The Cauchy integral, bounded and compact commutators

TL;DR

This paper characterizes BMO and VMO function spaces on the real line through the boundedness and compactness of the commutator of the Cauchy integral operator, linking operator theory with function space properties.

Contribution

It provides new characterizations of BMO and VMO spaces on the real line using the boundedness and compactness of the Cauchy integral commutator.

Findings

BMO space characterized via bounded commutator

VMO space characterized via compact commutator

Links between operator properties and function space regularity

Abstract

We study the commutator of the well-known Cauchy integral operator with a locally integrable function $b$ on $\mathbb R$, and establish the characterisation of the BMO space on $\mathbb R$ via the $L^p$ boundedness of this commutator. Moreover, we also establish the characterisation of the VMO space on $\mathbb R$ via the compactness of this commutator.