Trilinear smoothing inequalities and a variant of the triangular Hilbert transform

TL;DR

This paper establishes Lebesgue space inequalities for a curvature-involving variant of the triangular Hilbert transform, introducing a key trilinear smoothing inequality and exploring related maximal functions and pattern theorems.

Contribution

It develops a novel trilinear smoothing inequality crucial for analyzing a curvature-based triangular Hilbert transform and related operators.

Findings

Proved Lebesgue space inequalities for the transform.

Established bounds for an anisotropic twisted paraproduct.

Derived a nonlinear Roth-type theorem for Euclidean plane patterns.

Abstract

Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on a crucial trilinear smoothing inequality developed herein, and on bounds for an anisotropic variant of the twisted paraproduct. The trilinear smoothing inequality also leads to Lebesgue space bounds for a corresponding maximal function and a quantitative nonlinear Roth-type theorem concerning patterns in the Euclidean plane.