On the boundedness of certain bilinear Fourier integral operators

TL;DR

This paper establishes the boundedness of certain bilinear Fourier integral operators on L^2 spaces, extending classical results for pseudodifferential operators to a broader class of Fourier integral operators.

Contribution

It proves the global L^2 x L^2 to L^1 boundedness for bilinear Fourier integral operators with specific phase functions and amplitude classes, extending prior pseudodifferential operator results.

Findings

Proves boundedness of bilinear Fourier integral operators in L^2 spaces.

Extends classical pseudodifferential operator results to Fourier integral operators.

Requires phase functions to be decomposable and satisfy strong non-degeneracy conditions.

Abstract

We prove the global $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier integral operators with amplitudes in $S^0_{1,0} (n,2)$. To achieve this, we require that the phase function can be written as $(x,\xi,\eta) \mapsto \phase_1(x,\xi) + \phase_2(x,\eta)$ where each $\phase_j$ belongs to the class $\Phi^2$ and satisfies the strong non-degeneracy condition. This result extends that of R. Coifman and Y. Meyer regarding pseudodifferential operators to the case of Fourier integral operators.