Acta Mathematica

Volume 223 (2019)

Number 2

Sharp estimates for oscillatory integral operators via polynomial partitioning

Pages: 251 – 376



Larry Guth (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

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)


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 $n \geqslant 4$.

Received 6 November 2017

Accepted 5 May 2019

Published 19 December 2019