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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) (Gp): Lp-estimate for the gradient of the associated heat semigroup;
(ii) (RHp): Lp-reverse Hölder inequality for the gradients of harmonic functions;
(iii) (Rp): Lp-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  and Auscher-Coulhon . 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.
- Harmonic functions
- Heat kernels
- Li-Yau estimates
- Riesz transform