Weighted Jump and Variational Inequalities for Hypersingular Integrals with Rough Kernels

Abstract

In this paper, we study the weighted jump function and variation of hypersingular integral operators with rough kernels which are defined as

$${T_{\Omega,\alpha,\varepsilon}}f\left(x \right) = \int_{\left| y \right|>\varepsilon} {{{\Omega ({y^\prime})} \over {{{\left| y \right|}^{n + \alpha}}}}f(x - y)dy,} $$

where α ≥ 0, Ω is an integrable function on the unit sphere ${\mathbb S^{n - 1}}$ satisfying certain cancellation conditions. More precisely, we show that for 1 < p < ∞, the jump function and variation of the family of truncated hypersingular integrals {T_{Ω,α, ε}}_ε>0 extends to bounded operators from the weighted Sobolev space L ^p_α (w) to the weighted Lebesgue space L^p(w) with $\Omega \,\,{L^q}({\mathbb S^{n - 1}})$ where q > 1.