[Paper Review] On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited

Bang-Xian Han, Andrea Pinamonti|arXiv (Cornell University)|Oct 12, 2021

Nonlinear Partial Differential Equations34 references4 citations

TL;DR

This paper generalizes the Bourgain-Brezis-Mironescu asymptotic formula for fractional Sobolev seminorms to metric measure spaces, establishing convergence to the classical Sobolev seminorm under the condition that the tangent space at almost every point is a finite-dimensional Banach space or Carnot group. The key contribution is a unified proof framework valid for Euclidean spaces, weighted Riemannian manifolds, sub-Riemannian manifolds, and non-collapsed RCD(K,N) spaces, with explicit identification of the limiting constant K_{p,N}.

ABSTRACT

We generalize Bourgain-Brezis-Mironescu's asymptotic formula for fractional Sobolev functions, in the setting of abstract metric measure spaces, under the assumption that at almost every point the tangent space in the measured Gromov-Hausdorff sense is a finite dimensional Banach space or a Carnot group. Our result not only covers the known results concerning Euclidean spaces, weighted Riemannian manifolds and finite dimensional Banach spaces, but also extends Bourgain-Brezis-Mironescu's formula to $ cdkn$ spaces and sub-Riemannian manifolds.

Motivation & Objective

• To extend the Bourgain-Brezis-Mironescu asymptotic formula for fractional Sobolev seminorms beyond Euclidean spaces to general metric measure spaces.
• To establish the convergence of non-local seminorms to the classical Sobolev seminorm under minimal geometric assumptions on the tangent structure.
• To unify and generalize prior results on Euclidean spaces, finite-dimensional Banach spaces, weighted Riemannian manifolds, and sub-Riemannian manifolds under a single analytical framework.
• To prove the formula for a broad class of mollifiers, enabling new characterizations of Sobolev functions via non-local functionals.
• To extend the validity of the BBM formula to non-collapsed RCD(K,N) metric measure spaces, which model Riemannian spaces with Ricci curvature bounded below.

Proposed method

• Utilizes measured Gromov-Hausdorff convergence to analyze the asymptotic behavior of fractional seminorms at the level of tangent cones.
• Applies a non-smooth version of Rademacher's theorem, ensuring almost everywhere weak differentiability of Lipschitz functions on metric measure spaces.
• Employs a family of mollifiers ρ_n(x,y) depending only on distance d(x,y), increasing polynomially with order related to the dimension and exponent p.
• Constructs a sequence of bi-Lipschitz charts (φ_δ^i) to approximate the local geometry near regular points, ensuring the tangent cone is isometric to R^N or a Carnot group.
• Uses a density-based selection of regular points R_∞ where the tangent space is unique and the approximation holds at all scales.
• Applies a measure-theoretic argument to control the measure of exceptional sets (e.g., 𝒩_k) where the chart approximation fails, showing their density vanishes at small scales.

Experimental results

Research questions

• RQ1 Under what geometric conditions on a metric measure space does the fractional Sobolev seminorm converge to the classical Sobolev seminorm as s↑1?
• RQ2 Can the classical Bourgain-Brezis-Mironescu formula be extended to sub-Riemannian manifolds and RCD(K,N) spaces with lower Ricci curvature bounds?
• RQ3 What is the precise limiting constant K_{p,N} in the asymptotic formula for non-Euclidean metric measure spaces with finite-dimensional tangent cones?
• RQ4 How can the BBM formula be generalized to arbitrary mollifiers satisfying polynomial growth and radial dependence?
• RQ5 Is the convergence of the non-local seminorm to the Sobolev seminorm stable under measured Gromov-Hausdorff convergence of the underlying space?

Key findings

• The asymptotic formula holds for all metric measure spaces where the tangent cone at almost every point is a finite-dimensional Banach space or a Carnot group.
• The limiting constant K_{p,N} in the asymptotic formula is explicitly identified and matches the known Euclidean value, independent of the underlying geometry.
• The result applies to non-collapsed RCD(K,N) spaces, where the measure is the N-dimensional Hausdorff measure, extending the formula to spaces with synthetic lower Ricci curvature bounds.
• The convergence holds for a broad class of mollifiers satisfying Assumption 3.4, including those used in heat semigroup characterizations of Sobolev functions.
• The proof establishes that the exceptional set where the local chart approximation fails has vanishing density at small scales, ensuring measure-theoretic convergence.
• The framework unifies and generalizes prior results on Euclidean spaces, Banach spaces, weighted Riemannian manifolds, and sub-Riemannian manifolds under a single analytical principle.