an:07161056
Zbl 1439.53041
Coulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam
Gradient estimates for heat kernels and harmonic functions
EN
J. Funct. Anal. 278, No. 8, Article ID 108398, 67 p. (2020).
0022-1236
2020
j
53C23 58J05 58J35 31C05 31E05 35K08 31C25 43A85
harmonic function; heat kernel; Li-Yau estimate; Riesz transform; metric measure space; Dirichlet form
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 \(p \in(2, \infty]\):\begin{itemize}\item \(( G_p)\): \( L^p\)-estimate for the gradient of the associated heat semigroup;\item \((R H_p)\): \( L^p\)-reverse Hölder inequality for the gradients of harmonic functions;\item \(( R_p)\): \( L^p\)-boundedness of the Riesz transform \((p < \infty )\);\item \((G B E)\): a generalised Bakry-Émery condition.\end{itemize} 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 \textit{P. Auscher} et al. [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 911--957 (2004; Zbl 1086.58013)] and \textit{P. Auscher} and \textit{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.
Eleutherius Symeonidis (Bucureşti)
1086.58013; 1116.58023