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

TL;DR

This paper investigates the $L^1$-smoothing effects of semigroups derived from ground state transformations of Schr"odinger operators with non-local Lévy operators, highlighting their dependence on potentials and measures, and illustrating their long-term regularization.

Contribution

It provides new $L^1$-smoothing estimates for a broad class of non-local Schr"odinger semigroups, including fractional and relativistic Laplacians, with explicit dependence on potentials and Lévy measures.

Findings

$L^1$-regularization effects strengthen as time increases.

Estimates depend explicitly on the potential and Lévy measure.

Framework includes fractional and relativistic Laplacians.

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 $t \uparrow \infty$.