we-solve (beta)

Math-2941 · mathematics · Analysis

The Bochner-Riesz conjecture in dimension three

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $1<p<infinity$, is the Fourier multiplier $(1-|\xi |^2/R^2)_+^\delta $ uniformly bounded on $L^p(R^3)$ for all $R>0$ whenever $\delta >max(3|1/p-1/2|-1/2,0)$, with the order-zero $L^2$ case also bounded?

Short English statement

For every $1<p<infinity$, is the Fourier multiplier $(1-|\xi |^2/R^2)_+^\delta $ uniformly bounded on $L^p(R^3)$ for all $R>0$ whenever $\delta >max(3|1/p-1/2|-1/2,0)$, with the order-zero $L^2$ case also bounded?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2941-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2941

Three-dimensional Bochner--Riesz multipliers: The primary statement says: Bochner-Riesz means in R3 are uniformly Lp bounded for every 1<p<infinity and every delta>max(3|1/p-1/2|-1/2,0), by the stated full-range corollary of the L3 estimate. The target retains these quantified objects and hypotheses, so its positive assertion is covered.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

The Bochner-Riesz conjecture in dimension three

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

Go to contribution →