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THE MEAN-VALUE PROPERTY AND (α,β)-HARMONICITY

Part of: Higher-dimensional theory

Published online by Cambridge University Press:  14 October 2011

DAN LI  and
CONGWEN LIU
DAN LI*
Affiliation:
Chern Institute of Mathematics, Nankai University, Tianjin 300071, PR China (email: [email protected])
CONGWEN LIU
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this article we consider whether, for integrable functions on the unit ball of ℂn, the mean-value property implies (α,β)-harmonicity. We find that the answer is affirmative when 0<n+α+β≤ρ0, but is negative when n+α+β>ρ0. Here ρ0 is a constant between 11.025 and 11.069.

Keywords

(α,β)-harmonic functionsBerezin-type transformsmean-value property

MSC classification

Secondary: 31B05: Harmonic, subharmonic, superharmonic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 2 , October 2011 , pp. 189 - 206
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This work was supported by the National Natural Science Foundation of China (No. 10601025, 11071230) and the Anhui Provincial Natural Science Foundation (No. 090416233).

References

[1]Ahern, P., Bruna, J. and Cascante, C., ‘H p-theory for generalized M-harmonic functions in the unit ball’, Indiana Univ. Math. J. 45 (1996), 103–135.CrossRefGoogle Scholar
[2]Ahern, P. and Cascante, C., ‘Exceptional sets for Poisson integrals of potentials on the unit sphere in ℂn, p≤1’, Pacific J. Math. 153 (1992), 1–14.CrossRefGoogle Scholar
[3]Ahern, P., Flores, M. and Rudin, W., ‘An invariant volume-mean-value property’, J. Funct. Anal. 111 (1993), 380–397.CrossRefGoogle Scholar
[4]Andrews, G. E., Askey, R. and Roy, R., Special Functions (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[5]Arazy, J., Fisher, S. D. and Peetre, J., ‘Hankel operators on weighted Bergman spaces’, Amer J. Math. 110 (1988), 989–1053.CrossRefGoogle Scholar
[6]Arazy, J. and Zhang, G., ‘Invariant mean value and harmonicity in Cartan and Siegel domains’, in: Interaction between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), Lecture Notes in Pure and Applied Mathematics, 175 (Dekker, New York, 1996), pp. 19–40.Google Scholar
[7]Axler, S. and C̆uc̆ković, Z̆., ‘Commuting Toeplitz operators with harmonic symbols’, Integral Equations Operator Theory 14 (1991), 1–12.CrossRefGoogle Scholar
[8]Ben Natan, Y. and Weit, Y., ‘Integrable harmonic functions on ’, J. Funct. Anal. 150 (1997), 471–477.CrossRefGoogle Scholar
[9]Ben Natan, Y. and Weit, Y., ‘Integrable harmonic functions on symmetric spaces of rank one’, J. Funct. Anal. 160 (1998), 141–149.CrossRefGoogle Scholar
[10]Engliš, M., ‘Functions invariant under the Berezin transform’, J. Funct. Anal. 121 (1994), 233–254.CrossRefGoogle Scholar
[11]Engliš, M., ‘A mean value theorem on bounded symmetric domains’, Proc. Amer. Math. Soc. 127 (1999), 3259–3268.CrossRefGoogle Scholar
[12]Erdélyi, A., Higher Transcendental Functions (McGraw-Hill, New York, 1953).Google Scholar
[13]Geller, D., ‘Some results in H p theory for the Heisenberg group’, Duke Math. J. 47 (1980), 365–390.CrossRefGoogle Scholar
[14]Jevtić, M., ‘Fixed points of an integral operator’, J. Anal. Math. 91 (2003), 123–141.CrossRefGoogle Scholar
[15]Korányi, A., ‘On a mean value property for hyperbolic spaces’, Contemp. Math. 191 (1995), 107–116.CrossRefGoogle Scholar
[16]Nagel, A. and Rudin, W., ‘Moebius-invariant function spaces on balls and spheres’, Duke Math. J. 43 (1976), 841–865.CrossRefGoogle Scholar
[17]Rudin, W., Function Theory in the Unit Ball of ℂn (Springer, New York–Berlin, 1980).CrossRefGoogle Scholar
[18]Rudin, W., Functional Analysis, 2nd edn (McGraw-Hill, New York, 1991).Google Scholar
[19]Weit, Y., ‘On Furstenberg’s characterization of harmonic functions on symmetric spaces’, Israel J. Math. 114 (1999), 265–269.CrossRefGoogle Scholar
[20]Yi, Y., ‘A characterization of the functions fixed by a class of integral operators’, PhD Thesis, University of Wisconsin-Madison, 1995.Google Scholar