Local \(L^2\)-boundedness of rough Fourier integral operators with the rough corank condition

Abstract

In this paper, we consider a rough Fourier integral operator defined as

$$T_{\phi ,a}f(x)=\int \limits _{\mathbb {R}^{n}}e^{i\phi (x,\xi )}a(x,\xi )\hat{f}(\xi )d\xi ,$$

where the amplitude \(a\in L^{\infty }S^{m}_{\rho }\) and the phase \(\phi \in L^{\infty }\Phi ^{2}\) satisfy the rough k-corank condition. The motivation for this problem stems from the regularity of the maximal wave operator. We prove that this operator is bounded from \(L^{2}\) to \(L_{\text {loc}}^{2}\) provided

$$m<\min \left\{ \frac{n(\rho -1)}{2},\frac{\rho }{2}-\frac{n+1}{4}\right\} -\frac{k\rho }{2}.$$