  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.
