A class of Möbius invariant function spaces
Let \(\mathbb D\) be the open unit disk in the complex plane \(\mathbb C\) and let \(\text{Aut}(\mathbb D)\) denote the group of all Möbius maps of the unit disk. For \(0<p<\infty, -1<\alpha<\infty\), and \(n\) a positive integer, let \(Q(n,p,\alpha)\) denote the space of analytic functions \(f\) in \(\mathbb D\) with the property \[ \| f\|_{n,p,\alpha}^p= \sup_{\varphi\in\text{Aut}(\mathbb D)}\int_{\mathbb D}| (f\circ\varphi)^{(n)}(z)| ^p(1-| z| ^2)^{\alpha}\,dA(z)<\infty, \] where \(dA\) is the area measure on \(\mathbb D\), normalized so that the unit disk has area equal to \(1\). The author in the paper under review proves several results on the \(Q(n,p,\alpha)\) spaces, that is, \(Q(n,p,\alpha)\) is trivial when \(np>\alpha+2\), it contains all polynomials when \(np\leq\alpha+2\) and it coincides with the Besov space \(B_p\) when \(np=\alpha+2\). Further, \(Q(n,p,\alpha)\) is just the Bloch space for \(np<\alpha+1\). If \(\gamma=(\alpha+2)-np>0\), then an analytic function \(f\) in \(\mathbb D\) belongs to \(Q(n,p,\alpha)\) if and only if the measure \(| f^{(n)}(z)| ^p(1-| z| ^2)^{\alpha}dA(z)\) is \(\gamma\)-Carleson. Suppose \(\alpha+1\leq pn\leq\alpha+2\) and \(f(z)=\sum_{k=0}^{\infty}a_kz^{n_k}\) is a lacunary series in \(\mathbb D\). Then the following conditions are equivalent: (a) \(f\in Q(n,p,\alpha)\). (b) \(\int_{\mathbb D}| f^{(n)}(z)| ^p(1-| z| ^2)^{\alpha}dA(z)<\infty\). (c) The Taylor coefficients of \(f\) satisfy \[ \sum_{k=0}^{\infty}\frac{| a_k| }{n^{\alpha+1-np}} <\infty. \] Finally it is shown that if \(f\) is a lacunary series satisfying \(\int_{\mathbb D}| f(z)| ^p(1-| z| ^2)^{\alpha}\,dA(z)<\infty\), then \(\lim_{z\to1^-}(1-| z| ^2)^{\alpha+1}| f(z)| ^p=0\). It is worth noticing that the papers by \textit{R. Zhao} [Ann. Acad. Sci. Fenn. Math. Diss. 56 (1996; Zbl 0851.30017)] and by \textit{J. Rättyä} [Ann. Acad. Sci. Fenn. Math. Diss. 73 (2001; Zbl 0984.30019)] study a similar class of function spaces \(F(p,q,s)\), where \(p>0, q>-2\) and \(s\geq0\).
scientific article; zbMATH DE number 5316803
scientific article