The Structure of Functions

by Hans Triebel

Overview

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. It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and, in particular, how they are interrelated.

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The book under review can be regarded as a continuation of [his book on "Fractals and spectra", 1997] (...) There are many sections named: comments, preparations, motivations, discussions and so on. These parts of the book seem to be very interesting and valuable. They help the reader to deal with the main course.
(Mathematical Reviews)

Product Details

ISBN-13:	9783764365462
Publisher:	Birkhäuser Basel
Publication date:	09/06/2001
Series:	Monographs in Mathematics , #97
Edition description:	2001
Pages:	425
Product dimensions:	7.01(w) x 10.00(h) x (d)

About the Author

Hans Triebel was professor of mathematics at the Friedrich-Schiller-University Jena, Germany.

Table of Contents

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.