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Abstract
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 scaleinvariant L^{2}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 BakryÉmery 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 L^{2}Poincaré inequality.
Our result gives a characterisation of LiYau'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 AuscherCoulhonDuongHofmann [7] and AuscherCoulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and subRiemannian manifolds as well as to nonsmooth spaces, and to degenerate elliptic/parabolic equations in these settings.
Original language  English 

Article number  108398 
Pages (fromto)  167 
Number of pages  67 
Journal  Journal of Functional Analysis 
Volume  278 
Issue number  8 
DOIs  
Publication status  Published  1 May 2020 
Keywords
 Harmonic functions
 Heat kernels
 LiYau estimates
 Riesz transform
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Projects
 1 Active

Heat kernel and Riesz transform on noncompact metric measure spaces
Sikora, A. & Coulhon, T.
1/02/13 → …
Project: Research