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

Proves maximal and annular $r$-variation bounds, for every $r>2$, from complex $L^3({\mathbb{R}}^2)\times L^3({\mathbb{R}}^2)$ to $L^{3/2}({\mathbb{R}}^2)$. 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.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.