Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

INVARIANT MEAN VALUE PROPERTY AND 𝓜-HARMONICITY ON THE HALF-SPACE

  • Choe, Boo Rim (Department of Mathematics Korea University) ;
  • Nam, Kyesook (Faculty of Liberal Education Seoul National University)
  • Received : 2020.01.07
  • Accepted : 2020.05.07
  • Published : 2021.05.31
Download PDF
⟨ Previous Next ⟩

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.

Keywords

Acknowledgement

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

References

  1. P. Ahern, M. Flores, and W. Rudin, An invariant volume-mean-value property, J. Funct. Anal. 111 (1993), no. 2, 380-397. https://doi.org/10.1006/jfan.1993.1018
  2. L. V. Ahlfors, Mobius Transformations in Several Dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, MN, 1981.
  3. H. Leutwiler, Best constants in the Harnack inequality for the Weinstein equation, Aequationes Math. 34 (1987), no. 2-3, 304-315. https://doi.org/10.1007/BF01830680
  4. C. Liu and L. Peng, Boundary regularity in the Dirichlet problem for the invariant Laplacians Δγ on the unit real ball, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3259-3268. https://doi.org/10.1090/S0002-9939-04-07582-3
  5. C. Liu and J. H. Shi, Invariant mean-value property and M-harmonicity in the unit ball of Rn, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 1, 187-200. https://doi.org/10.1007/s10114-002-0203-9
  6. W. Rudin, Function theory in the unit ball of Cn, Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, New York, 1980.