$L^2$-boundedness of the $n$-th Calderón commutator on Lipschitz graphs

Abstract

This paper investigates the asymptotic behavior of the norm, as a bounded operator in $L^2(\mathbb{R})$, of the $n$-th Calderón commutator $T_{A,n}$ on the graph of a Lipschitz function $A:\mathbb{R}\to \mathbb{R}$. We prove the estimate $\Vert T_{A,n}{\Vert }_{L^2\to L^2}\le Cn\Vert A^{\prime }{\Vert }_{\infty }^n$, thus formalizing a claim by Mateu and Verdera via a symmetrization strategy and the $T1$ 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 $\sqrt{n}\Vert A^{\prime }{\Vert }_{\infty }^n$ under a Dini condition on $A^{\prime }$, or if $A^{\prime }$ belongs to the logarithmic Besov space $B_{1,1}^{1,0}(\mathbb{R})$. This space contains all compactly supported functions in the Sobolev spaces $H^s(\mathbb{R})$ for $0<s<1,$ as well as functions of bounded variation. These refined estimates are established through an alternative framework based on Hörmander-type conditions and interpolation, bypassing the standard $T1$ approach. Counterexamples are provided to demonstrate that the Dini and Sobolev fractional regularity conditions are incomparable.