Hardy decomposition of higher order Lipschitz classes by polymonogenic functions

TL;DR

This paper develops a multidimensional Hardy decomposition for higher order Lipschitz functions using Clifford analysis, providing a new way to represent these functions as traces of polymonogenic functions and solving related Riemann-Hilbert problems.

Contribution

It introduces a novel multidimensional Hardy decomposition for Lipschitz functions via Clifford analysis and solves associated Riemann-Hilbert problems.

Findings

Decomposition of higher order Lipschitz functions into polymonogenic traces

Use of Cliffordian Cauchy-type operator as an involution

Multidimensional sharpening of Hardy decomposition

Abstract

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.