EstimationsLpdes opérateurs de Schrödinger sur les groupes nilpotents

RésuméDans cet article, on se propose d'étudier la continuité dans les espacesLpdes groupes nilpotents des opérateurs: ∇2(−Δ+W)−1,W1/2nabla;(−Δ+W)−1et ∇(−Δ+W)−1/2oùΔest un sous-Laplacien etWun potentiel qui vérifie d'assez bonnes conditions

Weak type (1,1)(1, 1) of Riesz transform on some direct product manifolds with exponential volume growth

In this paper we are concerned with the Riesz transform on the direct product manifold ${\mathbb{H}}^n \times M$, where ${\mathbb{H}}^n$ is the $n$-dimensional real hyperbolic space and $M$ is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type $(1,1)$ property. In addition, we obtain the weak type $(1, 1)$ of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type $(1,1)$ boundedness of some operators considered in the work of Li, Sj\"ogren and Wu [27].Comment: This paper has already been accepted by Science China Mathematic

Estimates for operators related to the sub-Laplacian with drift in Heisenberg groups

In the Heisenberg group of dimension 2n+1, we consider the sub-Laplacian witha drift in the horizontal coordinates. There is a related measure for whichthis operator is symmetric.The corresponding Riesz transforms are known to be L^p boundedwith respect to this measure.We prove that the Riesz transforms of order 1 are also of weak type (1,1),and that this is false for order 3 and above. Further, we consider the relatedmaximal Littlewood-Paley-Stein operators and prove the weak type (1,1) forthose of order 1 and disprove it for higher orders

Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space.

Let $v \ne 0$ be a vector in $\R^n$. Consider the Laplacian on $\R^n$ with drift $\Delta_{v} = \Delta + 2v\cdot \nabla$ and the measure $d\mu(x) = e^{2 \langle v, x \rangle} dx$, with respect to which $\Delta_{v}$ is self-adjoint. %Let $d$ and $\nabla$ denote the Euclidean distance and the gradient operator on $\R^n$. Consider the space $(\R^n, d,d\mu)$, which has the property of exponential volume growth. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood-Paley-Stein functions associated with the heat and the Poisson semigroups