Invariant mean value property and $\mathcal M$-harmonicity on the half-space
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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.
Abstract
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.
참고문헌 (6)
- P. Ahern, "An Invariant Volume-Mean-Value Property", Journal of Functional Analysis, 111(2), pp.380~397 (1993), Elsevier BV[DOI]
- L. V. Ahlfors, "M¨obius Transformations in Several Dimensions, Ordway Professorship Lectures in Mathematics" (1981), University of Minnesota, School of Mathematics
- Heinz Leutwiler, "Best constants in the Harnack inequality for the Weinstein equation", Aequationes Mathematicae, 34(2-3), pp.304~315 (1987), Springer Science and Business Media LLC[DOI]
- Congwen Liu, "Boundary regularity in the Dirichlet problem for the invariant Laplacians ∆γ on the unit real ball", Proceedings of the American Mathematical Society, 132(11), pp.3259~3268 (2004), American Mathematical Society (AMS)[DOI]
- Cong Wen Liu, "Invariant Mean-Value Property and M-Harmonicity in the Unit Ball of ℝ n", Acta Mathematica Sinica, English Series, 19(1), pp.187~200 (2003), Springer Science and Business Media LLC[DOI]
- W. Rudin, "Function theory in the unit ball of Cn" (1980), Springer-Verlag