An extension of the Beckner's type Poincare inequality to convolution measures on abstract Wiener spaces

We generalize the Beckner's type Poincare inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397-400) to a large class of probability measures on an abstract Wiener space of the form , where is the reference Gaussian measure and is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincare and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.