Musielak–Orlicz Campanato spaces and applications

Abstract

Let φ:Rn×[0,∞)→[0,∞) be such that φ(x,⋅) is an Orlicz function and φ(⋅,t) is a Muckenhoupt A∞(Rn) weight uniformly in t. In this article, the authors introduce the Musielak–Orlicz Campanato space Lφ,q,s(Rn); as an application, the authors prove that some of them is the dual space of the Musielak–Orlicz Hardy space Hφ(Rn), which, in the case when q=1 and s=0, was obtained by L.D. Ky [ arXiv:1105.0486]. The authors also establish a John–Nirenberg inequality for functions in Lφ,1,s(Rn) and, as an application, the authors also obtain several equivalent characterizations of Lφ,q,s(Rn), which, in return, further induce the φ-Carleson measure characterization of Lφ,1,s(Rn).