulation ..........................138 14.6 Ageneralization. .. .. .139 15 Quasisymmetric mappings 141 15.1Basicnotions ...........................141 15.2Examples .............................144 vi CONTENTS 15.3Cantorsets ............................145 15.4 Bounds in terms of C ta .....................147 Bibliography 150 Chapter 1 Notation and conventions If a and b are real numbers with a ≤ b, then the following are the intervals in the real line R with endpoints a and b: [a, b] = {x ∈ R : a ≤ x ≤ b}; (a, b) = {x ∈ R : a<x<b}; [a, b) = {x ∈ R : a ≤ x < b}; (a, b] = {x ∈ R : a < x ≤ b}. All but the first is the empty set when a = b, while [a, b] consists of the one point a = b. In general, the first of these intervals is called the closed interval with endpoints a and b, and the second is the open interval with endpoints a and b. The third and fourth are half-open, half-closed intervals, with the third being left-closed and right-open, and the fourth left-open and right-closed. The length of each of these intervals is defined to be b − a. If an interval is denoted I, we may write |I| for the length of I. For the record, see Chapter 1 in [Rud1] concerning detailed properties of the real numbers (as well as the complex numbers C). In particular, let us recall the “least upper bound” or “completeness” property, to the effect that a nonempty set E of real numbers which has an upper bound has a least upper bound. The least upper bound is also called the supremum of E, and is denoted sup E. Similarly, if F is a nonempty set of real numbers which has a lower bound, then F has a greatest lower bound, or infimum, which is denoted inf F . We shall sometimes use the extended real numbers (as in [Rud1]), with ∞ and −∞ added to the real numbers, and write sup E = ∞ 1 2 CHAPTER 1. NOTATION AND CONVENTIONS and inf F = −∞ if E and F are nonempty sets of real numbers such that E does not have an upper bound and F does not have a lower bound. If A is a subset of some specified set X (like the real line), we let 1A(x) denote the indicator function of A on X (sometimes called the characteristic function associated to A, although in other contexts this name can be used for something quite different). This is the function which is equal to 1 when x ∈ A, and is equal to 0 when x ∈ X\A. Definition 1.1 (Step functions) A function on the real line, or on an in- terval in the real line, is called a step function if it is a finite linear combi- nation of indicator functions of intervals. This is equivalent to saying that there is a partition of the domain into intervals on which the function is constant. In these notes, one is normally welcome to assume that a given function on the real line, or on an interval in the real line, is a step function. In fact, one is normally welcome to assume that a given function is a dyadic step function, as defined in the next chapter. For step functions, it is very easy to define the integral over an interval in the domain of definition, by reducing to linear combinations of lengths of intervals. An exception to this convention occurs when we consider convex or mono- tone functions, which we do not necessarily wish to ask to be step functions. When dealing with integrals, typically the function being integrated can be taken to be a step function. (This function might be the composition of a non-step function with a step function, which is still a step function.) Chapter 2 Dyadic intervals 2.1 The unit interval and dyadic subintervals Normally, a reference to “the unit interval” might suggest the interval [0, 1] in the real line. It will be convenient to use [0, 1) instead, for minor technical reasons (and one could easily work around this anyway). Definition 2.1 (Dyadic intervals in [0, 1)) The dyadic subintervals of the unit interval [0, 1) are the intervals of the form [j 2−k, (j +1)2−k), where j and k are nonnegative integers, and j +1 ≤ 2k. (Thus the length of such an interval is of the form 2−k, where k is a nonnegative integer.) In general one can define the dyadic intervals in R to be the intervals of the same form, except that j and k are allowed to be arbitrary integers. The half-open, half-closed condition leads to nice properties in terms of disjointness, as in the following lemmas. (With closed intervals one could get disjointness of interiors in similar circumstances. This would be fine in terms of integrals, measures, etc.) Lemma 2.2 (Partitions of [0, 1)) For each nonnegative integer k, [0, 1) is the union of the dyadic subintervals of itself of length 2−k, and these intervals are pairwise disjoint. Lemma 2.3 (Comparing pairs of intervals) If J1 and J2 are two dyadic subintervals of [0, 1), then either J1 ⊆ J2, or J2 ⊆ J1, or J1 ∩ J2 = ∅. (The first two possibilities are not mutually exclusive, as one could have J1 = J2.) 3 4 CHAPTER 2.
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