# Trilinear Smoothing Inequalities and a Class of Bilinear Maximal Functions

The HADES seminar on Tuesday, March 1 will be at 3:30 pm in Room 740.

Speaker: Zirui Zhou

Abstract: In this talk, we will present a trilinear smoothing inequality of the form
$$\left|\int_{\mathbb R^2} \prod_{j=0}^2 (f_j\circ\varphi_j)(x)\,\eta(x)\,dx\right| \leq C \prod_{j=0}^2 \|f_j\|_{W^{p,\sigma}}$$and two of its applications. Lebesgue space bounds are established for certain maximal bilinear functions. The proof combines the degenerate-case trilinear smoothing inequality with Calderón-Zygmund theory.

The second application gives a quantitative nonlinear Roth theorem, which recovers Roth-type theorems proved by Bourgain and Christ-Durcik-Roos. This talk is based on joint work with Michael Christ.