A dichotomy for Hörmander-type oscillatory integral operators

Abstract

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 \(L^{p}\) bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain’s condition fails, then the \(L^{\infty } \to L^{q}\) boundedness always fails for some \(q= q(n) > \frac{2n}{n-1}\), 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 \(L^{p}\) bounds for Hörmander-type oscillatory integral operators for a range of \(p\) 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 \(\mathbb{R}^{n}\), the Bochner-Riesz problem on spheres \(S^{n}\), etc.