Geodesic
Inequalities of Littlewood--Paley Type for $n$-Harmonic Functions on the Polydisk
Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 483-492.

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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. 
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K. L. Avetisyan. Inequalities of Littlewood--Paley Type for $n$-Harmonic Functions on the Polydisk. Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 483-492. http://geodesic.mathdoc.fr/item/MZM_2004_75_4_a0/

[1] Littlewood J. E., Paley R. E. A. C., “Theorems on Fourier series and power series. I; II”, J. London Math. Soc., 6 (1931), 230–233 ; Proc. London Math. Soc. Ser. 2, 42 (1936), 52–89 | DOI | Zbl | DOI | Zbl

[2] Zigmund A., Trigonometricheskie ryady, T. II, Mir, M., 1965 | MR

[3] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[4] Flett T. M., “Mean values of power series”, Pacific J. Math., 25 (1968), 463–494 | MR | Zbl

[5] Littlewood J. E., Some Problems in Real and Complex Analysis, Raytheon Education Company, Massachusetts, 1968 | MR | Zbl

[6] Zygmund A., “On the boundary values of functions of several complex variables, I”, Fund. Math., 36 (1949), 207–235 | MR | Zbl

[7] Gundy R., Stein E. M., “$H^p$ theory for the poly-disc”, Proc. Nat. Acad. Sci. USA, 76:3 (1979), 1026–1029 | DOI | MR | Zbl

[8] Benedek A., Panzone R., “The spaces $L^p$ with mixed norm”, Duke Math. J., 28 (1961), 301–324 | DOI | MR | Zbl

[9] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR | Zbl