Fourier integral operators on Hardy spaces with Hormander class

Abstract

In this note, we consider a Fourier integral operator defined by \begin{align*} T_{\phi,a}f(x) = \int_{\mathbb{R}^{n}}e^{i\phi(x,\xi)}a(x,\xi)\widehat{f} \xi)d\xi, \end{align*}here $a$ is the amplitude, and $\phi$ is the phase. Let $0\leq\rho\leq 1,n\geq 2$ or $0\leq\rho<1,n=1$ and $$m_p=\frac{\rho-n}{p}+(n-1)\min\{\frac 12,\rho\}.$$ If $a$ belongs to the forbidden H\"{o}rmander class $S^{m_p}_{\rho,1}$ and $\phi\in \Phi^{2}$ satisfies the strong non-degeneracy condition, then for any $\frac {n}{n+1}<p\leq 1$, we can show that the Fourier integral operator $T_{\phi,a}$ is bounded from the local Hardy space $h^p$ to $L^p$. Furthermore, if $a$ has compact support in variable $x$, then we can extend this result to $0<p\leq 1$. As $S^{m_p}_{\rho,\delta}\subset S^{m_p}_{\rho,1}$ for any $0\leq \delta\leq 1$, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for $a\in S^{m}_{\rho,\delta}$ when $\delta$ is close to 1. As an important special case, when $n\geq 2$, we show that $T_{\phi,a}$ is bounded from $H^1$ to $L^1$ if $a\in S^{(1-n)/2}_{1,1}$ which is a generalization of the well-known Seeger-Sogge-Stein theorem for $a\in S^{(1-n)/2}_{1,0}$. This result is false when $n=1$ and $a\in S^{0}_{1,1}$.