A direct obstacle proof of a dimension-free weak-type $(1,1)$ bound for the Gaussian Riesz transforms

Abstract

We give a detailed, direct proof of a dimension-free weak-type $(1,1)$ estimate for the first-order Gaussian vector Riesz transform, using the fractional obstacle method of Ouyang, Spector, and Stockdale. For $dγ_n=π^{-n/2}e^{-|x|^2}\,dx$, let $L=-\tfrac12Δ+x\cdot\nabla$, $D=2^{-1/2}\nabla$, and $R_γ=DL^{-1/2}Π_0$, where $Π_0$ removes the mean. We prove that $R_γ$ has weak-type constant at most $2$ on real-valued inputs, independently of $n$. The main construction is the unique unshifted decomposition $F=g+L^{1/2}u$, for nonnegative $F\in L^2(γ_n)$ and $a>\int F\,dγ_n$, with $u\geq0$, $\min\{F,a\}\leq g\leq a$, preservation of mass, and $g=a$ on $\{u>0\}$. We retain the details of coercivity, the Lewy--Stampacchia estimate, the upgrade from the form domain to the operator domain, and locality. We also identify the $L^1$ extension with the principal-value transform and obtain the constant $8/π$ for complex-valued inputs.