Operators on some analytic function spaces and their dyadic counterparts

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.