Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Abstract

Let $$L=-\mathrm{div}(A\nabla )$$ be a second order divergence form elliptic operator and A an accretive $$n\times n$$ matrix with bounded measurable complex coefficients in $${\mathbb R}^n$$ . Let $$\nabla b\in L^n({\mathbb R}^n)\,(n>2)$$ . In this paper, we prove that the commutator generated by b and the square root of L, which is defined by $$[b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)$$ , is bounded from the homogenous Sobolev space $${\dot{L}}_1^2({\mathbb R}^n)$$ to $$L^2({\mathbb R}^n)$$ .