Global $L^{2}$-boundedness of a New Class of Rough Fourier Integral Operators

Abstract

In this paper, we investigate the $L^{2}$ boundedness of Fourier integral operator $T_{\phi,a}$ with rough symbol $a \in L^{\infty} S^{m}_{\rho}$ and rough phase $\phi \in L^{\infty} \Phi^{2}$ which satisfies $\big| \{ x: |\nabla_{\xi} \phi(x,\xi) - y| \leq r \} \big| \leq C(r^{n-1}+r^{n})$ for any $\xi,y \in \mathbb{R}^{n}$ and $r \gt 0$. We obtain that $T_{\phi,a}$ is bounded on $L^2$ if $m \lt \rho(n-1)/2 - n/2$ when $0 \leq \rho \leq 1/2$ or $m \lt -(n+1)/4$ when $1/2 \leq \rho \leq 1$. When $\rho = 0$ or $n = 1$, the condition of $m$ is sharp. Moreover, the maximal wave operator is a special class of $T_{\phi,a}$ which is studied in this paper. Thus, our main theorem substantially extends and improves some known results about the maximal wave operator.