Gradient estimates for heat kernels and harmonic functions

Let $(X,d,\mu )$ be a doubling metric measure space endowed with a Dirichlet form $\text{E}$ deriving from a “carr'e du champ”. Assume that $(X,d,\mu ,\text{E})$ 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 $p\in (2,\mathrm{\infty }]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(R{H}_p)$: $L^p$-reverse H"older inequality for the gradients of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\mathrm{\infty }$); (iv) $(GBE)$: a generalised Bakry-'Emery condition. We show that, for $p\in (2,\mathrm{\infty })$, (i), (ii) (iii) are equivalent, while for $p=\mathrm{\infty }$, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the $L^2$-Poincar'e inequality. Our result gives a characterisation of Li-Yau’s gradient estimate of heat kernels for $p=\mathrm{\infty }$, while for $p\in (2,\mathrm{\infty })$ it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.