Smoothing on $L^1$ for ground state transformed semigroups in non-local settings

Abstract

We study the $L^1$-smoothing properties for a broad class of semigroups arising from the ground state transformation of Schr\"odinger semigroups with confining potentials associated with non-local L\'evy operators, for which (asymptotic) ultracontractivity and hypercontractivity fail. Our work is inspired by Talagrand's convolution conjecture in the discrete cube setting, as well as by subsequent developments on the classical Ornstein--Uhlenbeck semigroup. The estimates we provide exhibit a clear dependence on the potential and the L\'evy measure defining the kinetic term operator, and they yield a description of the semigroups' action on $L^1$ in terms of Orlicz spaces. Our framework is quite general, encompassing fractional and relativistic Laplacians as kinetic operators. The results are illustrated by numerous examples demonstrating that the $L^1$-regularizing effects become stronger as $t \uparrow \infty$.