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

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 Schrödinger 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.

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