The optimal symmetric quasi-Banach range of the discrete Hilbert transform

TL;DR

This paper characterizes the optimal symmetric quasi-Banach range of the discrete Hilbert transform and Calderón operator on sequence spaces, providing specific examples in weak-$ ext{l}_1$ spaces.

Contribution

It identifies the symmetric quasi-Banach range of these operators on sequence spaces, including an explicit example for weak-$ ext{l}_1$ spaces.

Findings

Determined the symmetric quasi-Banach range of the discrete Hilbert transform.

Provided an example of the optimal range for weak-$ ext{l}_1$ space.

Extended the understanding of operator ranges on sequence spaces.

Abstract

We identify symmetric quasi-Banach range of the discrete Calder\'{o}n operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.