In this paper we prove a characterization of the $L^p$-to-$L^q$ boundedness of commutators to the Cauchy transform. Our work presents both new results and new proofs for established results. In particular, we show that the Campanato space characterizes boundedness of commutators for a certain range of $p$ and $q$.
Rectifiable measures, square functions involving densities, and the Cauchy transform
Characterization of compactness of commutators of bilinear singular integral operators
Characterization of Lipschitz spaces via commutators of the Hardy-Littlewood maximal function
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