  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, while
for $p=\infty$, (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 $p=\infty$, while for $p\in (2,\infty)$ 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.
