Jump and variational inequalities for hypersingular integrals with rough kernels

In this paper, we study the jump function and variation of hypersingular integral operators with rough kernels T-Omega,T- (alpha,epsilon)f(x) = integral(vertical bar y vertical bar>epsilon) Omega(y)/vertical bar y vertical bar(n+alpha) f(x - y) dy, where alpha >= 0, Omega is an integrable function on the unit sphere Sn-1 satisfying certain cancellation conditions. More precisely, we first show that for 1 < p < infinity, the jump function and variation of the family of truncated hypersingular integrals {T-Omega,T- (alpha,epsilon)}(epsilon>0) extends to a bounded operator from the Sobolev space L-alpha(p) to the Lebesgue space L-p with Omega belonging to the Hardy space H-q(Sn-1) where q = n-1/n-1+alpha, which gives a positive answer to an open problem proposed by Ding-Hong-Liu [15]. (C) 2022 Elsevier Inc. All rights reserved.