Recent progress and open problems in the Bergman space

Ann. Acad. Sci. Fenn. Math, 2008

We obtain several new characterizations for the standard weighted Bergman spaces A p α on the unit ball of C n in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.

Integral Equations and Operator Theory, 1999

Let II be the upper half-plane in C, consider the Bergman space A2(II), the subspace of all analytic functions from Lz(II). The complete decomposit~on of L2(II) onto Bergman and Bergman type spaces of poly-analytic and poly-antianalytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szego projections are established. where Pn-l in the orthogonal projection (4.6) of LI(R+) onto the one-dimensional space Ln-1, generated by the function I,-l(y), defined in (4.2). PROOF. Follows directly from Corollaries 4.2 and 4.4.

TDX (Tesis Doctorals en Xarxa), 2016

with whom I have shared the office. I am also extremely grateful to Josep Vives from the Department of Probability, Logic and Statistics. All of them were available when I wanted to talk maths as well as those times when I would have preferred to speak about anything else. To my close friends and family I will be eternally grateful. Without your encouragement I would never have completed this work. You will never know how much your love, kindness and support has meant to me these past years and I just hope that I can pay it all back in the years to come. v vi

Glasgow Mathematical Journal, 2009

It was shown in [2] that a holomorphic function f in the unit ball B n of C n belongs to the weighted Bergman space A p α , p > n + 1 + α, if and only if the function |f

Pacific Journal of Mathematics, 1991

By using a natural localization method, one describes the finite codimensional invariant subspaces of the Bergman //-tuple of operators associated to some bounded pseudoconvex domains in C" , with a sufficiently nice boundary. with the structure and classification of the invariant subspaces of the Bergman n-tuple of operators, cf. Agrawal-Salinas [2], Axler-Bourdon [4], Bercovici [5], Douglas [7], Douglas-Paulsen . Due to the richness of this lattice of invariant subspaces, the additional assumption on finite codimension was naturally adopted by the above mentioned authors as a first step towards a better understanding of its properties. The present note arose from the observation that, when the L 2bounded evaluation points of a pseudoconvex domain lie in the Fredholm resolvent set of the associated Bergman rc-tuple, then the description of finite codimensional invariant subspaces is, at least conceptually, a fairly simple algebraic matter. This simplification requires only the basic properties of the sheaf model for systems of commuting operators introduced in [11]. The main result below is also available by some other recent methods. First is the quite similar technique of localizing Hubert modules over function algebras, due to Douglas and Douglas and Paulsen , and secondly is the study of the so-called canonical subspaces of some Hubert spaces with reproducing kernels, developed by Agrawal and Salinas . Both points of view will be discussed in §2 of this note. In fact the Bergman space of a pseudoconvex domain is only an example within a class of abstract Banach ^(C^-modules, whose finite codimensional submodules turn out to have a similar structure. The precise formulation of this remark ends the note. We would like to thank the referee, whose observations pointed out some bibliographical omissions in a first version of the manuscript. Let Ω be a bounded pseudoconvex domain in C n , n > 1, and let L%(Q) denote the corresponding Bergman

2007

Journal of Functional Analysis, 2004

On the setting of general bounded smooth domains in R n ; we construct L 1 -bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev L p -estimates of any order even for p ¼ 1: Among applications are Gleason's problems for the harmonic Bergman-Sobolev and (little) Bloch functions on star-shaped domains with strong reference points. r 2004 Elsevier Inc. All rights reserved. MSC: primary 31B05; secondary 31B10

Bulletin of the American Mathematical Society, 1981

If D is a bounded open subset of C", the set H = {ƒ: D-> C| ƒ is holomorphic and S D \f\ 2 < +°°} is a separable infinite-dimensional Hubert space relative to the inner product <ƒ, g) = f D fg. The completeness of H can be seen from Cauchy integral estimates. Similar estimates show that for any p E D the functional ƒ H* ƒ(/?),ƒ£ H, is continuous. Thus there is a unique element K D (z, p) E f/ (as a function of z) such that, f(p) = f f(z)K D (z, p)dV(z) for all ƒ G H. The function K D is called the Bergman kernel function. If {y i }™ = , l is an orthonormal basis for f/ then K D (z, p) = ^.^.(z)^/?). The convergence of the series is absolute, uniformly on compact subsets ofD x D. For any z ED, K D (z, z) > 0 and log K D (z, z) is a real analytic function on D. The Hermitian form 3 2 ^âTÂF log K D(Z > z^dz i 0 dz~j i,j oz i oz j

We define a set of projections on the Bergman space A 2 parameterized by an affine closed space of a Banach space. This family is defined from an affine space of a Banach space of holomorphic functions in the disk and includes the classical Forelli-Rudin projections.

Journal of Operator …, 2001

Abstract. In this paper we study mapping properties of the Bergman pro-jection P, ie which function spaces or classes are preserved by P. It is shown that the Bergman projection is of weak type (1, 1) and bounded on the Orlicz space Lϕ(D, dA) iff Lϕ(D, dA) is reflexive. So the dual ...