Riesz transform and and harmonic functions

Abstract

Let $\left(X,d,\mu \right)$ be a doubling metric measure space endowed with a Dirichlet form satisfying a scale-invariant $L^2$-Poincaré inequality. We show that, for $p\in \left(2,\infty \right)$, the following conditions are equivalent:

(i) $\left(G_p\right)$: $L^p$-estimate for the gradient of the associated heat semigroup;

(ii) $\left(R{H}_p\right)$: $L^p$-reverse Hölder inequality for the gradients of harmonic functions;

(iii) $\left(R_p\right)$: $L^p$-boundedness of the Riesz transform ($p<\infty$).

This is joint work with Thierry Coulhon, Renjin Jiang and Pekka Koskela.