Heat Kernel Bounds on Metric Measure Spaces and Some Applications (original) (raw)

The Li–Yau inequality and heat kernels on metric measure spaces

Journal de Mathématiques Pures et Appliquées, 2014

Let (X, d, µ) be a RCD * (K, N) space with K ∈ R and N ∈ [1, ∞). Suppose that (X, d) is connected, complete and separable, and supp µ = X. We prove that the Li-Yau inequality for the heat flow holds true on (X, d, µ) when K ≥ 0. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on (X, d, µ) for general K ∈ R. Large time behaviors of heat kernels are also studied.

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

Advances in Mathematics, 2015

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies-Gaffney estimates.

2 O ct 2 01 7 Gradient estimates for heat kernels and harmonic functions

2018

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-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2,∞]: (i) (Gp): L -estimate for the gradient of the associated heat semigroup; (ii) (RHp): L -reverse Hölder inequality for the gradients of harmonic functions; (iii) (Rp): L -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-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-...

Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Bulletin des Sciences Mathématiques, 2006

Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimensionfree Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by −c(1 + ρ 2 o ), where c > 0 is a constant and ρ o is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li-Yau type heat kernel bound is presented for such semigroups.

Sharp explicit lower bounds of heat kernels

The Annals of Probability, 1997

By using logarithmic transformations, an explicit lower bound estimate of heat kernels is obtained for diffusion processes on Riemannian manifolds. This estimate is sharp for both short and long times, especially for heat kernels on a compact manifold, and is extended to manifolds with unbounded curvature.

Gradient estimates for heat kernels and harmonic functions

Journal of Functional Analysis, 2019

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

Journal de Mathématiques Pures et Appliquées, 2019

Let M be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of L pboundedness of the Riesz transform, p ∈ (2, ∞). We also provide counterexamples regarding instability for L p-boundedness of Riesz transform. 0 and ∇L −1/2 are isometries on L 2 (M, µ 0) and L 2 (M, µ), respectively. For the case p ∈ (1, 2), it was shown by Coulhon and Duong [17] that the Riesz operator is weakly L 1-bounded under a doubling condition and a Gaussian upper bound for the heat kernel. The L p-boundedness then follows from an interpolating, for all p ∈ (1, 2).

Riesz transform on manifolds and heat kernel regularity

Annales Scientifiques de l’École Normale Supérieure, 2004

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.