Abstract and Applied Analysis
Volume 2010 (2010), Article ID 354961, 23 pages
doi:10.1155/2010/354961
Research Article

Weighted Anisotropic Integral Representations of Holomorphic Functions in the Unit Ball of 𝐶 𝑛

Arman Karapetyan

Institute of Mathematics, National Academy of Sciences of Armenia, Marshal Bagramian Avenue 24 B, Yerevan 375019, Armenia

Received 15 November 2010; Accepted 27 December 2010

Academic Editor: Elena Litsyn

Copyright © 2010 Arman Karapetyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain weighted integral representations for spaces of functions holomorphic in the unit ball 𝐵 𝑛 and belonging to area-integrable weighted 𝐿 𝑝 -classes with “anisotropic” weight function of the type ∏ 𝑛 𝑖 = 1 ( 1 − | 𝑤 1 | 2 − | 𝑤 2 | 2 − ⋯ − | 𝑤 𝑖 | 2 ) 𝛼 𝑖 , 𝑤 = ( 𝑤 1 , 𝑤 2 , … , 𝑤 𝑛 ) ∈ 𝐵 𝑛 . The corresponding kernels of these representations are estimated, written in an integral form, and even written out in an explicit form (for 𝑛 = 2 ).