Weighted Norm Inequalities for Commutators of the Kato Square Root of Second Order Elliptic Operators on ℝn

Let L = −div(A∇) be a second order divergence form elliptic operator with bounded measurable coefficients in ℝn. We establish weighted Lp norm inequalities for commutators generated by L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt L $$\end{document} and Lipschitz functions, where the range of p is different from (1, ∞), and we isolate the right class of weights introduced by Auscher and Martell. In this work, we use good-λ inequality with two parameters through the weighted boundedness of Riesz transforms ∇L−1/2. Our result recovers, in some sense, a previous result of Hofmann.