Fourier integral operators on Hardy spaces with Hormander class

TL;DR

This paper proves boundedness of Fourier integral operators on Hardy spaces under Hormander class conditions, extending previous results and including special cases like the Seeger-Sogge-Stein theorem.

Contribution

It establishes new boundedness results for Fourier integral operators on Hardy spaces with Hormander class amplitudes, generalizing and improving prior theorems.

Findings

Boundedness from local Hardy space to L^p for certain amplitude classes.

Extension of results to cases with compactly supported amplitudes.

Generalization of the Seeger-Sogge-Stein theorem for n ≥ 2.

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 $a\in S^{0}_{1,1}$.