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Decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

Lianet De la Cruz Toranzo, Ricardo Abreu Blaya, Swanhild Bernstein

Abstract

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on a simple closed curve in the complex plane.

Decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

Abstract

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on a simple closed curve in the complex plane.
Paper Structure (8 sections, 5 theorems, 55 equations)

This paper contains 8 sections, 5 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Clifford algebras and Clifford analysis
  4. Higher order Lipschitz functions and Whitney extension theorem
  5. Auxiliary results
  6. Main results
  7. Concluding remarks
  8. Acknowledgments

Key Result

Theorem 1

Let $f\in{\hbox{Lip}}(k+\alpha,{\bf E})$. Then, there exists a function $\tilde{f}\in{\hbox{Lip}}(k+\alpha,\mathbb{R}^m)$ satisfying

Theorems & Definitions (8)

  • Definition 1: Higher order Lipschitz functions
  • Theorem 1: Whitney Extension Theorem
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4