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

Abstract

AbstractLet $v \ne 0$ 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 $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , 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.