The Structure of Functions

This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces, domains, manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces, spectral theory of fractal elliptic operators, and a regularity theory of some semi-linear equations. The book is self-contained, although some parts may be considered as a continuation of the author's book "Fractals and Spectra" (MMA 91). It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated. Front Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 12 1 Introduction, heuristics, and preliminaries ......Page 14 2 Spaces on R^n: the regular case......Page 23 3 Spaces on R^n: the general case......Page 40 4 An application: the Fubini property ......Page 47 5 Spaces on domains: localization and Hardy inequalities ......Page 54 6 Spaces on domains: decompositions ......Page 84 7 Spaces on manifolds ......Page 94 9 fraces on sets, related function spaces and their decompositions ......Page 133 10 Introduction: Outline of methods and results ......Page 174 11 Classical inequalities ......Page 180 12 Envelopes ......Page 194 13 The critical case ......Page 215 14 The super-critical case ......Page 231 15 The sub-critical case ......Page 242 16 Hardy inequalities ......Page 248 17 Complements ......Page 256 18 Introduction ......Page 264 19 Spectral theory for the fractal Laplacian ......Page 266 20 The fractal Dirichiet problem ......Page 307 21 Spectral theory on miinifolds ......Page 323 22 Isotropic fractals and related function spaces ......Page 342 23 Isotropic fractal drums ......Page 361 24 Introduction ......Page 368 25 Truncations ......Page 370 26 The Q-operator ......Page 398 References ......Page 416 Symbols ......Page 432 Index ......Page 436 I. Decompositions Of Functions -- Ii. Sharp Inequalities -- Iii. Fractal Elliptic Operators -- Iv. Truncations And Semi-linear Equations. Hans Triebel. Includes Bibliographical References (p. [403]-417) And Index.