Kato square root problem with unbounded leading coefficients

We prove the Kato conjecture for elliptic operators, $L=- ablacdotleft((mathbf A+mathbf D) abla right)$, 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 $|sqrt{L}, f|_2lesssim | abla f|_2$.