Nevanlinna counting function and Carleson function of analytic maps

Lefèvre, Pascal; Li, Daniel; Queffélec, Hervé; Rodrı́guez-Piazza, Luis

doi:10.1007/s00208-010-0596-1

Cited by 30 publications

(48 citation statements)

References 27 publications

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“…Let ∆ = {z : |z − 1 2 | < 1 2 } and let ϕ(z) = (1 + z)/2, a conformal mapping of D onto ∆. We will show that (16) with C = C ϕ fails, while This yields divergence of the series on the right hand side of (16). Therefore C ϕ is not a Hilbert-Schmidt operator.…”

Section: Integral Condition Insufficient When ϑ Is Not One-componentmentioning

confidence: 95%

“…Nevanlinna counting function. In this subsection we implement the approach of [16,17] in our more general setting, with the goal of showing that (9) is equivalent to…”

Section: 2mentioning

confidence: 99%

“…We combine the geometric approach with recent results [16,17] that clarify the connection between the Nevanlinna counting function N ϕ and the measure µ ϕ , in order to obtain the following characterization. The article is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Trace ideal criteria for embeddings and composition operators on model spaces

Aleman¹,

Lyubarskii²,

Malinnikova³

et al. 2016

Journal of Functional Analysis

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Let K ϑ be a model space generated by an inner function ϑ. We study the Schatten class membership of embeddings I : K ϑ ֒→ L 2 (µ), µ a positive measure, and of composition operators Cϕ :In the case of onecomponent inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I and Cϕ to the Hardy-Smirnov space E 2 (D) in some domain D ⊃ D. In particular, we obtain a characterization of Schatten membership of Cϕ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.

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Section: Integral Condition Insufficient When ϑ Is Not One-componentmentioning

confidence: 95%

“…Nevanlinna counting function. In this subsection we implement the approach of [16,17] in our more general setting, with the goal of showing that (9) is equivalent to…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Trace ideal criteria for embeddings and composition operators on model spaces

Aleman¹,

Lyubarskii²,

Malinnikova³

et al. 2016

Journal of Functional Analysis

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show abstract

“…Then we use the characterization of the membership of C ϕ to the Schatten classes, given by Luecking and Zhu [LZ] and involving the Nevanlinna counting function. We could also use directly the results on equivalence between the measure of Carleson's windows and the Nevanlinna counting function, see [LLQR2] and also [EK] for a recent new proof.…”

Section: Characterization Of Absolutely Summing Composition Operatorsmentioning

confidence: 99%

Absolutely summing Carleson embeddings on Hardy spaces

Lefèvre¹,

Rodrı́guez-Piazza²

2018

Advances in Mathematics

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We consider the Carleson embeddings of the classical Hardy spaces (on the disk) into a L p (µ) space, where µ is a Carleson measure on the unit disk. This includes the case of composition operators. We characterize such operators which are r-summing on H p , where p > 1 and r ≥ 1. This completely extends the former results on the subject and solves a problem open since the early seventies.Mathematics Subject Classification. Primary: 30H10; 47B10 -Secondary: 30H20; 47B33 Key-words. composition operator -absolutely summing operators -Hardy spaces -Carleson measuresHere λ stands for the normalized Haar measure on the torus (it is the normalized arc length), and f r (z) = f (rz) with r ∈ (0, 1) and z ∈ D. Now, let us turn to our main subject. Given a positive Borel measure µ on the closed unit disk D, we consider the formal identity J µ from the Hardy space H p into L p (µ) (we keep the notation J µ instead of J p,µ in the sequel for sake of lightness):Thanks to a famous result of Carleson (see [C]), this is well defined and bounded if and only if µ is a Carleson measure, i.e. sup ξ∈T µ W(ξ, h) = O(h), when h → 0, * Partially supported by the project MTM2015-63699-P (Spanish MINECO and FEDER funds)Actually the best admissible C is (π r (T )) r .Very few results are known on absolutely summing composition operators: there is a characterization of r-summing composition operators on H p due to Shapiro and Taylor in [ST] only when r = p ≥ 2. The same result (with an obviously adapted proof) is actually valid for general Carleson embeddings:( 1.2) Moreover, for every p ≥ 1, the condition (1.2) is sufficient to ensure that J µ is a p -summing operator on H p . When 1 ≤ p ≤ 2, J µ is actually even absolutely summing since H p has cotype 2.A natural question then arises: is (1.2) the good condition (i.e. a necessary condition) when 1 ≤ p ≤ 2? This is false in general: Domenig proved in [Do] that, given p ∈ [1, 2) there exists an absolutely summing composition operator on H p which is not order bounded. He was able to give some sufficient condition for the construction of his example, but without any characterization. Let us mention that, in this case, it is equivalent to be an order bounded Carleson embedding and to verify condition (1.2). Indeed, the following is known from the specialists (see for instance [LLQR1] in the case of composition operators). Proposition 1.3 Let µ be a Carleson measure on the open unit disk D and p ≥ 1. J µ : H p → L p (µ) is order bounded if and only if (1.2) is satisfied.Proof. By definition, J µ is order bounded if and only if there exists some h ∈ L p (µ) such that for every f in the unit ball of H p , we have |f | ≤ h a.e. on D. Since H p is separable, it suffices to test this control on a dense countable subset of the unit ball de H p . Hence J µ is order bounded if and only if

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“…We shall make a crucial use of the following result (see [14],Th1.1): Let φ : D → D an analytic self-map. For every β > 1, there exists a universal constant C β > 0 such that:…”

Section: Compactness and Carleson Measuresmentioning

confidence: 99%

Composition operators on weighted Bergman spaces of Dirichlet series

Bailleul

2015

Journal of Mathematical Analysis and Applications

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We study boundedness and compactness of composition operators on weighted Bergman spaces of Dirichlet series. Particularly, we obtain in some specific cases, upper and lower bounds of the essential norm of these operators and a criterion of compactness on classicals weighted Bergman spaces. Moreover, a sufficient condition of compactness is obtained using the notion of Carleson's measure. Mathematics Subject Classification.Primary: 47B33 -Secondary: 46E15.

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Nevanlinna counting function and Carleson function of analytic maps

Abstract: Abstract. We show that the maximal Nevanlinna counting function and theCarleson function of analytic self-maps of the unit disk are equivalent, up to constants.

Cited by 30 publications

References 27 publications

Trace ideal criteria for embeddings and composition operators on model spaces

Trace ideal criteria for embeddings and composition operators on model spaces

Absolutely summing Carleson embeddings on Hardy spaces

Composition operators on weighted Bergman spaces of Dirichlet series

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