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.

设\(\Gamma \)为\(\mathbb {R}^m\)中的d可求和曲面，即\( \mathbb {R}^m\)中乔丹域的边界，使得\ (\int \nolimits _{0}^{1}N_{\Gamma }(\tau )\tau ^{d-1}\textrm{d}\tau <+\infty \)，其中\(N_{\ Gamma }(\tau )\)是覆盖\(\Gamma \)和\(m-1<d<m\)所需的半径为\(\tau \)的球的数量。在本文中，我们考虑与迭代方程\({\mathcal {D}}_{\underline{x}}^kf=0\)关联的奇异积分算子\(S_\Gamma ^*\)，其中\ ({\mathcal {D}}_{\underline{x}}\)代表以正交基\( \mathbb {R}^m\)构造的狄拉克算子。获得的基本结果表明，如果\(\alpha >\frac{d}{m}\)，算子\(S_\Gamma ^*\)变换高阶 Lipschitz 类的函数\(\text{ Lip }( \Gamma , k +\alpha )\)转换为\(\text{ Lip }(\Gamma , k +\beta )\)类的函数，对于\(\beta =\frac{m\alpha -d}{ md}\)。此外，还获得了对其范数的估计。