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 < infinity, and from B-p(s+m,q) to B-p(s,q) if s > d/min (1, p) - d for 0 < p < infinity. In this work we extend the F-boundedness result to p = infinity. Additionally, we prove that the operators map F-infinity(m,1) into bmo when s = 0, and consider Hormander's twisted diagonal condition for arbitrary s is an element of R. We also prove that the restrictions on s are necessary for the boundedness to hold.