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 $s \gt d/\!\min{(1,p,q)}-d$ 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\inf