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

TL;DR

This paper derives explicit Gaussian bounds for heat kernels of certain semigroups dominated by the free heat semigroup and provides $L_1$-estimates for eigenfunctions below the essential spectrum, with applications to heat content and trace comparisons.

Contribution

It introduces new $L_1$-estimates for eigenfunctions and explicit heat kernel bounds for semigroups dominated by the free heat semigroup, extending previous results.

Findings

Established global Gaussian upper bounds for heat kernels.

Derived $L_1$-norm estimates for eigenfunctions below the essential spectrum.

Compared heat content with heat trace using these estimates.

Abstract

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.