Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

摘要

In this article, we consider the Marcinkiewicz integrals with variable kernels defined byμΩ(f)(x)=(∫0∞|∫|x?y|≤tΩ(x,x?y)|x?y|n?1f(y)dy|2dtt3)1/2, whereΩ(x,z)∈L∞(?n)×Lq(Sn?1)forq> 1. We prove that the operatorμΩis bounded from Hardy space,Hp(?n), toLp(?n)space; and is bounded from weak Hardy space,Hp,∞(?n), to weakLp(?n)space formax{2n2n+1,nn+α}<1, ifΩsatisfies theL1,α-Dini condition with any0<α≤1.