Bulletin of the
Korean Mathematical Society BKMS

It is well known that every invariant harmonic function on the unit ball of the multi-dimensional complex space has the volume version of the invariant mean value property. In 1993 Ahern, Flores and Rudin first observed that the validity of the converse depends on the dimension of the underlying complex space. Later Lie and Shi obtained the analogues on the unit ball of multi-dimensional real space. In this paper we obtain the half-space analogues of the results of Liu and Shi.

Keywords: Laplace-Beltrami operator, $\mathcal M$-harmonic, invariant mean value property, invariant volume mean value property, half-space

MSC numbers: Primary 31B05; Secondary 31B10, 30D45, 30D55

Supported by: B. R. Choe was supported by NRF(2018R1D1A1B07041183) of Korea and K. Nam was supported by Faculty of Liberal Education Seoul National University.