The Hausdorff operator on the Hardy space \(H^1(\mathbb{R}^1)\) (Q2400102)

In this paper the authors study the Hausdorff operator \[ H_{\phi}f(x)=\int_{0}^{\infty}\phi(u)f(ux)\;du \] where \(\phi\) is a function on (\(0,\;\infty\)). The authors give a sufficient condition on \(\phi\) such that the Hausdorff operator \(H_{\phi}\) is bounded on the Hardy space \(H^1(\mathbb{R})\) and it negates that \(\int_{0}^{\infty}\frac{| \phi(u)|}{u} < \infty\) is a necessary and sufficient condition of boundedness of \(H_{\phi}\) on \(H^1\). The sufficient condition given in this paper is slightly weaker than the condition \(\frac{| \phi(u)|}{u} \in L^1(0,\;\infty)\).