$L_{\mathrm{\infty }}$-estimates for the torsion function and $L_{\mathrm{\infty }}$-growth of semigroups satisfying Gaussian bounds

We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_{\mathrm{\infty }}$ operator norm of the semigroup. As an application we prove a new $L_{\mathrm{\infty }}$-bound for the torsion function of a Euclidean domain that is close to optimal.