The commutator of the Kato square root for second order elliptic operators on ℝn

Let L = −div(A∇) be a second order divergence form elliptic operator, and A be an accretive, n×n matrix with bounded measurable complex coefficients in ℝn. We obtain the Lp bounds for the commutator generated by the Kato square root 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 a Lipschitz function, which recovers a previous result of Calderón, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.