Extremal of Log-Sobolev Functionals and Li-Yau Estimate on $\text{RCD}^*(K,N)$ Spaces

TL;DR

This paper investigates extremal functions of the log-Sobolev functional on RCD* spaces, establishing their existence, regularity, positivity, and deriving Li-Yau type estimates, Harnack inequalities, and bounds.

Contribution

It provides the first analysis of extremal functions on RCD* spaces, including existence, regularity, positivity, and Li-Yau estimates, advancing the understanding of functional inequalities in this setting.

Findings

Existence and positivity of extremal functions

Li-Yau type estimate for extremal functions

Harnack inequality and bounds for extremal functions

Abstract

In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the $\mathrm{RCD}^*(K,N)$ condition for $K$ in $\mathbb{R}$ and $N$ in $(2,\infty)$. We show the existence, regularity and positivity of non-negative extremal functions. Based on these results, we prove a Li-Yau type estimate for the logarithmic transform of any non-negative extremal functions of the log-Sobolev functional. As applications, we show a Harnack type inequality as well as lower and upper bounds for the non-negative extremal functions.