we-solve (beta)

Math-2956 · mathematics · Analysis

Sharp L-infinity extension from the two-sphere

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let E $g(x)=integral_S2$ g(omega) exp(2pi i x dot omega) d sigma(omega). For every $p>3$, is there $C_p$ with ||E $g||_Lp(R3)<=C_p||g||_Linfinity(S2)$ for all bounded measurable g?

Short English statement

Let E $g(x)=integral_S2$ g(omega) exp(2pi i x dot omega) d sigma(omega). For every $p>3$, is there $C_p$ with ||E $g||_Lp(R3)<=C_p||g||_Linfinity(S2)$ for all bounded measurable g?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2956-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2956

Sphere extension and packet capacity: The primary statement says: The sphere extension operator maps L-infinity(S2) to Lp(R3) for every p>3; a corollary gives the three-dimensional Kakeya maximal 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.

Sharp L-infinity extension from the two-sphere

OpenAI's claim

Sphere extension and packet capacity: For bounded data on the two-sphere, the announced extension bound has every output exponent larger than three. The threshold three is also asserted to be the sharp endpoint of this open range.

Mathematical statement of the claimed result

Mathematical claim 1.
For surface area $\sigma$ on $S^2$ and $Eg(x)=\int_{S^2}e^{2\pi ix\cdot\omega}g(\omega)\,d\sigma(\omega)$, every bounded measurable complex $g$ satisfies \[\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2,\sigma)}\quad(p>3).\] The constant depends only on $p$; the endpoint $p=3$ and other ambient dimensions are outside this theorem.

Outline of the argument

Sphere extension and packet capacity: Joint tests of position and velocity use ellipse weights that penalize eccentricity. A geometric propagation estimate controls how this capacity evolves along lines, using overlap savings from differently directed tests. Exact packet analysis and synthesis retain the oscillatory information needed to convert the positive capacity estimate into Fourier extension bounds.

Formal verification

Sphere extension and packet capacity: 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 →