The paper characterizes boundedness and compactness of Beurling-Ahlfors commutators on weighted Morrey spaces via BMO and CMO, and applies the compactness result to solve Beltrami equations.
Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces
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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.
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2019 1verdicts
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Buerling-Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations
The paper characterizes boundedness and compactness of Beurling-Ahlfors commutators on weighted Morrey spaces via BMO and CMO, and applies the compactness result to solve Beltrami equations.