Gradient estimates for heat kernels and harmonic functions

Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carr\'e du champ". Assume that $(X,d,\mu,\E)$ supports a scale-invariant $L^2$-Poincar\'e inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse H\"older inequality for the gradients of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalised Bakry-\'Emery condition. We show that, for $p\in (2,\infty)$, (i), (ii) (iii) are equivalent, wh…