Stein’s dimension-free weak-(1,1) Riesz transform problem

The Riesz transforms $R_1,\ldots,R_n$ on $\mathbb{R}^n$ are the Fourier multipliers $-i\xi_j/|\xi|$ , the natural higher-dimensional Hilbert transforms. Stein proved in 1983 that their $L^p$ bounds can be taken independent of the dimension for every $1 < p < \infty$ . At the 1986 ICM he asked whether the same holds at the endpoint $p=1$ : is there an absolute constant $C$ , independent of $n$ , with $|\{x : |Rf(x)| > \lambda\}| \le \frac{C}{\lambda}\,\|f\|_{L^1(\mathbb{R}^n)}$ for every $\lambda > 0$ ? The Calderon-Zygmund route gives a constant that grows with the dimension, and the best known was Janakiraman's $c\log n$ .

This paper answers yes, with $C = 2$ , for the vector transform $R = (R_1,\ldots,R_n)$ - so the same constant serves every single component $R_j$ uniformly in $n$ .

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Argument
Field: Harmonic analysis
Posed by: Elias M. Stein
Year posed: 1986
Years open: 40y
Solved: 2026-08-18
Model: Claude Opus 5.0, GPT-5.6 Sol
Vendor: Anthropic, OpenAI
Collaborators: Yuyuan Ouyang, Daniel Spector, Cody B. Stockdale
Verification: Unreviewed
Publication: Preprint
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

The theorem is the vector-valued endpoint bound $\|Rf\|_{L^{1,\infty}} \le 2\|f\|_{L^1}$ for $R = (R_1,\ldots,R_n)$ , so the constant 2 also serves each component $R_j$ uniformly in the dimension; the best previously known component bound grew like $c\log n$ .

The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative $f \in L^1 \cap L^2$ and every $\lambda > 0$ , write $f = \mu + (-\Delta)^{\alpha/2}u$ with $\mu \le \lambda$ and $u$ in the fractional Sobolev space $H^\alpha$ , obtained from an obstacle problem for the fractional Laplacian together with a Lewy-Stampacchia type estimate on an unbounded domain. That replaces the Calderon-Zygmund decomposition, whose cube geometry is where the dimensional loss enters.

What the AI did

From the paper's "Artificial Intelligence Statement": "The proof strategy was developed by Large Language Models (LLMs), through a combination of ChatGPT (GPT-5.6 Sol), Codex CLI and web interface, mathematical reasoning agents, and Claude Opus 5.0, in dialogues with the authors."

The sequence is specific. The first author started from the second and third authors' 2020 paper and ran a coordinated attempt - a Sol agent with the Danus and Rethlas automated reasoning agents (both deployed on OpenAI Sol agents) and a Polya "How to Solve It" style agent - which reached only partial results. Further dialogue with ChatGPT Sol gave an attempted complete solution via variational inequalities on the torus and a transference principle, much more complicated than what was published. The second and third authors judged that a direct Euclidean proof should be possible; the second author prompted Claude Opus 5.0 to try it, and "the response was a longer document that provided the basis for the proof idea used in the present paper".

The authors then checked and rewrote the proofs, did the literature review, wrote the introduction, and state they "independently verified, validated, and rewritten all parts of the paper influenced by LLM-generated material" and take full responsibility for the mathematics.

Verification

A one-day-old arXiv preprint, unrefereed, with no independent endorsement on record, so this stays Unreviewed. Nothing was checked here either: the proof is twelve pages of obstacle-problem and fractional-Sobolev analysis, with no finite certificate to re-run.

The setting was checked and holds up. Stein's question is real and was open, and the trail is unusually clean: the second and third authors wrote "On the dimensional weak-type $(1,1)$ bound for Riesz transforms" (arXiv:2004.03382, Comm. Contemp. Math. 23, 2021), which reduces this exact question to finite sums of Dirac masses and records Janakiraman's $c\log n$ as the best known. This paper is a continuation of their own program, and the AI statement says the first author started from it. The claim is also unusually falsifiable for its kind: an absolute constant 2, not an asymptotic. It clears the known lower bound - in $n=1$ the transform is the Hilbert transform, whose weak-type $(1,1)$ norm is Davis's constant, about 1.347.

Against that: forty-year-old endpoint problems do not usually fall in twelve pages, and the strategy came from a model, so the argument has had less human incubation than its length suggests.

Sources

PaperA dimension-free weak-type (1,1) bound for the vector Riesz transform on R^nSpector and Stockdale (2020), the reduction to Dirac masses this paper grew out of

Submitted by VibeGene on 19 Aug 2026

Stein’s dimension-free weak-(1,1) Riesz transform problem

What was actually shown

What the AI did

Verification

Sources

Discussion