Dimension-free weak $(1,1)$ estimates for Riesz transforms with constant drift

Abstract

Let $L_v=-\Delta -2v\cdot \nabla$ on $L^2({\mathbb{R}}^n,e^{2v\cdot x}dx)$, where $v\in {\mathbb{R}}^n\setminus \{0\}$ is a constant drift vector. We prove that the full vector Riesz transform $\nabla L_v^{-1/2}$ satisfies a weak-type $(1,1)$ estimate for real-valued functions with constant at most $2$, uniformly in both the dimension and the drift vector. Our proof extends the fractional obstacle method of Ouyang, Spector and Stockdale (https://arxiv.org/abs/2608.18068) to the exponentially weighted setting. Interpolation then yields strong $L^p$ bounds for $1<p\le 2$, with constants depending only on $p$.

arXiv comment19 pages