E. Harboure

Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup

We find necessary and sufficient conditions on a Banach spaceX in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak type (1, 1) or strong type (p, p) in the range 1<p<∞. In this setting, we consider the Riesz transforms and the Littlewood-Paleyg-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure.In all cases, we show that the condition onX is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property. In doing so, we also find some new equivalences even for the harmonic case.

Oscillation and variation for the Gaussian Riesz transforms and Poisson integral

For the family of truncations of the Gaussian Riesz transforms and Poisson integral we study their rate of convergence through the oscillation and variation operators. More precisely, we search for their L p (d γ )-boundedness properties, where d γ denotes the Gauss measure. We achieve our results by looking at the oscillation and variation operators from a vector-valued point of view.

On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

Abstract. In this paper, for each given

Commutators of Riesz Transforms Related to Schrödinger Operators

p, 1 < p < \infty,

Extrapolation Results for Classes of Weights

we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in

Riesz transforms related to Schrödinger operators acting on BMO type spaces

L^p(vd\gamma)

Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from

Classes of weights related to Schrödinger operators

L^p(vd\gamma)

Weighted inequalities for negative powers of Schrödinger operators

into