Endpoint regularity of general Fourier integral operators

Let $n\ge 1,0<\rho <1,\max \{\rho ,1-\rho \}\le \delta \le 1$ and $$m_1=\rho -n+(n-1)\min \{\frac{1}{2},\rho \}+\frac{1-\delta }{2}\mathrm{.}$$ If the amplitude $a$ belongs to the Hörmander class $S_{\rho ,\delta }^{m_1}$ and $\phi \in {\mathrm{\Phi }}^2$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{\phi ,a}$ 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*} is bounded from the local Hardy space $h^1({\mathbb{R}}^n)$ to $L^1({\mathbb{R}}^n)$. As a corollary, we can also obtain the corresponding $L^p({\mathbb{R}}^n)$-boundedness when $1<p<2$. These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When $0\le \rho \le 1,\delta \le \max \{\rho ,1-\rho \}$, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.