Norm of the Hausdorff operator on the real Hardy space H¹(mathbb R)

Let $\varphi$ be a nonnegative integrable function on $(0,\infty)$. It is well-known that the Hausdorff operator $\mathcal H_\varphi$ generated by $\varphi$ is bounded on the real Hardy space $H^1(\mathbb R)$. The aim of this paper is to give the exact norm of $\mathcal H_\varphi$. More precisely, we prove that $$\|\mathcal H_\varphi\|_{H^1(\mathbb R)\to H^1(\mathbb R)}= \int_0^\infty \varphi(t)dt.$$