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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.

Distribution and rearrangement estimates of the maximal function and interpolation

72%

U. Asekritova I., Krugljak N. Y., Maligranda L., Persson L.-E.

Studia Mathematica

1997

tom 124

nr 2

107-132

There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Studia Mathematica

1 Karapetyan A.

1 Krugljak N. Y.

1 Maligranda L.

1 Persson L.-E.

1 U. Asekritova I.

1 1997

1 1995

Wyniki wyszukiwania

Bounded projections in weighted function spaces in a generalized unit disc

Distribution and rearrangement estimates of the maximal function and interpolation