Abstract:
We obtain a new general sufficient condition for the continuity of the Bergman projection in tube domains over symmetric cones using multifunctional embeddings.We also obtain some sharp embedding relations between the generalized Hilbert-HArdy spaces and the mixed norm Bergman spaces in this setting.

Abstract:
We characterize bounded Toeplitz and Hankel operators from weighted Bergman spaces to weighted Besov spaces in tube domains over symmetric cones. We deduce weak factorization results for some Bergman spaces of this setting.

Abstract:
We remark that a dyadic version of the Carleson embedding theorem for the Bergman space extends to vector-valued functions and operator-valued measures. This is in contrast to a result by Nazarov, Treil, Volberg in the context of the Hardy space. We also discuss some embeddings for analytic vector-valued functions.

Abstract:
New sharp estimates concerning distance function in Bergman - type analytic function spaces on tube domains over symmetric cones are obtained. These are first results of this type in tube domains over symmetric cones.

Abstract:
let be a bounded strictly convex domain in , and the tube domain over . in this paper, we show that the bergman kernel of can be expressed easily by an integral formula.

Abstract:
This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. We also consider little Hankel operators on these Bergman spaces. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases.

Abstract:
It is shown how results on Carleson embeddings induced by the Laplace transform can be use to derive new and more general results concerning the weighted admissibility of control and observation operators for linear semigroup systems with q-Riesz bases of eigenvectors. Next, a new Carleson embedding result is proved, which gives further results on weighted admissibility for analytic semigroups. Finally, controllability by smoother inputs is characterised by means of a new result about weighted interpolation.

Abstract:
Let be a finite, positive measure on , the polydisc in , and let be 2n-dimensional Lebesgue volume measure on . For an Orlicz function , a necessary and sufficient condition on is given in order that the identity map is bounded. 1. Introduction We denote by the unit polydisc in and by the distinguished boundary of . We will use to denote the -dimensional Lebesgue volume measure on , normalized so that . We use to describe rectangles on , and we use to denote the corona associated to these sets. In particular, if I is an interval on of length centered at , Then, if , with intervals having length and having centers , is given by , and let If is any open set in , we define where runs through all rectangles in . An Orlicz function is a real-valued, nondecreasing, convex function such that and . To avoid pathologies, we will assume that we work with an Orlicz function having the following additional properties: is continuous and strictly convex (hence increasing), such that The Orlicz space is the space of all (equivalence classes of) measurable functions for which there is a constant such that and then (the Luxemburg norm) is the infimum of all possible constant such that this integral is . It is well known that is a Banach space under the Luxemburg norm . For , let The Bergman-Orlicz space consists of all analytic functions in , which is a closed subspace of , so it is an analytic Banach space also. A theorem of Carleson [1, 2] characterizes those positive measure on for which the Hardy space norm dominates the norm of elements of . Since then, there is a long history of the development and application of Carleson measures, see [3]. This rich area of research contains a large body of literature on characterizations of different classes of operators in different spaces and their applications. Chang [4] has characterized the bounded measures on using a two-line proof referring to a result of Stein. Characterization of the bounded identity operators on Hardy spaces is an immediate consequence of Chang's proof using standard arguments. Hastings [5] has given a similar result for unweighted Bergman spaces. MacCluer [6] has obtained a Carleson measure characterization of the identity operators on Hardy spaces of the unit ball in using the well-known results of Hormander. Lefèvre et al. [7] have introduced an adapted version of Carleson measure in Hardy-Orlicz spaces. Xiao [8], Ortiz, and Fernandez [9] have got a characterization of the Carleson measure in Bergman-Orlicz spaces of the unit disc. A finite, positive measure on is called a Carleson measure if there

Abstract:
Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of K\"ahler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be those metrics induced by such embeddings. Our main result is to identify a natural sequence of Bergman metrics approximating any given Riemannian metric. In particular we have constructed finite dimensional symmetric space approximations to the space of all Riemannian metrics. Moreover the construction induces a Riemannian metric on that infinite dimensional manifold which we compute explicitly.

Abstract:
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Ces\`aro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.