Gradient estimates for heat kernels and harmonic functions

Let (X,d,μ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a “carré du champ”. Assume that (X,d,μ,E) supports a scale-invariant L²-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p∈(2,∞]: (i) (G_p): L^p-estimate for the gradient of the associated heat semigroup; (ii) (RH_p): L^p-reverse Hölder inequality for the gradients of harmonic functions; (iii) (R_p): L^p-boundedness of the Riesz transform (p<∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p∈(2,∞), (i), (ii) (iii) are equivalent, while for p=∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L²-Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p=∞, while for p∈(2,∞) 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.