Wavelets and Singular Integrals on Curves and Surfaces

Abstract

INTRODUCTIONThese notes are the transcript of a series of lectures that were held in the Nankai Institute of Mathematics, in June 1988, as a part of the program on harmonic analysis. This book consists of three parts devoted to the following topics : wavelets, Calder6n-Zygmund operators, and singular integral operators on some curves and rectifiable subsets of IR n .Our aims for the three parts are slightly different. The first part is intended to be an introduction to the theory of wavelets, and will insist mostly on the construction of various orthonormal bases of L2 . The second part is centered on a necessary and sufficient condition for the L2-lxmndedness of certain singular integral operators on IRn (the so-called T(b)-theorem), and some of its applications. Although the result is not very recent, the techniques to prove it have not been described, up to now, in too many sources. Thanks to a recent work of R. Coifman and S. Semmes, we shall be able to give a full, conceptually very simple proof of that theorem. The final part is a survey of the author's main area of interest: questions related to the description of those kdimensional subsets of IR n on which analogues of the Cauchy kernel define bounded operators on L2 . The area is quite recent, but there are already sufficiently many results, often of a fairly technical nature, to justify the description attempted in Part 3. Unfortunately, there will not be enough room for a complete description of all recent results, but we shall try to allude to most of them, and to give a precise account of the fundamental techniques involved.This book is not intended to be exhaustive. We shall try instead to explain in details a few of the "real-variable methods" that have appeared after the books [St], [CMI] and [Je]. The lectures in Nankai and, to a lesser extent, these notes, were prepared for an audience whose main center of interest would be close to classical analysis. This means that, in many cases, we shall avoid spending time on some of the classical results of harmonic analysis. A fair knowledge of these results, although not really required, will often help the reader understand our motivations, for instance.The first part is about wavelets. Actually, we shall only consider orthonormal wavelets, i.e. wavelets that form orthonormal bases of L2(IRn) . These bases are obtained from one, or'a finite number of functions '1', by dyadic translations and dilations. For instance, the Haar system is a basis of this type, but the corresponding functions 'I' are not, in this case, continuous functions. The interesting wavelets will be asked to have both a good decay at infinity and some reasonable amount of smoothness. The main advantage, from our point of view, is that the VI decomposition of a function f in such a basis then looks a lot like a Littlewood-Paley decomposition of f , but of course is simpler and also uniquely determined. In practical terms, this will imply that the size of the coefficients of f, as well as the type and speed of convergence of the seri...

Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

“…One should note that in the case of one dimensional Uniformly Rectifiable sets (see [DS93] for definition of Uniformly Rectifiable) these results are obtained with much less difficulty by combining [Dav91,DS93,Jon88,Jon90,Oki92]. The key idea is that using [Jon88] one gets that Ahlfors regular curves contain what is called 'big pieces of chord-arc curves' (see [DS93] for a definition).…”

“…We make use of this by employing ideas of Okikiolu, as well as ideas similar to ones of G. David and M. Christ (see e.g. [Chr90], and [Dav91] page 93 for a simple version). Note however that one main difference with the case of Christ and David is the lack of homogeneity assumption on Γ!…”

“…[Chr90], and [Dav91] page 93 for a simple version). One should note that condition (2) replaces a doubling condition.…”

“…To see the origins of the idea for the statement of Lemma 3.19, see [Dav91] page 93, or [Chr90]. (1) For every x ∈ X n there exists a unique k such that cQ ⊂ U c,k n for some family, where radius(…”

Условия $C^m$-приближаемости функций решениями эллиптических уравнений

“…Идея доказательства этой леммы состоит в том, что сначала строится "слабое" (L 1 с весом) приближение, из которого нужное равномерное приближение получа-ется с помощью выпуклых комбинаций "слабо" приближающих функций на ос-нове известной теоремы об отделимости выпуклых множеств в нормированном пространстве. Центральным моментом доказательства теоремы 1.8 является лемма 2.15 из [12], где, наряду с технически очень сложным геометрическим построением и идеей выпуклых комбинаций из [12; лемма 2.9], применяется теория сингулярных интегралов на липшицевой поверхности [27].…”