The unboundedness of Hausdorff operators on Quasi-Banach spaces

TL;DR

This paper demonstrates that the Hausdorff operator is generally unbounded on many Quasi-Banach spaces unless it is the zero operator, highlighting limitations of these operators in such spaces.

Contribution

It establishes the unboundedness of Hausdorff operators on a broad class of Quasi-Banach spaces, except when the operator is trivial.

Findings

Hausdorff operator is unbounded on many Quasi-Banach spaces

Unboundedness holds unless the operator is zero

Highlights limitations of Hausdorff operators in Quasi-Banach spaces

Abstract

In this note, we show that the Hausdorff operator $H_{\Phi}$ is unbounded on a large family of Quasi-Banach spaces, unless $H_{\Phi}$ is a zero operator.