Fractional Orlicz-Sobolev embeddings into Campanato type spaces

TL;DR

This paper establishes optimal embeddings of fractional Orlicz-Sobolev spaces into Campanato spaces, including BMO and VMO, highlighting differences from classical fractional Sobolev embeddings.

Contribution

It provides new optimal embedding results for fractional Orlicz-Sobolev spaces into generalized Campanato spaces, including characterizations for vanishing spaces and sharp embeddings into BMO and VMO.

Findings

Optimal embeddings into Campanato spaces are characterized.

Embeddings into vanishing Campanato spaces are described.

Sharp embeddings into BMO and VMO are derived.

Abstract

Optimal embeddings for fractional Orlicz-Sobolev spaces into (generalized) Campanato spaces on the Euclidean space are exhibited. Embeddings into vanishing Campanato spaces are also characterized. Sharp embeddings into $\operatorname{BMO}(\mathbb R^n)$ and $\operatorname{VMO}(\mathbb R^n)$ are derived as special instances. Dissimilarities to corresponding embeddings for classical fractional Sobolev spaces are pointed out.