Sharp heat kernel bounds and entropy in metric measure spaces

Abstract

We establish sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying RCD(0, N) ( equivalently, RCD * (0, N)) condition with N ∈ N \ {1} and having maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in Riemannian manifolds and also in metric measure spaces obtained recently by the author with R. Jiang and H. Zhang in [21].

Weighted Littlewood–Paley inequalities for heat flows in RCD spaces

“…) and the same inequality holds if we interchange the roles of x and y. In addition, if the dimension N is required to be an integer no less than 2, then sharper heat kernel estimates can be established; see [32,Theorem 3.12]. Let Z = (Z t ) t≥0 , (P x ) x∈M \N be the µ-symmetric Hunt process corresponding to the Dirichlet form (D, W 1,2 (M )), where N is a properly exceptional set in the sense that µ(N ) = 0 and P x (Z t ∈ N for some t > 0) = 0 for all x ∈ M \ N .…”

On the Shannon entropy power on Riemannian manifolds and Ricci flow

“…The above argument can be extended to general case of weighted complete Riemnanian manifolds with the CD(0, m) and maximal volume growth conditions. Indeed, by similar argument as used for the proof of Proposition 8.1, and based on two-sides heat kernel estimates and the maximal volume growth property, H. Li [13] extended Proposition 8.1 to the so-called RCD(0, N ) metric measure spaces with maximum volume growth condition for N ∈ N with N ≥ 2. Thus, as it is well-known that all weighted complete Riemannian manifolds with the CD(0, m)-condition are RCD(0, N ) metric measure space with N = m ≥ 2, we can use the entropy dissipation formula in Theorem 3.1, the entropy power concavity inequality in Theorem 2.2 and the extended version of Proposition 8.1 on weighted complete Riemannian manifolds with CD(0, m) and maximal volume growth conditions to prove the following isoperimetric inequality for Shannon entropy power on weighted complete Riemannian manifolds.…”

Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces

“…This problem is considered first in [26] when the underlying space is compact. The noncompact case is discussed in [30] by following an argument in [13] on Riemannian manifolds. However, it seems that some technical details are not well described in the latter case.…”