A note on parameterized Marcinkiewicz integrals with variable kernels

In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by are investigated.It is proved that if Ω∈ L∞(Rn) × Lr(Sn-1)(r>(n-n1)p) is an odd function in the second variable y,then the operator μρΩ is bounded from Lp(Rn) to Lp(Rn) for 1 < p ≤ max{(n+2 1),2}.It is also proved that,if Ω satisfies the L1-Dini condition,then μρΩ is of type(p,p) for 1 < p ≤ 2,of the weak type(1,1) and bounded from H1 to L1.