Embeddings for spaces of Lorentz-Sobolev type

TL;DR

This paper characterizes embeddings of Besov and Triebel-Lizorkin spaces with Lorentz space metrics, extending classical function space theory and including results on Fourier multipliers and quasi-norm constants.

Contribution

It provides a comprehensive characterization of embeddings for Lorentz-Sobolev type spaces, expanding understanding of these spaces and their multiplier properties.

Findings

Characterized all embeddings for Lorentz-Sobolev type Besov and Triebel-Lizorkin spaces.

Included analysis of endpoint Fourier multipliers of Mikhlin-H"ormander type.

Examined constants in the triangle inequality for $L^{p,r}$ when $p<1$.

Abstract

The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation between classes of endpoint Mikhlin-H\"ormander type Fourier multipliers, and one on the constant in the triangle inequality for the spaces $L^{p,r} $ when $p<1$.