Uniform weak type $(1,1)$ bounds for Riesz transforms on stratified Lie groups

Let $G$ be a stratified Lie group and $\mathcal{L}$ its sub-Laplacian. We prove that the full horizontal Riesz transform ${\mathrm{\nabla }}_H{\mathcal{L}}^{-1\mathrm{/}2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $G$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068 [math.CA], with the same universal constant. Our proof constitutes a fractional obstacle problem formulated through the heat semigroup and the associated Dirichlet form, adapted to the stratified structure. This extends the method of Ouyang, Spector, and Stockdale arXiv:2608.18068 [math.CA] without using the Fourier analytic ingredients.