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

## TL;DR

This paper determines the exact operator norm of the Hausdorff operator on the real Hardy space $H^1(
eal)$, showing it equals the integral of the generating function $\

## Contribution

It provides the precise norm of the Hausdorff operator on $H^1(
eal)$, extending known boundedness results to an exact norm calculation.

## Key findings

- The norm of $\\mathcal{H}_\varphi$ equals $\int_0^\infty \varphi(t) dt$.
- The Hausdorff operator is bounded on $H^1(\real)$ with a known exact norm.
- The result clarifies the operator's behavior on the real Hardy space.

## Abstract

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 $$\|\mathcal H_\varphi\|_{H^1(\mathbb R)\to H^1(\mathbb R)}= \int_0^\infty \varphi(t)dt.$$