Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

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 $b\in \mathrm{BMO}(\mathbb{R})$ (resp., $\mathrm{CMO}(\mathbb{R})$). 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.

26 pages