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

## TL;DR

This paper establishes a new multilinear Fourier extension identity in higher dimensions, generalizing classical bilinear identities and extending known results for Schrödinger solutions to broader oscillatory integral operators.

## Contribution

It introduces a novel multilinear identity for Fourier extension operators in R^n, extending classical bilinear identities and applying to general oscillatory integrals with transversality conditions.

## Key findings

- Derived a multilinear extension identity in R^n.
- Extended Ozawa and Tsutsumi's identity to higher dimensions.
- Applied the identity to general oscillatory integral operators.

## Abstract

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 Schr\"odinger 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.