Function spaces and potential theory in the Orlicz setting

TL;DR

This paper extends classical potential theory and function spaces to the Orlicz setting, establishing new equivalences, inclusion results, and atomic decompositions for these generalized spaces.

Contribution

It generalizes Bessel and Lizorkin-Triebel spaces to Orlicz spaces, recovering classical results and establishing new inclusion and decomposition theorems.

Findings

Bessel-Orlicz spaces of integer order coincide with Orlicz-Sobolev spaces

Inclusion results for fractional order spaces are established

Atomic decompositions for Orlicz-Lizorkin-Triebel spaces are provided

Abstract

In this article, we study certain transcendental function spaces arising in potential theory within the framework of Orlicz spaces. Specifically, we generalize Bessel and Lizorkin-Triebel spaces to the nonstandard setting of Orlicz spaces. We recover classical results from potential theory, such as the fact that Bessel-Orlicz spaces of integer order coincide with Orlicz-Sobolev spaces (Calder\'on type theorem), and we establish inclusion results for fractional orders. Moreover, we prove a Strauss-type lemma for potential spaces. In the last sections, we show that certain Orlicz-Lizorkin-Triebel spaces coincide with Bessel-Orlicz spaces, and we provide a useful atomic decomposition for these spaces.