Journal of Function Spaces and Applications (Jan 2010)
Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
Abstract
In this article, we consider the Marcinkiewicz integrals with variable kernels defined by μΩ(f)(x)=(∫0∞|∫|x−y|≤tΩ(x,x−y)|x−y|n−1f(y)dy|2dtt3)1/2, where Ω(x,z)∈L∞(ℝn)×Lq(Sn−1) for q > 1. We prove that the operator μΩ is bounded from Hardy space, Hp(ℝn), to Lp(ℝn) space; and is bounded from weak Hardy space, Hp,∞(ℝn), to weak Lp(ℝn) space for max{2n2n+1,nn+α}<p<1, if Ω satisfies the L1,α-Dini condition with any 0<α≤1.