The Bochner–Riesz conjecture in three dimensions

The Bochner–Riesz conjecture asks for which orders δ the Bochner–Riesz means, defined by the Fourier multiplier (1 − |ξ|²)₊^δ, are bounded on L^p(Rⁿ). The conjectured range is δ > max{n|1/p − 1/2| − 1/2, 0}. In three dimensions the critical exponent is p = 3, where every positive order is conjectured to suffice. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims the conjecture in three dimensions in its strict-order formulation. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

Background

• As recounted in the introduction of the manuscript:
  • Riesz's one-dimensional summation methods preceded Bochner's spherical means for multiple Fourier series, and Herz and Welland studied radial inversion and summability. Using a Kakeya construction, Fefferman proved that in dimension at least two the order-zero ball multiplier is unbounded on L^p for p ≠ 2. Carleson and Sjölin established the conjectured range in the plane.
  • Córdoba connected the problem with Kakeya maximal functions, and Bourgain obtained improved restriction and multiplier bounds in three dimensions. The bilinear restriction theorems of Wolff and Tao led, through Lee's multiplier reduction, to sharp-order estimates for max{p, p′} ≥ 10/3. Wu obtained the strict-order range for max{p, p′} ≥ 13/4 (arXiv:2008.13043). Guo, Oh, Wang, Wu and Zhang developed a pseudoconformal approach for transferring restriction methods to the multiplier (arXiv:2104.11188). Gao, Wu and Xi established the strict-order range max{p, p′} ≥ 22/7 (arXiv:2509.01116, Corollary 1.3).
  • Tao showed that the Bochner–Riesz conjecture implies the restriction conjecture for the sphere.
  • Zipeng Wang has proposed a proof of the Bochner–Riesz conjecture in all dimensions (arXiv:2501.12742, version 6 of 9 June 2026, still the current version on 26 October 2026).

The original result

• Bochner–Riesz multipliers in three dimensions (24 September 2026). Theorem 1.1 of the manuscript states that for every δ > 0 the multiplier (1 − |ξ|²)₊^δ is bounded on L³(R³). By interpolation and duality its corollary gives boundedness on L^p(R³), 1 ≤ p ≤ ∞, for δ > max{3|1/p − 1/2| − 1/2, 0}, together with bounds for complex orders. Through Tao's implication this gives the spherical extension estimate ‖Eg‖_L^p(R³) ≤ C_p‖g‖_L^∞(S²) for p > 3.
  • Method, as described: the main estimate is an L³ oscillatory-integral bound with arbitrarily small power loss, stated uniformly over a class of phase functions, and the Bochner–Riesz kernel is brought into this class by the pseudoconformal change of variables of Guo, Oh, Wang, Wu and Zhang. The bound comes from entropy growth estimates, first for positive masses moving along lines and then for oscillatory wave packets. The single geometric input is the sharp planar Furstenberg estimate of Ren and Wang (arXiv:2308.08819).
  • The manuscript states that the collection's companion manuscripts on spherical restriction and on local smoothing are not inputs to its argument.
• Related results in the collection. The local-smoothing manuscript claims the same strict range by an independent route, uniformly in the radius, together with maximal Bochner–Riesz bounds and almost-everywhere convergence for 3 ≤ p < ∞ and δ > 1 − 3/p; see Sogge's local smoothing conjecture in three dimensions. Spherical restriction for p > 3 is also claimed directly; see Fourier extension for positively curved surfaces in three dimensions.
• Source. Commit 1581d4c of the openai/math repository, folder Bochner–Riesz multipliers in three dimensions.
• Formalization. The collection lists no Lean formalization for this result.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 26 October 2026)
2026-10-13	MColbrook/AIM, problem 209 (GitHub; AI for Mathematical Problems)	Catalogue	Lists Sharp Bochner–Riesz summability in three dimensions as solved by the manuscript. Its status review states that the primary L3 estimate and the interpolation corollary give exactly the problem's multiplier and full strict range, after an AI statement-level comparison of the theorem, hypotheses and reductions, and that no corresponding complete Lean development was identified.	Open-problem collection entry, not affiliated with the American Institute of Mathematics; the review describes itself as not an independent audit of the complete proof. Status unchanged at repository commit 8729eab (22 October 2026).
2026-10-16	J. Jardine, Lattice24, referee review rank_244 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reports no gap found. The reviewer states that it checked Section 12 in full, including the kernel expansion, the shell rescaling, the pseudoconformal change of variables, the global convexification and the deduction of the multiplier bound, and that it only spot-checked the growth theorems of Sections 4–11. Its one minor note is a coverage caveat, not an identified error: a dependency of the boundary alternative in Section 10 on the alignment lemma was not traced end to end.	Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.

Notes

• The earlier partial results are due to the authors named in the Background section, as recounted by the manuscript.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.