Möbius invariant Dirichlet type spaces
Let \((Y, \|\,\|_Y)\) be a Banach space of analytic functions containing all constant functions on the unit disc \(\mathbb D\). As in [\textit{A. Aleman} and \textit{A.-M. Simbotin}, Complex Variables, Theory Appl. 49, No. 7--9, 487--510 (2004; Zbl 1076.30037)] \(M(Y)\) denotes the Möbius invariant space consisting of all analytic functions \(f\) on \(\mathbb D\) satisfying \[ \|f\|_{M(Y)} = \sup_{\phi\in\mathrm{Aut}(\mathbb D)} \|f\circ\phi - f(\phi(0))\|_Y <\infty. \] For \(0<p<\infty\) and \(\mu\) a positive measure on \(\mathbb D\), the Dirichlet space \(\mathcal D_{\mu,p}\) consists of the analytic functions \(f\) on \(\mathbb D\) for which \[ \|f\|_{\mathcal D_{\mu,p}}^2 = \int_{\mathbb D} |f^\prime(z)|^2 U_{\mu,p}(z) dA(z) <\infty, \] where \[ U_{\mu,p}(z) = \int_{\mathbb D} (1- |\sigma_z(w)|^2)^p d\mu(w). \] In the above, \(\sigma_z\) is the automorphism of \(\mathbb D\) that interchanges \(0\) and \(z\). In [the author et al., Can. Math. Bull. 60, No. 4, 690--704 (2017; Zbl 1386.30053)] it is proved that if \(\mu\) is a finite measure on \(\mathbb D\) then \(M(\mathcal D_{\mu,p})\) is the well-known Möbius invariant space \(\mathcal Q_p\). In the present paper the author investigates the spaces \(\mathcal D_{\mu,p}\) and \(M(\mathcal D_{\mu,p})\) when the measures \(\mu\) are not necessarily finite. It is shown that for \(0<p<\infty\), \(M(\mathcal D_{\mu,p})\) is always a subset of the Bloch space \(\mathcal B\) with \(M(\mathcal D_{\mu,p})\subsetneq\mathcal B\). If \(p>1\) then \(M(\mathcal D_{\mu,p}) = \mathcal B\) if and only if \(\mu(\mathbb D)<\infty\). In the paper the author also characterizes the inner functions in \(M(\mathcal D_{\mu,p})\).
scientific article; zbMATH DE number 6910521
scientific article