  We investigate selfadjoint positivity preserving $C_0$-semigroups that are
dominated by the free heat semigroup on $\mathbb R^d$. Major examples are
semigroups generated by Dirichlet Laplacians on open subsets or by
Schr\"odinger operators with absorption potentials. We show explicit global
Gaussian upper bounds for the kernel that correctly reflect the exponential
decay of the semigroup. For eigenfunctions of the generator that correspond to
eigenvalues below the essential spectrum we prove estimates of their $L_1$-norm
in terms of the $L_2$-norm and the eigenvalue counting function. This estimate
is applied to a comparison of the heat content with the heat trace of the
semigroup.
