An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: Da Pelo, Paolo; Lanconelli, Alberto; Stan, Aurel I.
• Is Part Of: Stochastic analysis and applications. Volume 34:Issue 1(2016)
• Page Start: 47
• Page End: 64
• Publication Date: 2016-01-02
• Subjects: Beckner's type Poincaré inequality -- Ornstein-Uhlenbeck semigroup -- Wick product -- convolution measures; 60H07 -- 60H30; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2015.1099443
• Languages: English
• ISSNs: 0736-2994

Abstract

ABSTRACT: We generalize the Beckner's type Poincaré 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 μ⋆ν, where μ is the reference Gaussian measure and ν 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. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.