Uniform $L^2$-bounds for the $n$-th Calderón commutator on Lipschitz graphs

We prove that, for every real Lipschitz function $A$ and every integer $n\ge 1$, $\mathrm{\parallel }{T}_{A,n}{\mathrm{\parallel }}_{L^2(\mathbb{R})\to L^2(\mathbb{R})}\le C\mathrm{\parallel }{A}^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}^n$, where $T_{A,n}$ is the $n$-th Calderón 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 yields an $O(\log n)$ bound and motivates the choice of contour.