Dimension-free weak-type bound for the vector Riesz transform
The best constant in the weak-type (1,1) bound for the vector Riesz transform on n-dimensional space is at most 2 whatever the dimension, answering a question Stein raised in 1986.
- Lab
- Independent
- Model
- GPT-5.6 Sol and Claude Opus 5.0, with Danus and Rethlas agents
- Field
- Mathematics
- Date
- 2026-08-18
- Human collaborators
- Yuyuan Ouyang, Daniel Spector, Cody B. Stockdale
- Problem posed
- 1986 · open 40 yrs
What was found
Dimension-free bounds for Riesz transforms are known in the strong-type range, and Stein asked at the 1986 ICM whether the weak-type (1,1) endpoint behaves the same way. The paper answers yes with an explicit constant: the best weak-type (1,1) constant for the vector Riesz transform is at most 2, independent of n. The proof goes through a new decomposition of the input data rather than through the torus-transference route the models first proposed, which the authors judged more complicated than necessary and replaced with a direct argument in Euclidean space.
Novelty check
Stein posed the question at the 1986 ICM and the paper cites it directly, along with the authors' own 2021 Communications in Contemporary Mathematics paper on the dimensional weak-type (1,1) bound, which fixes the prior state of the art as dimensional rather than dimension-free. Dimension-free strong-type bounds are long established and are not what is claimed here. No prior dimension-free weak-type constant appears in the literature. The result is new.
Caveats and known objections
A preprint, unrefereed and not formalized, with no independent check recorded. Autonomy graded collaborative rather than ai-led, and the disclosure is detailed enough to justify the weaker reading: the statement opens by saying the proof strategy was developed by large language models in dialogue with the authors, but the same paragraph records that the first attempt failed and produced only partial results, that a second attempt yielded a solution the authors judged significantly more complicated than what was published, that the human authors then assessed feasibility and redirected the work into Euclidean space, and that they checked and rewrote the proofs where the presentation was unnatural, performed the literature review, and take full responsibility. Humans chose the route that was kept. The weaker defensible reading applies.
Also recorded at
vibemathedstein-s-dimension-free-weak-1-1-riesz-transform-problem
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and Collaborative.
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Graded author verified for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Dimension-free weak-type bound for the vector Riesz transform. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-18-stein-riesz-weak-typeBibTeX
@misc{whataifound-independent-2026-type,
title = {Dimension-free weak-type bound for the vector Riesz transform},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-08-18-stein-riesz-weak-type}
}