Sharp Li–Yau-Type Gradient Estimates on Hyperbolic Spaces

Abstract

In this paper, motivated by the works of Bakry et. al in finding sharp Li-Yau type gradient estimate for positive solutions of the heat equation on complete Riemannian manifolds with nonzero Ricci curvature lower bound, we first introduce a general form of Li-Yau type gradient estimate and show that the validity of such an estimate for any positive solutions of the heat equation reduces to the validity of the estimate for the heat kernel of the Riemannian manifold. Then, a sharp Li-Yau type gradient estimate on the three dimensional hyperbolic space is obtained by using the explicit expression of the heat kernel and some optimal Li-Yau type gradient estimates on general hyperbolic spaces are obtained.2010 Mathematics Subject Classification. Primary 35K05; Secondary 53C44.

Li--Yau Inequalities for Dunkl Heat Equations

“…Recently, in [27], sharp Li-Yau inequalities for the Laplace-Beltrami operator on hyperbolic spaces were obtained by employing the explicit formula for the corresponding heat kernel. Very recently, in [26], similar to the idea of [27], the Li-Yau inequality in the sense of (1.1) for the fractional Laplacian has been proved; however, the Li-Yau inequality of gradient type in the sense of (1.2) has not been mentioned, where the "gradient" should be understood as the the carré du champ operator induced by the fractional Laplacian (see also the conjectures at the end of [13,Section 21]). We should mention that there are works on the Li-Yau inequality in the setting of graphs via various curvature-dimension conditions in the sense of Barky-Emery [3]; see e.g.…”

“…Motivated by [27] and [26], in the present paper, we mainly study Li-Yau inequalities for the generalized heat equation and the heat kernel corresponding to the Dunkl Laplacian, which is a non-local operator parameterized by reflection groups and multiplicity functions.…”

Li-Yau gradient estimates on closed manifolds under bakry-emery ricci curvature conditions

“…Moreover, Li-Yau-type bounds were also got for weighted manifolds with Bakry-Émery Ricci curvature bounded below [12]. More information about Li-Yau-type bounds can be found in ( [8], [10], [16], [18], [27], [25], [24], [26]).…”

Revisiting Li-Yau type inequalities on Riemannian manifolds

“…on R n , in this case the inequality (1.1) is equality for K = 0, α = 1. There are a series work on improving this inequality for negative curvature and for small time and large time, see [7,13,8,5,1,17] and the references therein. We briefly recall them as follows:…”

“…See also [5] and [6] in this direction. The author himself in [11] provides a general form of gradient estimate which generalizes (1.5) and (1.6), see also [5,17].…”