The Hausdorff operator is bounded on the real Hardy space $H^{1} ({\mathbb{R}})$

We prove that the Hausdorff operator generated by a function $\varphi \in L^{1} ({\mathbb{R}})$ is bounded on the real Hardy space $H^{1} ({\mathbb{R}})$. The proof is based on the closed graph theorem and on the fact that if a function $f$ in $L^{1} ({\mathbb{R}})$ is such that its Fourier transform $\widehat {f} (t)$ equals $0$ for $t<0$ (or for $t>0$), then $f\in H^{1} ({\mathbb{R}})$.