Rearrangement estimates and limiting embeddings for anisotropic Besov spaces

TL;DR

This paper investigates the embeddings of anisotropic Besov spaces into Lorentz spaces, providing sharp asymptotic estimates for embedding constants as certain smoothness exponents approach 1, extending known isotropic results and linking to anisotropic Lipschitz spaces.

Contribution

It extends the Bourgain-Brezis-Mironescu estimate to anisotropic Besov spaces and establishes new rearrangement estimates involving partial moduli of continuity.

Findings

Sharp asymptotic behavior of embedding constants as exponents tend to 1

Extension of Bourgain-Brezis-Mironescu estimate to anisotropic spaces

Link between anisotropic Besov and Lipschitz space embeddings

Abstract

The paper is dedicated to the study of embeddings of the anisotropic Besov spaces $B^{\beta_1,...,beta_n}_{p;\theta_1,...,\theta_n}(\Bbb R^n)$ into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of the exponents $\beta_k$ tend to 1 ($\beta_k<1)$. In particular, these results give an extension of the estimate proved bt\'y Bourgain, Brezis, and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link with some known embeddings of anisotropic Lipschitz spaces. One of the key results of the paper is an anisotropic type estimate of rearrangements in terms of partial moduli of continuity.