On new sharp theorems on Toeplits operators in Lizorkin–Triebel spaces

In this short note we provided new sharp results concerning the action of so called integral Toeplits operators in holomorphic Lizorkin-Triebel function spaces in the unit polydisk. Holomorphic function spaces of Lizorkin-Triebel type in the unit polydisk extend well known Bergman and Hardy spaces simultuanously. Because of this it is interesting to study these new spaces in higher dimension. Our results may have various interesting applications in complex function theory. Of one and several complex variables. Our proofs in particular are based on several standard estimates of complex function theory of several complex variables. We also mention that Toeplits operators were mainly studied previously only in one dimension namely in the unit disk case. Our results may have also interesting applications in operator theory.