ESTIMATES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS ON MUSIELAK-ORLICZ HARDY SPACES

Let phi : R-n x [0, infinity) -> [0, infinity) satisfy that phi(x, .), for any given x is an element of R-n, is an Orlicz function and phi(., t) is a Muckenhoupt A(infinity) weight uniformly in t is an element of (0, infinity). The Musielak-Orlicz Hardy space H-phi(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-phi(R-n) and the weak Musielak-Orlicz space WL phi(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 mu(rho)(ohm) from to H-phi(R-n) to L-phi(R-n) (resp. WL phi(R-n)) under weaker smoothness condition (resp. some Lipschitz condition) assumed on ohm. These results are also new even when phi(x, t) := phi(t) for all (x, t) is an element of R-n x [0, infinity), where phi is an Orlicz function.