A Poincaré Inequality on Loop Spaces

We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L2 variance of an admissible function is controlled by the homogeneous H1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homogeneous H1 norm alone may not control the L2 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 Poincaré inequalities with precise estimates on the order of convergence. The order of convergence in the weak Sobolev inequalities are related to weak L1 estimates on the weight function. This and a relation between Logarithmic Sobolev inequalities and weak Poincaré inequalities lead to a Poincaré inequality on the loop space over certain manifolds.