Super Poincar\'e inequalities, Orlicz norms and essential spectrum

TL;DR

This paper explores the relationship between super Poincaré inequalities, Orlicz norms, and the spectrum of operators, providing new proofs and applications to Gaussian measures and spectral bounds.

Contribution

It introduces an alternative formulation of SPI using Orlicz norms and links spectral bounds to SPI, offering new proofs and insights.

Findings

SPI can be expressed with Orlicz norms instead of L1 norms

A bound on the essential spectrum implies a SPI

Spectral proof of the log Sobolev inequality for Gaussian measure

Abstract

We prove some results about the super Poincar\'e inequality (SPI) and its relation to the spectrum of an operator: we show that it can be alternatively written with Orlicz norms instead of L1 norms, and we use this to give an alternative proof that a bound on the bottom of the essential spectrum implies a SPI. Finally, we apply these ideas to give a spectral proof of the log Sobolev inequality for the Gaussian measure.