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Gradient estimates for heat kernels and harmonic functions. (English) Zbl 1439.53041
Authors’ abstract: Let \((X, d, \mu)\) be a doubling metric measure space endowed with a Dirichlet form \(\mathcal{E}\) deriving from a “carré du champ”. Assume that \((X, d, \mu, \mathcal{E})\) supports a scale-invariant \(L^2\)-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for \(p \in(2, \infty]\):
\(( G_p)\): \( L^p\)-estimate for the gradient of the associated heat semigroup;
\((R H_p)\): \( L^p\)-reverse Hölder inequality for the gradients of harmonic functions;
\(( R_p)\): \( L^p\)-boundedness of the Riesz transform \((p < \infty )\);
\((G B E)\): a generalised Bakry-Émery 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 \(L^2\)-Poincaré 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 P. Auscher et al. [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 911–957 (2004; Zbl 1086.58013)] and P. Auscher and T. Coulhon [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 3, 531–555 (2005; Zbl 1116.58023)]. 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.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
58J05 Elliptic equations on manifolds, general theory
58J35 Heat and other parabolic equation methods for PDEs on manifolds
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31E05 Potential theory on fractals and metric spaces
35K08 Heat kernel
31C25 Dirichlet forms
43A85 Harmonic analysis on homogeneous spaces
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