The three-dimensional Bochner–Riesz conjecture

1 · The essentials

This manuscript concerns a family of mathematical operations on functions in three-dimensional space. The question is whether these operations keep the size of their output under control compared with the size of their input. The answer depends on how size is measured and on a parameter called the order. The manuscript claims this control whenever the order is strictly above a specified threshold. Its central result concerns one particular way of measuring size, from which it obtains the full stated range. Equality at the threshold is not covered by this claim. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a processing machine whose output must stay within a fixed multiple of the size of what goes in. The manuscript's claim is like giving a sufficient setting for that control, with the required setting changing according to the measuring rule used.

Why it matters

The repository summary describes the result as resolving the three-dimensional Bochner–Riesz conjecture in its strict range. The manuscript reduces that range to a central claim about $L^{3}$ , with interpolation and duality supplying the remaining cases. The strict inequality is an essential part of the stated resolution.

2 · Going further

The precise problem is boundedness of Bochner–Riesz multipliers of order $δ$ on $L^{p} (R^{3})$ . The repository summary states the range as every $1 \leq p \leq \infty$ whenever $δ > max {3∣1/ p - 1/2∣ - 1/2, 0}$ . Here boundedness means that the size of the output, measured in the same $L^{p}$ space as the input, is controlled by a fixed multiple of the input's size. The source describes this as the three-dimensional conjecture in its strict-order formulation.

The main abstract identifies the central result as boundedness on $L^{3} (R^{3})$ for every $δ > 0$ . At $p = 3$ , the threshold in the displayed condition is zero, so this is precisely the positive-order claim at that exponent. The abstract says that interpolation and duality then yield the full conjectured strict range. It supplies no details of the argument establishing the central $L^{3}$ result. Only one manuscript is provided, so no relationship with companion manuscripts can be described from this source.

Useful notions

Functions on three-dimensional space
Measuring function size with $L^{p}$ norms
Bounded linear operators
Interpolation and duality

What was known before / what is new

Before

The source identifies the target statement as the three-dimensional Bochner–Riesz conjecture. It does not describe earlier partial results or give a historical account, so no more detailed prior state of knowledge can be extracted from the supplied text.

New

The manuscript claims boundedness of Bochner–Riesz multipliers on $L^{3} (R^{3})$ for every positive order. It states that interpolation and duality extend this to the full conjectured strict range for $1 \leq p \leq \infty$ .

Limits and assumptions

The statement is restricted to three dimensions and requires $δ > max {3∣1/ p - 1/2∣ - 1/2, 0}$ with $1 \leq p \leq \infty$ . Equality at the threshold is outside the stated result; the supplied text does not specify the status of those boundary cases. The abstract does not explain how the central $L^{3}$ estimate is obtained. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

$L^{p} (R^{3})$: A space of functions on three-dimensional space whose size is measured using the parameter $p$ .
Boundedness: A uniform bound on output size by a fixed multiple of input size.
Order: The parameter $δ$ indexing the Bochner–Riesz multipliers in the stated result.
Strict range: The parameter choices for which the order is strictly greater than the stated threshold, excluding equality.

Keywords : Bochner–Riesz multipliers · Three-dimensional analysis · Strict-order boundedness

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on $L^{p} (R^{3})$ for every $1 \leq p \leq \infty$ whenever $δ > max {3∣1/ p - 1/2∣ - 1/2, 0}$ .

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Bochner–Riesz multipliers in three dimensions

24/09/2026

To be verified

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

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