The optimal range of the Calder\`{o}n operator and its applications

TL;DR

This paper determines the optimal functional spaces for the Calderón operator and related transforms, impacting the understanding of operator bounds and functional calculus in analysis.

Contribution

It establishes the precise optimal range of the Calderón operator and Hilbert transform within symmetric quasi-Banach spaces, extending to related operators and estimates.

Findings

Identified the optimal range of the Calderón operator.

Extended results to the Hilbert transform and triangular truncation.

Provided new bounds for operator Lipschitz functions and commutator estimates.

Abstract

We identify the optimal range of the Calder\`{o}n operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the optimal range of the triangular truncation operator, operator Lipschitz functions and commutator estimates in ideals of compact operators.