Inclusion relations among fractional Orlicz-Sobolev spaces and a Littlewood-Paley characterization

TL;DR

This paper characterizes embeddings among fractional Orlicz-Sobolev spaces, establishing their norm equivalences via Littlewood-Paley decompositions and difference quotients, supported by a new optimal convolution inequality in Orlicz spaces.

Contribution

It provides a comprehensive analysis of inclusion relations and norm equivalences in fractional Orlicz-Sobolev spaces, introducing a novel convolution inequality.

Findings

Embeddings among fractional Orlicz-Sobolev spaces are characterized.

Norm equivalences via Littlewood-Paley, oscillations, and difference quotients are established.

A new optimal convolution inequality in Orlicz spaces is introduced.

Abstract

Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. The equivalence of their Gagliardo-Slobodeckij norms to norms defined via Littlewood-Paley decompostions, via oscillations, or via Besov type difference quotients is also established. These equivalences, of independent interest, are a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.