Riesz transforms through reverse Hölder and Poincaré inequalities

We study the boundedness of Riesz transforms in L for p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on L for p in some interval [2, 2 + ε), and that L gradient estimates for the semigroup imply boundedness of the Riesz transform in L for q ∈ [2, p). This improves results of [7] and [6], where the stronger assumption of a Poincaré inequality and the assumption e−tL(1) = 1 were made. The Poincaré inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms.