$$L^{1}$$-boundedness of rough Fourier integral operators

Abstract

In this paper, we study the $$L^{1}$$ boundedness of Fourier integral operator $$T_{\phi ,a}$$ with rough symbol $$a\in L^{\infty }S^{m}_{\rho }$$ and a new class of rough phase $$\phi $$ . In this class, we extend the $$L^{\infty }\Phi ^{2}$$ and non-degeneracy conditions to some generalized derivative estimation and some measure condition respectively. Our main result substantially extends and improves some known results about $$L^{1}$$ boundedness of Fourier integral operator. Moreover, the result in this paper can be used to prove the boundedness of the maximal wave operator.