On the rigidity and boundary regularity for Bakry-Emery-Kohn harmonic functions in Bergman metric on the unit ball in 𝐶ⁿ

Abstract

In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function u u in the unit ball B n B_n in C n \textit {C}^n , which satisfies ◻ ψ u = ∑ i , j = 1 n ( δ i j − z i z ¯ j ) ∂ 2 u ∂ z i ∂ z ¯ j + ( 1 − | z | 2 ) ψ ′ ( | z | 2 ) ∑ j = 1 n z ¯ j ∂ u ∂ z ¯ j = 0 \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 ψ \psi at 1 1 . We prove that any smooth B-E-K harmonic function on B ¯ n \overline {B}_n must be holomorphic in B n B_n . We study the regularity problem for the solution of the Dirichlet boundary value problem: { ◻ ψ u = 0 , if z ∈ B n , u = f , if z ∈ ∂ B n . \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*}