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Math-2957 · mathematics · Analysis

Annular variation of the triangular Hilbert transform

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For F,G in $L^3(R^2)$, write $B_{a,b}(F,G)(x,y)=integral_{a<|t|<b}$ F(x+t,y) G(x,y+t) dt/t. For every $r>2$, is the $L^{3/2}$ norm of the supremum of $(sum_j$ $|B_{a_j,b_j}(F,G)|^r)^{1/r}$, over finite disjoint ordered positive annuli, at most $C_r||F||_3||G||_3$?

Short English statement

For F,G in $L^3(R^2)$, write $B_{a,b}(F,G)(x,y)=integral_{a<|t|<b}$ F(x+t,y) G(x,y+t) dt/t. For every $r>2$, is the $L^{3/2}$ norm of the supremum of $(sum_j$ $|B_{a_j,b_j}(F,G)|^r)^{1/r}$, over finite disjoint ordered positive annuli, at most $C_r||F||_3||G||_3$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2957-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2957

Annular triangular variation: The primary statement says: For every r>2 the ordered-disjoint-annulus r-variation of the triangular Hilbert transform maps L3(R2) times L3(R2) to L^(3/2)(R2). 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.

Annular variation of the triangular Hilbert transform

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

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