Some Möbius Invariant Spaces of Analytic Functions. Spectrum of the Cesàro Operator

Abstract: This thesis consists of three papers in which different topics in spaces of analytic functions are considered. These papers are: I. "Estimates in Möbius Invariant Spaces of Analytic Functions." II. "Preduals of Q_p-spaces and Carleson Embeddings of Weighted Dirichlet Spaces." III. "On the Spectrum of the Cesàro Operator on Spaces of Analytic Functions." The first two papers are closely related and they concern some new developments in the theory of Möbius invariant spaces of analytic functions. The first article is written together with Alexandru Aleman and ithas been published in Complex Variables, Vol. 49, No 7-9, (2004) (Special Issue, A Tribute to Matts Essén). In this article weproved a number of new estimates för Q_K spaces of analytic functions which extend the classic inequalities of John and Nirenberg, Korenblum and Beurling. More precisely, we derived a general inequality for the seminorms of dilated functions, as well as radial growth estimates, embedding theorems in L^p spaces on the unit disc and integral estimates of exponentials of functions in such spaces. Also, we discussed some properties of the inner-outer factorization for certain Q_K spaces. The second paper, by Alexandru Aleman, Marcus Carlsson and myself, contains a description of the preduals of Q_p- spaces, mainly based on Möbius invariance. The problem is rather complicated and it requires a number of techniques from harmonic analysis. The conclusion gives though a very tractable characterization of these spaces, that is, we show that functions in the predual spaces can be written as sums of products of functions in given weighted Dirichlet and Bergman spaces with the usual control on the norms. This representation is used to investigate the relation between Q_p and Carleson inequalities for functions in weighted Dirichlet spaces. Our approach is based on imbeddings in vector-valued sequence spaces and yields also atomic decompositions of the predual of Q_p. The third paper is about another topic. It contains a unified approach to the study of the spectrum of the Cesàro Operator on Hardy, Bergman and Dirichlet spaces, which, in particular, enables us to derive new results concerning subdecomposability of the Cesàro operator on these spaces.