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.