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A CHARACTERIZATION OF M-HARMONICITY

  • Published : 2010.01.31

Abstract

If f is M-harmonic and integrable with respect to a weighted radial measure $\upsilon_{\alpha}$ over the unit ball $B_n$ of $\mathbb{C}^n$, then $\int_{B_n}(f\circ\psi)d\upsilon_{\alpha}=f(\psi(0))$ for every $\psi{\in}Aut(B_n)$. Equivalently f is fixed by the weighted Berezin transform; $T_{\alpha}f = f$. In this paper, we show that if a function f defined on $B_n$ satisfies $R(f\circ\phi){\in}L^{\infty}(B_n)$ for every $\phi{\in}Aut(B_n)$ and Sf = rf for some |r|=1, where S is any convex combination of the iterations of $T_{\alpha}$'s, then f is M-harmonic.

Keywords

M-harmonic function;weighted Berezin transform;Gelfand transform

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. S. Axler and Z. Cuckovic, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory 14 (1991), no. 1, 1–12. https://doi.org/10.1007/BF01194925
  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. A. Erdeli et. al., Higher Transcendental Functions Vol. I, McGraw-Hill, New York, 1953.
  6. H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. https://doi.org/10.2307/1970220
  7. A. Erdeli et. al., 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.
  8. S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progress in Mathematics, 13. Birkhauser, Boston, Mass., 1981.
  9. S. Helgason, Groups and Geometric Analysis, Academic Press, 1984.
  10. 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
  11. J. Lee, Properties of the Berezin transform of bounded functions, Bull. Austral. Math. Soc. 59 (1999), no. 1, 21–31. https://doi.org/10.1017/S0004972700032561
  12. W. Rudin, Function Theory in the Unit Ball of $C^n$, Springer-Verlag, New York Inc., 1980.