  When $L$ is the Hermite or the Ornstein-Uhlenbeck operator, we find minimal
integrability and smoothness conditions on a function $f$ so that the
fractional power $L^\sigma f(x_0)$ is well-defined at a given point $x_0$. We
illustrate the optimality of the conditions with various examples. Finally, we
obtain similar results for the fractional operators $(-\Delta+R)^\sigma$, with
$R>0$.
