Higher-Order Riesz Transforms in the Inverse Gaussian Setting and UMD Banach Spaces

In this paper, we study higher-order Riesz transforms associated with the inverse Gaussian measure given by π n / 2 e x 2 d x on ℝ n . We establish L p ℝ n , e x 2 d x -boundedness properties and obtain representations as principal values singular integrals for the higher-order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.

Boundedness of the Maximal Function of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure and consequences

The main result of this paper is the proof of the boundedness of the Maximal Function T* of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd, on Gaussian variable Lebesgue spaces Lp(.) (γd); under a condition of regularity on p(.) following [5] and [8]. As an immediate consequence of that result, the Lp(.) (γd)-boundedness of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd is obtained. Another consequence of that result is the Lp(.) (γd)-boundedness of the Poisson-Hermite semigroup and the Lp(.) (γd)- boundedness of the Gaussian Bessel potentials of order β > 0.

Intermediate spaces, Gaussian probabilities and exponential tightness

We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS, thus extending what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result, it is very simple to deduce a result of exponential tightness for Gaussian probabilities.

Domains of elliptic operators on sets in Wiener space

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.