Fourier integral operators on Hardy spaces with Hormander class

Abstract

In this note, we consider a Fourier integral operator defined by \begin{align*} T_{ϕ,a}f(x) = \int_{\mathbb{R}^{n}}e^{iϕ(x,ξ)}a(x,ξ)\widehat{f} ξ)dξ, \end{align*}here $a$ is the amplitude, and $\ddot{I}•$ is the phase. Let $0\le \ddot{I}\le 1,n\ge 2$ or $0\le \ddot{I}<1,n=1$ and $$m_p=\frac{\ddot{I}-n}{p}+(n-1)\min \{\frac{1}{2},\ddot{I}\}.$$ If $a$ belongs to the forbidden Hörmander class $S_{\ddot{I},1}^{m_p}$ and $\ddot{I}•\in \hat{I}{¦}^2$ satisfies the strong non-degeneracy condition, then for any $\frac{n}{n+1}<p\le 1$, we can show that the Fourier integral operator $T_{\ddot{I}•,a}$ 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 $0<p\le 1$. As $S_{\ddot{I},\hat{I}´}^{m_p}\subset S_{\ddot{I},1}^{m_p}$ for any $0\le \hat{I}´\le 1$, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for $a\in S_{\ddot{I},\hat{I}´}^m$ when $\hat{I}´$ is close to 1. As an important special case, when $n\ge 2$, we show that $T_{\ddot{I}•,a}$ is bounded from $H^1$ to $L^1$ if $a\in S_{1,1}^{(1-n)/2}$ which is a generalization of the well-known Seeger-Sogge-Stein theorem for $a\in S_{1,0}^{(1-n)/2}$. This result is false when $n=1$ and $a\in S_{1,1}^0$.

24 pages