Hardy decomposition of first order Lipschitz functions by Lam\'e-Navier solutions

Abstract

The Clifford algebra language allows us to rewrite the Lam\'e-Navier system in terms of the Euclidean Dirac operator. In this paper, the main question we shall be concerned with is whether or not a higher order Lipschitz function on the boundary $\Gamma$ of a Jordan domain $\Omega\subset\mathbb{R}^m$ can be decomposed into a sum of the two boundary values of a solution of the Lam\'e-Navier system with jump across $\Gamma$. Our main tool are the Hardy projections related to a singular integral operator arising in the context of Clifford analysis, which turns out to be an involution operator on the first order Lipschitz classes.