A note on the equivalence of super-Poincar\'e inequality

TL;DR

This paper investigates the equivalence between super-Poincaré inequalities and various log-Sobolev type inequalities, establishing explicit relations between their rate functions.

Contribution

It provides a detailed analysis of the equivalence and explicit relations between super-Poincaré and log-Sobolev inequalities.

Findings

Established the equivalence between super-Poincaré and certain log-Sobolev inequalities.

Derived explicit relations between their associated rate functions.

Abstract

In this paper we will study the equivalence between super-Poincar\'e inequality and some log-Sobolev type inequalities, including weak log-Sobolev inequality and super log-Sobolev inequality. The explicit relations between associated rate functions will also be established.