Operators on some analytic function spaces and their dyadic counterparts

Sehba, Benoit Florent (2009) Operators on some analytic function spaces and their dyadic counterparts. PhD thesis, University of Glasgow.

Abstract

In this thesis we consider several questions on harmonic and analytic functions spaces and
some of their operators. These questions deal with Carleson-type measures in the unit
ball, bi-parameter 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 integral-type 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 bi-parameter 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
Hardy-type 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
• Supervisor's Name: Pott, Dr. Sandra
• Date of Award: 2009
• Depositing User: Mr Benoit Florent Sehba
• Unique ID: glathesis:2009-1189
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 30 Sep 2009
• Last Modified: 10 Dec 2012 13:35
• URI: https://theses.gla.ac.uk/id/eprint/1189