Fourier integral operators on Hardy spaces with amplitudes in forbidden Hörmander classes

In this note, we consider a Fourier integral operator defined by $T_{\varphi ,a}f\left(x\right)={\int }_{R^n}{e}^{i\varphi (x,\xi )}a(x,\xi )\hat{f}\left(\xi \right)d\xi ,$where $a$ is the amplitude, and $\varphi$ is the phase.

Let $0\le \rho \le 1,n\ge 2$ or $0\le \rho <1,n=1$ and $m_p=\frac{\rho -n}{p}+(n-1)min\{\frac{1}{2},\rho \}.$If $a$ belongs to the forbidden Hörmander class $S_{\rho ,1}^{m_p}$ and $\varphi \in {\Phi }^2$ satisfies the strong non-degeneracy condition, then for any $\frac{n}{n+1}<p\le 1$, we can show that the Fourier integral operator $T_{\varphi ,a}$ 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 $S_{\rho ,\delta }^{m_p}\subset S_{\rho ,1}^{m_p}$ for any $0\le \delta \le 1$, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for $a\in S_{\rho ,\delta }^m$ when $\delta$ is close to 1.

As an important special case, when $n\ge 2$, we show that $T_{\varphi ,a}$ 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 $a\in S_{1,1}^0$.