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Bounded projections in weighted function spaces in a generalized unit disc

100%

Karapetyan A.

Annales Polonici Mathematici

1995

tom 62

nr 3

193-218

Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br> $R_{m,n} = {Z ∈ M_{m,n}: I^{(m)} - ZZ* is positive definite}$. Here $I^{(m)} ∈ M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L^{p}_{α}(R_{m,n})$ is defined to be the space $L^p{R_{m,n}; [det(I^{(m)} - ZZ*)]^α dμ_{m,n}(Z)}$, where $μ_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H^p_α(R_{m,n}) ⊂ L^{p}_{α}(R_{m,n})$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Reβ > (α+1)/p -1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then $f(𝒵)= T^{β}_{m,n}(f)(𝒵), 𝒵 ∈ R_{m,n}, where $T^{β}_{m,n}$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p < ∞, we find conditions on α and β for $T^{β}_{m,n}$ to be a bounded projection of $L^p_α(R_{m,n})$ onto $H^p_α(R_{m,n})$. Some applications of this result are given.

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

63%

Djrbashian M., Karapetyan A.

Annales Polonici Mathematici

1991

tom 55

nr 1

87-94

In 1945 the first author introduced the classes $H^p(α)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk 𝔻 with finite integral (1) ∬_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 ∬_𝔻 f(ζ) ((1-|ζ|²)^α)/((1-zζ̅)^{2+α}) dξdη, z∈ 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.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Karapetyan A.

1 Djrbashian M.

1 1995

1 1991

Wyniki wyszukiwania

Bounded projections in weighted function spaces in a generalized unit disc

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