Sharp estimates for oscillatory integral operators via polynomial partitioning

Jonathan Hickman (Department of Mathematics, University of Chicago, Illinois, U.S.A.; and School of Mathematics, University of Edinburgh, United Kingdom)

Marina Iliopoulou (Department of Mathematics, University of California at Berkeley; and School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, United Kingdom)

Abstract

The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments. The main result implies improved bounds for the Bochner–Riesz conjecture in dimensions $n \geqslant 4$.

Full Text (PDF format)

Received 6 November 2017

Accepted 5 May 2019

Published 19 December 2019