Duality for some large spaces of analytic functions (Q2758148)

Denote by \(H(\Delta)\) the space of all holomorphic functions on the open unit disk \(\Delta\) in \(\mathbb{C}\). For \(-1<\alpha <\infty\) consider the probability measure \(m_\alpha\) on \(\Delta\) defined by \(dm_\alpha (x+iy): ={ \alpha+1 \over\pi} (1-|x+iy |^2)^\alpha dxdy\). For \(1\leq p<\infty\) let NEWLINE\[NEWLINE{\mathcal N}^p_\alpha: =\bigl\{f\in H(\Delta): \log^+|f|\in L^p(m_\alpha) \bigr\}NEWLINE\]NEWLINE endowed with its standard \(F\)-norm, and define \({\mathcal N}^p_{-1}\) as the Hardy-Orlicz space \(LH^p\) for \(1<p <\infty\) and as the Smirnow class \({\mathcal N}^+\) for \(p=1\). Furthermore define the weighted Bergman spaces \({\mathcal A}^p_\beta: =L^q(m_\beta) \cap H(\Delta)\) for \(\beta>-1\) and \(0<q < \infty\) and \(A^q_{-1}: =H^q\), the classical Hardy space.NEWLINENEWLINENEWLINEThe authors present three main results. The first one characterizes the sequences \((\lambda_n)_{n \in\mathbb{N}_0}\) which have the property that for given \(\alpha\geq -1\) and \(1\leq p<\infty\) there are \(\beta\geq -1\) and \(0<q <\infty\) such that for each function \(\sum_{n\in \mathbb{N}_0} a_nz^n\) in \({\mathcal N}^p_\alpha\) the function \(\sum_{n\in \mathbb{N}_0} \lambda_n a_nz^n\) belongs to \({\mathcal A}^q_\alpha\), in terms of growth properties of \((\lambda_n)_{n\in \mathbb{N}_0}\). As second result the Fréchet envelope of \(N^p_\alpha\), and the dual space of \({\mathcal N}^p_\alpha\) are computed. Finally it is characterized when for a given holomorphic function \(\varphi: \Delta\to \Delta\) the composition operator \(C_\varphi: f\to f \circ \varphi\) is a bounded (nuclear) operator \(C_\varphi:{\mathcal N}^p_\alpha \to{\mathcal A}^p_\beta\).