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arXiv · 2610.20489

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

Joan Hernández·Joan Mateu·Laura Prat
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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 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 gives $O(\log n)$ and explains why the contour has the shape it has.

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Joan Hernández, Joan Mateu, Laura Prat. 2026-10-14. Uniform $L^2$-bounds for the $n$-th Calderón commutator on Lipschitz graphs. https://arxiv.org/abs/2610.20489

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