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

Authors

Liming Yin and Samuel Drapeau

Date

2023

Journal

Forthcoming in Potential Analysis

Abstract

In this work, we study the extremal functions of log-Sobolev functional in metric measure space satisfying $\text{RCD}^*(K,N)$ condition for $K>0$ 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 all non-negative extremal functions of log-Sobolev functional. As applications, we show a Harnack type inequality as well as lower and upper bounds for the non-negative extremal functions.

Keywords

Log-Sobolev functional, Metric measure space, Li-Yau inequality, Curvature-dimension condition, Extremal function

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