Analytic spaces defined by symmetric norming functions (Q2493606)

The authors define certain spaces of analytic functions on the unit disc which generalize the analytic Besov spaces. Recall that \(B_p\) consists of analytic functions such that \[ \int_{\mathbb D} | f''(z)| ^p(1-| z| ^2)^{2p-2}dA(z)<\infty. \] It is not difficult to see that the integrability condition can be replaced by \[ \sum_k \left(\sup_{z\in D_k}| f''(z)| (1-| z| ^2)^{2}\right)^p<\infty, \] where \({\mathcal D}=(D_k)\) is a cover of the unit disc adapted to the hyperbolic metric. With this starting point, they define \(\lambda_k(f)= \sup_{z\in D_k}| f''(z)| (1-| z| ^2)^{2}\) and consider the spaces \(B_{{\mathcal D},\Phi}\), where the assumption is now that \((\lambda_k(f))\in c_\Phi\), where \(\Phi\) is a symmetric norming function. They show that \(B_{{\mathcal D},\Phi}\) is a Banach space and it is independent of the cover \({\mathcal D}\). Also, some partial results about separability are presented.