Boundedness of functions in fractional Orlicz-Sobolev spaces

TL;DR

This paper characterizes when functions in fractional Orlicz-Sobolev spaces are bounded and continuous, providing optimal embeddings and extending classical results to more general spaces.

Contribution

It establishes necessary and sufficient conditions for embeddings into $L^ Infty$ in fractional Orlicz-Sobolev spaces, including optimal target spaces and continuity results.

Findings

Characterization of embedding conditions into $L^ Infty$

Identification of optimal Orlicz and rearrangement-invariant target spaces

Extension of classical fractional Sobolev embedding theorems

Abstract

A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^\infty(\mathbb R^n)$ fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.