The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces

By Renjin Jiang. 2014

63 citations recorded by the source corpus.

Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in mathbb{R}$ and $N\in [1,\infty)$. Suppose that $(X,d)$ is connected, complete and separable, and $\supp \mu=X$. We prove that the Li-Yau inequality for the heat flow holds true on $(X,d,\mu)$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on $(X,d,\mu)$ for general $K\in \mathbb{R}$. Large time behaviors of heat kernels are also studied.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/14e16faa7576caecd3184474ea014eb412a35ffb

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, unknown): “This, combined with the Li–Yau inequality [24,32] gives a gradient estimate: |∇x p(x, y, t)| ≤ C3 tm(Bt1/2 (x)) exp ( − 2(x, y) (4+ )t + C4t ) (3.”
Heat Kernel Bounds on Metric Measure Spaces and Some Applications (2014, methodology): “, for all t >0 and all x,y ∈ X.This implies the desired estimate. Proof of Corollary 1.1. It follows from the Li–Yau inequality |∇log pt(x,·)|2(y) − ∂ ∂t log pt(x,y) ≤ N 2t , for µ-a.e.x,y ∈ X, in [18, 22] and Theorem 1.1 that, for each ǫ>0, |∇pt(x,·)|2(y) ≤ N 2t pt(x,y)2 + pt(x,y)|∆pt(x,·)(y)”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “The weak Bakry-Émery curvature condition is satisfied in the following examples: • Complete Riemannian manifolds with non-negative Ricci curvature and more generally, the RCD(0,+∞) spaces (see [50]); • Carnot groups (see [16]); • Complete sub-Riemannian manifolds with generalized non-negative Ricci ”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, background): “A further result that we shall need soon is the Li-Yau inequality in the form proved by Baudoin and Garofalo (see [22] for the case of finite mass and [35] for the general one).”
Gradient estimates for heat kernels and harmonic functions (2017, methodology): “To prove (RHp) =⇒ (GLYp) for p ∈ (2,∞), we follow some ideas from [73, 75].”