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

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.

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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