Hausdorff operators on weighted Banach spaces of type \(H^\infty\) (Q2064068)

Let \(\mu\) a positive measure on \((0,\infty)\) and \(K\) be a \(\mu\)-measurable function on \(\mathbb{D}\). The paper is devoted to study the continuity of Hausdorff operators induced by the measure \(\mu\), \(\mathcal{H}_{\mu}\), and associated with \(\mu\) and \(K\), \(\mathcal{H}_{K,\mu}\), on weighted Banach spaces of holomorphic functions of type \(H^{\infty}\) which are \begin{align*} H_v^{\infty} (G) = & \{ f \in H(G): ||f||_v = \sup_{z \in \mathbb{D}} v(z) |f(z)| < \infty \}, \\ H_v^0 (G) = & \{ f \in H(G): v|f| \text{ vanishes at infinity on }G \}, \end{align*} where \(G\) is \(\mathbb{D}\) or \(\mathbb{C}\) and \(v\) is a weight function with some conditions. Among other results, it was proved that the continuity of \(\mathcal{H}_{\mu} : H_v^{\infty} (\mathbb{C}) \rightarrow H_v^{\infty} (\mathbb{C})\) implies the continuity of \(\mathcal{H}_{\mu} : H_v^{0} (\mathbb{C}) \rightarrow H_v^{0} (\mathbb{C})\) and both imply that \[ \sup_{n \in \mathbb{N}_0} \int_0^{\infty} \frac{1}{t^{n+1}} d \mu (t) \leq ||H_{\mu}|| < \infty. \]