Lipschitz Norm Estimate for a Higher Order Singular Integral Operator

Abstract

Let \(\Gamma \) be a d-summable surface in \(\mathbb {R}^m\), i.e., the boundary of a Jordan domain in \( \mathbb {R}^m\), such that \(\int \nolimits _{0}^{1}N_{\Gamma }(\tau )\tau ^{d-1}\textrm{d}\tau <+\infty \), where \(N_{\Gamma }(\tau )\) is the number of balls of radius \(\tau \) needed to cover \(\Gamma \) and \(m-1<d<m\). In this paper, we consider a singular integral operator \(S_\Gamma ^*\) associated with the iterated equation \({\mathcal {D}}_{\underline{x}}^k f=0\), where \({\mathcal {D}}_{\underline{x}}\) stands for the Dirac operator constructed with the orthonormal basis of \( \mathbb {R}^m\). The fundamental result obtained establishes that if \(\alpha >\frac{d}{m}\), the operator \(S_\Gamma ^*\) transforms functions of the higher order Lipschitz class \(\text{ Lip }(\Gamma , k +\alpha )\) into functions of the class \(\text{ Lip }(\Gamma , k +\beta )\), for \(\beta =\frac{m\alpha -d}{m-d}\). In addition, an estimate for its norm is obtained.