Atomic and Wavelet Characterization of Musielak–Orlicz Hardy Spaces for Generalized Orlicz Functions

Abstract

The goal of this paper is to obtain atomic and wavelet characterizations of Musielak–Orlicz Hardy spaces. Recently, in 2018, Fu and Yang obtained wavelet characterizations of Musielak–Orlicz Hardy spaces for growth functions of uniformly upper type $$p_-$$ and of uniformly lower type $$p_+$$ with $$0<p_- \le p_+\le 1$$ . What is different from the existing works is that we merely assume $$0<p_- \le p_+<\infty $$ . One of the important tools that make it possible is to refine the convexity of Orlicz functions by obtaining canonical equivalent functions. As applications of the atomic characterization, we investigate the boundedness property of singular integral operators. Especially, we obtain the boundedness property of Marcinkiewicz integral operators acting on some Musielak–Orlicz Hardy spaces which are quasi-Banach spaces.