Rearrangement Estimates and Limiting Embeddings for Anisotropic Besov Spaces

The paper is dedicated to the study of embeddings of the anisotropic Besov spaces Bp,θ1,…,θnβ1,…,βn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{p,{\theta _1}, \ldots ,{\theta _n}}^{{\beta _1}, \ldots ,{\beta _n}}$$\end{document} (ℝn) into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of the exponents βk tend to 1 (βk < 1). In particular, these results give an extension of the estimate proved by Bourgain, Brezis, and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link with some known embeddings of anisotropic Lipschitz spaces.