Korean J. Math.  Vol 27, No 2 (2019)  pp.437-444
DOI: https://doi.org/10.11568/kjm.2019.27.2.437

Characterizing functions fixed by a weighted Berezin transform in the bidisc

Jaesung Lee


For $c>-1$, let $\nu_{c}$ denote a weighted radial measure on $\mathbb{C}$ normalized so that $\nu_{c}(D)=1.$ For $c_1, c_2 >-1$ and $f \in L^{1}(D^{2},\ \nu_{c_1} \times \nu_{c_2})$, we define the weighted Berezin transform $B_{c_1,c_2}f$ on $D^2$ by
$$\big(B_{c_1, c_2}\big)f(z,w) = \int_{D}\int_{D} f\big( \varphi_{z}(x) , \varphi_{w}(y) \big)\ d\nu_{c_1}(x) d\nu_{c_2}(y).$$
This paper is about the space $M_{c_1, c_2}^{p}$ of function $f \in L^{p}(D^{2},\ \nu_{c_1} \times \nu_{c_2})$ satisfying $B_{c_1,c_2}f=f$ for $1 \le p< \infty$. We find the identity operator on $M_{c_1, c_2}^{p}$ by using invariant Laplacians and we characterize some special type of functions in $M_{c_1, c_2}^{p}$.


weighted Berezin transform, invariant Laplacian, joint eigenfunction

Subject classification

32A70, 47G10.


Full Text:



P. Ahern, M. Flores and W. Rudin, An invariant volume-mean-value property, J. Funct. Anal. 111 (1993) (2), 380–397. (Google Scholar)

J.P Ferrier, Spectral Theory and Complex Analysis, North-Holland, 1973. (Google Scholar)

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. (Google Scholar)

J. Lee, Weighted Berezin transform in the polydisc, J. Math. Anal. Appl. 338 (2) (2008), 1489-1493. (Google Scholar)

J. Lee, Some properties of the Berezin transform in the bidisc, Comm. Korean Math. Soc. 32 (3) (2017), 779–787. (Google Scholar)

J. Lee, Some properties of the weighted Berezin transform in the unit disc and bidisc, Global Journal of Pure and Applied Mathematics, 14 (2) (2018), 275–283 (Google Scholar)

W. Rudin, Function theory in the unit ball of Cn, Springer-Verlag, New York Inc., 1980. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr