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Pseudo-differential operators of type $(1,1)$ and order $m$ are continuous from $F_p^{s+m,q}$ to $F_p^{s,q}$ if $s>\max{(0,d({1}/{p}-1),d({1}/{q}-1))}$ for $0 \max{(0,d({1}/{p}-1))}$ for $0<p\leq\infty$. In this work we prove the sharpness of the conditions on $s$, and extend the $F$-boundedness result to $p=\infty$. We also prove that the operators map $F_{\infty}^{m,1}$ into $bmo$ when $s=0$. The boundedness of the operators also hold for arbitrary $s$ with H\"ormander's condition on the twisted diagonal.

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