Semi-harmonicity, Integral Means and Euler Type 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.

This paper is a corrected version of "Semi-Harmonicity, Integral Means and Euler Type Vector Fields, Adv. Appl. Clifford Alg. 17 (2007), 555-573". This publication is available at Springer via http://dx.doi.org/ DOI 10.1007/s00006-007-0036-9