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 $dγ<em>n=π<sup>{-n/2}e<sup>{-|x|<sup>2}\,dx$, let $L=-\tfrac12Δ+x\cdot\nabla$, $D=2<sup>{-1/2}\nabla$, and $R</em>γ=DL<sup>{-1/2}Π_0$, where $Π<em>0$ removes the mean. We prove that $R</em>γ$ has weak-type constant at most $2$ on real-valued inputs, independently of $n$. The main construction is the unique unshifted decomposition $F=g+L<sup>{1/2}u$, for nonnegative $F\in L<sup>2(γ_n)$ and $a&gt;\int F\,dγ_n$, with $u\geq0$, $\min{F,a}\leq g\leq a$, preservation of mass, and $g=a$ on ${u&gt;0}$. 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 $L<sup>1$ extension with the principal-value transform and obtain the constant $8/π$ for complex-valued inputs.