Some generalizations of Littlewood-Paley inequality in the polydisc

Abstract

The paper generalizes the well-known inequality of Littlewood-Paley in the polydisc.We establish a family of inequalities which are analogues and extensions of Littlewood-Paleytype inequalities proved by Sh.\ Yamashita and D. Luecking in the unit disk.Some other generalizations of the Littlewood-Paley inequality are statedin terms of anisotropic Triebel-Lizorkin spaces.With the help of an extension of Hardy-Stein identity, we also obtain area inequalities and representationsfor quasi-norms in weighted spaces of holomorphic functions in the polydisc.