Beatriz Viviani

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.

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

Abstract. In this paper, for each given

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

p, 1 < p < \infty,

Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

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

A Look at BMOϕ(ω) through Carleson Measures

L^p(vd\gamma)

A sharp weighted transplantation theorem for Laguerre function expansions

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

Power weighted Lp-inequalities for Laguerre–Riesz transforms

L^p(vd\gamma)

A Look at BMO ? (?) through Carleson Measures

into

Global W 2;p estimates for nondivergence elliptic operators with potentials satisfying a reverse Holder condition

L^p(ud\gamma).