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## Annales Polonici Mathematici

1995 | 62 | 3 | 193-218
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### Bounded projections in weighted function spaces in a generalized unit disc

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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})$. Some applications of this result are given. Słowa kluczowe EN Kategorie tematyczne Czasopismo Rocznik Tom Numer Strony 193-218 Opis fizyczny Daty wydano 1995 otrzymano 1993-04-14 Twórcy autor • Institute of Mathematics, Armenian Academy of Sciences, Marshal Bagramian Ave. 24 B, Erevan 375019, Armenia Bibliografia •  M. Andersson, Formulas for the L²-minimal solutions of the ∂∂̅-equation in the unit ball of$ℂ^N$, Math. Scand. 56 (1985), 43-69. •  A. E. Djrbashian and A. H. Karapetyan, Integral inequalities between conjugate pluriharmonic functions in multidimensional domains, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 23 (1988), 216-236 (in Russian). •  A. E. Djrbashian and F. A. Shamoian, Topics in the Theory of$A_α^p$Spaces, Teubner-Texte Math. 105, Teubner, Leipzig, 1988. •  M. M. Djrbashian, On the representability of certain classes of functions meromorphic in the unit disc, Dokl. Akad. Nauk Armyan. SSR 3 (1945), 3-9 (in Russian). •  M. M. Djrbashian, On the problem of representability of analytic functions, Soobshch. Inst. Mat. Mekh. Akad. Nauk Armyan. SSR 2 (1948), 3-40 (in Russian). •  M. M. Djrbashian, Survey of some achievements of Armenian mathematicians in the theory of integral representations and factorization of analytic functions, Mat. Vesnik 39 (1987), 263-282. •  M. M. Djrbashian, A brief survey of the results obtained by Armenian mathematicians in the field of factorization theory of meromorphic functions and its applications, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 23 (1988), 517-545 (in Russian). •  M. M. Djrbashian and A. H. Karapetyan, Integral representations in a generalized unit disc, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 24 (1989), 523-546 (in Russian). •  M. M. Djrbashian and A. H. Karapetyan, Integral representations in a generalized unit disc, Dokl. Akad. Nauk SSSR 312 (1990), 24-27 (in Russian). •  M. M. Djrbashian and A. H. Karapetyan, Integral representations in a generalized upper half-plane, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 25 (1990), 507-533 (in Russian). •  F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974), 593-602. •  L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in Classical Domains, Inostr. Liter., Moscow, 1959 (in Russian). •  A. H. Karapetyan, On computation of Cauchy type determinants, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 26 (1991), 343-350 (in Russian). •  F. D. Murnaghan, The Theory of Group Representations, Inostr. Liter., Moscow, 1950 (in Russian). •  W. Rudin, Function Theory in the Unit Ball of$ℂ^n\$, Springer, New York, 1980.
•  M. Stoll, Mean value theorems for harmonic and holomorphic functions on bounded symmetric domains, J. Reine Angew. Math. 290 (1977), 191-198.
•  H. Weyl, The Classical Groups, Inostr. Liter., Moscow, 1947 (in Russian).
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