On some new sharp results on Toeplits operators in the unit disk

We provide new sharp results on the action of Toeplitz operators from Triebel and Besov spaces to new BMOA-type function spaces on the unit disk. In this paper we consider $s\ge 1$ case in previous papers $s<1$ was covered for $BMO{A}_{s,q}$ and $BMO{A}_s^p$ spaces. We modify a little our previously known proofs.