Weighted weak-type (1, 1) inequalities for pseudo-differential operators with symbol in $S_{0,\delta }^m$

Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in Hörmander class $S_{0,\delta }^m$ with $m\in \mathbb{R}$ and $0\le \delta \le 1$. It is well-known that the weak type (1,1) behavior of $T_a$ is not fully understood when the index $m$ is equal to the possibly optimal value $-\frac{n}{2}-\frac{n}{2}\delta$ for $0\le \delta <1$, and that $T_a$ is not of weak type (1,1) when $m=-n$ and $\delta =1$. In this note, we prove that $T_a$ is of weighted weak type (1,1) if $a\in S_{0,\delta }^{-n}$ with $0\le \delta <1$. Additionally, we show that the dual operator $T_a^{\ast }$ is of weighted weak type (1,1) if $a\in L^{\mathrm{\infty }}{S}_0^{-n}$. We also identify $m=-n$ as a critical index for these weak type estimates. As applications, we derive weighted weak type (1,1) estimates for certain classes of Fourier integral operators.