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

Abstract: Pseudo-differential operators of type $(1,1)$ and order $m$ are continuous from $F_p^{s+m,q}$ to $F_p^{s,q}$ if $s>d/\min{(1,p,q)}-d$ for $0<p<\infty$, and from $B_p^{s+m,q}$ to $B_{p}^{s,q}$ if $s>d/\min{(1,p)}-d$ for $0<p\leq\infty$. In this work we extend the $F$-boundedness result to $p=\infty$. Additionally, we prove that the operators map $F_{\infty}^{m,1}$ 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.