Bounds on oscillatory integral operators based on multilinear estimates

TL;DR

This paper uses multilinear restriction estimates to improve bounds on oscillatory integral operators and restriction problems, especially in higher dimensions, and relates these to Kakeya set dimensions.

Contribution

It applies multilinear restriction estimates to obtain new bounds on oscillatory integrals and restriction operators, with implications for Kakeya set dimensions in even dimensions.

Findings

Improved L^p estimates for restriction operators in dimension ≥5

Enhanced bounds for Hormander-type oscillatory integrals with positive definite quadratic phase in dimension ≥5

New estimates for oscillatory integrals in even dimensions related to Kakeya sets

Abstract

We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L^p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hormander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hormander-type oscillatory integral operators in even dimensions. These last oscillatory estimates are related to improved bounds on the dimensions of curved Kakeya sets in even dimensions.