Uniform $L^2$-bounds for the $n$-th Calder\'on commutator on Lipschitz graphs

Abstract

We prove that, for every real Lipschitz function $A$ and every integer $n\ge1$, $\|T_{A,n}\|_{L^2(\mathbb{R})\to L^2(\mathbb{R})}\le C\,\|A'\|_\infty^n$, where $T_{A,n}$ is the $n$-th Calder\'on commutator and $C$ is an absolute constant.