Norm of the Hausdorff operator on the real Hardy space $H^1(\mathbb{R})$

Abstract

Let $\phi$ be a nonnegative integrable function on $(0,\infty )$. It is well-known that the Hausdorff operator ${\mathcal{H}}_{\phi }$ generated by $\phi$ is bounded on the real Hardy space $H^1(\mathbb{R})$. The aim of this paper is to give the exact norm of ${\mathcal{H}}_{\phi }$. More precisely, we prove that $$\Vert {\mathcal{H}}_{\phi }{\Vert }_{H^1(\mathbb{R})\to H^1(\mathbb{R})}={\int }_0^{\infty }\phi (t)dt.$$

Complex Anal. Oper. Theory (to appear)