Bochner-Riesz conjecture

Bochner-Riesz conjecture

OpenAI's claim

Three-dimensional Bochner--Riesz multipliers: The manuscript claims boundedness on $L^3(\mathbb R^3)$ of every positive-order Bochner--Riesz multiplier. Interpolation and duality give the remaining strict conjectured range in dimension three.

Mathematical statement of the claimed result

Mathematical claim 1.

Define the Bochner--Riesz multiplier by $\widehat{T_\delta f}(\xi)=(1-|\xi|^2)_+^\delta\widehat f(\xi)$ on $\mathbb R^3$. The main theorem gives $\|T_\delta f\|_3\le C_\delta\|f\|_3$ for every $\delta>0$. Its stated corollary supplies the full strict range \[1\le p\le\infty,\qquad\delta>\max\{0,\,3|1/p-1/2|-1/2\}.\] In that range $T_\delta$ is bounded on $L^p(\mathbb R^3)$, uniformly under a positive-radius dilation. No equality case at the displayed threshold is included.

Outline of the argument

Three-dimensional Bochner--Riesz multipliers: The first stage bounds the information revealed by observing positive line masses across scales. The second stage transfers this to oscillatory wave packets, keeping track of the positional uncertainty of earlier packets. Excessive concentration growth would require more fine velocity information than the model can carry, yielding the strict-order multiplier estimates.

Formal verification

Three-dimensional Bochner--Riesz multipliers: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

• OpenAI, Bochner–Riesz multipliers in three dimensions, 2026-09-24; citation key OAI:Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:bochner-riesz; Statement source: cor:full-bochner-riesz.
• OpenAI catalogue entry.

Bochner-Riesz conjecture

Definitions and mathematical statement

For $R>0$, define the Euclidean Bochner--Riesz operator on Schwartz functions by \[ \widehat{S_R^\delta f}(\xi)=\left(1-\frac{|\xi|^2}{R^2}\right)_+^\delta\widehat f(\xi), \] where at $\delta=0$ the multiplier is the indicator of the ball, up to its irrelevant boundary. Uniform $L^p$ boundedness means $\sup_{R>0}\|S_R^\delta\|_{L^p\to L^p}<\infty$. For $d\ge2$ and $1<p<\infty$, the conjectured strong-type range is $\delta\ge0$ when $p=2$, and $\delta>\max\{d|1/p-1/2|-1/2,0\}$ when $p\ne2$. The problem is this full characterization in every dimension. Strong operator boundedness is distinct from weak-type endpoint estimates or almost-everywhere convergence, even when those topics appear in the source title.

Formulation and concept references: [S1].

Short English statement

For exactly which summation orders and integrability exponents do spherical Fourier cutoffs remain uniformly bounded on Euclidean Lp spaces?

Research attempt

Current frontier.

Editorial source-fit note. The retained formulation reference concerns almost-everywhere convergence on Heisenberg-type groups. That is not the Euclidean strong-operator characterization requested here [S1]. For the latter, Guo--Oh--Wang--Wu--Zhang give an explicit restriction-based sufficient range in Theorem 1.3 and explain a barrier to treating the multiplier as a completely general positive-curvature oscillatory integral [E1], Introduction. Their result supplies substantial partial ranges, not all exponents in the question. Kim's 2024 manuscript, published in 2025, gives improved almost-everywhere convergence conditions in dimensions two and three; that conclusion also must not be substituted for uniform strong-type boundedness [E3]. The attempt below concentrates on the precise loss incurred by estimating one thin spherical shell absolutely.

Attack route.

A restriction transfer needs sharp estimates for the relevant oscillatory operators rather than just their arbitrary-phase analogues. Polynomial partitioning needs to control the narrow contributions without losing the critical exponent. We pursue a simpler diagnostic: decompose at distance $2^{-j}$ from the sphere, compute the exact loss of the elementary kernel argument, and state the uniform shell improvement that would close the positive part of the conjecture. The point is to isolate cancellation that a proposed proof must actually deliver, not to hide the desired bound inside an unspecified interpolation step.

Attempt: reduction to one radius and thin shells.

If $D_Rf(x)=f(Rx)$, Fourier transformation gives $S_R^\delta D_R=D_RS_1^\delta$. The $L^p$ scaling factors cancel on conjugation, so the operator norm is independent of $R$. It suffices to study $R=1$; this already handles the uniform-radius quantifier.

Fix a smooth dyadic partition on $0<t<1/4$. Up to a smooth multiplier supported away from the unit sphere, write \begin{equation}\label{wsref-0-65-71-31-38-30-2d-73-68-65-6c-6c-73} (1-|\xi|^2)_+^\delta =m_{\mathrm{int}}(\xi)+\sum_{j\ge j_0}2^{-j\delta} a_\delta\left(2^j(1-|\xi|^2)\right), \end{equation} where $a_\delta(s)=s^\delta\psi(s)$ with $\psi\in C_c^\infty((1/2,2))$. Finitely many transition terms can be absorbed in $m_{\mathrm{int}}$. For each fixed $\delta\ge0$, all the rescaled profile derivatives are bounded independently of $j$. Constants may depend on $\delta$; no uniformity as $\delta$ tends to an endpoint is claimed. Let $P_j$ be the multiplier with the profile in the sum.

Elementary shell bound. For this fixed profile and $t_p=|1/p-1/2|$, \begin{equation}\label{wsref-0-65-71-31-38-30-2d-65-6c-65-6d-65-6e-74-61-72-79} \|P_j\|_{L^p\to L^p}\lesssim_{p,d,\delta} 2^{j(d-1)t_p}. \end{equation} Here is the kernel calculation identifying the loss. Put $h=2^{-j}$ and $K_j=\mathcal F^{-1}(a_\delta(h^{-1}(1-|\xi|^2)))$. Polar coordinates express it as an integral of the Fourier transform of surface measure over a radial interval of length $O(h)$ around radius one. For $|x|\ge1$, the spherical integral is a sum of the two phases $e^{\pm2\pi i r|x|}|x|^{-(d-1)/2}$ with symbol amplitudes; their radial derivatives obey the corresponding symbol bounds. Repeated integration by parts in $r$ outside $h|x|\lesssim1$, and the absolute radial bound inside, give, for every fixed $N$, \[ |K_j(x)|\le C_N h(1+|x|)^{-(d-1)/2}(1+h|x|)^{-N}. \] The bound for $|x|<1$ follows directly from the volume of the frequency shell. Integrating first over $|x|\le h^{-1}$ and then its complement, with $N>d$, yields \[ \|K_j\|_1\lesssim h^{-(d-1)/2}. \] Plancherel gives $\|P_j\|_{2\to2}\lesssim1$. Interpolation between $L^1$ and $L^2$, followed by duality for $p>2$, proves (eq180-elementary). This uses the standard spherical stationary-phase expansion underlying the reductions in [E1], Section 2; it does not assume the restriction conjecture.

Summing the operator norms in (eq180-shells) proves the classical sufficient condition \begin{equation}\label{wsref-0-65-71-31-38-30-2d-63-6c-61-73-73-69-63-61-6c} \delta>(d-1)|1/p-1/2|. \end{equation} The sum is genuinely convergent in operator norm, so no interchange of an uncontrolled infinite series with the $L^p$ norm is involved.

Attempt: why the conjectured exponent is smaller.

The full kernel is, up to a nonzero dimensional constant, \[ K^\delta(x)=|x|^{-d/2-\delta} J_{d/2+\delta}(2\pi|x|). \] The Bessel asymptotic has leading term a nonzero multiple of $|x|^{-(d+1)/2-\delta}\cos(2\pi|x|-\theta)$, with an error smaller by one power of $|x|$. Choose a Schwartz function whose Fourier transform is one on the closed unit ball. Its image is exactly $K^\delta$. Integrating on periodic radial subintervals where the cosine is bounded away from zero proves that this image is not in $L^p$ unless \[ \delta>\frac d p-\frac{d+1}{2}. \] At equality the radial integral diverges logarithmically; replacing the oscillatory kernel by its average would incorrectly lose that obstruction. Duality gives the corresponding condition at $p>2$. Combined, these yield the familiar necessary strict bound when it is nonnegative, \[ \delta>d|1/p-1/2|-\tfrac12. \] For the remaining $\delta=0$, $p\ne2$ cases, the ball-multiplier obstruction is Fefferman's separate theorem [E4], not a consequence of this integrability test. At $p=2$, all $\delta\ge0$ are bounded by Plancherel.

Set $\delta_c(p)=\max\{dt_p-1/2,0\}$. The loss in (eq180-classical), relative to the desired positive-order range, is \begin{equation}\label{wsref-0-65-71-31-38-30-2d-6c-6f-73-73} (d-1)t_p-\delta_c(p)= \begin{cases}(d-1)t_p,&dt_p\le1/2,\\ 1/2-t_p,&dt_p\ge1/2. \end{cases} \end{equation} It is strictly positive for $1<p<\infty$, $p\ne2$. Thus neither optimizing the dyadic summation nor keeping logarithmic factors more carefully can repair the elementary argument: its error is a power of the shell width.

Attempt: an explicit sufficient missing estimate.

For each fixed smooth profile of the kind in (eq180-shells), consider \begin{equation}\label{wsref-0-65-71-31-38-30-2d-6d-69-73-73-69-6e-67} \forall\eta>0\quad \|P_j\|_{p\to p}\le C_{p,d,\eta,a}\, 2^{j(\delta_c(p)+\eta)} \quad(j\ge j_0). \end{equation} If this estimate holds for every $d,p$ under consideration, then every $\delta>\delta_c(p)$ in the target follows. Indeed, fix $\delta$ first, choose $0<\eta<\delta-\delta_c(p)$, apply the estimate to $a=a_\delta$, and sum the convergent geometric series $\sum_j2^{-j(\delta-\delta_c(p)-\eta)}$. Together with the necessary results and the separate $p=2$ case this would give the full stated characterization. This is a sufficient reduction; no converse to this particular shell formulation is asserted.

The estimate is where angular interaction must enter. Applying absolute values to the kernel before summation recovers only (eq180-elementary). A square-function argument that merely combines already bounded shells does not, by itself, improve the estimate for a single shell: it must also prove new information about the angular pieces within that shell. Nor can one replace the phase by an arbitrary positive-curvature phase and demand the same range. The sharpness examples for that larger class are precisely a barrier explained in [E1], Introduction.

Stress test.

At $p=2$ the elementary shell exponent is already zero, but the sum of norms at $\delta=0$ diverges. The actual ball projection is nevertheless bounded by Plancherel. Thus failure of absolute summation is not evidence of unboundedness; the endpoint must be handled separately. At $p\ne2$, weak-type endpoint information and almost-everywhere convergence do not supply the missing strong shell estimate. In particular the 2025 weighted decoupling application [E3] cannot be inserted into (eq180-missing) simply by changing the words describing its norm.

There is also a dependency warning. The full Bochner--Riesz conjecture implies the corresponding restriction conjecture by Tao's theorem [E2]. A proposed elementary cancellation argument proving (eq180-missing) universally must therefore be checked against that major additional consequence. No such cancellation has been proved here.

Outcome.

Outcome: Conditional reduction.

The attempt verifies the uniform-radius reduction, derives the elementary shell loss, and proves that (eq180-missing) would supply the entire positive-order sufficient range. It also separates the two different necessary obstructions and the $p=2$ endpoint. These are standard analytic ingredients assembled and audited here, not a claimed new improvement to the best Bochner--Riesz bounds.

What remains is the sharp shell estimate, uniformly in the shell scale. The present calculation exposes exactly the power saving needed and why absolute-kernel interpolation does not produce it. It neither proves the full characterization nor constructs an example contradicting it.

Sources

• S1 Almost everywhere convergence of Bochner-Riesz means on Heisenberg-type groups. https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.12401. Catalogue formulation source.
• E1 Shaoming Guo, Changkeun Oh, Hong Wang, Shukun Wu and Ruixiang Zhang, The Bochner--Riesz problem: An old approach revisited, arXiv:2104.11188, Introduction, Theorem 1.3, and Section 2. https://arxiv.org/html/2104.11188. Consulted 27 September 2026; Euclidean operator bounds and the arbitrary-phase barrier, not the group setting of [S1].
• E2 Terence Tao, The Bochner--Riesz conjecture implies the restriction conjecture, Duke Mathematical Journal 96$2$ $1999$, 363--375. https://doi.org/10.1215/S0012-7094-99-09610-2. Dependency theorem; also explicitly discussed in [E1], Introduction. Bibliographic record checked 27 September 2026.
• E3 Jongchon Kim, Weighted decoupling estimates and the Bochner--Riesz means, arXiv:2406.03741; Forum of Mathematics, Sigma 13 $2025$, e167. https://arxiv.org/abs/2406.03741. Consulted 27 September 2026; the cited application is almost-everywhere convergence, not the full strong-type target.
• E4 Charles Fefferman, The multiplier problem for the ball, Annals of Mathematics 94$2$ $1971$, 330--336. https://annals.math.princeton.edu/1971/94-2/p05. Primary theorem at order zero; publication record checked 27 September 2026.