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Smoothing inequalities for transport metrics in compact spaces

Bence Borda, Jean-Claude Cuenin

TL;DR

This work develops a comprehensive smoothing framework to bound the Wasserstein distance $W_p$ between probability measures on compact spaces in terms of Fourier data. By combining a kernel-based smoothing procedure with Kantorovich duality via the Hopf–Lax semigroup and dual Sobolev norms, it yields explicit, implementable inequalities for all $1\le p\le \infty$ across the torus, compact Lie groups, homogeneous spaces, spheres, and general compact Riemannian manifolds. The results unify and extend previous smoothing inequalities (which were known mainly for $p\le 1$ or specific settings) and provide concrete applications, notably showing that spherical designs are optimally close to the uniform measure in Wasserstein distance. The framework also supplies dispersion and Riesz-transform estimates that are broadly applicable to problems in optimal transport, quantization, and irregularities of distribution on manifolds. Overall, the paper delivers explicit, sharp tools for comparing measures via their Fourier content in a broad geometric panorama, with direct implications for distribution design and numerical integration on manifolds.

Abstract

We prove general upper estimates for the distance between two Borel probability measures in Wasserstein metric in terms of the Fourier transforms of the measures. We work in compact manifolds including the torus, the Euclidean unit sphere, compact Lie groups and compact homogeneous spaces, and treat the Wasserstein metric $W_p$ in the full range $1 \le p \le \infty$ for the first time. The proofs are based on a comparison between the Wasserstein metric and a dual Sobolev norm, Riesz transform estimates and Hausdorff--Young inequalities on compact manifolds. As an application, we show that spherical designs are optimally close to the uniform measure on the sphere in Wasserstein metric.

Smoothing inequalities for transport metrics in compact spaces

TL;DR

This work develops a comprehensive smoothing framework to bound the Wasserstein distance between probability measures on compact spaces in terms of Fourier data. By combining a kernel-based smoothing procedure with Kantorovich duality via the Hopf–Lax semigroup and dual Sobolev norms, it yields explicit, implementable inequalities for all across the torus, compact Lie groups, homogeneous spaces, spheres, and general compact Riemannian manifolds. The results unify and extend previous smoothing inequalities (which were known mainly for or specific settings) and provide concrete applications, notably showing that spherical designs are optimally close to the uniform measure in Wasserstein distance. The framework also supplies dispersion and Riesz-transform estimates that are broadly applicable to problems in optimal transport, quantization, and irregularities of distribution on manifolds. Overall, the paper delivers explicit, sharp tools for comparing measures via their Fourier content in a broad geometric panorama, with direct implications for distribution design and numerical integration on manifolds.

Abstract

We prove general upper estimates for the distance between two Borel probability measures in Wasserstein metric in terms of the Fourier transforms of the measures. We work in compact manifolds including the torus, the Euclidean unit sphere, compact Lie groups and compact homogeneous spaces, and treat the Wasserstein metric in the full range for the first time. The proofs are based on a comparison between the Wasserstein metric and a dual Sobolev norm, Riesz transform estimates and Hausdorff--Young inequalities on compact manifolds. As an application, we show that spherical designs are optimally close to the uniform measure on the sphere in Wasserstein metric.
Paper Structure (20 sections, 25 theorems, 388 equations)

This paper contains 20 sections, 25 theorems, 388 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Smoothing inequalities on the torus
  4. Smoothing inequalities on compact Lie groups
  5. Smoothing inequalities on compact homogeneous spaces
  6. An application to spherical designs
  7. Smoothing inequalities on compact Riemannian manifolds
  8. Approach
  9. Smoothing procedure
  10. Sobolev norms and Ledoux's lemma
  11. Riesz transform estimates
  12. Dispersion rate of the heat kernel
  13. Proofs of the main results
  14. Torus with $1 \le p < \infty$
  15. Torus with $p=\infty$
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mu, \nu \in \mathcal{P}(\mathbb{R}^d / \mathbb{Z}^d)$, and assume there exists a constant $c \ge 0$ such that $\mu \ge c \mathrm{Vol}$. Let $1 \le p < \infty$ and $1<q \le \infty$ be such that $1/p+1/q=1$. Assume either $p=1$ or $c>0$. For any integer $H>H_0$, we have where $q_0=\min \{ q, 2 \}$,

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • proof
  • ...and 41 more