On the curved Trilinear Hilbert transform

Building on the (Rank I) LGC-methodology introduced by the second author and on the novel perspective employed in the time-frequency discretization of the non-resonant bilinear Hilbert--Carleson operator, we develop a new, versatile method -- referred to as Rank II LGC -- that has as a consequence the resolution of the $L^p$ boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator \begin{equation*} H_{C}(f_1, f_2, f_3)(x):= \textrm{p.v.}\,\int_{\mathbb{R}} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb{R}\,, \end{equation*} 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 H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1<p_1,p_3<\infty$, $1<p_2\leq \infty$ and $1\leq r <\infty$. A crucial difficulty in approaching this problem is the lack of absolute summability for the linearized discretized model (derived via Rank I LGC method) of the quadrilinear form associated to $H_{C}$. In order to overcome this, we develope a so-called correlative time-frequency model whose control is achieved via the following interdependent elements: (1) a sparse-unform 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", and (3) a level set analysis with respect to the time-frequency correlation set.