[Paper Review] Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

Fabrice Baudoin, Michel Bonnefont|arXiv (Cornell University)|Jun 2, 2011

Geometric Analysis and Curvature Flows44 references79 citations

TL;DR

This paper establishes log-Sobolev, Poincaré, and logarithmic isoperimetric inequalities for invariant measures of subelliptic diffusions under a generalized curvature-dimension condition CD(ρ₁, ρ₂, κ, ∞). By leveraging Γ-calculus and sub-Riemannian geometry, it proves dimension-free functional inequalities, including a Gaussian-type isoperimetric inequality with explicit constants depending on ρ₁, ρ₂, and κ, extending classical results to non-elliptic settings like the Heisenberg group and Sasakian manifolds.

ABSTRACT

Let $\\M$ be a smooth connected manifold endowed with a smooth measure $\\mu$ and a smooth locally subelliptic diffusion operator $L$ which is symmetric with respect to $\\mu$. We assume that $L$ satisfies a generalized curvature dimension inequality as introduced by Baudoin-Garofalo \\cite{BG1}. Our goal is to discuss functional inequalities for $\\mu$ like the Poincar\\'e inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.

Motivation & Objective

• To establish functional inequalities—Poincaré, log-Sobolev, and isoperimetric—for invariant measures of subelliptic diffusions.
• To extend the Bakry-Emery criterion beyond elliptic settings by introducing a generalized curvature-dimension inequality CD(ρ₁, ρ₂, κ, ∞).
• To derive dimension-free bounds on log-Sobolev constants and isoperimetric profiles in sub-Riemannian geometry.
• To unify and generalize results from Heisenberg groups, Sasakian manifolds, and Hörmander-type operators under a common curvature framework.
• To prove that the quadratic transportation cost inequality implies a modified log-Sobolev inequality with explicit dependence on ρ₁, ρ₂, and κ.

Proposed method

• Introduce a generalized curvature-dimension condition CD(ρ₁, ρ₂, κ, ∞) involving the carré du champ Γ and a second-order form ΓZ, with ρ₁ ∈ ℝ, ρ₂ > 0, κ ≥ 0, and d = ∞.
• Use Γ₂ calculus and the assumption (H.1)–(H.2) to ensure essential self-adjointness and structural consistency of the diffusion operator L.
• Apply reverse Poincaré and log-Harnack inequalities to control the heat semigroup and derive entropy bounds.
• Adapt Otto-Villani and Bobkov-Gentil-Ledoux methods to prove a modified HWI inequality linking entropy, Wasserstein distance, and energy.
• Use hypercontractivity and semigroup estimates to bound the L¹ norm of Pt1A − 1A, enabling isoperimetric estimates.
• Derive the perimeter bound via approximation of characteristic functions in BV(M) and apply the reverse Poincaré inequality to control the L¹ norm of the semigroup difference.

Experimental results

Research questions

• RQ1 Can log-Sobolev inequalities be established for subelliptic diffusions without ellipticity, under a generalized curvature-dimension condition?
• RQ2 What are the sharp, dimension-free bounds on the log-Sobolev constant for such operators?
• RQ3 How does the quadratic transportation cost inequality relate to modified log-Sobolev inequalities in sub-Riemannian settings?
• RQ4 Can a Gaussian-type logarithmic isoperimetric inequality be derived under CD(ρ₁, ρ₂, κ, ∞) and a log-Sobolev inequality?
• RQ5 What is the role of the ΓZ form in extending classical inequalities to non-elliptic operators?

Key findings

• Under CD(ρ₁, ρ₂, κ, ∞) with ρ₁ > 0, ρ₂ > 0, the Poincaré inequality holds with constant (κ + ρ₂)/(ρ₁ρ₂), independent of dimension.
• For probability measures, a modified log-Sobolev inequality is proven with constants depending on ρ₁, ρ₂, κ, and the exponential moment of d²(x₀, x), yielding a dimension-free bound.
• If the quadratic transportation cost inequality holds with c < 2/ρ₋₁, then a modified log-Sobolev inequality holds with constants C₁, C₂ depending only on c, ρ₁, ρ₂, and κ.
• A Gaussian-type logarithmic isoperimetric inequality is established: P(A) ≥ (ln 2 / 4)(3 + 2κ/ρ₂) min(√ρ₀, ρ₀/√|ρ₋₁|) μ(A)(ln(1/μ(A)))¹ᐟ² for μ(A) ≤ 1/2.
• The constant in the isoperimetric inequality is independent of dimension, unlike classical results, due to the use of reverse Poincaré instead of Li-Yau estimates.
• The results apply to sub-Laplacians on Heisenberg groups, Sasakian manifolds, and Hörmander-type operators, generalizing known functional inequalities in sub-Riemannian geometry.