Bounded projections in weighted function spaces in a generalized unit disc

A. Karapetyan¹

¹

Annales Polonici Mathematici, Tome 62 (1995) no. 3, pp. 193-218.

Résumé

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(

Détail

DOI : 10.4064/ap-62-3-193-218

Keywords: generalized unit disc, holomorphic and pluriharmonic functions, weighted spaces, integral representations, bounded integral operators