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

Miłosz Baraniewicz, Kamil Kaleta

Abstract

We study the $L^1$-smoothing properties for a broad class of semigroups arising from the ground state transformation of Schrödinger semigroups with confining potentials associated with non-local Lévy 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 Lévy 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 $t \uparrow \infty$.

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

Abstract

We study the -smoothing properties for a broad class of semigroups arising from the ground state transformation of Schrödinger semigroups with confining potentials associated with non-local Lévy 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 Lévy measure defining the kinetic term operator, and they yield a description of the semigroups' action on 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 -regularizing effects become stronger as .
Paper Structure (13 sections, 9 theorems, 169 equations)

This paper contains 13 sections, 9 theorems, 169 equations.

Table of Contents

  1. Motivation, introduction, setting and results
  2. Motivation and introduction
  3. Non-local Schrödinger operators and ground state-transformed semigroups
  4. Formulation of the problem and presentation of results
  5. Dependence on parameters and sharpness of the results
  6. Notation
  7. Assumption \ref{['A']} and its consequences
  8. Proof of Theorem \ref{['thm:main']}
  9. Proof of Theorem \ref{['prop:main2']} and Lemma \ref{['lemma:lem2']}
  10. Proof of Theorem \ref{['prop:main3']}, Corollary \ref{['cor:cor1']} and Lemma \ref{['lemma:lem3']}
  11. Applications and examples
  12. Intrinsic semigroups associated with fractional Schrödinger operators
  13. Intrinsic semigroups associated with fractional relativistic Schrödinger operators

Key Result

Theorem 1.1

Assume A. If eq:decr_to_zero holds, then for every $t>0$ there exists a constant $C(t)>0$ such that for every $h \in L^1(\mu)$ with $\left\|h\right\|_{L^1(\mu)} = 1$ and $u \geqslant \kappa(t)$ we have where with the constant $K$ that comes from the upper bound in eq:est_noniuc, and $\alpha_t(u)$ is defined by def:alpha.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Theorem \ref{['prop:main2']}
  • ...and 12 more