Stein’s vector Riesz transform weak-type problem

Let $\mathcal{R}f=(R_1f,\ldots,R_nf)$ be the vector Riesz transform on $\mathbb{R}^n$ , where $R_j$ is the Fourier multiplier with symbol $-i\xi_j/|\xi|$ . Does there exist a universal constant $C<\infty$ , independent of $n$ , such that for every $n\ge 1$ , every $f\in L^1(\mathbb{R}^n)$ , and every $\lambda>0$ , $\bigl|\{x\in\mathbb{R}^n:|\mathcal{R}f(x)|>\lambda\}\bigr|\le \frac{C}{\lambda}\|f\|_{L^1(\mathbb{R}^n)}$ ?

References

Primary source

arXiv

Additional references

A dimension-free weak-type (1,1) bound for the vector Riesz transform on R^n — arXiv — Pierre Bénilan, D. Lamberton

harmonic-analysis

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 preprint claims a dimension-independent weak-type bound, but the proof has not been independently verified.

Elias Stein posed the question at the 1986 International Congress of Mathematicians. It asks whether the vector Riesz transform has a weak-type bound with one constant in every dimension.

Known results

Janakiraman obtained componentwise weak-type estimates with logarithmic dimension dependence.
Spector–Stockdale (2020) reduced relevant estimates to finite positive atomic measures, but did not resolve the dimension-free question.

August 18, 2026 claimed solution

Ouyang, Spector, and Stockdale’s preprint claims $\|\mathcal{R}f\|_{L^{1,\infty}}\le 2\|f\|_{L^1}$ uniformly in the dimension, using a fractional-Laplacian obstacle decomposition and a Lewy–Stampacchia estimate. This would settle Stein’s problem with $C=2$ , but the claim remains unverified. A September 3, 2026 submitted proof expansion develops the proposed argument while explicitly acknowledging that a theorem-dependent decomposition is not established.

Community submission (unverified)

A September 3, 2026 submission expands atomic calculations and the proposed fractional-obstacle strategy, but does not establish the missing decomposition or independently verify the claimed theorem.

Current status (as of September 2026): The preprint claims the bound with $C=2$ , but independent verification is absent, so the problem remains formally open.

Sources

ProofSTEIN'S DIMENSION-FREE WEAK-(1,1) RIESZ TRANSFORM PROGRAM Fully Expanded Tutor-Exam Handnotes and Line-by-Line Analytical Derivation Based strictly on the two uploaded prototype reports 20 August 2026See full solutionHide full solution

This document expands the mathematical steps in both uploaded reports into a tutor/exam-handnotes format. Every displayed identity is followed by the algebraic operation that produces it, the geometric meaning, and the role it plays in the endpoint argument. Important research-status warning: the uploaded reports themselves distinguish elementary calculations from the proposed fractional-obstacle proof architecture. Therefore this expansion does NOT silently turn the prototype into a completed theorem. Wherever a step is only proposed or theorem-dependent, it is explicitly marked as such.

Stein_Riesz_Tutor_Exam_Handnotes_Full_Analytical_Expansion-21-8-2026.pdf409,390 bytesOpen
Stein_Riesz_Transform_Prototype_Research_Report_Refined-21-8-2026.pdf370,871 bytesOpen