Inequalities of Littlewood–Paley Type for $n$-Harmonic Functions on the Polydisk

For $n$-harmonic functions on the unit polydisk in the space $\mathbb C^n$ we define $g$-functions of Littlewood–Paley type and establish $L^p$-inequalities related to them. In the present paper, the main theorems deal with the extension of results of Littlewood, Paley, and Flett to the polydisk and their generalizion to fractional derivatives of arbitrary order. This gives an answer to a question posed by Littlewood.