  If $X$ is a non-degenerate vector field on ${\bf R}$ and $H=-X^2$ we examine
conditions for the closure of $H$ to generate a continuous semigroup on
$L_\infty$ which extends to the $L_p$-spaces. We give an example which cannot
be extended and an example which extends but for which the real part of the
generator on $L_2$ is not lower semibounded.
