AI-Assisted Proof Gives a Dimension-Free Weak-Type Bound for the Vector Riesz Transform

In a preprint first uploaded on August 18, 2026, Yuyuan Ouyang, Daniel Spector, and Cody B. Stockdale prove a dimension-free weak-type $(1,1)$ estimate for the full vector Riesz transform on $\mathbb{R}^n$. Their theorem gives the explicit bound

$$\|Rf\|_{L^{1,\infty}(\mathbb{R}^n)}\leq 2\|f\|_{L^1(\mathbb{R}^n)}$$

in every dimension, and therefore also gives the same uniform constant for each component Riesz transform. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians; the previously cited componentwise bound grew like $\log n$.

The proof replaces dimension-dependent Calderón–Zygmund estimates with a decomposition adapted to $R=\nabla(-\Delta)^{-1/2}$. It obtains the decomposition from a fractional-Laplacian obstacle problem and a Lewy–Stampacchia estimate on an unbounded domain, then combines a support estimate with the $L^2$ isometry of the vector transform.

The authors state that the proof strategy was developed through dialogue with ChatGPT using GPT-5.6 Sol, Codex, the Danus and Rethlas mathematical-reasoning agents, and Claude Opus 5.0. They report that the initial AI attempts were incomplete or more complicated, while a Claude response supplied the basis for the Euclidean-space proof; the authors then checked, rewrote, and situated the argument in the literature.