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

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