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. The main result is for the sharp truncated singular integrals itself $$\Vert V_q\{T_{\Omega ,\epsilon }{\}}_{\epsilon >0}{\Vert }_{L^p(w)\to L^p(w)}\le c_{p,q,n}\Vert \Omega {\Vert }_{L^{\infty }}(w{)}_{A_p}^{1+1/q}\{w{\}}_{A_p},$$ where the quantity $(w{)}_{A_p}$, $\{w{\}}_{A_p}$ will be recalled in the introduction; we do not know whether this is sharp, but it is the best known quantitative result for this class of operators, since when $q=\infty$, it coincides with the best known quantitative bounds by Di Pilino--Hytönen--Li or Lerner. In the course of establishing the above estimate, we obtain several quantitative weighted bounds which are of independent interest. We hereby highlight two of them. The first one is $$\Vert V_q\{{\varphi }_k\ast T_{\Omega }{\}}_{k\in \mathbb{Z}}{\Vert }_{L^p(w)\to L^p(w)}\le c_{p,q,n}\Vert \Omega {\Vert }_{L^{\infty }}(w{)}_{A_p}^{1+1/q}\{w{\}}_{A_p},$$ where ${\varphi }_k(x)=\frac{1}{2^{kn}}\varphi (\frac{x}{2^k})$ with $\varphi \in C_c^{\infty }({\mathbb{R}}^n)$ being any non-negative radial function, and the sharpness for $q=\infty$ is due to Lerner; the second one is $$\Vert {\mathcal{S}}_q\{T_{\Omega ,\epsilon }{\}}_{\epsilon >0}{\Vert }_{L^p(w)\to L^p(w)}\le c_{p,q,n}\Vert \Omega {\Vert }_{L^{\infty }}(w{)}_{A_p}^{1/q}\{w{\}}_{A_p},$$ and the sharpness for $q=\infty$ follows from the Hardy--Littlewood maximal function.

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