Pointwise convergence of fractional powers of Hermite type operators

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 $(-\mathrm{\Delta }+R{)}^{\sigma }$, with $R>0$.