Envelopes and Sharp Embeddings of Function Spaces
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Summary
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.
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 Figures
Editorial Reviews
“This interesting book is devoted to two new concepts of the theory of function spaces: growth envelopes and continuity envelopes. … After some nice preliminaries, the author introduces the new concepts, proves their basic properties and calculates growth and continuity envelopes for some classical function spaces … helps one understand better the differences between these cases and can be useful in dealing with a number of problems. ”
— Leszek Skrzypczak, in Mathematical Reviews, Issue 2007
"The approach, built upon an impressive series of the author’s results, has turned into a worthwhile general theory of beautiful, deep results, interesting examples and plenty of applications . . . All this the reader will find in the text. On top of that, the book is more reader-friendly than the standard . . . Truly delightful stuff!"
– In EMS Newsletter, September 2007
