Bounds on oscillatory integral operators based on multilinear estimates
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.
Cite
@article{arxiv.1012.3760,
title = {Bounds on oscillatory integral operators based on multilinear estimates},
author = {Jean Bourgain and Larry Guth},
journal= {arXiv preprint arXiv:1012.3760},
year = {2011}
}
Comments
69 pages, 7 figures. Improved following suggestions from the referee: slightly sharper results and more references. To be published in Geometric and Functional Analysis. Reference added: Some of our estimates for Hormander-type oscillatory integral operators with positive-definite phase function were proven earlier by Lee