Bounds of riesz transforms on L^p spaces for second order elliptic operators

Let ℒ = -div (A(x) ∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ℝⁿ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(ℒ)^-1/2 on the L^p space. As an application, for 1 < p < 3 +ε, we establish the L ^p boundedness of Riesz transforms on Lipschitz domains for operators with VMO coefficients. The range of p is sharp. The closely related boundedness of ∇(ℒ)^-1/2 on weighted L² spaces is also studied.