Sehba, Benoit Florent
(2009)
Operators on some analytic function spaces and their dyadic counterparts.
PhD thesis, University of Glasgow.
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Abstract
In this thesis we consider several questions on harmonic and analytic functions spaces and
some of their operators. These questions deal with Carlesontype measures in the unit
ball, biparameter paraproducts and multipliers problem on the bitorus, boundedness of
the Bergman projection and analytic Besov spaces in tube domains over symmetric cones.
In part I of this thesis, we show how to generate Carleson measures from a class of
weighted Carleson measures in the unit ball. The results are used to obtain boundedness
criteria of the multiplication operators and Ces`aro integraltype operators between
weighted spaces of functions of bounded mean oscillation in the unit ball.
In part II of this thesis, we introduce a notion of functions of logarithmic oscillation
on the bitorus. We prove using Cotlar’s lemma that the dyadic version of the set of
such functions is the exact range of symbols of bounded biparameter paraproducts on the
space of functions of dyadic bounded mean oscillation. We also introduce the little space of
functions of logarithmic mean oscillation in the same spirit as the little space of functions of
bounded mean oscillation of Cotlar and Sadosky. We obtain that the intersection of these
two spaces of functions of logarithmic mean oscillation and L1 is the set of multipliers of
the space of functions of bounded mean oscillation in the bitorus.
In part III of this thesis, in the setting of the tube domains over irreducible symmetric
cones, we prove that the Bergman projection P is bounded on the Lebesgue space Lp if
and only if the natural mapping of the Bergman space Ap0 to the dual space (Ap) of
the Bergman space Ap, where 1
p + 1
p0 = 1, is onto. On the other hand, we prove that for
p > 2, the boundedness of the Bergman projection is also equivalent to the validity of an
Hardytype inequality. We then develop a theory of analytic Besov spaces in this setting
defined by using the corresponding Hardy’s inequality. We prove that these Besov spaces
are the exact range of symbols of Schatten classes of Hankel operators on the Bergman
space A2.
Item Type: 
Thesis
(PhD)

Qualification Level: 
Doctoral 
Keywords: 
Hardy spaces, Bergman spaces,Besov spaces, BMO, LMO, Bloch spaces, Carleson measures, Paraproducts, Multiplication operators, Hankel operators, Bergman projection, Hardy's inequalities, Unit ball, bidisc, symmetric cones 
Subjects: 
Q Science > QA Mathematics 
Colleges/Schools: 
College of Science and Engineering > School of Mathematics and Statistics > Mathematics 
Funder's Name: 
UNSPECIFIED 
Supervisor's Name: 
Pott, Dr. Sandra 
Date of Award: 
2009 
Depositing User: 
Mr Benoit Florent Sehba

Unique ID: 
glathesis:20091189 
Copyright: 
Copyright of this thesis is held by the author. 
Date Deposited: 
30 Sep 2009 
Last Modified: 
10 Dec 2012 13:35 
URI: 
http://theses.gla.ac.uk/id/eprint/1189 
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