OpenAI · 082-01 · 2026-10-05 · Real and complex analysis

Annular variation of the triangular Hilbert transform at the symmetric point

preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026

Family 082. Maximal and variational bounds for the triangular Hilbert transform.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves maximal and annular $r$ -variation bounds, for every $r > 2$ , from complex $L^{3} (R^{2}) \times L^{3} (R^{2})$ to $L^{3/2} (R^{2})$ . The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^{3} \times L^{3} \times L^{3}$ point.

We prove the annular r-variation estimate for the triangular Hilbert transform from complex $L^{3} \times L^{3}$ to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.

Preprint source and Lean scope

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A Lean scope note is stored for family 082, but it does not name this preprint. It names 082-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.