Pseudo-differential operators with forbidden symbols on Triebel-Lizorkin spaces

In this note, we consider a pseudo-differential operator $T_a$ defined as \begin{align*} T_a f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}a(x,\xi)\widehat{f}(\xi)d\xi. \end{align*} It is well-known that $T_a$ is not bounded on $L^2$ in general when $a$ belongs to the forbidden H\"{o}rmander class $S^{n(\rho-1)/2}_{\rho,1},0\leq \rho \leq 1$. In this note, when $s>0,0\leq \rho \leq 1,1\leq r\leq 2$ and $a\in S^{n(\rho-1)/r}_{\rho,1}$, we prove that $T_a$ is bounded on the Triebel-Lizorkin space $F^s_{p,q}$ if $r 0$, if $2 Mathematics Subject Classification: 42B20 · 35S30.