Suites D'Interpolation Pour les Classes de Bergman de la Boule et du Polydisque de C<sup>n</sup>

Amar, Éric

doi:10.4153/cjm-1978-062-6

Cited by 38 publications

(34 citation statements)

References 5 publications

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“…It is a result of Eric Amar [1] (but see also [10]) that if {a,} is a separated sequence, then it is the union of finitely many interpolation sequences. Specifically, the following was shown.…”

Section: Definition a Sequence {A} In U Is Said To Be Separated If mentioning

confidence: 99%

Multipliers of Bergman spaces into Lebesgue spaces

Luecking

1986

Proceedings of the Edinburgh Mathematical Society

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Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.

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Section: Definition a Sequence {A} In U Is Said To Be Separated If mentioning

confidence: 99%

Multipliers of Bergman spaces into Lebesgue spaces

Luecking

1986

Proceedings of the Edinburgh Mathematical Society

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“…)* = b q , see [8]. In view of interpolation results of [1], [9] for holomorphic Bergman functions on various domains, a good candidate condition for "onto" is "sufficient separation." In this section we prove that the same phenomenon persists to hold on the setting of the present paper.…”

Section: Representation On B and Bomentioning

confidence: 99%

“…These two properties of holomorphic Bergman spaces were studied on various settings. See [5], [7] for the representation and [1], [9] for the interpolation. In [5], the representation properties of harmonic Bergman functions, as well as harmonic Bloch functions, are also proved on the unit ball of R n .…”

Section: Jhmentioning

confidence: 99%

“…Let H = H n (n > 2) denote the upper half-space R 71 " 1 x R+ where R+ denotes the set of all positive real numbers. For 1 < p < oo, we will write Ψ for the harmonic Bergman space consisting of all harmonic functions u on H such that P where V denotes the volume measure on H. The space IP turns out to be a closed subspace of IP', the Lebesgue space on iϊ, and thus bP is a Banach space.…”

Section: §1 Introductionmentioning

confidence: 99%

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Representations and interpolations of harmonic Bergman functions on half-spaces

Choe¹,

Huang²

1998

Nagoya Mathematical Journal

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Abstract. On the setting of the half-space of the euclidean n-space, we prove representation theorems and interpolation theorems for harmonic Bergman functions in a constructive way. We also consider the harmonic (little) Bloch spaces as limiting spaces. Our results show that well-known phenomena for holomorphic cases continue to hold. Our proofs of representation theorems also yield a uniqueness theorem for harmonic Bergman functions. As an application of interpolation theorems, we give a distance estimate to the harmonic little Bloch space. In the course of the proofs, pseudohyperbolic balls are used as substitutes for Bergman metric balls in the holomorphic case. §1. Introduction Let H = H n (n > 2) denote the upper half-space R 71 " 1 x R+ where R+ denotes the set of all positive real numbers. For 1 < p < oo, we will write Ψ for the harmonic Bergman space consisting of all harmonic functions u on H such that P where V denotes the volume measure on H. The space IP turns out to be a closed subspace of IP', the Lebesgue space on iϊ, and thus bP is a Banach space. In particular, b 2 is a Hubert space. Hence, there is a unique Hubert space orthogonal projection R: L 2

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“…It is a result of Amar [1] (see also [12]) that if a sequence of points in the unit ball of C n is separated enough with respect to the pseudo-hyperbolic distance then the sequence is an interpolating sequence of holomorphic Bergman spaces (the results of [1] and [12] are more general). Seip [13] gave a characterization of interpolating sequences of Bergman spaces on the unit disk.…”

mentioning

confidence: 99%