On locally convex extension of H<sup>∞</sup>in the unit ball and continuity of the Bergman projection

Jasiczak, M.

doi:10.4064/sm156-3-4

Cited by 5 publications

(9 citation statements)

References 7 publications

(12 reference statements)

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“…The proof of the next proposition is exactly the same as in the case of the unit ball (see [22] v . Open problem.…”

Section: It Remains To Show That the Last Integral Is O(|log | (·)| |mentioning

confidence: 88%

“…The dual projective sequence (Hv n ) b is compact and its limit is a reflexive Fréchet space (an excellent reference book is [29]; see also the survey [10] and [22]). …”

Section: Preliminariesmentioning

confidence: 99%

“…This statement is just continuity with respect to a weight of the form v(t) = t α . The next lemma, with an easy proof in [22], allows us to reduce the case of each weight to this particular one.…”

Section: Preliminariesmentioning

confidence: 99%

“…can be extended to an operator Φ: H 2 1, (Ω) ∩ C 0, → L 2 satisfying BΦ = id on its domain (we refer the reader to [22] for the notation). …”

Section: )])mentioning

confidence: 99%

“…This was the starting point for [30] in the unit disc D as well as in [22] in the case of the unit ball, where we have formulated the problem of extending the result to strongly pseudoconvex domains.…”

Section: Introductionmentioning

confidence: 99%

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On boundary behaviour of the Bergman projection on pseudoconvex domains

Jasiczak

2005

Studia Math.

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Abstract. It is shown that on strongly pseudoconvex domains the Bergman projection maps a space Lv k of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character.Let Ω be a bounded, pseudoconvex set with C 3 boundary. We show that if the Bergman projection is continuous on a space E ⊃ L ∞ (Ω) defined by weighted-sup seminorms and equipped with the topology given by these seminorms, then E must contain the spaces Lv k for each natural k. As a result, in the case of strongly pseudoconvex domains the inductive limit of this sequence of spaces is the smallest extension of L ∞ in the class of spaces defined by weighted-sup seminorms on which the Bergman projection is continuous. This is a generalization of the results of J. Taskinen in the case of the unit disc as well as of the previous research of the author concerning the unit ball.

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“…The proof of the next proposition is exactly the same as in the case of the unit ball (see [22] v . Open problem.…”

Section: It Remains To Show That the Last Integral Is O(|log | (·)| |mentioning

confidence: 88%

“…The dual projective sequence (Hv n ) b is compact and its limit is a reflexive Fréchet space (an excellent reference book is [29]; see also the survey [10] and [22]). …”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…can be extended to an operator Φ: H 2 1, (Ω) ∩ C 0, → L 2 satisfying BΦ = id on its domain (we refer the reader to [22] for the notation). …”

Section: )])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On boundary behaviour of the Bergman projection on pseudoconvex domains

Jasiczak

2005

Studia Math.

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Atomic Decomposition of a Weighted Inductive Limit in $$\mathbb{C}^{n} $$

Hänninen

Taskinen

2005

MedJM

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Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

Jasiczak

2006

Arch. Math.

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We study regularity of Bergman and Szegö projections on Sobolev type weightedsup spaces. The paper covers the case of strongly pseudoconvex domains with C 4 boundary and, partially, domains of finite type in the sense of D'Angelo. Introduction.Let be a bounded domain in C n defined by a differentiable, nondegenerate function r. A fundamental problem in several complex variables is to prove sharp kernel and mapping properties for Bergman and Szegö projections. The problem of determining such estimates is closely connected with regularity of canonical solutions of ∂ and ∂ b , respectively.It is an immediate consequence of the definition that the Bergman projection is bounded on L 2 . However, the situation on other L p , Lipschitz or Sobolev spaces is more complicated. Importantly, regularity of the Bergman projection on some spaces is of interest, because of its application to problems of extension of biholomorphic maps up to the boundary.It is a classic fact that even in the case of the unit disc the Bergman projection B does not map L ∞ into itself. However, B maps boundedly the space L ∞ into BMO. This property is often recognized as a substitute of continuity on L ∞ .The aim of this paper is to study mapping properties of the Bergman projection B on weighted-sup (Sobolev) spaces. We point out a space, which is another subsitute of BMO, as far as continuity of the Bergman projection is concerned. This space is in some sense minimal.Let us start with necessary definitions. Assume that W is a family of continuous functions on with values in R + . We will refer to W as the weight family, provided:

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On locally convex extension of H^∞in the unit ball and continuity of the Bergman projection

Cited by 5 publications

References 7 publications

On boundary behaviour of the Bergman projection on pseudoconvex domains

On boundary behaviour of the Bergman projection on pseudoconvex domains

Atomic Decomposition of a Weighted Inductive Limit in $$\mathbb{C}^{n} $$

Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

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On locally convex extension of H∞in the unit ball and continuity of the Bergman projection

Cited by 5 publications

References 7 publications

On boundary behaviour of the Bergman projection on pseudoconvex domains

On boundary behaviour of the Bergman projection on pseudoconvex domains

Atomic Decomposition of a Weighted Inductive Limit in $$\mathbb{C}^{n} $$

Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

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On locally convex extension of H^∞in the unit ball and continuity of the Bergman projection