Double integral characterizations of harmonic Bergman spaces

Recently Li et al. have characterized, except for a critical case, the weighted Bergman spaces over the complex ball by means of integrability conditions of double integrals associated with difference quotients of holomorphic functions. In this paper we extend those characterizations to the case of weighted harmonic Bergman spaces over the real ball and complement their results by providing a characterization for the missing critical case. We also investigate the possibility of extensions to the half-space setting. Our observations reveal an interesting half-space phenomenon caused by the unboundedness of the half-space.