Sparse Domination and Weighted Inequalities for the ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-Variation of Singular Integrals and Commutators

This paper gives pointwise sparse domination results for variation operators of singular integrals and commutators with kernels satisfying Lr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^r$$\end{document}-Hörmander conditions. As applications, we obtain strong-type quantitative weighted bounds for such variation operators, weak-type quantitative weighted bounds for the variation operators of singular integrals and quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the un-weighted case. In addition, we also obtain local exponential decay estimates for such variation operators.