## On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in $\textit {C}^n$

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**145**(2017), 2971-2979 Request permission

## Abstract:

In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function $u$ in the unit ball $B_n$ in $\textit {C}^n$, which satisfies \begin{equation*} \Box ^{\psi } u=\sum _{i, j=1}^n (\delta _{ij}-z_i \overline {z}_j){\partial ^2 u \over \partial z_i \partial \overline {z}_j}+(1-|z|^2) \psi ’(|z|^2) \sum _{j=1}^n \overline {z}_j {\partial u\over \partial \overline {z}_j}=0 \end{equation*} with some restriction of the coefficients of Taylor expansion for $\psi$ at $1$. We prove that any smooth B-E-K harmonic function on $\overline {B}_n$ must be holomorphic in $B_n$. We study the regularity problem for the solution of the Dirichlet boundary value problem: \begin{equation*} \begin {cases} \Box ^\psi u=0, \hbox { if } z\in B_n,\cr \quad \ u=f, \hbox { if } z\in \partial B_n.\cr \end{cases} \end{equation*}## References

- Patrick Ahern, Joaquim Bruna, and Carme Cascante,
*$H^p$-theory for generalized $M$-harmonic functions in the unit ball*, Indiana Univ. Math. J.**45**(1996), no. 1, 103–135. MR**1406686**, DOI 10.1512/iumj.1996.45.1961 - C. Robin Graham,
*The Dirichlet problem for the Bergman Laplacian. I*, Comm. Partial Differential Equations**8**(1983), no. 5, 433–476. MR**695400**, DOI 10.1080/03605308308820275 - C. Robin Graham,
*The Dirichlet problem for the Bergman Laplacian. II*, Comm. Partial Differential Equations**8**(1983), no. 6, 563–641. MR**700730**, DOI 10.1080/03605308308820279 - C. Robin Graham and John M. Lee,
*Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains*, Duke Math. J.**57**(1988), no. 3, 697–720. MR**975118**, DOI 10.1215/S0012-7094-88-05731-6 - Song-Ying Li and Lei Ni,
*On the holomorphicity of proper harmonic maps between unit balls with the Bergman metrics*, Math. Ann.**316**(2000), no. 2, 333–354. MR**1741273**, DOI 10.1007/s002080050015 - Song-Ying Li and Ezequias Simon,
*Boundary behavior of harmonic functions in metrics of Bergman type on the polydisc*, Amer. J. Math.**124**(2002), no. 5, 1045–1057. MR**1925342** - Song-Ying Li and DongHuan Wei,
*On the rigidity theorem for harmonic functions in Kähler metric of Bergman type*, Sci. China Math.**53**(2010), no. 3, 779–790. MR**2608333**, DOI 10.1007/s11425-010-0040-8 - Congwen Liu and Lizhong Peng,
*Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma$ on the unit real ball*, Proc. Amer. Math. Soc.**132**(2004), no. 11, 3259–3268. MR**2073300**, DOI 10.1090/S0002-9939-04-07582-3 - L. K. Hua,
*Harmonic analysis of functions of several complex variables in the classical domains*, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR**0171936** - Walter Rudin,
*Function theory in the unit ball of $\textbf {C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594**

## Additional Information

**QiQi Zhang**- Affiliation: School of Mathematics and Computer Science, FuJian Normal University, Fuzhou 350007, People’s Republic of China
- Email: zqq123_{good}@163.com
- Received by editor(s): August 8, 2016
- Published electronically: February 22, 2017
- Communicated by: Lei Ni
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 2971-2979 - MSC (2010): Primary 32A50
- DOI: https://doi.org/10.1090/proc/13501
- MathSciNet review: 3637945