Quantitative weighted bounds for the $q$-variation of singular integrals with rough kernels (2008.13071v2)

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 $$ |V_q{T_{\Omega,\varepsilon}}{\varepsilon>0}|{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} |\Omega|{ L^\infty}(w){A_p}^{1+1/q}{w}_{A_p},$$ 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--Hyt\"{o}nen--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 $$ |V_q{\phi_k\ast T_{\Omega}}{k\in\mathbb Z}|{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} |\Omega|{ L^\infty}(w){A_p}^{1+1/q}{w}_{A_p},$$ where $\phi_k(x)=\frac1{2^{kn}}\phi(\frac x{2^k})$ with $\phi\in C^\infty_c(\mathbb R^n)$ being any non-negative radial function, and the sharpness for $q=\infty$ is due to Lerner; the second one is $$ |\mathcal{S}q{T{\Omega,\varepsilon}}{\varepsilon>0}|{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} |\Omega|{ L^\infty}(w){A_p}^{1/q}{w}_{A_p},$$ and the sharpness for $q=\infty$ follows from the Hardy--Littlewood maximal function.