A dichotomy for H\"ormander-type oscillatory integral operators

TL;DR

This paper introduces a curvature condition called Bourgain's condition for H"ormander-type oscillatory integral operators, linking their boundedness properties to the Fourier restriction and Bochner-Riesz problems, and extends known results in high dimensions.

Contribution

It generalizes Bourgain's work by defining Bourgain's condition, and shows that operators satisfying this condition have improved $L^p$ bounds, advancing understanding of the Fourier restriction and Bochner-Riesz conjectures.

Findings

Bourgain's condition is satisfied by phase functions in key problems.

Failure of Bourgain's condition implies failure of certain $L^{ olinebreak}^ ext{infinity} o L^q$ bounds.

Operators satisfying Bourgain's condition have extended $L^p$ bounds, improving current results.

Abstract

In this paper, we first generalize the work of Bourgain and state a curvature condition for H\"ormander-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 H\"ormander-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. On the other hand, if Bourgain's condition holds, then we prove $L^p$ bounds for H\"ormander-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. This gives new progress on the Fourier restriction problem and the Bochner-Riesz problem.