Lipschitz Norm Estimate for a Higher Order Singular Integral Operator

Let Γ be a d-summable surface in Rm, i.e., the boundary of a Jordan domain in Rm, such that ∫01NΓ(τ)τd-1dτ<+∞, where NΓ(τ) is the number of balls of radius τ needed to cover Γ and m-1<d<m. In this paper, we consider a singular integral operator SΓ∗ associated with the iterated equation Dx̲kf=0, where Dx stands for the Dirac operator constructed with the orthonormal basis of Rm.

The fundamental result obtained establishes that if α>dm, the operator SΓ∗ transforms functions of the higher order Lipschitz class Lip(Γ,k+α) into functions of the class Lip(Γ,k+β), for β=mα-dm-d. In addition, an estimate for its norm is obtained.