Haroske Envelopes and Sharp Embeddings of Function Spaces

1. Auflage 2010
ISBN: 978-1-58488-751-5
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 222 Seiten

Reihe: Chapman & Hall/CRC Research Notes in Mathematics Series

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.

Zielgruppe

Graduate students and researchers in functional analysis and approximation theory; researchers in partial differential equations.

Autoren/Hrsg.

Haroske, Dorothee D.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

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