Estimates For Parametric Marcinkiewicz Integrals On Musielak-Orlicz Hardy Spaces

Letφp:Rn ×[0,∞)→[0,∞)satisfy thatφ(x,·),for any given x ∈ Rn,is an Orlicz function andφ(·,t)is a Muckenhoupt A∞ weight uniformly in t ∈(0,∞).The Musielak-Orlicz Hardy space Hφ(Rn)generalizes both of the weighted Hardy space and the Orlicz Hardy space and hence this space not only has a wide generality but also sat-isfies the kernel properties of Hardy spaces and Orlicz spaces.In this paper,the au-thors first prove the completeness of both of Musielak-Orlicz space Lφ(Rn)and weak Musielak-Orlicz space WLφ(Rn).Then the authors obtain two boundedness criterions of some sublinear operators on Musielak-Orlicz spaces.As applications,the authors es-tablish the boundedness of parametric Marcinkiewicz integral μΩρ from Hφ(Rn)to Lφ(Rn)(resp.WLφ(Rn))under weaker smoothness condition(resp.some Lipschitz condition)assumed on function Ω:Rn→R.These results are also new even when φ(x,t):Φ(t)for all(x,t)∈Rn×[0,∞),where Φ is an Orlicz function.