Lipschitz Continuity for Harmonic Functions and Solutions of the <inline-formula><math display="inline"><semantics><mover accent="true"><mi>α</mi><mo>¯</mo></mover></semantics></math></inline-formula>-Poisson Equation

Abstract

In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.

Keywords

• Poisson’s kernel
• Green function
• harmonic functions
• gradient estimate
• lipschitz continuity