Hilbert matrix operator on spaces of analytic functions (Q2908738)

The Hilbert matrix \(\mathcal{H}=((n+k+1)^{-1})_{n\geq 0,k\geq 0}\) induces a transformation of the space \(H\) of analytic functions on the unit disk \(\mathbb{D}\subset\mathbb{C}\), defined by NEWLINE\[NEWLINE \mathcal{H}\left(\sum_{k=0}^\infty a_k z^k\right)=\sum_{n=0}^\infty \sum_{k=0}^\infty \frac{a_k}{n+k+1}z^n.NEWLINE\]NEWLINE It was proved in [\textit{E. Diamantopoulus} and \textit{A. G. Siskakis}, Stud. Math. 140, No. 2, 191--198 (2000; Zbl 0980.47029)] that \(\mathcal{H}\) is bounded on the Hardy spaces \(H^p\), \(1<p<\infty\), but neither on \(H^1\) nor on \(H^\infty\).NEWLINENEWLINEIn this paper, the authors study the action of \(\mathcal{H}\) on weighted Hardy spaces, weighted Bergman and Besov spaces and weighted Bloch spaces.NEWLINENEWLINEAmong the main results obtained by the authors, we have:NEWLINENEWLINE(1) If \(H^p_\alpha=\{f\in H: \sup_{0<r<1}(1-r)^{\alpha p} \int_{-\pi}^{\pi}|f(re^{i\theta})|^p\, d\theta <\infty\},\) \(1\leq p<\infty,\) \(\alpha>0\), then \(\mathcal{H}\) is bounded on \(H^p_\alpha\) if and only if \(\alpha +1/p<1\).NEWLINENEWLINE(2) If \(f\in H^1\) and \(\int_{-\pi}^\pi|f(e^{i\theta})|\log(\pi/|\theta|)\, d\theta<\infty\), then \(\mathcal{H}(f)\in H^1\).NEWLINENEWLINE(3) If \(f(z)=\sum_{n=0}^\infty a_n z^n\), \(a_n\geq 0\), then \(\mathcal{H}(f)\in H^1\) if and only if \(\sum_{n=0}^\infty \frac{a_n\log(n+2)}{n+1}<\infty\).NEWLINENEWLINE(4) Let \(\omega_\alpha(z)=\log^\alpha (2/(1-|z|^2))\). If \(\alpha >3\), then \(\mathcal{H}\) is bounded from the weighted Bergman space \(B^2(\omega_\alpha)=H\cap L^2(\omega_\alpha)\) to the Bergman space \(B^2\). If \(\alpha>1\), then \(\mathcal{H}\) is bounded from the weighted Bloch space NEWLINE\[NEWLINE \mathcal{B}(\omega_\alpha)=\left\{f\in H: \sup_{z\in\mathbb{D}}\{(1-|z|^2)\omega_\alpha(z)|f'(z)|\}<\infty\right\} NEWLINE\]NEWLINE to the Bloch space \(\mathcal{B}\).