Riesz transforms through reverse Hölder and Poincaré inequalities

We study the boundedness of Riesz transforms in L p 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 p for p in some interval [2, 2 + ), and that L p gradient estimates for the semigroup imply boundedness of the Riesz transform in Lq for q ∈ [2, p). This improves results of Auscher et al. (Ann Sci Ecole Norm Sup 37(4):911–957, 2004) and Auscher and Coulhon (Ann Scuola Norm Sup Pisa 4:531–555, 2005), where the stronger assumption of a Poincaré inequality and the assumption e−t L (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.