A dichotomy for Hörmander-type oscillatory integral operators

In this paper, we first generalize the work of Bourgain (Geom. Funct. Anal. 1(4):321–374, 1991) and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain’s condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain’s condition, they satisfy the same $L^p$ $L^{p}$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain’s condition fails, then the $L^{\mathrm{\infty }}\to L^q$ $L^{\infty } \to L^{q}$ boundedness always fails for some $q=q(n)>\frac{2n}{n-1}$ $q= q(n) > \frac{2n}{n-1}$ , extending Bourgain’s three-dimensional result (Geom. Funct. Anal. 1(4):321–374, 1991). On the other hand, if Bourgain’s condition holds, then we prove $L^p$ $L^{p}$ bounds for Hörmander-type oscillatory integral operators for a range of $p$ $p$ that extends the currently best-known range for the Fourier restriction conjecture in high dimensions, given by Hickman and Zahl (A note on Fourier restriction and nested polynomial wolff axioms, 2020, arXiv:2010.02251). This gives new progress on the Fourier restriction problem, the Bochner-Riesz problem on ${\mathbb{R}}^n$ $\mathbb{R}^{n}$ , the Bochner-Riesz problem on spheres $S^n$ $S^{n}$ , etc.