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

Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in H\"{o}rmander class $S^m_{0,\delta}$ with $m \in \mathbb{R}$ and $0 \leq \delta \leq 1$. 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 $0 \leq \delta < 1$, 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 $a \in S^{-n}_{0, \delta}$ with $0 \leq \delta < 1$. Additionally, we show that the dual operator $T_a^*$ is of weighted weak type (1,1) if $a \in L^\infty S^{-n}_0$. 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.