Riesz transforms through reverse Hölder and Poincaré inequalities

Abstract

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, $$\omega $$ -accretive operator L satisfying Davies–Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Holder inequality, the Riesz transform is always bounded on $$L^p$$ for p in some interval $$[2,2+\epsilon )$$ , and that $$L^p$$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $$L^q$$ for $$q \in [2,p)$$ . 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 Poincare inequality and the assumption $$e^{-tL}(1)=1$$ were made. The Poincare inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms.