Kato square root problem with unbounded leading coefficients

Abstract

We prove the Kato conjecture for elliptic operators, $L=-\nabla \cdot \left((\mathbf{A}+\mathbf{D})\nabla \right)$, with $\mathbf{A}$ a complex measurable bounded coercive matrix and $\mathbf{D}$ a measurable real-valued skew-symmetric matrix in ${\mathbb{R}}^n$ with entries in $BMO({\mathbb{R}}^n)$;\, i.e., the domain of $\sqrt{L}$ is the Sobolev space ${\dot{H}}^1({\mathbb{R}}^n)$ in any dimension, with the estimate $\Vert \sqrt{L}f{\Vert }_2\lesssim \Vert \nabla f{\Vert }_2$.