Browse › Real and complex analysis › 078 The three-dimensional Bochner–Riesz conjecture
Bochner–Riesz multipliers in three dimensions
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Abstract
We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on $L^3(\mathbb R^3)$ for every positive order. Interpolation and duality then give the full conjectured strict range.
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Possibly relevant literature (found by the agent, not checked)
- Jean Bourgain, Larry Guth, Bounds on oscillatory integral operators based on multilinear estimates (arXiv:1012.3760, Dec 2010; GAFA). An earlier multilinear approach to restriction and Hörmander-type oscillatory integral operators with positive-definite phase; the introduction's history of three-dimensional multiplier ranges (Lee, Wu, Gao–Wu–Xi) does not cite it.
- Larry Guth, Jonathan Hickman, Marina Iliopoulou, Sharp estimates for oscillatory integral operators via polynomial partitioning (arXiv:1710.10349, Oct 2017; Acta Math.). Gives the sharp $L^p$ range for general positive-definite Hörmander-type phases, relevant to the paper's oscillatory estimate (Sections 9 and 12), which is stated for the restricted phase class $\partial_v\Phi=\zeta(v,x-cv)$ of Section 9.
- Shengwen Gan, Yifan Jing, Shukun Wu, New bounds for Stein's square function in $\mathbb{R}^3$ (arXiv:2104.14731, Apr 2021). Earlier maximal Bochner–Riesz progress in $\mathbb R^3$ ($p\ge3.25$), relevant to the introduction's statement that the companion local-smoothing manuscript gives maximal Bochner–Riesz bounds for $3\le p<\infty$, $\delta>1-3/p$.
- Shengwen Gan, Shukun Wu, A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem (arXiv:2403.05017, Mar 2024). Prior maximal Bochner–Riesz result in $\mathbb R^3$ beyond the real interpolation exponents, relevant to the same maximal-operator statement in the introduction.
Digests and expositions
- Latent Seminar, Bochner–Riesz multipliers in three dimensions (Latent Seminar, 2026-10-06). Explains the claimed strict-order range, positive transport and entropy arguments, packet composition at the uncertainty boundary, and shell summation, while explicitly stating that it does not independently verify the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 078
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 078: 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 $L^p(\mathbb R^3)$ for every $1\le p\le\infty$ whenever $\delta\gt \max\{3|1/p-1/2|-1/2,0\}$.
This is the only paper in this family.