Envelopes and Sharp Embeddings of Function Spaces: 1st Edition (Hardback) book cover

Envelopes and Sharp Embeddings of Function Spaces

1st Edition

By Dorothee D. Haroske

Chapman and Hall/CRC

222 pages | 15 B/W Illus.

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pub: 2006-09-22
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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.


“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

Table of Contents




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


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


Examples: Lipschitz spaces, Sobolev spaces


Function Spaces and Embeddings

Spaces of type Bsp,q, Fsp,q


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


Hardy inequalities and limiting embeddings

Envelopes and lifts

Compact embeddings




List of Figures

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Functional Analysis