On the curved trilinear Hilbert transform

Analysis Seminar
Wednesday, November 15, 2023 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005
Bingyang Hu – Auburn University – bzh0108@auburn.edu
Michael Lacey

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


H_C(f_1, f_2, f_3)(x):=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


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


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 approch -- 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 deomposition 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.