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The Structure of Functions

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概要 This book deals with the symbiotic relationship between I Quarkonial decompositions of functions, on the one hand, and II Sharp inequalities and embeddings in function spaces, III Fractal elliptic ope...rators, IV Regularity theory for some semi-linear equations, on the other hand. Accordingly, the book has four chapters. In Chapter I we present the Weier strassian approach to the theory of function spaces, which can be roughly described as follows. Let 'IjJ be a non-negative Coo function in]R. n with compact support such that {'ljJe - m) : m E zn} is a resolution of unity in ]R. n. Let 'IjJ!3(x) = x!3'IjJ(x) where x E ]R. n and {3 E N~. One may ask under which circumstances functions and distributions f in ]R. n admit expansions 00 (0. 1) f(x) = L L L ). . ~m'IjJ!3(2jx - m), x E ]R. n, n !3ENg j=O mEZ with the coefficients ). . ~m E C. This resembles, at least formally, the Weier strassian approach to holomorphic functions (in the complex plane), combined with the wavelet philosophy: translations x 1---4 x - m where m E zn and dyadic j dilations x 1---4 2 x where j E No in ]R. n. Such representations pave the way to constructive definitions offunction spaces.続きを見る
目次 I Decompositions of Functions
1 Introduction, heuristics, and preliminaries
2 Spaces on ?n: the regular case
3 Spaces on ?n: the general case
4 An application: the Fubini property
5 Spaces on domains: localization and Hardy inequalities
6 Spaces on domains. decompositions
7 Spaces on manifolds
8 Taylor expansions of distributions
9 Traces on sets, related function spaces and their decompositions
II Sharp Inequalities
10 Introduction: Outline of methods and results
11 Classical inequalities
12 Envelopes
13 The critical case
14 The super-critical case
15 The sub-critical case
16 Hardy inequalities
17 Complements
III Fractal Elliptic Operators
18 Introduction
19 Spectral theory for the fractal Laplacian
20 The fractal Dirichlet problem
21 Spectral theory on manifolds
22 Isotropic fractals and related function spaces
23 Isotropic fractal drums
IV Truncations and Semi-linear Equations
24 Introduction
25 Truncations
26 The Q-operator
27 Semi-linear equations; the Q-method
References
Symbois.
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登録日 2020.06.27
更新日 2020.06.28