One radius theorem for the eigenfunctions of the invariant Laplacian

Abstract:

Let $B$ be the open unit ball in ${{\mathbf {C}}^n}$ with its boundary $S$. Suppose that $\alpha \geq \tfrac {1}{2}$ and $u(z) = {(1 - {\left | z \right |^2})^{n(1 - \alpha )}}F(z)$ for some $F(z) \in C(\overline B )$. If for every $z \in B$ there corresponds an $r(z):0 < r(z) < 1$ and an automorphism ${\psi _z}$ with ${\psi _z}(0) = z$ such that \[ u(z) = \frac {1}{{{g_\alpha }(r(z))}}\int _S {u \circ {\psi _z}(r(z)\zeta )d\sigma (\zeta )} ,\] then $\tilde \Delta u(z) = - 4{n^2}\alpha (1 - \alpha )u(z),z \in B$. Here $\tilde \Delta$ is the invariant Laplacian and ${g_\alpha }(r)$ is the hypergeometric function $F(n - n\alpha ,n - n\alpha ,n;{r^2})$.