# Integral representations for some weighted classes of functions holomorphic in matrix domains

M. M. Djrbashian; A. H. Karapetyan

Annales Polonici Mathematici (1991)

- Volume: 55, Issue: 1, page 87-94
- ISSN: 0066-2216

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topM. M. Djrbashian, and A. H. Karapetyan. "Integral representations for some weighted classes of functions holomorphic in matrix domains." Annales Polonici Mathematici 55.1 (1991): 87-94. <http://eudml.org/doc/262402>.

@article{M1991,

abstract = {In 1945 the first author introduced the classes $H^p(α)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk with finite integral
(1) ∬ |f(ζ)|p (1-|ζ|²)α dξ dη < ∞ (ζ=ξ+iη)
and established the following integral formula for $f ∈ H^p(α)$:
(2) f(z) = (α+1)/π ∬ f(ζ) ((1-|ζ|²)α)/((1-zζ̅)2+α) dξdη, z∈ .
We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(Ω;[K(w)]^α dm(w))$, where:
1) $Ω = \{w = (w₁,...,w_n) ∈ ℂ^n: Im w₁ > ∑_\{k=2\}^n |w_k|²\}$, $K(w) = Im w₁ - ∑_\{k=2\}^n |w_k|²$;
2) Ω is the matrix domain consisting of those complex m × n matrices W for which $I^\{(m)\} - W·W*$ is positive-definite, and $K(W) = det[I^\{(m)\} - W·W*]$;
3) Ω is the matrix domain consisting of those complex n × n matrices W for which $Im W = (2i)^\{-1\} (W - W*)$ is positive-definite, and K(W) = det[Im W].
Here dm is Lebesgue measure in the corresponding domain, $I^\{(m)\}$ denotes the unit m × m matrix and W* is the Hermitian conjugate of the matrix W.},

author = {M. M. Djrbashian, A. H. Karapetyan},

journal = {Annales Polonici Mathematici},

keywords = {Siegel domain; matrix domains; generalized unit disk; generalized upper half-plane; weighted classes of holomorphic functions; integral representations; integral representation},

language = {eng},

number = {1},

pages = {87-94},

title = {Integral representations for some weighted classes of functions holomorphic in matrix domains},

url = {http://eudml.org/doc/262402},

volume = {55},

year = {1991},

}

TY - JOUR

AU - M. M. Djrbashian

AU - A. H. Karapetyan

TI - Integral representations for some weighted classes of functions holomorphic in matrix domains

JO - Annales Polonici Mathematici

PY - 1991

VL - 55

IS - 1

SP - 87

EP - 94

AB - In 1945 the first author introduced the classes $H^p(α)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk with finite integral
(1) ∬ |f(ζ)|p (1-|ζ|²)α dξ dη < ∞ (ζ=ξ+iη)
and established the following integral formula for $f ∈ H^p(α)$:
(2) f(z) = (α+1)/π ∬ f(ζ) ((1-|ζ|²)α)/((1-zζ̅)2+α) dξdη, z∈ .
We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(Ω;[K(w)]^α dm(w))$, where:
1) $Ω = {w = (w₁,...,w_n) ∈ ℂ^n: Im w₁ > ∑_{k=2}^n |w_k|²}$, $K(w) = Im w₁ - ∑_{k=2}^n |w_k|²$;
2) Ω is the matrix domain consisting of those complex m × n matrices W for which $I^{(m)} - W·W*$ is positive-definite, and $K(W) = det[I^{(m)} - W·W*]$;
3) Ω is the matrix domain consisting of those complex n × n matrices W for which $Im W = (2i)^{-1} (W - W*)$ is positive-definite, and K(W) = det[Im W].
Here dm is Lebesgue measure in the corresponding domain, $I^{(m)}$ denotes the unit m × m matrix and W* is the Hermitian conjugate of the matrix W.

LA - eng

KW - Siegel domain; matrix domains; generalized unit disk; generalized upper half-plane; weighted classes of holomorphic functions; integral representations; integral representation

UR - http://eudml.org/doc/262402

ER -

## References

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- [2] Š. A. Dautov and G. M. Henkin, The zeroes of holomorphic functions of finite order and weight estimates for solutions of the ∂̅-equation, Mat. Sb. 107 (1978), 163-174 (in Russian). Zbl0392.32001
- [3] A. È. Djrbashyan and F. A. Shamoyan, Topics in the Theory of ${A}_{\alpha}^{p}$ Spaces, Teubner-Texte zur Math. 105, Teubner, Leipzig 1988.
- [4] M. M. Djrbashian, On the representability of certain classes of functions meromorphic in the unit disk, Akad. Nauk Armyan. SSR Dokl. 3 (1945), 3-9 (in Russian).
- [5] M. M. Djrbashian, On the problem of representing analytic functions, Soobshch. Inst. Mat. Mekh. Akad. Nauk Armyan. SSR 2 (1948), 3-40 (in Russian).
- [6] M. M. Djrbashian, A survey of some achievements of Armenian mathematicians in the theory of integral representations and factorization of analytic functions, Mat. Vesnik 39 (1987), 263-282. Zbl0642.30026
- [7] M. M. Djrbashian, A brief survey of the results obtained by Armenian mathematicians in the field of factorization of meromorphic functions and its applications, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 23 (6) (1988), 517-545 (in Russian).
- [8] M. M. Djrbashian and A. È. Djrbashyan, Integral representations for some classes of functions analytic in the half-plane, Dokl. Akad. Nauk SSSR 285 (3) (1985), 547-550 (in Russian).
- [9] M. M. Djrbashian and A. H. Karapetyan, Integral representations for some classes of functions analytic in a Siegel domain, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (4) (1987), 399-405 (in Russian).
- [10] M. M. Djrbashian and A. H. Karapetyan, Integral representations in the generalized unit disk, ibid. 24 (6) (1989), 523-546 (in Russian).
- [11] M. M. Djrbashian and A. H. Karapetyan, Integral representations in the generalized upper half-plane, ibid. 25 (6) (1990) (in Russian).
- [12] F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (6) (1974), 593-602. Zbl0297.47041
- [13] S. G. Gindikin, Analysis in homogeneous domains, Uspekhi Mat. Nauk 19 (4) (1964), 3-92 (in Russian). Zbl0144.08101
- [14] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Inostr. Liter., Moscow 1959 (in Russian).
- [15] M. Stoll, Mean value theorems for harmonic and holomorphic functions on bounded symmetric domains, J. Reine Angew. Math. 290 (1977), 191-198. Zbl0342.32003

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