The weighted Kato square root problem of elliptic operators having a BMO anti-symmetric part

Abstract

Let n ≥ 2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝⁿ. 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 \(||{L^{1/2}}(f)|{|_{L_\omega ^p({\mathbb{R}^n})}} \le C||\nabla f|{|_{L_\omega ^p({\mathbb{R}^n};{\mathbb{R}^n})}}\) for any p ∈ (1, ∞) and ω ∈ A_p(ℝⁿ) (the class of Muckenhoupt weights), and that \(||\nabla f|{|_{L_\omega ^p({\mathbb{R}^n};{\mathbb{R}^n})}} \le C||{L^{1/2}}(f)|{|_{L_\omega ^p({\mathbb{R}^n})}}\) for any p ∈ (1, 2 + ε) and ω ∈ A_p(ℝⁿ) ∩ \(R{H_{({{2 + \varepsilon } \over p})\prime }}({\mathbb{R}^n})\) (the class of reverse Hölder weights), where ε ∈ (0, ∞) is a constant depending only on n and the operator L, and where \(({{2 + \varepsilon } \over p})\prime \) denotes the Hölder conjugate exponent of \({{2 + \varepsilon } \over p}\). Moreover, for any given q ∈ (2, ∞), we give a sufficient condition to obtain that \(||\nabla f|{|_{L_\omega ^p({\mathbb{R}^n};{\mathbb{R}^n})}} \le C||{L^{1/2}}(f)|{|_{L_\omega ^p({\mathbb{R}^n})}}\) for any p ∈ (1, q) and \(\omega \in {A_p}({\mathbb{R}^n}) \cap R{H_{({q \over p})\prime }}({\mathbb{R}^n})\). As an application, we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition, the Riesz transform ∇L^−1/2 is bounded on L ^p_ω (ℝⁿ) for any given p ∈ (1, ∞) and ω ∈ A_p(ℝⁿ). Furthermore, applications to the weighted L²-regularity problem with the Dirichlet or the Neumann boundary condition are also given.