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Bounded projections in weighted function spaces in a generalized unit disc

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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.

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1 Annales Polonici Mathematici

1 Karapetyan A.

1 1995

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Bounded projections in weighted function spaces in a generalized unit disc