Sharp local smoothing estimates for Fourier integral operators
Abstract
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.
Keywords
Cite
@article{arxiv.1812.11616,
title = {Sharp local smoothing estimates for Fourier integral operators},
author = {David Beltran and Jonathan Hickman and Christopher D. Sogge},
journal= {arXiv preprint arXiv:1812.11616},
year = {2019}
}
Comments
58 pages, 9 figures, 3 tables. Revised version incorporating referee suggestions. Survey article for the Proceedings of the conference 'Geometric Aspects of Harmonic Analysis', on the occasion of Fulvio Ricci's 70th birthday