L²-boundedness of the n-th Calder\'on commutator on Lipschitz graphs

This paper investigates the $L^2$-boundedness of the $n$-th Calder\'on commutator $T_{A,n}$ associated with a Lipschitz graph. We prove the estimate $\|T_{A,n}\|_{L^2\to L^2} \leq Cn\|A'\|_\infty^n$ for general Lipschitz functions, formalizing a claim by Verdera and Mateu via a symmetrization strategy and the local $T1$ theorem. We also show that additional regularity on $A$ yields sublinear growth in $n$. Specifically, for $A$ supported in $[0,1]$, the bound improves to a behavior of the form $\sqrt{n}\|A'\|_\infty^n$ under a Dini condition on $A'$, or if $A'$ belongs to the logarithmic Besov space $B^{1,0}_{1,1}(\R)$. This space contains all compactly supported functions in the Sobolev spaces $H^s(\R)$ for $0<s<1,$ as well as functions of bounded variation. These refined estimates are established through an alternative framework based on H\"ormander-type conditions and interpolation, bypassing the standard $T1$ approach. Counterexamples are provided to demonstrate that the Dini and Sobolev fractional regularity conditions are incomparable.