A sharp commutator estimate for all Riesz modulated energies

TL;DR

The paper delivers a sharp first-order commutator-type control for the Riesz modulated energy across all $0\le \mathsf{s}<\mathsf{d}$, achieving the optimal additive error $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$ in the sub-Coulomb and Coulomb/Riesz regimes. It introduces a wavelet-type truncation $\mathsf{g}_{\eta}$ via a Mellin-like integral representation and leverages Kato–Ponce commutator estimates (via Bessel-potentials and a dimension-extension framework) to reduce the core estimate to scale-averaged bounds. The result yields sharp rates of convergence in mean-field limits for first-order Hamiltonian/gradient dynamics, facilitates central limit-type descriptions for fluctuations, and supports quasi-neutral limits and related transport-CLT frameworks. The methods combine a robust potential truncation with an $L^2$ commutator calculus and a local, renormalized analysis of singular interactions, thereby completing the sharp, whole-range Riesz theory and enriching the transport approach to many-body limits. These advances provide precise quantification of microscopic-to-macroscopic convergence and open avenues for localized and higher-order commutator analyses in Riesz-type systems.

Abstract

We prove a functional inequality in any dimension controlling the derivative along a transport of the Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the third author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where this control is an essential ingredient. Previous work of the last two authors and Q.H. Nguyen arXiv:2107.02592 showed a similar functional inequality but with an additive $N$-dependent error (where $N$ is the number of particles, $\mathsf{d}$ the dimension, and $\mathsf{s}$ the inverse power of the Riesz potential) which was not sharp. In this paper, we obtain the optimal $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$ error, for all cases, including the sub-Coulomb case. Our method is conceptually simple and, like previous work, relies on the observation that the derivative along a transport of the modulated energy is the quadratic form of a commutator. Through a new potential truncation scheme based on a wavelet-type representation of the Riesz potential to handle its singularity, the proof reduces to averaging over a family of Kato-Ponce type estimates. The commutator estimate has applications to sharp rates of convergence for mean-field limits, quasi-neutral limits, and central limit theorems for the fluctuations of Coulomb/Riesz gases both at and out of thermal equilibrium. In particular, we show here for $\mathsf{s}<\mathsf{d}-2$ the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows. This complements the recent work arXiv:2407.15650 on the optimal rate for the (super-)Coulomb case $\mathsf{d}-2\le \mathsf{s}<\mathsf{d}$ and therefore resolves the entire potential Riesz case.

Theorems & Definitions (56)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10