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Math-2958 · mathematics · Analysis

The symmetric maximal triangular Hilbert transform

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For F,G in $L^3(R^2)$, is $||sup_{0<a<b<infinity}$ $|integral_{a<|t|<b}$ F(x+t,y)G(x,y+t) dt/t| $||_L(3/2)(R2)$ $<=$ $C||F||_3||G||_3$ with one absolute C?

Short English statement

For F,G in $L^3(R^2)$, is $||sup_{0<a<b<infinity}$ $|integral_{a<|t|<b}$ F(x+t,y)G(x,y+t) dt/t| $||_L(3/2)(R2)$ $<=$ $C||F||_3||G||_3$ with one absolute C?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2958-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2958

Maximal triangular Hilbert transform: The primary statement says: The hard-truncated maximal triangular Hilbert transform maps complex L3(R2) times L3(R2) to L^(3/2)(R2), uniformly in both annulus endpoints. 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 symmetric maximal triangular Hilbert transform

OpenAI's claim

Maximal triangular Hilbert transform: The announced symmetric triangular Hilbert estimate controls the supremum over both finite hard cutoffs for arbitrary complex $L^3$ inputs. It also gives joint principal-value convergence almost everywhere and in $L^{3/2}$.

Mathematical statement of the claimed result

Mathematical claim 1.
For $F,G\in L^3(\mathbb R^2;\mathbb C)$ let \[B_*(F,G)(x,y)=\sup_{0<a<b<\infty}\left|\int_{a<|t|<b}F(x+t,y)G(x,y+t)\,\frac{dt}{t}\right|.\] There is an absolute finite $C$ with $\|B_*(F,G)\|_{L^{3/2}(\mathbb R^2)}\le C\|F\|_3\|G\|_3$. Both finite hard-truncation endpoints vary in the pointwise supremum; this is the balanced exponent pair $(3,3)$.

Outline of the argument

Maximal triangular Hilbert transform: The cyclic form is represented by matrix traces and a positive energy evolving under Gaussian smoothing. Mixed trace estimates compare energy derivatives at adjacent variables. A further bound pays for entries switched on or off at separately chosen scales, so telescoping remains uniform under pointwise selection of the cutoffs.

Formal verification

Family 082 describes Lean coverage for the two-cutoff symmetric maximal estimate for complex inputs; later variation and dyadic absolute-sum refinements are separate.

Maximal triangular Hilbert transform: The manuscript links TriangularHilbert.lean (declaration OAI.TriangularHilbert.main_estimate).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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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