2507.04467
The Bilinear Hilbert–Carleson operator along curves. The purely non-zero curvature case
Árpád Bényi, Bingyang Hu, Victor Lie
correcthigh confidence
- Category
- math.DS
- Journal tier
- Top Field-Leading
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The uploaded paper proves the full quasi-Banach range L^{p1} × L^{p2} → L^r for every 1 < p1, p2 < ∞ and 1/2 < r < ∞ in the purely nonresonant case (pairwise distinct exponents), precisely the result the model labeled as likely open. The Main Theorem 1.1 states this result explicitly and the body of the paper develops a new Rank II LGC method, together with a sparse–uniform decomposition and a constancy-propagation argument, culminating in Section 6 with the quasi-Banach bounds. Therefore, the paper settles the question in the affirmative, contradicting the model’s assessment.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} top field-leading
\textbf{Justification:}
The paper settles a prominent open question in the purely nonresonant curved setting by establishing the full quasi-Banach range r>1/2 for the bilinear Hilbert–Carleson operator along curves. The work introduces the Rank II LGC method and a constancy-propagation bootstrap, overcoming single-scale non-absolute summability obstacles beyond classical size–energy methods. The exposition is detailed and largely clear; minor revisions could further streamline the reduction to general exponents and enhance readability of the technical sections.