On the curved trilinear Hilbert transform

\[ H_C(f_1, f_2, f_3)(x):= p.v. \int_{\mathbb{R}} f_1(x-t) f_2 \left( x + t × t \right) f_3\left( x+t^3 \right) \frac{d t}{t}, \quad x \in \mathbb{R} \] is bounded from \(L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \times L^{p_3}(\mathbb{R})\) into \(L^r(\mathbb{R})\) within the Banach Hölder range \(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}\) with \(1<p_1, p_3<\infty\), \(1<p_2 \le \infty\) and \(1 \le r <\infty\). The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approach -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:

  1. a sparse-uniform decomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
  2. a structural analysis of suitable maximal ''joint Fourier coefficients'';
  3. a level set analysis with respect to the time-frequency correlation set.

This is a joint work with my postdoc advisor Victor Lie from Purdue.