Harmonic Analysis Seminar: On multilinear oscillatory integral operator inequalities

Seminar | April 10 | 1:10-2 p.m. | 736 Evans Hall

 Michael Christ, UCB

 Department of Mathematics

The title refers to inequalities of the form $\int _{[0,1]^d} \prod _{j=1}^d f_j(x_j) \,e^{i\lambda \psi (x)}\,dx = O(|\lambda |^{-\gamma } \prod _j \|f_j\|_{L^{p_j}})$ for large $\lambda \in {\mathbb R}$. Here $\psi :{\mathbb R}^d\to {\mathbb R}$ is a smooth phase function, and the exponent γ depends on ψ and on the exponents $p_j$. These inequalities are well understood in the bilinear case $d=2$, and sharp bounds have been obtained by Phong-Stein-Sturm and Gilula-Gressman-Xiao for certain parameter ranges for $d >2$. Nonetheless, the case $d\ge 3$ remains largely mysterious. I will argue that the most basic question in this context remains unaddressed for $d\ge 3$, and will present recent partial results and examples for $d=3$ with an outline of the proofs.