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

Understand

We generalize the Beckner's type Poincar\'e inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $\mu\star\nu$, where $\mu$ is the reference Gaussian measure and $\nu$ is a probability measure satisfying a certain integrability condition.

• As the Beckner inequality interpolates between the Poincar\'e 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.
• Our dimension-independent results are compared with some very recent findings in the literature.