Sharp estimates for pseudo-differential operators of type (1,1) on Triebel-Lizorkin and Besov spaces

Pseudo-differential operators of type $(1,1)$ and order $m$ are continuous from $F_p^{s+m,q}$ to $F_p^{s,q}$ if $s>d\mathrm{/}\min (1,p,q)-d$ for $0<p<\mathrm{\infty }$, and from $B_p^{s+m,q}$ to $B_p^{s,q}$ if $s>d\mathrm{/}\min (1,p)-d$ for $0<p\le \mathrm{\infty }$. In this work we extend the $F$-boundedness result to $p=\mathrm{\infty }$. Additionally, we prove that the operators map $F_{\mathrm{\infty }}^{m,1}$ into $bmo$ when $s=0$, and consider H"ormander’s twisted diagonal condition for arbitrary $s\in \mathbb{R}$. We also prove that the restrictions on $s$ are necessary conditions for the boundedness to hold.