Variational inequalities for hypersingular integrals with variable kernels

Abstract

We systematically study variational inequalities for hypersingular integral operators. More precisely, we show the variational inequalities for the families 𝒯α:={Tα,𝜀}𝜀>0 of truncated hypersingular integrals with variable kernels, which are defined by Tα,𝜀f(x)=∫|x−y|>𝜀Ω(x,x−y)|x−y|n+αf(y)dy, where α≥0 and the kernel Ω belongs to L∞(ℝn)×L1(𝕊n−1). We first prove that the variation of the hypersingular integral with variable kernel is bounded from the Sobolev space L˙α2 to the Lebesgue space L2 when Ω∈L∞(ℝn)×Lq(𝕊n−1) for q>max{1,2(n−1)∕(n+2α)} and satisfies some cancellation condition in its second variable. The result is sharp in the sense that the (L˙α2,L2) boundedness of 𝒯α fails if q≤2(n−1)∕(n+2α). After strengthening the smoothness of Ω(x,z′) in its second variable, we give the weighted boundedness of the variation of the hypersingular integrals with smooth variable kernels from L˙αp(w) to Lp(w) for 1<p<∞ and w∈Ap. Finally, we extend the result to the Sobolev–Morrey spaces.