Estimates for parametric Marcinkiewicz integrals on Musielak-Orlicz Hardy spaces

摘要

Let . : R n × [0, ∞ ) → [0, ∞ ) satisfy that . (x, ·) , for any given x ∈ R n , is an Orlicz function and . (·,t) is a Muckenhoupt A ∞ weight uniformly in t ∈ (0, ∞ ) . The Musielak- Orlicz Hardy space H . (R n ) generalizes both of the weighted Hardy space and the Orlicz Hardy space and hence has a wide generality. In this paper, the authors first prove the completeness of both of the Musielak-Orlicz space L . (R n ) and the weak Musielak-Orlicz space WL . (R n ) . Then the authors obtain two boundedness criterions of operators on Musielak-Orlicz spaces. As applications, the authors establish the boundedness of parametric Marcinkiewicz integral μ ρ Ω from H . (R n ) to L . (R n ) (resp. WL . (R n )) under weaker smoothness condition (resp. some Lipschitz condition) assumed on Ω . These results are also new even when . (x, t) := φ (t) for all (x, t) ∈ R n × [0, ∞ ) ,where φ is an Orlicz function.