Gradient estimates for heat kernels and harmonic functions

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 scale-invariant 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 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 L 2 -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 [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.

Riesz transform under perturbations via heat kernel regularity

“…by [6,18], one further sees that the gradient estimates for heat kernels and harmonic functions are also stable under such metric perturbations. Coulhon-Dungey [16] has addressed the stability issue of Riesz transform under perturbations.…”

“…It was shown in [18] See [35] for the case R n , and [5,6,29] for earlier results and also further generalizations. Further, it was shown in [19] that, the local Riesz transform ∇(1 + L) −1/2 is bounded on L p (M, µ), p ∈ (2, ∞), if and only if, the above inequality (RH p ) holds for all balls B(x, r) with r < 1. By the perturbation result of Caffarelli and Peral [12], one has a good understanding of the local gradient estimates for elliptic equations on R n .…”

“…If we strengthen the assumption from (GU B) to two side bounds of the heat kernel, then by using the open-ended property of the Riesz transform (cf. [18]) we have the end-point estimate.…”

“…Zhang [38] had derived a sufficient condition on the perturbation of a manifold with nonnegative Ricci curvature for the stability of Yau's estimate (equivalent to (RH p ) with p = ∞, cf. [15,18,37]), which implies the boundedness of the Riesz transform for all p ∈ (1, ∞) by [18, Theorem 1.9]. We did not prove the stability of Yau's estimate, but the advantage of our result is that our condition (GD) is much more explicit and, it is convenient for applications.…”

“…The estimates for the heat kernel and its gradient (cf. [6,18]), together with (GD) are essential in our proofs.…”