Inner functions in \(Q_K\) type spaces
Let \({\mathcal H}(\mathbb D)\) be the algebra of all analytic functions in the unit disc \(\mathbb D\) of the complex plane, let \(g(z,a) = - \log|\phi_a(z)|\) be the Green function of \(\mathbb D\), where \(\phi_a(z) : = (a - z)/(1 - \overline{a}z)\) denotes the automorphism of \(\mathbb D\) which interchanges the origin and the point \(a \in \mathbb D\), let \(K : [0,\infty) \to [0,\infty)\) be an arbitrary function, and let \(0 < p < \infty\) and \(-2 < q < \infty\). The space \(Q_K(p,q)\) consists of those \(f \in {\mathcal H}(\mathbb D) \) such that \[ \sup_{a \in \mathbb D}\int_{\mathbb D}|f'(z)|^p(1 - |z|^2)^qK(g(z,a))dA(z) < \infty, \tag{1} \] where \(dA\) is the element of the Lebesgue area on \(\mathbb D\). The space \(Q_K(p,q)\) was introduced by \textit{H. Wulan} and \textit{J. Zhou} [J. Funct. Spaces Appl. 4, No. 1, 73--84 (2006; Zbl 1098.30027)] and it is strongly motivated by the \(Q_p\) theory and the studies on the \(Q_K\) and \(F(p,q,s)\) spaces which are different natural generalizations of \(Q_p\). The definition of \(Q_K(p,q)\), however, generalizes the concept of both of these spaces. Indeed, \(Q_K(2,0) = Q_K\) and \(Q_K(p,q) = F(p,q,s)\) when \(K(t) = t^s\). In the paper under review the inner functions in \(Q_K(p,q)\) and \(Q_{K,0}(p,q)\) are studied. \(Q_{K,0}(p,q)\) denotes the subspace of \(Q_{K}(p,q)\) that consist of those functions \(f \in {\mathcal H}(\mathbb D)\) for which the integral expression in (1) tends to zero as \(a\) goes to the unit circle \(\mathbb T\). For certain values of \(p\) and \(q\), and assuming that the weight \(K\) satisfies some conditions, Theorem 2 gives a characterization for singular inner functions \(S\) belonging to \(Q_K(p,q)\). The proof, given in Section 3, distinguishes between atomic cases and the non-atomic case. Another main result of the paper on inner functions is Theorem 6 which gives a complete characterization of inner function in \(Q_K(p,p-2)\). Specifically, provided \(p \geq 1\) and that the function \(\varphi_K(s): = \sup_{0 < t \leq 1}K(st)/K(t)\) satisfies the regularity conditions {(a)} \( \int_1^\infty \frac{\varphi_K(s)}{s^2}ds < \infty\) and {(b)} \( \int_0^1 \frac{\varphi_K(s)}{s}ds < \infty\), an inner function \(S\) belongs to \(Q_K(p,p-2)\) if and only if \(S\) is the Blaschke product associated to a sequence \(\{z_n\}\) which satisfies \[ \sup_{a \in \mathbb D} \sum_{n=1}^\infty K\left( 1 - |\varphi_a(z_n)|^2 \right) < \infty\,. \] This theorem extends a result due to \textit{M. Essén, H. Wulan} and \textit{J. Xiao} [J. Funct. Anal. 230, No. 1, 78--115 (2006; Zbl 1089.30049)].
scientific article; zbMATH DE number 5997920