The three-dimensional Bochner–Riesz conjecture

A claimed proof of the Bochner–Riesz conjecture in three dimensions, in its strict-order form.

1 paper September 24, 2026 Explained by AI

Bochner–Riesz means rebuild a function from its frequencies while softening the cut-off at the edge of a ball. The conjecture says how little softening keeps the size of the function under control. This paper claims the three-dimensional case, starting from the hardest exponent, p = 3. There any positive amount of softening is claimed to be enough. Interpolation then gives the full conjectured range.

First step

1935

Salomon Bochner, spherical means for multiple Fourier series

Significance

Landmark

The Bochner–Riesz conjecture in three dimensions is a central open problem in harmonic analysis, and it implies the sphere restriction conjecture.

Trust

Claimed, not checked

Field

Real and complex analysis

1 paper

The explanation

Take a function on three-dimensional space and split it into its frequencies, using the Fourier transform. Keep only frequencies of size below 1, but soften the edge. Multiply each frequency ξ inside the unit ball by (1 − |ξ|²)^δ, and drop those outside. The order δ sets how soft the edge is. With δ = 1, a frequency of size 0.9 keeps 1 − 0.81 = 0.19 of its weight. With δ = 0.1 it keeps 0.19^0.1, about 0.847. At size 0.99 it keeps 0.0199^0.1, about 0.676. So a small δ gives an edge that is almost sharp.

This operation is the Bochner–Riesz mean T_δ. The question is how much softening keeps the function's Lᵖ size under controlSource 1. The Lᵖ norm is the p-th root of the integral of |f|ᵖ. The conjecture says that δ > max{3|1/p − 1/2| − 1/2, 0} is enough in three dimensions. At p = 3, the first term is 3 × 1/6 − 1/2 = 0, so any δ > 0 should work. At p = 4 the threshold is 3/4 − 1/2 = 1/4. At p = 1 it is 3/2 − 1/2 = 1.

Riesz and Bochner introduced such means to sum Fourier series. In 1971, Fefferman showed that the sharp edge, δ = 0, fails for every p ≠ 2 in two or more dimensionsSource 2. His proof used a Kakeya construction, a set packed with thin tubes pointing in many directions. Carleson and Sjölin settled the plane in 1972Source 3. In three dimensions, Lee reached the conjectured range when max{p, p′} ≥ 10/3Source 4. Here p′ is the dual exponent, with 1/p + 1/p′ = 1. Wu lowered the bound to 13/4Source 5. Gao, Wu and Xi lowered it to 22/7Source 6. Tao showed that the Bochner–Riesz conjecture implies the restriction conjecture for the sphereSource 7. Zipeng Wang has also proposed a proof of the conjecture in all dimensionsSource 8.

This paper claims that T_δ is bounded on L³ for every δ > 0Source 9. Interpolation and duality then give the full conjectured rangeSource 10. Interpolation fills in exponents between two known bounds, and duality swaps p with p′. The hard case is p = 3, where the conjectured order drops all the way to zero. For 3/2 ≤ p ≤ 3, the paper claims any mild spherical smoothing recovers the functionSource 11.

The key geometric input is the sharp planar Furstenberg estimate of Ren and WangSource 12. A Furstenberg estimate limits how small a set can be when many lines each meet it in a large piece. Another paper in the release reaches the same range by a separate route, through local smoothing for wavesSource 13.

How sure is this? There is no Lean scope note for this result, so no computer proof checker has tested it. The paper is long and depends on the recent Ren–Wang estimate. No specialist review has been published.

What it doesn't show. This does not cover endpoint orders, dimensions other than three, or maximal Bochner–Riesz operators.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark if the proof holds, since it claims a central conjecture in three dimensions. It is one long paper with no Lean scope note. No specialist review is published.

Words used

Fourier transform: The split of a function into frequencies, each with its own strength.
Bochner–Riesz mean: The operation that keeps frequencies inside the unit ball, weighted by (1 − |ξ|²) to the power δ.
Order δ: The power that sets how softly the frequency cut-off falls to zero at the edge of the ball.
Lᵖ norm: A measure of a function's size: the p-th root of the integral of its absolute value to the power p.
Interpolation: A method that turns bounds at two exponents into bounds at the exponents between them.
Kakeya set: A set containing a thin tube, or line segment, in many directions.
Furstenberg estimate: A lower bound on the size of a set that many lines each meet in a large piece.

The original paper

Bochner–Riesz multipliers in three dimensions (PDF, opens in a new tab)
September 24, 2026

All quoted lines (13)

The multiplier $(1-|\xi|^2)_+^\delta$ softens the boundary of the frequency ball; the question is how much smoothing is needed to preserve an $L^p$ norm.
Bochner–Riesz multipliers in three dimensions Back to text
Using a Kakeya construction, Fefferman proved that in dimension at least two the order-zero ball multiplier is unbounded on $L^p$ for $p\ne2$ , while Carleson and Sj\"olin established the conjectured positive-order range in the plane.
Bochner–Riesz multipliers in three dimensions Back to text
Using a Kakeya construction, Fefferman proved that in dimension at least two the order-zero ball multiplier is unbounded on $L^p$ for $p\ne2$ , while Carleson and Sj\"olin established the conjectured positive-order range in the plane.
Bochner–Riesz multipliers in three dimensions Back to text
The bilinear restriction theorems of Wolff for the cone and Tao for the paraboloid and elliptic surfaces led, through Lee's multiplier reduction, to sharp-order Bochner–Riesz estimates for $\max\{p,p'\}\ge10/3$ ; here $p'$ is the H\"older conjugate of $p$ .
Bochner–Riesz multipliers in three dimensions Back to text
For the multiplier itself, Wu obtained the conjectured strict-order range when $\max\{p,p'\}\ge13/4$ .
Bochner–Riesz multipliers in three dimensions Back to text
More recently, Gao, Wu, and Xi established the strict-order multiplier range $\max\{p,p'\}\ge22/7$ .
Bochner–Riesz multipliers in three dimensions Back to text
The connection also runs in the other direction: Tao showed that the Bochner–Riesz conjecture implies the spherical restriction conjecture.
Bochner–Riesz multipliers in three dimensions Back to text
Zipeng Wang has also proposed a proof of the Bochner–Riesz conjecture in all dimensions.
Bochner–Riesz multipliers in three dimensions Back to text
For every $\delta>0$ , there is a finite constant $C_\delta$ such that $\norm{T_\delta f}_{L^3(\RR^3)} \le C_\delta\norm f_{L^3(\RR^3)} \qquad(f\in\mathcal S(\RR^3)).$
Bochner–Riesz multipliers in three dimensions Back to text
The main estimate is boundedness on $L^3$ for every positive order; interpolation and duality then give the full range of exponents.
Bochner–Riesz multipliers in three dimensions Back to text
This recovers Fourier inversion by arbitrarily mild spherical smoothing at the central exponents $3/2\le p\le3$ .
Bochner–Riesz multipliers in three dimensions Back to text
The geometric input here is the sharp planar Furstenberg estimate of Ren and Wang.
Bochner–Riesz multipliers in three dimensions Back to text
An independent route through critical local smoothing gives the same full strict fixed-radius range in (…), uniformly in the radius.
Bochner–Riesz multipliers in three dimensions Back to text

Other sources

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