Optimality of embeddings in Orlicz spaces

TL;DR

This paper investigates the optimality of Orlicz spaces in Sobolev embeddings, revealing nonexistence results in critical cases related to isoperimetric properties of Euclidean domains.

Contribution

It establishes nonexistence of optimal Orlicz spaces in certain critical Sobolev embeddings, extending understanding of function space optimality in analysis.

Findings

Nonexistence of optimal Orlicz spaces in specific embeddings

Results apply to critical cases including Brezis-Wainger embedding

Highlights limitations of Orlicz spaces in optimality problems

Abstract

Working with function spaces in various branches of mathematical analysis introduces optimality problems, where the question of choosing a function space both accessible and expressive becomes a nontrivial exercise. A good middle ground is provided by Orlicz spaces, parameterized by a single Young function and thus accessible, yet expansive. In this work, we study optimality problems on Sobolev embeddings in Mazya classes of Euclidean domains which are defined through their isoperimetric behavior. In particular, we prove the nonexistence of optimal Orlicz spaces in certain Orlicz Sobolev embeddings in a limiting, or critical, state whose pivotal special case is the celebrated embedding of Brezis and Wainger for John domains.