Abstract: Let $n\ge2$ and let $L$ be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a $\mathrm{BMO}$ anti-symmetric part in $\mathbb{R}^n$. In this article, we consider the weighted Kato square root problem for $L$. More precisely, we prove that the square root $L^{1/2}$ satisfies the weighted $L^p$ estimates $\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}\le C\|\nabla f\|_{L^p_\omega (\mathbb{R}^n;\mathbb{R}^n)}$ for any $p\in(1,\infty)$ and $\omega\in A_p{(\mathbb{R}^n)}$ (the class of Muckenhoupt weights), and that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)}\le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,2+\varepsilon)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{2+\varepsilon}{p})'}(\mathbb{R}^n)$ (the class of reverse Hölder weights), where $\varepsilon\in(0,\infty)$ is a constant depending only on $n$ and the operator $L$, and where $(\frac{2+\varepsilon}{p})'$ denotes the Hölder conjugate exponent of $\frac{2+\varepsilon}{p}$. Moreover, for any given $q\in(2,\infty)$, we give a sufficient condition to obtain that $\|\nabla f\|_{L^p_\omega(\mathbb{R}^n;\mathbb{R}^n)} \le C\|L^{1/2}(f)\|_{L^p_\omega(\mathbb{R}^n)}$ for any $p\in(1,q)$ and $\omega\in A_p(\mathbb{R}^n)\cap RH_{(\frac{q}{p})'}(\mathbb{R}^n)$. As an application, we prove that when the coefficient matrix $A$ that appears in $L$ satisfies the small $\mathrm{BMO}$ condition, the Riesz transform $\nabla L^{-1/2}$ is bounded on $L^p_\omega(\mathbb{R}^n)$ for any given $p\in(1,\infty)$ and $\omega\in A_p(\mathbb{R}^n)$. Furthermore, applications to the weighted $L^2$-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Key words: elliptic operator, Kato square root problem, Muckenhoupt weight, Riesz transform, reverse Hölder inequality