函数结构(影印版)

　　 《函数结构》内容简介：This book deals with the symbiotic relationship betweenI Quarkonial decompositions of functions，on the one hand，andII Sharp inequalities and embeddings in function spaces，III Fractal elliptic operators，IV Regularity theory for some semi-linear equations，on the other hand.
　　

preface
i decompositions of functions
1 introduction, heuristics, and preliminaries
2 spaces on rn: the regular case
3 spaces on rn: 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

　　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 Ψ be a non-negative C∞ function in Rn with compact support such that {Ψ(. - m): m E Zn) is a resolution of unity in Rn. Let Ψβ(x) - xβΨ(x) where x E Rn and /3 E Nn0. One may ask under which circumstances functions and distributions f in Rn admit expansions with the coefficients λβjm 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→x-m where m E Zn and dyadic dilations x→2jx where j E No in Rn. Such representations pave the way to constructive definitions of function spaces. We are mainly interested in the two scales B3pq and Fpq with s C R, 0＜p≤∞, 0＜q≤∞, which cover many well-known classical spaces, such as (fractional) Sobolev spaces, HSlder-Zygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. The theory of these spaces has been developed systematically by many mathematicians since the early 1960s. The first chapter in [Trir] is a historically-minded survey of this subject with many references covering the period up to 1990. In Chapter I of the present book we offer the indicated fresh constructive approach to these spaces on IRn, domains, fractals and some manifolds. Chapters II, III, and IV deal with various applications of the results of Chapter I. In Chapter II we contribute to one of the main topics in the theory of function spaces: embed-dings and inequalities. We are mostly interested in delicate limiting situations,asking for necessary and sufficient conditions. Chapter III deals with elliptic operators, preferably the Laplacian, in diverse fractal settings, such as fractal boundaries of underlying domains, measure-valued coefficients or potentials,etc. We wish to demonstrate the symbiotic relationship between some basic notation of fractal geometry and spectral theory. This chapter might be con-sidered as the continuation of the Chapters IV and V in [Triδ]. Finally, in Chapter IV of the present book we study truncations in function spaces and we use these results, combined with the quarkonial decompositions indicated above, to develop a new regularity theory for some semi-linear integral and dif-ferential equations. Each chapter begins with a separate introduction, where we outline in somewhat greater detail what can be expected.
　　This book is mainly based on the results of the author and his co-workers Ob-tained in the last few years. We tried to present the material in such a way that the main ideas can be understood independently of the existing literature. On the other hand, after proving in Chapter I that the function spaces introduced via quarkonial decompositions coincide with the well-established spaces B8pq and Fspq we feel free to use known results about these spaces, especially when we have nothing new to say about the assertions used. A reader who is mostly interested in the material presented in one of the Chapters II, III, or IV, which are largely independent of each other, may skip Chapter I, at the first glance.But most of the related proofs in these chapters depend substantially on the theory developed in the first chapter.
　　It is a pleasure to acknowledge the great help I have received from my col-laborators in Jena, in particular Dorothee Haroske and Winfried Sickel, who made valuable suggestions which have been incorporated in the text. I am especially indebted to Dorothee Haroske for producing all the figures in this book. Last, but not least, I wish to thank my friend and colleague David Edmunds in Brighton who looked through the whole manuscript and offered many comments.
　　Jena, Spring 2001 Hans Triebel