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 (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

Poincaré inequality 3/2 on the Hamming cube

“…Inequality (8) was studied in different settings, for different measures and in different spaces as well. For possible references we refer the reader to [1,18,2,4,5,6,7,9,10,11,12,14,16,17,19].…”

Improving Beckner’s bound via Hermite functions

“…for any smooth bounded function f : R n → R. Later William Beckner [7] generalized (1) for any real power p, 1 ≤ p ≤ 2 as follows for any smooth bounded f : R n → (0, ∞). We caution the reader that in [7] inequality (2) was formulated in a slightly different but equivalent form (see Theorem 1,inequality (3) in [7]). It should be also mentioned that in case p = 2 inequality (2) does coincide with (1) for all f ≥ 0 but it does not imply the Poincaré inequality for the functions taking the negative values, especially when R n f dγ n = 0.…”

“…Later Beckner's inequality (2) was studied by many mathematicians for different measures, in different settings and for different spaces as well. For possible references we refer the reader to [1,2,4,5,6,8,9,10,11,26,21,20].…”

“…An analysis done in [18] indicates that the right hand side (RHS) of (2) can be improved. In the present paper we address this issue: what is the precise estimate of the difference given in the left hand side (LHS) of (2), and whether the requirement p ∈ [1,2] can be avoided by slightly changing the RHS of (2).…”

Functional inequalities for convolution probability measures

“…Of course, for functional inequalities having the sub-additivity property, it is possible to derive multiplicity-free estimates on the optimal constant, see e.g. the recent paper [15] for Beckner-type inequalities of convolution measures on the abstract Wiener space.…”