Result 078, Real and complex analysis

The three-dimensional Bochner–Riesz conjecture

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on Lp(R3)L^p(\mathbb R^3) for every 1≤p≤∞1\le p\le\infty whenever δ>max⁡{3∣1/p−1/2∣−1/2,0}\delta\gt \max\{3|1/p-1/2|-1/2,0\}.

Proof

The bigger picture

Why it matters

Breaking a function into frequencies is useful only if the chosen frequency cutoff does not distort its size uncontrollably. This manuscript reports the conjectured stability bounds for a softened frequency cutoff in three-dimensional space.

What changes?

Bochner-Riesz multipliers retain frequencies inside a ball, weighting them down toward its boundary; the order delta controls that weighting. The reported bound holds on L^p(R^3), spaces that measure function size, for every p from 1 to infinity whenever delta > max{3|1/p - 1/2| - 1/2, 0}. Boundedness means the output size is at most a fixed multiple of the input size. The boundary orders are not included.

What does that help mathematicians do?

The key claim is boundedness on L^3 for every positive order, so even arbitrarily small positive smoothing suffices at that exponent. The abstract says interpolation and duality extend this to the full stated strict range. Researchers can therefore use these frequency truncations in three dimensions without losing L^p control within that range. This does not settle the boundary orders or dimensions beyond three.

Are there practical applications?

Its immediate value is foundational: the claimed estimates identify when a particular Fourier cutoff preserves control of function size. Such control supports mathematical arguments that separate functions into frequency components. The supplied material reports no computational method or practical performance gain; the contribution is an analytic guarantee, not a demonstrated engineering application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Bochner–Riesz multipliers in three dimensions

September 24, 2026 114 pages

We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on L3(R3)L^3(\mathbb R^3) for every positive order. Interpolation and duality then give the full conjectured strict range.

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Cite (BibTeX) 
@misc{OAI:Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026,
  author = {{OpenAI}},
  title = {{Bochner--Riesz multipliers in three dimensions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf}{OAI:Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.