The Structure of Functions

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.