CLASSICAL PROPERTIES OF COMPOSITION OPERATORS ON HARDY–ORLICZ SPACES ON PLANAR DOMAINS

2018 ◽  
Vol 107 (02) ◽  
pp. 256-271
Author(s):  
MICHAŁ RZECZKOWSKI
Keyword(s):  
Composition Operators ◽  
Orlicz Spaces ◽  
Weak Compactness ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Jordan Curves ◽  
Planar Domains

In this paper we study composition operators on Hardy–Orlicz spaces on multiply connected domains whose boundaries consist of finitely many disjoint analytic Jordan curves. We obtain a characterization of order-bounded composition operators. We also investigate weak compactness and the Dunford–Pettis property of these operators.

scholarly journals Carleson Measure Theorems for Large Hardy-Orlicz and Bergman-Orlicz Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Stéphane Charpentier ◽  
Benoît Sehba
Keyword(s):  
Unit Ball ◽  
Orlicz Space ◽  
Carleson Measure ◽  
Composition Operators ◽  
Orlicz Spaces ◽  
Growth Functions

We characterize those measuresμfor which the Hardy-Orlicz (resp., weighted Bergman-Orlicz) spaceHΨ1(resp.,AαΨ1) of the unit ball ofCNembeds boundedly or compactly into the Orlicz spaceLΨ2(BN¯,μ)(resp.,LΨ2(BN,μ)), when the defining functionsΨ1andΨ2are growth functions such thatL1⊂LΨjforj∈{1,2}, and such thatΨ2/Ψ1is nondecreasing. We apply our result to the characterization of the boundedness and compactness of composition operators fromHΨ1(resp.,AαΨ1) intoHΨ2(resp.,AαΨ2).

scholarly journals On the Riemann-Hilbert problem in multiply connected domains

2016 ◽  
Vol 14 (1) ◽  
pp. 13-18 ◽  
Author(s):  
Vladimir Ryazanov
Keyword(s):  
Boundary Data ◽  
Hilbert Problem ◽  
Multiply Connected Domains ◽  
Natural Parameter ◽  
Multiply Connected ◽  
Jordan Curves ◽  
Measurable Coefficients ◽  
Asymptotic Values ◽  
Riemann Hilbert Problem ◽  
Number Of Branches

AbstractWe proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthermore, it is shown that the dimension of the spaces of these solutions is infinite.

scholarly journals Secondary Schottky–Klein prime functions associated with multiply connected planar domains

2015 ◽  
Vol 471 (2173) ◽  
pp. 20140688 ◽  
Author(s):  
Giovani L. Vasconcelos ◽  
Jonathan S. Marshall ◽  
Darren G. Crowdy
Keyword(s):  
Mixed Type ◽  
Compact Riemann Surface ◽  
Mathematical Framework ◽  
Multiply Connected Domains ◽  
Additional Function ◽  
Multiply Connected ◽  
Planar Domains ◽  
Recent Interest ◽  
Schottky Double ◽  
Prime Function

In recent years, a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the Schottky–Klein (S–K) prime function of an underlying compact Riemann surface known as the Schottky double of the domain. In this paper, we describe additional function-theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary S–K prime functions. The basic idea develops, and extends, an observation of Burnside dating back to 1892. Applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given. Other possible applications are also surveyed.

scholarly journals -Carleson Measures for a Class of Hardy-Orlicz Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Benoît Florent Sehba
Keyword(s):  
Composition Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Alternative Interpretation ◽  
Carleson Measures ◽  
Weighted Composition Operators ◽  
Weak Factorization ◽  
Type Integral ◽  
Weighted Composition

An alternative interpretation of a family of weighted Carleson measures is used to characterize -Carleson measures for a class of Hardy-Orlicz spaces admitting a nice weak factorization. As an application, we provide with a characterization of symbols of bounded weighted composition operators and Cesàro-type integral operators from these Hardy-Orlicz spaces to some classical holomorphic function spaces.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

Conformal and Quasiconformal Mappings of Close Multiply-Connected Domains

2002 ◽  
Vol 9 (2) ◽  
pp. 367-382
Author(s):  
Z. Samsonia ◽  
L. Zivzivadze
Keyword(s):  
Integral Equations ◽  
Quasiconformal Mappings ◽  
Multiply Connected Domains ◽  
Multiply Connected

Abstract Doubly-connected and triply-connected domains close to each other in a certain sense are considered. Some questions connected with conformal and quasiconformal mappings of such domains are studied using integral equations.

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

scholarly journals A characterization of internal functions on multiply connected regions

1984 ◽  
Vol 96 ◽  
pp. 23-28
Author(s):  
Lee A. Rubel
Keyword(s):  
Hardy Spaces ◽  
Multiply Connected ◽  
Multiply Connected Regions ◽  
Internal Function

The notion of internal function enters naturally in the study of factorization of function in Lumer’s Hardy spaces—see [RUB], where this aspect is developed in some detail.

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