Semi-harmonicity, Integral Means and Euler Type Vector Fields

TL;DR

This paper introduces semi-harmonicity and generalized Euler vector fields on Riemann domains, establishing mean-value characterizations and extending Weyl's Lemma to complex spaces.

Contribution

It develops the concept of semi-harmonicity on complex spaces and extends classical harmonic analysis results to Riemann domains.

Findings

Semi-harmonicity characterized by local mean-value properties

Extension of Weyl's Lemma to Riemann domains

Introduction of generalized Euler vector fields

Abstract

The Dirichlet product of functions on a semi-Riemann domain and generalized Euler vector fields, which include the radial, $\bar \partial$-Euler, and the $\bar \partial$-Neumann vector fields, are introduced. The integral means and the harmonic residues of functions on a Riemann domain are studied. The notion of semi-harmonicity of functions on a complex space is introduced. It is shown that, on a Riemann domain, the semi-harmonicity of a locally forwardly $L^2$-function is characterized by local mean-value properties as well as by weak-harmonicity. In particular, the Weyl's Lemma is extended to a Riemann domain.