Sharp Bochner–Riesz summability in three dimensions
For $\delta > 0$ and a Schwartz function $f$ on $\mathbb{R}^3$, define $B^\delta f$ through its Fourier transform by $$\widehat{B^\delta f}(\xi) = (1 - |\xi|^2)_+^\delta\, \widehat{f}(\xi), \qquad r_+ = \max\{r, 0\}.$$ Fefferman proved that the multiplier with $\delta = 0$, the indicator of the ball, is unbounded on $L^p(\mathbb{R}^n)$ for $n \ge 2$ and $p \ne 2$. Carleson and Sjölin proved the two-dimensional analogue of the bound below, and in three dimensions Wu proved it whenever $\max\{p, p/(p-1)\} \ge 13/4$.
Problem. For every $1 < p < \infty$ and every $$\delta > \max\left\{ 3\left| \frac1p - \frac12 \right| - \frac12,\ 0 \right\},$$ does there exist $C_{p,\delta} < \infty$ such that $\|B^\delta f\|_{L^p} \le C_{p,\delta} \|f\|_{L^p}$ for all such $f$?
Source: The Bochner–Riesz conjecture, in three dimensions; in this form Wu (2023), Conjecture 1.1 with n = 3. Stated by AIM — Open Applied Problems for 1 < p < ∞.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, Bochner–Riesz means, Fourier multipliers
References
@article{wu2023,
author = {Wu, Shukun},
title = {On the {B}ochner--{R}iesz operator in $\mathbb{R}^3$},
journal = {J. Anal. Math.},
volume = {149},
number = {2},
year = {2023},
pages = {677--718},
doi = {10.1007/s11854-022-0263-y},
eprint = {2008.13043},
archivePrefix = {arXiv},
}
@article{guo2025,
author = {Guo, Shaoming and Oh, Changkeun and Wang, Hong and Wu, Shukun and Zhang, Ruixiang},
title = {The {B}ochner--{R}iesz problem: an old approach revisited},
journal = {Peking Math. J.},
volume = {8},
number = {2},
year = {2025},
pages = {201--270},
doi = {10.1007/s42543-023-00082-4},
eprint = {2104.11188},
archivePrefix = {arXiv},
}
@article{fefferman1971,
author = {Fefferman, Charles},
title = {The multiplier problem for the ball},
journal = {Ann. of Math. (2)},
volume = {94},
number = {2},
year = {1971},
pages = {330--336},
doi = {10.2307/1970864},
}
@article{carleson1972,
author = {Carleson, Lennart and Sj{\"o}lin, Per},
title = {Oscillatory integrals and multiplier problem for the disc},
journal = {Studia Math.},
volume = {44},
number = {3},
year = {1972},
pages = {287--299},
doi = {10.4064/sm-44-3-287-299},
}
@misc{aim-bochner-riesz-three-dimensions,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp Bochner–Riesz summability in three dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/209-bochner-riesz-three-dimensions.md},
note = {Entry 209 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove the three-dimensional Bochner--Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner--Riesz multipliers on $L^3(\RR^3)$ for every positive order. Interpolation and duality then give the full conjectured strict range.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00667, "Sharp Bochner–Riesz summability in three dimensions", clopen.solutions.