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 $ϕ$ is the phase. Let $0\leqρ\leq 1,n\geq 2$ or $0\leqρ<1,n=1$ and $$m_p=\frac{ρ-n}{p}+(n-1)\min\{\frac 12,ρ\}.$$ If $a$ belongs to the forbidden Hörmander class $S^{m_p}_{ρ,1}$ and $ϕ\in Φ^{2}$ satisfies the strong non-degeneracy condition, then for any $\frac {n}{n+1}<p\leq 1$, we can show that the Fourier integral operator $T_{ϕ,a}$ 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 $S^{m_p}_{ρ,δ}\subset S^{m_p}_{ρ,1}$ for any $0\leq δ\leq 1$, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for $a\in S^{m}_{ρ,δ}$ when $δ$ is close to 1. As an important special case, when $n\geq 2$, we show that $T_{ϕ,a}$ 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 $a\in S^{0}_{1,1}$.