### On some properties of a differential operator on the polydisk

Songxiao Li and Romi Shamoyan
Source: Banach J. Math. Anal. Volume 3, Number 1 (2009), 68-84.

#### Abstract

We study the action and properties of a differential operator in the polydisk, extending some classical results from the unit disk. Using so called dyadic decomposition of the polydisk we find precise connections between quazinorms of holomorphic function in the polydisk with quazinorms on the subframe and the unit disk. All our results were previously well-known in the unit disk.

First Page:
Primary Subjects: 32A18
Secondary Subjects: 32A36
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.bjma/1240336425
Mathematical Reviews number (MathSciNet): MR2461748
Zentralblatt MATH identifier: 05379951

### References

K. L. Avetisyan and R. F. Shamoyan, Some generalization of Littlewood-Paley inequality in the polydisk, Matematicki Vesnik, 58 (3-4) (2006), 97–110.
Mathematical Reviews (MathSciNet): MR2318224
S. M. Buckley, P. Koskela and D. Vukotic, Fractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Cambridge Phil. Soc., 126 (1999), 369–385.
Mathematical Reviews (MathSciNet): MR1670257
Digital Object Identifier: doi:10.1017/S030500419800334X
Zentralblatt MATH: 0930.42007
W. Cohn, Weighted Bergman projections and tangential area integrals, Studia Math., 106 (1993), 59–76.
Mathematical Reviews (MathSciNet): MR1226424
Zentralblatt MATH: 0811.32001
A. E. Djrbashian and F. A. Shamoian, Topics in the Theory of $A\sp p\sb \alpha$ Spaces, Leipzig, Teubner, 1988.
Mathematical Reviews (MathSciNet): MR1021691
Zentralblatt MATH: 0667.30032
P. L. Duren, Theory of $H\spp$ Spaces, Academic Press, New York, 1970.
Mathematical Reviews (MathSciNet): MR268655
Zentralblatt MATH: 0215.20203
V. S. Guliev and P. I. Lizorkin, Spaces of holomorphic functins and harmonic functions in the polydisk and their connections with boundary values, Trudy Mat. Inst. Steklov RAN, 204 (1993), 137–159.
Mathematical Reviews (MathSciNet): MR1320022
M. I. Gvaradze, Multipliers of a class of analytic functions defined on a polydisc, Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze, 66 (1980), 15–21.
Mathematical Reviews (MathSciNet): MR613150
H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, 199, Springer, New York, 2000.
Mathematical Reviews (MathSciNet): MR1758653
Zentralblatt MATH: 0955.32003
V. G. Mazya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
Mathematical Reviews (MathSciNet): MR817985
J. M. Ortega and J. Fàbrega, Hardy's inequality and embeddings in holomorphic Triebel-Lizorkin spaces, Illinois J. Math., 43 (1999), 733–751.
Mathematical Reviews (MathSciNet): MR1712520
Zentralblatt MATH: 0936.32004
R. F. Shamoyan, On multipliers from spaces of Bergman type to Hardy spaces in polydisk, Ukrainian Math. J., 52 (10) (2000), 1606–1617.
Mathematical Reviews (MathSciNet): MR1830671
Digital Object Identifier: doi:10.1023/A:1010457218795
R. F. Shamoyan, On quasinorms of functions from holomorphic Lizorkin Triebel type spaces on subframe and polydisk, Sympozium Fourier series and application, Conference materials, Rostov Na Donu, in Russian, 2002, 54-55.
A. L. Shields and D. L. Williams, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162 (1971), 287–302.
Mathematical Reviews (MathSciNet): MR283559
Digital Object Identifier: doi:10.2307/1995754
R. M. Trigub, Multipliers in the Hardy spaces $H^p(D^m)$ with $p \in (0,1]$ and approxiamation properties of summability methods for power series, Mat. Sb., 188 (4) (1997), 145–160.
Mathematical Reviews (MathSciNet): MR1462032
W. Rudin, Function Theory in the Polydisk, Benjamin, New York, 1969.
Mathematical Reviews (MathSciNet): MR255841
K. Zhu, Weighted Bergman projections on the polydisk, Houston J. Math., 20 (2) (1994), 275–292.
Mathematical Reviews (MathSciNet): MR1283276
Zentralblatt MATH: 0818.32007
K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, New York, 2005.
Mathematical Reviews (MathSciNet): MR2115155
A. Zygmund, Trigonometric Series, (2nd.ed.), Vol.2, Cambridge Univ. Press, Cambridge, 1959.
Mathematical Reviews (MathSciNet): MR107776

2013 © Tusi Mathematical Research Group