$(L^{\infty},{\rm BMO})$ estimates and $(H^{1},L^{1})$ estimates for Fourier integral operators with symbol in $S^{m}_{0,δ}$

Abstract: Let $T_{a,\varphi}$ be a Fourier integral operator defined with $a\in S^{m}_{0,\delta}$ with $0\leq\delta<1$ and $\varphi\in \Phi^{2}$ satisfying the strong non-degenerate condition. It is showed that $T_{a,\varphi}$ is a bounded operator from $L^{\infty}(\mathbb{R}^n)$ to ${\rm BMO}(\mathbb{R}^n)$, if $$m\leq -\frac{n}{2},$$ and from $H^{1}(\mathbb{R}^n)$ to $L^{1}(\mathbb{R}^n)$, if $$m\leq -\frac{n}{2}-\frac{n}{2}\delta.$$