Optimal Sobolev embeddings for generalized Lorentz-Zygmund spaces

TL;DR

This paper characterizes the optimal Sobolev embeddings into generalized Lorentz-Zygmund spaces, providing explicit descriptions of the best possible target spaces for various rearrangement-invariant classes on bounded domains.

Contribution

It offers a comprehensive and explicit characterization of the optimal target spaces in Sobolev embeddings for generalized Lorentz-Zygmund spaces, filling gaps in existing literature.

Findings

Identifies optimal target spaces for Sobolev embeddings.

Provides explicit scenarios for embeddings into various function spaces.

Enhances understanding of embeddings in Lorentz-Zygmund and related spaces.

Abstract

A comprehensive description of Sobolev-type embeddings for generalized Lorentz-Zygmund spaces on bounded open subsets of the $n$-dimensional Euclidean space is offered. Rearrangement-invariant spaces, H\"older, Morrey and Campanato type spaces are the classes of range spaces taken into account. Within each of these classes, the optimal target spaces in the relevant embeddings are identified and discussed. This way neat explicit scenarios are developed, which seem to be still missing in the existing literature.