Mathematical Research Letters

Volume 25 (2018)

Number 4

A multilinear Fourier extension identity on $\mathbb{R}^n$

Pages: 1089 – 1108



Jonathan Bennett (School of Mathematics, University of Birmingham, United Kingdom)

Marina Iliopoulou (Department of Mathematics, University of California at Berkeley)


We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa and Tsutsumi for solutions to the free time-dependent Schrödinger equation. We conclude with a similar treatment of more general oscillatory integral operators whose phase functions collectively satisfy a natural multilinear transversality condition. The perspective we present has its origins in work of Drury.

This work was supported by the European Research Council [grant number 307617] and a Postdoctoral Fellowship at the Mathematical Sciences Research Institute, Berkeley, CA, USA.

Received 3 February 2017

Accepted 14 May 2017

Published 16 November 2018