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 of truncated hypersingular integrals with variable kernels, which are defined by

where $\alpha \ge 0$ and the kernel $\mathrm{\Omega }$ belongs to $L^{\infty }({ℝ}^n)\times L^1({\mathbb{𝕊}}^{n-1})$. We first prove that the variation of the hypersingular integral with variable kernel is bounded from the Sobolev space ${\dot{L}}_{\alpha }^2$ to the Lebesgue space $L^2$ when $\mathrm{\Omega }\in L^{\infty }({ℝ}^n)\times L^q({\mathbb{𝕊}}^{n-1})$ for and satisfies some cancellation condition in its second variable. The result is sharp in the sense that the $({\dot{L}}_{\alpha }^2,L^2)$ boundedness of ${\mathcal{𝒯}}_{\alpha }$ fails if $q\le 2(n-1)∕(n+2\alpha )$. After strengthening the smoothness of $\mathrm{\Omega }(x,z^{\prime })$ in its second variable, we give the weighted boundedness of the variation of the hypersingular integrals with smooth variable kernels from ${\dot{L}}_{\alpha }^p(w)$ to $L^p(w)$ for and $w\in A_p$. Finally, we extend the result to the Sobolev–Morrey spaces.