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.

Inhalt
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.