Boundedness of Marcinkiewicz integrals with rough kernels on Musielak-Orlicz Hardy spaces

Let [Formula: see text] satisfy that [Formula: see text] , for any given [Formula: see text] , is an Orlicz function and [Formula: see text] is a Muckenhoupt [Formula: see text] weight uniformly in [Formula: see text] . The Musielak-Orlicz Hardy space [Formula: see text] is defined to be the set of all tempered distributions such that their grand maximal functions belong to the Musielak-Orlicz space [Formula: see text] . In this paper, the authors establish the boundedness of Marcinkiewicz integral [Formula: see text] from [Formula: see text] to [Formula: see text] under weaker smoothness conditions assumed on Ω. This result is also new even when [Formula: see text] for all [Formula: see text] , where ϕ is an Orlicz function.