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

M. 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 -