A Poincar\'e Inequality on Loop Spaces

TL;DR

This paper establishes a Poincaré inequality on loop spaces by connecting modified Logarithmic Sobolev inequalities with weak Poincaré inequalities, providing new insights into measures in infinite-dimensional spaces.

Contribution

It demonstrates that Aida's weighted Logarithmic Sobolev inequality implies weak inequalities, leading to a Poincaré inequality on loop spaces over certain manifolds.

Findings

Weak Logarithmic Sobolev Inequalities imply weak Poincaré inequalities.

Order of convergence relates to weak L^1 estimates on the weight.

Established a Poincaré inequality on loop spaces over specific manifolds.

Abstract

We investigate properties of measures in infinite dimensional spaces in terms of Poincar\'e inequalities. A Poincar\'e inequality states that the $L^2$ variance of an admissible function is controlled by the homogeneous $H^1$ norm. In the case of Loop spaces, it was observed by L. Gross that the homogeneous $H^1$ norm alone may not control the $L^2$ norm and a potential term involving the end value of the Brownian bridge is introduced. Aida, on the other hand, introduced a weight on the Dirichlet form. We show that Aida's modified Logarithmic Sobolev inequality implies weak Logarithmic Sobolev Inequalities and weak Poincar\'e inequalities with precise estimates on the order of convergence. The order of convergence in the weak Sobolev inequalities are related to weak $L^1$ estimates on the weight function. This and a relation between Logarithmic Sobolev inequalities and weak Poincar\'e inequalities lead to a Poincar\'e inequality on the loop space over certain manifolds.