On $$L^2$$ boundedness of rough Fourier integral operators

Abstract

In this paper, let \(T_{a,\varphi }\) be a Fourier integral operator with rough amplitude \(a \in {L^\infty }S_\rho ^m\) and rough phase \(\varphi \in {L^\infty }{\Phi ^2}\) which satisfies a new class of rough non-degeneracy condition. When \(0 \leqslant \rho \leqslant 1\), if \(m < \frac{{n(\rho - 1)}}{2} - \frac{{\rho (n - 1)}}{4}\), we obtain that \(T_{a,\varphi }\) is bounded on \({L^2}\). Our main result extends and improves some known results about \({L^2}\) boundedness of Fourier integral operators.