Function spaces of logarithmic smoothness : embeddings and characterizations / Óscar Domínguez, Sergey Tikhonov.

Format:

Book

Author/Creator:

Domínguez, Óscar (Domínguez Bonilla), author.

Tikhonov, Sergey, 1976- author.

Series:

Memoirs of the American Mathematical Society ; no. 1393.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1393

Language:

English

Subjects (All):

Function spaces.

Smoothness of functions.

Logarithms.

logarithms.

Physical Description:

vii, 166 pages : illustrations ; 26 cm

Place of Publication:

Providence, RI : AMS, 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:

Preliminaries

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

Characterizations and embedding theorems for general monotone functions

Characterizations and embedding theorems for lacunary Fourier series

Optimality of Propositions 1.2 and 1.3

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

Comparison between different kinds of smoothness spaces involving only logarithmic smoothness

Optimality of embeddings between Besov spaces

Various characterizations of Besov spaces

Besov and Bianchini norms

Functions and their derivatives in Besov spaces

Lifting operators in Besov spaces

Regularity estimates of the fractional Laplace operator.

Notes:

Includes bibliographical references.

ISBN:

9781470455385

1470455382

OCLC:

1371972517