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

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
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