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. The essential tool is the Coifman--McIntosh--Meyer theorem, applied to a holomorphic family of complex curves, together with the Cauchy integral formula on a contour surrounding the interval of secant slopes. We also include a simpler version of the argument that gives $O(\log n)$ and explains why the contour has the shape it has.