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Logarithmic Sobolev inequalities on infinite-dimensional reduced Heisenberg groups

Maria Gordina, Liangbing Luo

TL;DR

The paper addresses establishing logarithmic Sobolev inequalities for hypoelliptic heat kernel measures on infinite-dimensional reduced Heisenberg groups modeled on abstract Wiener spaces. The authors develop a general invariance framework for LSI under quasi-homeomorphisms of quasi-regular Dirichlet spaces and construct the infinite-dimensional reduced Heisenberg groups along with their Dirichlet-form and hypoelliptic-operator structures. They then transfer the known LSI from the ambient infinite-dimensional Heisenberg group to its reduced quotient via the quotient map, obtaining a dimension-free LSI constant that does not depend on the underlying symmetry parameter $\omega$, and provide a direct argument as well. The work highlights a novel approach to infinite-dimensional functional inequalities on non-linear homogeneous spaces and showcases the robustness of LSI under quotient-type group actions, with potential extensions to hypercontractivity and related inequalities.

Abstract

We construct a family of infinite-dimensional reduced Heisenberg groups which can be viewed as infinite-dimensional homogeneous spaces. Such a space is an analogue of finite-dimensional reduced Heisenberg groups in infinite dimensions. We study properties of the hypoelliptic heat kernel measure on this space, including hypoelliptic logarithmic Sobolev inequalities there.

Logarithmic Sobolev inequalities on infinite-dimensional reduced Heisenberg groups

TL;DR

The paper addresses establishing logarithmic Sobolev inequalities for hypoelliptic heat kernel measures on infinite-dimensional reduced Heisenberg groups modeled on abstract Wiener spaces. The authors develop a general invariance framework for LSI under quasi-homeomorphisms of quasi-regular Dirichlet spaces and construct the infinite-dimensional reduced Heisenberg groups along with their Dirichlet-form and hypoelliptic-operator structures. They then transfer the known LSI from the ambient infinite-dimensional Heisenberg group to its reduced quotient via the quotient map, obtaining a dimension-free LSI constant that does not depend on the underlying symmetry parameter , and provide a direct argument as well. The work highlights a novel approach to infinite-dimensional functional inequalities on non-linear homogeneous spaces and showcases the robustness of LSI under quotient-type group actions, with potential extensions to hypercontractivity and related inequalities.

Abstract

We construct a family of infinite-dimensional reduced Heisenberg groups which can be viewed as infinite-dimensional homogeneous spaces. Such a space is an analogue of finite-dimensional reduced Heisenberg groups in infinite dimensions. We study properties of the hypoelliptic heat kernel measure on this space, including hypoelliptic logarithmic Sobolev inequalities there.

Paper Structure

This paper contains 16 sections, 16 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Logarithmic Sobolev inequalities under quasi-homeomorphisms
  3. Infinite-dimensional reduced Heisenberg groups
  4. Abstract Wiener spaces
  5. Reduced Heisenberg groups
  6. Examples
  7. Properties of infinite-dimensional reduced Heisenberg groups
  8. Finite-dimensional projection groups
  9. Left-invariant vector fields
  10. Distance on the Cameron-Martin subgroup
  11. Sub-Laplacian and horizontal gradient
  12. Hypoelliptic Brownian motion and hypoelliptic heat kernel measure
  13. Dirichlet forms
  14. Dirichlet form on $G$
  15. Dirichlet form on $\widetilde{G}$
  16. ...and 1 more sections

Key Result

Theorem 2.4

Given two quasi-regular Dirichlet spaces $(E_i,\mu_i,\mathcal{E}_i,\mathcal{D}(\mathcal{E}_i))$, suppose $\varphi: E_1 \longrightarrow E_2$ is a quasi-homeomorphism between these two Dirichlet spaces. If $LSI_C(E_1,\mu_1,\mathcal{E}_1,\mathcal{D}(\mathcal{E}_1))$ holds on $E_1$, then $LSI_C(E_2, \mu

Theorems & Definitions (57)

  • Definition 2.2
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • ...and 47 more