Differentiability criteria and harmonic functions on $B^{n}$

Abstract:

Let ${B^n}$ be the unit ball in ${{\mathbf {F}}^n}$ where ${\mathbf {F}}$ is either ${\mathbf {R}}$, ${\mathbf {C}}$ or ${\mathbf {H}}$. The space ${B^n}$ is a classical rank one symmetric space with respect to the action of a Lie group $G$. Suppose $f$ is a $G$-harmonic function on ${B^n}$ all of whose derivatives of order $\leqslant k$ are bounded. Our main result obtains restrictions on $f$ depending on $k$.