  In this paper, we study an extension problem for the Ornstein-Uhlenbeck
operator $L=-\Delta+2x\cdot\nabla +n$ and we obtain various characterisations
of the solution of the same. We use a particular solution of that extension
problem to prove a trace Hardy inequality for $L$ from which Hardy's inequality
for fractional powers of $L$ is obtained. We also prove an isometry property of
the solution operator associated to the extension problem. Moreover, new
$L^p-L^q$ estimates are obtained for the fractional powers of the Hermite
operator.
