Some remarks on the operator of diagonal map and on multifunctional analytic spaces

This article has nature of announcement of new results.
We consider the action of the operator of diagonalization and we assert the boundedness of this operator from one holomorphic space to another one. Assertions in the bidisc extend previously known onedimensional estimates. Also new sharp theorems on diagonal map are given. These results essentially compliment previously known subtle results on diagonal map in polydisc. Some estimates for multifunctional analytic spaces in higher dimension are also given. These inequalities were previously known in the case of unit disc.