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

Abstract

Let $v\ne 0$ be a vector in ${\mathbb{R}}^n$. Consider the Laplacian on ${\mathbb{R}}^n$ with drift ${\Delta }_v=\Delta +2v\cdot \nabla$ and the measure $d\mu (x)=e^{2⟨v,x⟩}dx$, with respect to which ${\Delta }_v$ is self-adjoint. 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.