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We investigate selfadjoint positivity preserving $C_0$-semigroups that aredominated by the free heat semigroup on $\mathbb R^d$. Major examples aresemigroups generated by Dirichlet Laplacians on open subsets or bySchr\"odinger operators with absorption potentials. We show explicit globalGaussian upper bounds for the kernel that correctly reflect the exponentialdecay of the semigroup. For eigenfunctions of the generator that correspond toeigenvalues below the essential spectrum we prove estimates of their $L_1$-normin terms of the $L_2$-norm and the eigenvalue counting function. This estimateis applied to a comparison of the heat content with the heat trace of thesemigroup.

$L_1$-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup