On Generalized Integral Means and Euler Type Vector Fields

Formulas for the Euler vector fields, the Neumann derivatives, and the Euler as well as Dirichlet product are derived. Extensions to a Riemann domain of the Gauss operator, the Gauss’ lemma and the related jump formulas are given, and the Gauss–Helmholtz representation with ramifications proved. Examples of elementary solutions to certain modified Laplace operators, applications to pseudospherical harmonics, and characterizations of pseudoradial, pseudospherical, nearly holomorphic, and holomorphic functions, are obtained, and constancy criterion for locally Lipschitz, semiharmonic, respectively, weakly holomorphic functions are given.