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

让 $v \ne 0$ 成为 ${\mathbb {R}}^n$ 中 的向量 。考虑 ${\mathbb {R}}^n$ 上的拉普拉斯算子 ，漂移 $\Delta _{v} = \Delta + 2v\cdot \nabla $ 和度量 $d\mu (x) = e^{2 \ langle v, x \rangle } dx$ ，关于 $\Delta _{v}$ 是自伴随的。该度量相对于欧几里得距离呈指数增长。我们研究了任何阶 Riesz 变换的弱类型 $(1, 1)$ 和其他尖锐端点估计，以及与热和泊松半群相关的垂直和水平 Littlewood-Paley-Stein 函数。