Jump and variational inequalities for hypersingular integrals with rough kernels

Abstract

In this paper, we study the jump function and variation of hypersingular integral operators with rough kernelsTΩ,α,εf(x)=∫|y|>εΩ(y)|y|n+αf(x−y)dy, where α≥0, Ω is an integrable function on the unit sphere Sn−1 satisfying certain cancellation conditions. More precisely, we first show that for 1<p<∞, the jump function and variation of the family of truncated hypersingular integrals {TΩ,α,ε}ε>0 extends to a bounded operator from the Sobolev space Lαp to the Lebesgue space Lp with Ω belonging to the Hardy space Hq(Sn−1) where q=n−1n−1+α, which gives a positive answer to an open problem proposed by Ding-Hong-Liu [15].