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Interpolating sequences, Carleson measures and Wirtinger inequality

100%

Amar E.

Annales Polonici Mathematici

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2008

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tom 94

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nr 1

79-87

EN

Let S be a sequence of points in the unit ball 𝔹 of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure $μ_{S}:= ∑_{a∈S} (1-|a|²)ⁿ δ_{a}$ is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of 𝔹 such that any δ -separated sequence S has its associated measure $μ_{S}$ bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of 𝔹. As an easy consequence, we prove that if S is a dual bounded sequence in $H^{p}(𝔹)$, then $μ_{S}$ is a Carleson measure, which gives a short proof in one variable of a theorem of L. Carleson and in several variables of a theorem of P. Thomas.

2

On linear extension for interpolating sequences

100%

Amar E.

Studia Mathematica

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2008

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tom 186

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nr 3

251-265

EN

Let A be a uniform algebra on X and σ a probability measure on X. We define the Hardy spaces $H^{p}(σ)$ and the $H^{p}(σ)$ interpolating sequences S in the p-spectrum $ℳ _{p}$ of σ. We prove, under some structural hypotheses on A and σ, that if S is a "dual bounded" Carleson sequence, then S is $H^{s}(σ)$-interpolating with a linear extension operator for s < p, provided that either p = ∞ or p ≤ 2. In the case of the unit ball of ℂⁿ we find, for instance, that if S is dual bounded in $H^{∞}(𝔹)$ then S is $H^{p}(𝔹)$-interpolating with a linear extension operator for any 1 ≤ p < ∞. Already in this case this is a new result.

3

Sur un problème de W. Rudin concernant les fonctions holomorphes et borneés

32%

Amar E.

Studia Mathematica

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1982

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tom 74

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nr 1

49-56

Ograniczanie wyników

2 Studia Mathematica

1 Annales Polonici Mathematici

3 Amar E.

2 2008

1 1982

Interpolating sequences, Carleson measures and Wirtinger inequality

On linear extension for interpolating sequences

Sur un problème de W. Rudin concernant les fonctions holomorphes et borneés