  It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck
operator $$ L= \frac{1}{2}\text{ tr} (QD^2 ) + \langle Ax, D \rangle =
\frac{1}{2}\text{ div} (Q D ) + \langle Ax, D \rangle,\;\; x \in R^N, $$ all
(globally) bounded solutions of $Lu=0$ on $R^N$ are constant if and only if all
the eigenvalues of $A$ have non-positive real parts (i.e., $s(A) \le 0)$. We
show that if $Q$ is positive definite and $s(A) \le 0$, then any non-negative
solution $v$ of $Lv=0$ on $R^N$ which has at most an exponential growth is
indeed constant. Thus under a non-degeneracy condition we relax the boundedness
assumption on the harmonic functions and maintain the sharp condition on the
eigenvalues of $A$. We also prove a related one-side Liouville theorem in the
case of hypoelliptic Ornstein-Uhlenbeck operators.
