$L^p$-to-$L^q$ boundedness for commutators of the Cauchy transform

TL;DR

This paper characterizes the $L^p$-to-$L^q$ boundedness of commutators of the Cauchy transform, providing new proofs and linking boundedness to Campanato spaces for specific $p$ and $q$ ranges.

Contribution

It offers a new characterization of boundedness for these commutators using Campanato spaces, with novel proofs and extending known results.

Findings

Campanato space characterizes boundedness for certain $p$ and $q$ ranges

New proofs for established boundedness results

Extension of boundedness criteria for commutators of the Cauchy transform

Abstract

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 $q$.