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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 respectto which $\Delta_{v}$ is self-adjoint. This measure has exponential growth withrespect to the Euclidean distance. We study weak type $(1, 1)$ and other sharpendpoint estimates for the Riesz transforms of any order, and also for thevertical and horizontal Littlewood-Paley-Stein functions associated with theheat and the Poisson semigroups.

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