Annular variation and dyadic absolute bounds for the triangular Hilbert transform

What OpenAI claims

Proves maximal and annular r-variation bounds, for every r > 2, from complex L³(ℝ²)× L³(ℝ²) to L^3/2(ℝ²). 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 L³× L³× L³ point. (Lean)

Papers (3)

1. Annular variation of the triangular Hilbert transform at the symmetric point

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove the annular r-variation estimate for the triangular Hilbert transform from complex L³× L³ 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.

2. An L³ bound for the dyadic triangular Hilbert form

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove a uniform L³× L³× L³ estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.

3. The maximal triangular Hilbert transform at the symmetric point

   24 September 2026 · Preprint, not peer reviewed · PDF

   For arbitrary complex inputs in L³(ℝ²), we prove the pointwise maximal L³× L³ → L^3/2 estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.