we-solve (beta)

Math-2479 · mathematics · Analysis

Uniform Bounds for Two-Dimensional Bilinear Hilbert Transforms

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Can one obtain uniform bounds for the forms associated with the entire family of two-dimensional bilinear Hilbert transforms?

Formulation references: [S1], [S2].

Short English statement

Can one obtain uniform bounds for the forms associated with the entire family of two-dimensional bilinear Hilbert transforms?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2479-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2479

Annular triangular variation: The annular-variation estimate implies the L3 times L3 to L^(3/2) bound for the triangular member of the two-dimensional bilinear Hilbert family. It does not give the full family or exponent range.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Uniform Bounds for Two-Dimensional Bilinear Hilbert Transforms

OpenAI's claim

Annular triangular variation: The paper announces an $r$-variation estimate for every $r>2$ from two complex $L^3$ inputs to $L^{3/2}$. Output-dependent partitions over positive annular scales are allowed, and the result implies maximal and principal-value conclusions.

Mathematical statement of the claimed result

Mathematical claim 1.
For complex $F,G\in L^3(\mathbb R^2)$ set $B_{a,b}(F,G)(x,y)=\int_{a<|t|<b}F(x+t,y)G(x,y+t)\,dt/t$. For $r>2$ define \[V_r(F,G)=\sup_{J\ge1,\ 0<t_0<\cdots<t_J}\left(\sum_{j=1}^J|B_{t_{j-1},t_j}(F,G)|^r\right)^{1/r}.\] The supremum may be taken over rational endpoints and imposes no bound on $J$. The claim is $\|V_r(F,G)\|_{3/2}\le C_r\|F\|_3\|G\|_3$, with quantities defined off one common null set.

Outline of the argument

Annular triangular variation: The proof first bounds the sum of absolute contributions from finitely many disjoint annuli, selected separately at each output location. Quantitative growth control for this counting quantity yields the variation inequality. Approximation and the maximal consequence then give common almost-everywhere limits and convergence in the output norm.

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.

Annular triangular variation: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

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 →