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 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 |

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