Lp-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on Double-struck capital Rn

Let L = -div(A backward difference ) be a second-order divergent-form elliptic operator, where A is an accretive n x n matrix with bounded and measurable complex coefficients on Double-struck capital R-n: Herein, we prove that the commutator [b; 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}] of 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 b with backward difference b is an element of L-n(Double-struck capital R-n)(n > 2), is bounded from the homogenous Sobolev space L & x2d9;1p(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot L_1<^>p(\mathbb{R}<^>n)$$\end{document} to L-p(Double-struck capital Rn) (p-(L) < p < p+(L)).