Function spaces and potential theory in the Orlicz setting

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.