The Structure of Functions

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 operators, 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.

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.