Envelopes and Sharp Embeddings of Function Spaces

Envelopes and Sharp Embeddings of Function Spaces

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Description

Until now, no book has systematically presented the recently developed concept of envelopes in function spaces. Envelopes are relatively simple tools for the study of classical and more complicated spaces, such as Besov and Triebel-Lizorkin types, in limiting situations. This theory originates from the classical result of the Sobolev embedding theorem, ubiquitous in all areas of functional analysis. Self-contained and accessible, Envelopes and Sharp Embeddings of Function Spaces provides the first detailed account of the new theory of growth and continuity envelopes in function spaces. The book is well structured into two parts, first providing a comprehensive introduction and then examining more advanced topics. Some of the classical function spaces discussed in the first part include Lebesgue, Lorentz, Lipschitz, and Sobolev. The author defines growth and continuity envelopes and examines their properties. In Part II, the book explores the results for function spaces of Besov and Triebel-Lizorkin types. The author then presents several applications of the results, including Hardy-type inequalities, asymptotic estimates for entropy, and approximation numbers of compact embeddings. As one of the key researchers in this progressing field, the author offers a coherent presentation of the recent developments in function spaces, providing valuable information for graduate students and researchers in functional analysis.show more

Product details

  • Electronic book text | 222 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 15 Illustrations, black and white
  • 1584887516
  • 9781584887515

Table of contents

Preface DEFINITION, BASIC PROPERTIES, AND FIRST EXAMPLES Introduction Preliminaries, Classical Function Spaces Non-increasing rearrangements Lebesgue and Lorentz spaces Spaces of continuous functions Sobolev spaces Sobolev's embedding theorem The Growth Envelope Function EG Definition and basic properties Examples: Lorentz spaces Connection with the fundamental function Further examples: Sobolev spaces, weighted Lp-spaces Growth Envelopes EG Definition Examples: Lorentz spaces, Sobolev spaces The Continuity Envelope Function EC Definition and basic properties Some lift property Examples: Lipschitz spaces, Sobolev spaces Continuity Envelopes EC Definition Examples: Lipschitz spaces, Sobolev spaces RESULTS IN FUNCTION SPACES AND APPLICATIONS Function Spaces and Embeddings Spaces of type Bsp,q, Fsp,q Embeddings Growth Envelopes EG Growth envelopes in the sub-critical case Growth envelopes in sub-critical borderline cases Growth envelopes in the critical case Continuity Envelopes EC Continuity envelopes in the super-critical case Continuity envelopes in the super-critical borderline case Continuity envelopes in the critical case Envelope Functions EG and EC Revisited Spaces on R+ Enveloping functions Global versus local assertions Applications Hardy inequalities and limiting embeddings Envelopes and lifts Compact embeddings References Symbols Index List of Figuresshow more