Operators on some analytic function spaces and their dyadic counterparts

QA Mathematics

Sehba, Benoit Florent ; Pott, Sandra

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: 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;   

Others  :  http://theses.gla.ac.uk/1189/1/2009sehbaphd.pdf

来源: University of Glasgow

【 摘 要 】

In this thesis we consider several questions on harmonic and analytic functions spaces andsome of their operators. These questions deal with Carleson-type measures in the unitball, bi-parameter paraproducts and multipliers problem on the bitorus, boundedness ofthe 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 ofweighted Carleson measures in the unit ball. The results are used to obtain boundednesscriteria of the multiplication operators and Ces`aro integral-type operators betweenweighted 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 oscillationon the bitorus. We prove using Cotlar’s lemma that the dyadic version of the set ofsuch functions is the exact range of symbols of bounded bi-parameter paraproducts on thespace of functions of dyadic bounded mean oscillation. We also introduce the little space offunctions of logarithmic mean oscillation in the same spirit as the little space of functions ofbounded mean oscillation of Cotlar and Sadosky. We obtain that the intersection of thesetwo spaces of functions of logarithmic mean oscillation and L1 is the set of multipliers ofthe 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 symmetriccones, we prove that the Bergman projection P is bounded on the Lebesgue space Lp ifand only if the natural mapping of the Bergman space Ap0 to the dual space (Ap) ofthe Bergman space Ap, where 1p + 1p0 = 1, is onto. On the other hand, we prove that forp > 2, the boundedness of the Bergman projection is also equivalent to the validity of anHardy-type inequality. We then develop a theory of analytic Besov spaces in this settingdefined by using the corresponding Hardy’s inequality. We prove that these Besov spacesare the exact range of symbols of Schatten classes of Hankel operators on the Bergmanspace A2.