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

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...

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Bibliographic Details
Published in:arXiv.org 2014-09
Main Authors: Paolo Da Pelo, Lanconelli, Alberto, Stan, Aurel I
Format: Article
Language:English
Subjects:
Convolution
Covariance
Gaussian distribution
Inequality
Integral calculus
Metric space
Multiplication
Quadratic forms
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Summary: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.
ISSN:2331-8422