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Some Möbius Invariant Spaces of Analytic Functions. Spectrum of the Cesàro Operator.

Persson, Anna-Maria LU (2006)
Abstract (Swedish)
Popular Abstract in Swedish

Avhandlingen består sammanlagt av tre vetenskapliga artiklar:



I. "Estimates in Möbius Invariant Spaces of Analytic Functions."



II. "Preduals of Q_p-spaces and Carleson Embeddings of Weighted Dirichlet Spaces."



III. "On the Spectrum of the Cesàro Operator on Spaces of Analytic Functions. "



De första två arbeten är nära relaterade och de handlar om nya utvecklingar inom teorin kring Möbiusinvarianta rum av analytiska funktioner.



Den första artikeln är skriven tillsammans med Alexandru Aleman och den innehåller ett antal helt nya uppskattningar för Q_K-rum som utvidgar John och Nirenberg, samt Korenblum... (More)
Popular Abstract in Swedish

Avhandlingen består sammanlagt av tre vetenskapliga artiklar:



I. "Estimates in Möbius Invariant Spaces of Analytic Functions."



II. "Preduals of Q_p-spaces and Carleson Embeddings of Weighted Dirichlet Spaces."



III. "On the Spectrum of the Cesàro Operator on Spaces of Analytic Functions. "



De första två arbeten är nära relaterade och de handlar om nya utvecklingar inom teorin kring Möbiusinvarianta rum av analytiska funktioner.



Den första artikeln är skriven tillsammans med Alexandru Aleman och den innehåller ett antal helt nya uppskattningar för Q_K-rum som utvidgar John och Nirenberg, samt Korenblum och Beurlings klassiska olikheter.



I den andra artikeln, skriven tillsammans med Alexandru Aleman och Marcus Carlsson, presenteras en beskrivning av de preduala rummen till Q_p-rum huvudsakligen baserad på Möbiusinvarians. Problemet är rätt komplicerat och tillvägagångssättet kräver ett antal tekniker från harmonisk analys. Slutresultatet ger dock en väldigt hanterbar beskrivning av dessa rum.



Den tredje artikeln handlar om ett annat ämne. Den innehåller en enhetlig metod för att studera spektrumet av Cesàro-operatorn på Hardy-, Bergman- och Dirichletrum, som i synnerhet leder till andra intressanta resultat. (Less)
Abstract
This thesis consists of three papers in which different topics in spaces of analytic functions are considered. These papers are:



I. "Estimates in Möbius Invariant Spaces of Analytic Functions."



II. "Preduals of Q_p-spaces and Carleson Embeddings of Weighted Dirichlet Spaces."



III. "On the Spectrum of the Cesàro Operator on Spaces of Analytic Functions."



The first two papers are closely related and they concern some new developments in the theory of Möbius invariant spaces of analytic functions. The first article is written together with Alexandru Aleman and ithas been published in Complex Variables, Vol. 49, No 7-9, (2004) (Special Issue, A Tribute to Matts... (More)
This thesis consists of three papers in which different topics in spaces of analytic functions are considered. These papers are:



I. "Estimates in Möbius Invariant Spaces of Analytic Functions."



II. "Preduals of Q_p-spaces and Carleson Embeddings of Weighted Dirichlet Spaces."



III. "On the Spectrum of the Cesàro Operator on Spaces of Analytic Functions."



The first two papers are closely related and they concern some new developments in the theory of Möbius invariant spaces of analytic functions. The first article is written together with Alexandru Aleman and ithas been published in Complex Variables, Vol. 49, No 7-9, (2004) (Special Issue, A Tribute to Matts Essén).



In this article weproved a number of new estimates för Q_K spaces of analytic functions which extend the classic inequalities of John and Nirenberg, Korenblum and Beurling. More precisely, we derived a general inequality for the seminorms of dilated functions, as well as radial growth estimates, embedding theorems in L^p spaces on the unit disc and integral estimates of exponentials of functions in such spaces. Also, we discussed some properties of the inner-outer factorization for certain Q_K spaces. The second paper, by Alexandru Aleman, Marcus Carlsson and myself, contains a description of the preduals of Q_p- spaces, mainly based on Möbius invariance. The problem is rather complicated and it requires a number of techniques from harmonic analysis. The conclusion gives though a very tractable characterization of these spaces, that is, we show that functions in the predual spaces can be written as sums of products of functions in given weighted Dirichlet and Bergman spaces with the usual control on the norms. This representation is used to investigate the relation between Q_p and Carleson inequalities for functions in weighted Dirichlet spaces. Our approach is based on imbeddings in vector-valued sequence spaces and yields also atomic decompositions of the predual of Q_p.



The third paper is about another topic. It contains a unified approach to the study of the spectrum of the Cesàro Operator on Hardy, Bergman and Dirichlet spaces, which, in particular, enables us to derive new results concerning subdecomposability of the Cesàro operator on these spaces. (Less)
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author
supervisor
opponent
  • Prof. Ross, William T., University of Richmond, Richmond, Virginia, USA
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Mathematics, Matematik, subdecomposable, Cesàro operator, atomic decompositions, Hankel operators, duality, integral estimates, Möbius invariant spaces of analytic functions, growth estimates
publisher
Mathematics Centre for Mathematical Sciences Lund University Lund
defense location
Matematikcentrum, Sölvegatan 18, sal MH:C
defense date
2006-10-25 13:15
ISBN
978-91-628-6950-2
language
English
LU publication?
yes
id
23cf2fdc-ec71-4e67-8cd5-a3852de1a013 (old id 547348)
date added to LUP
2007-09-27 16:05:42
date last changed
2016-09-19 08:45:09
@misc{23cf2fdc-ec71-4e67-8cd5-a3852de1a013,
  abstract     = {This thesis consists of three papers in which different topics in spaces of analytic functions are considered. These papers are:<br/><br>
<br/><br>
I. "Estimates in Möbius Invariant Spaces of Analytic Functions."<br/><br>
<br/><br>
II. "Preduals of Q_p-spaces and Carleson Embeddings of Weighted Dirichlet Spaces."<br/><br>
<br/><br>
III. "On the Spectrum of the Cesàro Operator on Spaces of Analytic Functions."<br/><br>
<br/><br>
The first two papers are closely related and they concern some new developments in the theory of Möbius invariant spaces of analytic functions. The first article is written together with Alexandru Aleman and ithas been published in Complex Variables, Vol. 49, No 7-9, (2004) (Special Issue, A Tribute to Matts Essén).<br/><br>
<br/><br>
In this article weproved a number of new estimates för Q_K spaces of analytic functions which extend the classic inequalities of John and Nirenberg, Korenblum and Beurling. More precisely, we derived a general inequality for the seminorms of dilated functions, as well as radial growth estimates, embedding theorems in L^p spaces on the unit disc and integral estimates of exponentials of functions in such spaces. Also, we discussed some properties of the inner-outer factorization for certain Q_K spaces. The second paper, by Alexandru Aleman, Marcus Carlsson and myself, contains a description of the preduals of Q_p- spaces, mainly based on Möbius invariance. The problem is rather complicated and it requires a number of techniques from harmonic analysis. The conclusion gives though a very tractable characterization of these spaces, that is, we show that functions in the predual spaces can be written as sums of products of functions in given weighted Dirichlet and Bergman spaces with the usual control on the norms. This representation is used to investigate the relation between Q_p and Carleson inequalities for functions in weighted Dirichlet spaces. Our approach is based on imbeddings in vector-valued sequence spaces and yields also atomic decompositions of the predual of Q_p.<br/><br>
<br/><br>
The third paper is about another topic. It contains a unified approach to the study of the spectrum of the Cesàro Operator on Hardy, Bergman and Dirichlet spaces, which, in particular, enables us to derive new results concerning subdecomposability of the Cesàro operator on these spaces.},
  author       = {Persson, Anna-Maria},
  isbn         = {978-91-628-6950-2},
  keyword      = {Mathematics,Matematik,subdecomposable,Cesàro operator,atomic decompositions,Hankel operators,duality,integral estimates,Möbius invariant spaces of analytic functions,growth estimates},
  language     = {eng},
  publisher    = {ARRAY(0xb7422e0)},
  title        = {Some Möbius Invariant Spaces of Analytic Functions. Spectrum of the Cesàro Operator.},
  year         = {2006},
}