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ON HARMONICITY IN A DISC AND n-HARMONICITY

  • Received : 2009.03.02
  • Published : 2010.07.31

Abstract

Let ${\tau}\;{\neq}\;\delta_0$ be either a power bounded radial measure with compact support on the unit disc D with $\tau(D)\;=\;1$ such that there is a $\delta$ > 0 so that ${\mid}\hat{\tau}(s){\mid}\;{\neq}\;1$ for every $s\;{\in}\;\Sigma(\delta)$ \ {0,1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {$h\;{\in}\;L^{\infty}(D)\;{\mid}\;lim_{n{\rightarrow}{\infty}}\;h\;*\;\tau^n$ exists}. Let $\tau_1$, ..., $\tau_n$ be measures on D with above mentioned properties. Here, we prove that if $f\;{in}\;L^{\infty}(D^n)$ satisfies an invariant volume mean value property with respect to $\tau_1$, ..., $\tau_n$, then f is n-harmonic.

Acknowledgement

Supported by : Sogang 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. J. Arazy and M. Englis, Iterates and the boundary behavior of the Berezin transform, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 4, 1101–1133. https://doi.org/10.5802/aif.1847
  3. Y. Benyamini and Y. Weit, Harmonic analysis of spherical functions on SU(1, 1), Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 671–694. https://doi.org/10.5802/aif.1305
  4. M. Englis, Functions invariant under the Berezin transform, J. Funct. Anal. 121 (1994), no. 1, 233–254. https://doi.org/10.1006/jfan.1994.1048
  5. H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. https://doi.org/10.2307/1970220
  6. H. Furstenberg, Boundaries of Riemannian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), pp. 359–377. Pure and Appl. Math., Vol. 8, Dekker, New York, 1972.
  7. Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313–328. https://doi.org/10.1016/0022-1236(86)90101-1
  8. J. Lee, Weighted Berezin transform in the polydisc, J. Math. Anal. Appl. 338 (2008), no. 2, 1489–1493. https://doi.org/10.1016/j.jmaa.2007.06.048
  9. W. Rudin, Function Theory in the Unit Ball of $C^n$, Springer-Verlag, New York Inc., 1980.