# Optimal domain spaces in Orlicz-Sobolev embeddings

## TL;DR

This paper characterizes the conditions for the existence of the largest Orlicz-Sobolev space that can be embedded into a given Orlicz space, including boundary and measure-related cases.

## Contribution

It establishes necessary and sufficient conditions for optimal Orlicz-Sobolev embeddings and explicitly describes the optimal spaces when they exist.

## Key findings

- Characterization of optimal Orlicz-Sobolev embeddings
- Explicit description of optimal spaces when they exist
- Extension to embeddings involving Frostman measures and boundary traces

## Abstract

We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces with respect to a Frostman measure, and, in particular, for trace embeddings on the boundary.