On some new estimates for integrals of the Lusin's square function in the unit polydisk

The purpose of the note is to obtain new estimates for the quasinorm of Hardy's analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables.
Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in $\mathbb{C}^n$. Let further $D$ be a bounded or unbounded domain in $\mathbb{C}^n$. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to $D\times\dots\times D$. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk.