Quantitative Weighted Bounds for the q-Variation of Singular Integrals with Rough Kernels

Abstract

In this paper, we study the quantitative weighted bounds for the q-variational singular integral operators with rough kernels, a stronger nonlinearity than the maximal truncations. The main result is for the truncated singular integrals itself $$\begin{aligned} \Vert V_q\{T_{\Omega ,\varepsilon }\}_{\varepsilon >0}\Vert _{L^p(w)\rightarrow L^p(w)}\lesssim \Vert \Omega \Vert _{ L^\infty }(w)_{A_p}^{1+1/q}\{w\}_{A_p}, \end{aligned}$$ it is the best known quantitative result for this class of operators. In the course of establishing the above estimate, we obtain several quantitative weighted bounds which are of independent interest.