Weighted anisotropic integral representations of holomorphic functions in the unit ball of \(\mathbb C^{n}\). (English) Zbl 1225.32008

Author’s abstract: “We obtain 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 \(\prod_{i=1}^n (1-|w_1|^2-|w_2|^2- \dots - |w_i|^2)^{a_i}\), \(w=(w_1, \dots , w_n)\in B_n\). The corresponding kernels of these representations are estimated, written in an integral form, and even written out in an explicit form (for \(n=2\)).”


32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
Full Text: DOI EuDML


[1] M. M. Djrbashian, “On the representability of certain classes of functions meromorphic in the unit disc,” Doklady. Izdatel’stvo Akademii Nauk Armyanskoj SSR, vol. 3, no. 1, pp. 3-9, 1945 (Russian). · Zbl 0774.30037
[2] M. M. Djrbashian, “On the problem of representability of analytic functions,” Soobshch. Inst. Mat. Mekh. Akad. Nauk Arm. SSR, vol. 2, pp. 3-40, 1948 (Russian).
[3] F. Forelli and W. Rudin, “Projections on spaces of holomorphic functions in balls,” Indiana University Mathematics Journal, vol. 24, pp. 593-602, 1974/75. · Zbl 0297.47041 · doi:10.1512/iumj.1974.24.24044
[4] M. M. Djrbashian, “Survey of some achievements of Armenian mathematicians in the theory of integral representations and factorization of analytic functions,” Matematicheski Vesnik, vol. 39, no. 3, pp. 263-282, 1987. · Zbl 0642.30026
[5] M. M. Djrbashian, “A brief survey of the results obtained by Armenian mathematicians in the field of factorization theory of meromorphic functions and its applications,” Journal of Contemporary Mathematical Analysis, vol. 23, no. 6, pp. 8-34, 1988. · Zbl 0683.01009
[6] A. E. Djrbashian and F. A. Shamoyan, Topics in the Theory of Ap\alpha Spaces, vol. 105 of Teubner-Texte zur Mathematik, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, Germany, 1988. · Zbl 0667.30032
[7] W. Rudin, Function Theory in the Unit Ball of Cn, vol. 241 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1980. · Zbl 0495.32001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.