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

TL;DR

This paper studies the long-term behavior of certain semigroups on Euclidean domains, providing new bounds on the torsion function and growth estimates for semigroups with Gaussian bounds, relevant for elliptic operators with potentials.

Contribution

It introduces new $L_ty$-bounds for the torsion function and analyzes the $L_ty$-growth of semigroups satisfying Gaussian bounds, extending previous results.

Findings

Established near-optimal $L_ty$-bounds for the torsion function.

Analyzed the long-time $L_ty$-growth of semigroups with Gaussian bounds.

Applied results to semigroups generated by elliptic operators with potentials.

Abstract

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 $L_\infty$ operator norm of the semigroup. As an application we prove a new $L_\infty$-bound for the torsion function of a Euclidean domain that is close to optimal.