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

Abstract

In 1945 the first author introduced the classes $H^p(α)$, 1 ≤ p -1, of holomorphic functions in the unit disk with finite integral (1) $∬_\mathbb{D} |f(ζ)|^p (1-|ζ|²)^α dξ dη < ∞ (ζ=ξ+iη)$ and established the following integral formula for $f ∈ H^p(α)$: (2) f(z) = (α+1)/π ∬_\mathbb{D} f(ζ) ((1-|ζ|²)^α)//((1-zζ̅)^{2+α}) dξdη, z∈ \mathbb{D} . 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