Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

In this article,we consider the marcinkiewicz integrals with variable kernels defined by μω(f)(x)= ( ∫ 0 ∞|∫ |x-y|≤t ω(x,x-y)/|x-y| n-1f(y)dy| 2 dt/t 3) 1/2,where ω(x,z) ∈ l ∞(ℝ n)×l q(double-struck s sign n-1) for q > 1. we prove that the operator μω is bounded from hardy space,h p(ℝ n),to l p(ℝ n) space; and is bounded from weak hardy space,h p,∞(ℝ n),to weak l p(ℝ n) space for max{2n/2n+1,n/n+α}<p<1,if ω satisfies the l 1,α -dini condition with any 0<≤1. copyright © 2010 hindawi publishing corporation.