  In this paper we find a decomposition of higher order Lipschitz functions
into the traces of a polymonogenic function and solve a related Riemann-Hilbert
problem. Our approach lies in using a cliffordian Cauchy-type operator, which
behaves as an involution operator on higher order Lipschitz spaces. The result
obtained is a multidimensional sharpened version of the Hardy decomposition of
H\"older continuous functions on a simple closed curve in the complex plane.
