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

Abstract

We generalize the Beckner's type Poincaré 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 Poincaré 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. In addition, we prove that in the finite dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.

18 pages. arXiv admin note: text overlap with arXiv:1409.3447

References in corpus (1)

• A Hölder--Young--Lieb inequality for norms of Gaussian Wick products