Optimal Orlicz domains in Sobolev embeddings into Marcinkiewicz spaces

TL;DR

This paper develops a method to identify the largest Orlicz space that can be embedded into Marcinkiewicz endpoint spaces via Sobolev embeddings, advancing understanding of optimal function spaces in these embeddings.

Contribution

It introduces a novel technique for determining the optimal Orlicz domain spaces in Sobolev and trace embeddings into Marcinkiewicz spaces, applicable to domains with various regularities.

Findings

Established a criterion for the existence of maximal Orlicz spaces in Sobolev embeddings.

Extended the method to Sobolev trace embeddings on irregular domains.

Provided insights into the optimality conditions for Orlicz spaces in these embeddings.

Abstract

In this paper we present a method for determining whether there exists a largest Orlicz space $L^A(\Omega)$ satisfying the Sobolev embedding $W^mL^A(\Omega) \to Y(\Omega)$ where $Y(\Omega)$ stands for an arbitrary so-called Marcinkiewicz endpoint space. The tool developed in this work enables us to investigate the optimality of Orlicz domain spaces in Sobolev embeddings and also in Sobolev trace embeddings on domains $\Omega$ in $\mathbb{R}^n$ with various regularity.