  We consider second-order partial differential operators $H$ in divergence
form on $\Ri^d$ with a positive-semidefinite, symmetric, matrix $C$ of real
$L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only
if the associated semigroup kernel satisfies local lower bounds, or, if and
only if the kernel satisfies Gaussian upper and lower bounds.
