Lipschitz Type Characterizations of Harmonic Bergman Spaces

Abstract

Abstract. Wulan and Zhu [16] have characterized the weighted Bergman space in the setting of the unit ball of C n in terms of Lipschitz type conditions in three different metrics. In this paper, we study characterizations of the harmonic Bergman space on the upper half-space in R n . Furthermore, we extend harmonic analogues in the setting of the unit ball to the full range 0 < p < ∞. In addition, we provide the application of characterizations to showing the boundedness of a mapping defined by a difference quotient of harmonic function.

Bergman-type and Qk-type spaces of p-harmonic functions

“…It is well-known that integral estimate (1.1) plays an important role in the theory of holomorphic functions. For the generalizations and applications of (1.1) to the spaces of holomorphic functions, harmonic functions, and solutions to certain PDEs, see [3,4,5,9,15,11,14,21,25] and the references therein. In [18], Siskakis extended (1.1) to the setting of exponentially weighted Bergman space of holomorphic functions for 1 ≤ s < ∞.…”