DMS Analysis and Stochastic Analysis Seminar (SASA)

Speaker: Bingyang Hu (Auburn University)

Title: On the curved Trilinear Hilbert transform

Abstract: The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator

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.