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Math-2943 · mathematics · Analysis

Critical L3 local smoothing in three dimensions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $\epsilon >0$, does the half-wave propagator satisfy ||exp(it $\operatorname{sqrt}(-Delta))f||_L3(R3$ x $[1,2])<=C_\epsilon ||(1-Delta)^{\epsilon /2}f||_L3(R3)$ for every Schwartz f?

Short English statement

For every $\epsilon >0$, does the half-wave propagator satisfy ||exp(it $\operatorname{sqrt}(-Delta))f||_L3(R3$ x $[1,2])<=C_\epsilon ||(1-Delta)^{\epsilon /2}f||_L3(R3)$ for every Schwartz f?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2943-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2943

Wave local smoothing: The primary statement says: The three-dimensional half-wave propagator on R3 times [1,2] obeys the critical L3 local-smoothing estimate with every positive epsilon derivative loss. 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.

Critical L3 local smoothing in three dimensions

OpenAI's claim

Wave local smoothing: OpenAI claims the critical $L^3$ spacetime estimate for the wave equation in three dimensions with any positive Sobolev loss. This is the Euclidean three-dimensional smoothing assertion at that exponent.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Uf(x,t)=(2\pi)^{-3}\int e^{i(x\cdot\xi+t|\xi|)}\widehat f(\xi)\,d\xi$ and $J^\epsilon=(1-\Delta)^{\epsilon/2}$. The claimed three-dimensional critical local-smoothing estimate is \[\|Uf\|_{L^3(\mathbb R^3\times[1,2])}\le C_\epsilon\|J^\epsilon f\|_{L^3(\mathbb R^3)}\quad(\epsilon>0,\ f\in\mathcal S(\mathbb R^3)).\] Every positive Sobolev loss is allowed, but zero loss is not asserted.

Outline of the argument

Wave local smoothing: Null coordinates put light rays into a quadratic velocity model, and packet synthesis reduces the physical estimate to a cubic mass sum. Entropy records retain changing shear frames as well as direction and time. Planar Furstenberg and cone-envelope inputs constrain those records, giving a frequency-annulus estimate that can be summed with arbitrarily small loss.

Formal verification

Wave local smoothing: 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

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