# Michael Christ

Title: Multilinear oscillatory integral inequalities: Best of the best

Abstract: Consider $T_\lambda^\phi(f_1,\dots,f_d) = \int_{[0,1]^d} e^{i\lambda\phi(x)} \prod_{j=1}^d f_j(x_j)\,dx$ with $\phi$ a $C^\omega$ real-valued phase function. What is the largest exponent $\gamma$ for which an upper bound $O(\lambda^{-\gamma} \prod_j \|f_j\|_\infty)$ holds, uniformly for all functions $f_j$ and large positive parameters $\lambda$? There is a large and successful body of work for the ``linear'' case $d=2$. For the multilinear case, optimal exponents have been determined by Phong-Stein-Sturm, by Gilula-Gressman-Xiao, and by others, with $L^{p_j}$ norms on the right-hand side, for certain ranges of exponents under natural nondegeneracy hypotheses on $\phi$. The largest optimal exponent arises for $L^\infty$ norms, which typically lie outside the parameter ranges of earlier works. Exponents $\gamma$ obtained in those works do not exceed $\tfrac12$.

Examples demonstrate that certain basic themes of earlier works are violated in the $L^\infty$ (or $L^p$ for large $p$) regime. The main result breaks the barrier $\gamma = 1/2$ in substantial generality for $d=3$, without yielding optimal exponents. Themes include dichotomy between structure and pseudorandomness, notions of degeneracy of phases, multi-scale analysis, and a connection with certain sublevel sets. A variant of the familiar notion of a sublevel set arises.