Carleson embeddings for weighted Bergman spaces. (English) Zbl 1052.46013

Jarchow, Hans; Kollbrunner, Urs

Jarosz, Krzysztof (ed.), Function spaces. Proceedings of the 4th conference, Edwardsville, IL, USA, May 14–19, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3269-7/pbk). Contemp. Math. 328, 217-230 (2003).

Let \(D\) be the open unit disk in the complex plane. For \(p>0\) and \(\alpha>-1\) the weighted Bergman space \(A^p_\alpha\) consists of analytic functions \(f\) in \(D\) such that \[ \int_D\bigl| f(z)\bigr|^p \bigl(1-| z|^2)^\alpha dA(z) <\infty, \] where \(dA\) is area measure on \(D\). This paper discusses various problems about the embedding of \(A^p_\alpha\) into \(L^q(D,d\mu)\), where \(\mu\) is a positive Borel measure on \(D\). In particular, the compactness, the order-boundedness, and summing properties of this embedding are characterized.
For the entire collection see [Zbl 1015.00023].

Reviewer: Kehe Zhu (Albany)

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
47B38	Linear operators on function spaces (general)
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46B25	Classical Banach spaces in the general theory
30H05	Spaces of bounded analytic functions of one complex variable

Keywords:

Carleson measure; Bergman space