Function Spaces of Logarithmic Smoothness : Embeddings and Characterizations / Óscar Domínguez and Sergei Tikhonov.

Format:

Book

Author/Creator:

Domínguez, Óscar (Domínguez Bonilla), author.

Tikhonov, Sergei, author.

Series:

Memoirs of the American Mathematical Society ; Volume 282.

Memoirs of the American Mathematical Society Series ; Volume 282

Language:

English

Subjects (All):

Function spaces.

Logarithms.

Smoothness of functions.

Physical Description:

1 online resource (180 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: (1) Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; (2) Various new characterizations for Besov norms in terms of different Kfunctionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; (3) Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Besov, Sobolev, and Triebel-Lizorkin spaces

1.2. Applications of function spaces with smoothness zero

1.3. Goals and results

1.4. Structure of the paper

Acknowledgment

Chapter 2. Preliminaries

2.1. General notation

2.2. Function spaces

2.3. Interpolation methods

2.4. Hardy-type inequalities

Chapter 3. Embeddings between Besov, Sobolev and Triebel-Lizorkin spaces with logarithmic smoothness

3.1. Embeddings with smoothness zero

3.2. Sobolev embeddings for ^{0, }_{ , }(ℝ^{ })

Chapter 4. Characterizations and embedding theorems for general monotone functions

4.1. Definition and basic properties of general monotone functions

4.2. Characterization of spaces ^{ , }_{ , }(ℝ^{ })

4.3. Characterization of spaces ^{ , }_{ , }(ℝ^{ })

4.4. Characterization of the space ^{ , }_{ }(ℝ^{ })

4.5. Embeddings of ^{0, }_{ , }(ℝ^{ }) in _{ }(ℝ^{ })

4.6. Embeddings between ^{ , }_{ }(ℝ^{ }) and ^{ , }_{ , }(ℝ^{ })

4.7. Embeddings between ^{0, }_{ }(ℝ^{ }) and ^{0, }_{ , }(ℝ^{ })

4.8. Embeddings between ^{0, }_{ , }(ℝ^{ }) and ^{0, }_{ , }(ℝ^{ })

4.9. Sobolev embeddings

4.10. Franke-Jawerth embeddings

4.11. Characterization of spaces ^{ , }_{ , }( ),ℍ^{ , }_{ }( ), and ^{ , }_{ , }( )

Chapter 5. Characterizations and embedding theorems for lacunary Fourier series

5.1. Characterization of spaces ^{ , }_{ , }(ℝ^{ }), ^{ , }_{ . }(ℝ^{ }), and ^{ , }_{ , }(ℝ^{ })

5.2. Embeddings of ^{0, }_{ , }(ℝ^{ }) in _{ }(ℝ^{ })

5.3. Embeddings between ^{ , }_{ , }(ℝ^{ }) and ^{ , }_{ , }(ℝ^{ })

5.4. Embeddings between ^{0, }_{ , }(ℝ^{ }) and ^{0, }_{ , }(ℝ^{ })

5.5. Embeddings between ^{0, }_{ , }(ℝ^{ }) and ^{0, }_{ , }(ℝ^{ })

5.6. Characterization of spaces ^{ , }_{ , }( ), ^{ , }_{ , }( ), and ^{ , }_{ , }( ).

Chapter 6. Optimality of Propositions 1.2 and 1.3

Chapter 7. Optimality of embeddings between Sobolev and Besov spaces with smoothness close to zero

Chapter 8. Comparison between different kinds of smoothness spaces involving only logarithmic smoothness

Chapter 9. Optimality of embeddings between Besov spaces

9.1. Optimality of embeddings with smoothness near zero

9.2. Optimality of Sobolev embeddings for ^{0, }_{ , }(ℝ^{ })

9.3. Comparison between embeddings with smoothness near zero and Sobolev embeddings

9.4. New interpolation formulas and duality theorems

9.5. Proof of Proposition 9.1

9.6. Proof of Proposition 9.2

9.7. Proof of Theorem 9.6

9.8. Proof of Proposition 9.7

9.9. Proof of Proposition 9.8

9.10. Proofs of Proposition 9.10 and 9.12

Chapter 10. Various characterizations of Besov spaces

10.1. Characterizations involving Riesz potential space

10.2. Characterizations via averages on balls

10.3. Characterizations in terms of differences

10.4. Characterizations via approximation processes

10.5. Characterizations by means of heat and Poisson kernels

Chapter 11. Besov and Bianchini norms

11.1. Characterizations of various -functionals

11.2. Equivalences between Besov and Bianchini norms

11.3. Besov norm and functions of bounded p-variation

Chapter 12. Functions and their derivatives in Besov spaces

12.1. Inequalities for moduli of smoothness of functions and their derivatives

12.2. Sobolev-type characterizations of Besov spaces

Chapter 13. Lifting operators in Besov spaces

13.1. Inequalities for moduli of smoothness of Riesz and Bessel potentials

13.2. Lifting property on ^{0, }_{ , }(ℝ^{ })

Chapter 14. Regularity estimates of the fractional Laplace operator

14.1. Function spaces in PDE's

14.2. Local regularity for the Dirichlet fractional Laplacian.

14.3. ₂-Regularity estimates

14.4. _{ }-Regularity estimates

Appendix A. List of symbols

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Domínguez, Óscar Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

ISBN:

9781470473495

1470473496