Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

TL;DR

This paper characterizes when the commutator of the Cauchy integral operator on Lipschitz curves is bounded or compact on Morrey spaces, linking it to BMO and CMO functions, and applies this to Hardy space factorization.

Contribution

It provides necessary and sufficient conditions for boundedness and compactness of Cauchy commutators on Morrey spaces, connecting them to BMO and CMO, and offers a Hardy space factorization.

Findings

Boundedness of commutators characterized by BMO functions.

Compactness of commutators characterized by CMO functions.

Application to Hardy space $H^1$ factorization.

Abstract

Let $C_\Gamma$ be the Cauchy integral operator on a Lipschitz curve $\Gamma$. In this article, the authors show that the commutator $[b,C_\Gamma]$ is bounded (resp., compact) on the Morrey space $L^{p,\,\lambda}(\mathbb R)$ for any (or some) $p\in(1, \infty)$ and $\lambda\in(0, 1)$ if and only if $b\in {\rm BMO}(\mathbb R)$ (resp., ${\rm CMO}(\mathbb R)$). As an application, a factorization of the classical Hardy space $H^1(\mathbb R)$ in terms of $C_\Gamma$ and its adjoint operator is obtained.