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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 $s \gt d/\!\min{(1,p,q)}-d$ for $0 \lt p \lt \infty$, and from $B_p^{s+m,q}$ to $B_{p}^{s,q}$ if $s \gt d/\!\min{(1,p)}-d$ for $0 \lt p\leq\infty$. 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 Hörmander’s twisted diagonal condition for arbitrary $s\in\mathbb{R}$. We also prove that the restrictions on $s$ are necessary for the boundedness to hold.