On \(l^p\)-multipliers of functions analytic in the disk (Q2258225)

Let \(\mathbb{D}\) be the open unit disk in the complex plane \(\mathbb{C}\). For \(1\leq p\leq \infty\), let \(A_{p}^{+}(\mathbb{D})\) denote the space of all analytic functions \(f\) on \(\mathbb{D}\) such that the sequence \(\hat{f}= \{\hat{f}(n): n=0, 1,...\}\) of Taylor coefficients of \(f\) belongs to \(\ell^{p}\) with \(\|f\|_{A_{p}^{+}(\mathbb{D})}= \|\hat{f}\|^{}_{\ell^p}\). An analytic function \(m\) is called an \(\ell^p\)-multiplier if, for every \(f\in A_{p}^{+}(\mathbb{D})\), we have \(m\cdot f\in A_{p}^{+}(\mathbb{D})\). We denote the class of all those multipliers by \(M_{p}^{+}(\mathbb{D})\). This class is a Banach algebra with respect to the norm \[ \| m\|^{}_{M_{p}^{+}(\mathbb D)} = \sup_{\|f\|_{A_{p}^{+}(\mathbb{D})} \leq 1} \|m\cdot f\|_{A_{p}^{+}(\mathbb{D})}. \] Let \(\Omega\subseteq \mathbb{C}\) be an arbitrary domain and let \(F\) be an arbitrary closed subset of the boundary circle \(\partial\mathbb{D}\), we denote by \(\Omega_{F}\) a starlike domain generated by \(F\). In this paper, the author considers the Hardy space \(H^{\infty}(\Omega_{F})\) of all analytic functions \(f\) in \(\Omega_{F}\) with \(\displaystyle \|g\|_{H^{\infty}(\Omega_{F})}=\sup_{z\in\Omega_{F}}|f(z)|\), where \(F\) is a closed subset of measure zero in the boundary circle of \(\partial\mathbb{D}\) which has the Little-Paley property \(\text{LP}(p)\), \(1<p<\infty\). If \(F\) has the Little-Paley property \(\text{LP}(p)\) for all \(p\), \(1<p<\infty\), we say that \(F\) has property LP. The author shows that if \(F\subseteq\partial\mathbb{D}\) has property \(\text{LP}(p)\) and \(\Omega_{F}\) is a starlike domain generated by \(F\), then \(H^{\infty}(\Omega_{F}) \subseteq M_{p}^{+}(\mathbb{D})\). Moreover, if \(F\) has the property LP, then \(\displaystyle H^{\infty}(\Omega_{F}) \subseteq \bigcap_{1<p<\infty}M_{p}^{+}(\mathbb{D})\).