On the action of Toeplits operators into new BMOA type spaces 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. We modify little our previously known proofs. We show our theorems using standard estimates of complex function theory. The proof of sufficiency and neccesity parts of our theorems follow the same type arguments as in previous already known cases. In particular we use standart test function for the proof of neccesity part and the same line of arguments as in previous already known cases. Our theorems may have many applications in complex function theory. Previously such type theorems were known in less general classses of BMOA type in the unit disk.