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/min (1, p, q) - d for 0 < p < ∞, and from B_p^s+^m,q to B_p^s,q if s > d/min (1, p) - d for 0 < p ≤ ∞. In this work we extend the F-boundedness result to p = ∞. Additionally, we prove that the operators map F_∞^m,1 into bmo when s = 0, and consider Hörmander’s twisted diagonal condition for arbitrary s ∈ R. We also prove that the restrictions on s are necessary for the boundedness to hold.