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

Abstract: 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.