Smoothing inequalities for transport metrics in compact spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918169558384640 |
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| author | Borda, Bence Cuenin, Jean-Claude |
| author_facet | Borda, Bence Cuenin, Jean-Claude |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21380 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smoothing inequalities for transport metrics in compact spaces Borda, Bence Cuenin, Jean-Claude Classical Analysis and ODEs Optimization and Control Probability 43A77, 43A85, 49Q22, 60B15 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. |
| title | Smoothing inequalities for transport metrics in compact spaces |
| topic | Classical Analysis and ODEs Optimization and Control Probability 43A77, 43A85, 49Q22, 60B15 |
| url | https://arxiv.org/abs/2510.21380 |