Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Sharp heat kernel bounds and entropy in metric measure spaces

“…The second application is on the large-time asymptotics of the heat kernel, which is a strengthening of the result obtained recently by the author with R. Jiang and H. Zhang in [21,Theorem 4.1]. Corollary 3.16.…”

“…From now on, let (X, d, µ) be an RCD(K, N) space with K ∈ R and N ∈ (1, ∞). Then the measure µ satisfies the local doubling (global doubling, provided K ≥ 0) property, which we present in the next lemma (see e.g., [40], [17,Section 5] or [21,Section 2]). Lemma 2.7.…”

“…Thus we first need to prove the monotonicity of the W functional with respect to the time variable. In a very recent manuscript [22], R. Jiang and H. Zhang proved the monotonicity in the case when the metric measure space (X,…”

“…Let (X, d, µ) be an RCD * (K, N) space with K ≤ 0 and N ∈ (1, ∞). In a recent joint work [21], by using the comparison result (see e.g. Lemma 3.3 below), the author with R. Jiang and H. Zhang established the following heat kernel upper and lower bounds of Gaussian type.…”

Second order differentiation formula on RCD$(K,N)$ spaces

“…. Since recently such asymptotic behavior for the heat kernel on RCD * (K, N ) spaces has been proved by Jiang-Li-Zhang in [30], Léonard's result applies. Thus in this remark we simply wanted to show an alternative proof of such limiting property.…”