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Sharp commutator estimates of all order for Coulomb and Riesz modulated energies

Matthew Rosenzweig, Sylvia Serfaty

TL;DR

This work establishes sharp, all-order commutator estimates for the Coulomb and super-Coulomb Riesz modulated energies in arbitrary dimensions by reinterpreting transport derivatives as commutators and employing a stress-energy framework together with a degenerate elliptic (Caffarelli–Silvestre) extension. The authors develop a local regularity theory for higher-order commutators, prove localized and unlocalized functional inequalities with optimal $N^{ rac{ extsf{s}}{ extsf{d}}-1}$ additive errors, and implement a renormalization scheme that handles singular charges via higher-order smearings. These results yield optimal mean-field convergence rates, CLTs for fluctuations, and quasi-neutral limit insights for Coulomb/Riesz gases, while offering a versatile mesoscale energy control applicable to more general Riesz interactions. The methods combine a novel PDE viewpoint on commutators with a flexible localization strategy, enabling sharp, density-aware bounds and potential extensions to sub-Coulomb regimes in forthcoming work. Collectively, this advances the understanding of quantitative mean-field limits and fluctuations for singular particle systems with long-range interactions.

Abstract

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here 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.

Sharp commutator estimates of all order for Coulomb and Riesz modulated energies

TL;DR

This work establishes sharp, all-order commutator estimates for the Coulomb and super-Coulomb Riesz modulated energies in arbitrary dimensions by reinterpreting transport derivatives as commutators and employing a stress-energy framework together with a degenerate elliptic (Caffarelli–Silvestre) extension. The authors develop a local regularity theory for higher-order commutators, prove localized and unlocalized functional inequalities with optimal additive errors, and implement a renormalization scheme that handles singular charges via higher-order smearings. These results yield optimal mean-field convergence rates, CLTs for fluctuations, and quasi-neutral limit insights for Coulomb/Riesz gases, while offering a versatile mesoscale energy control applicable to more general Riesz interactions. The methods combine a novel PDE viewpoint on commutators with a flexible localization strategy, enabling sharp, density-aware bounds and potential extensions to sub-Coulomb regimes in forthcoming work. Collectively, this advances the understanding of quantitative mean-field limits and fluctuations for singular particle systems with long-range interactions.

Abstract

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected -rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows.
Paper Structure (27 sections, 26 theorems, 395 equations)

This paper contains 27 sections, 26 theorems, 395 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. New functional inequalities
  4. Proof method: stress tensor and commutators
  5. Applications
  6. Acknowledgments
  7. Notation
  8. The modulated energy
  9. Electric formulation
  10. Monotonicity and local energy control
  11. Mesoscale interaction control
  12. Commutators and stress-energy tensor
  13. Renormalization of the first order commutator estimate
  14. Regularity theory and commutator estimates
  15. Finding the PDE solved by $\kappa^{(n),f}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

There exists a constant $C>0$ depending only $\mathsf{d},\mathsf{s}$ such that the following holds. Let $\mu \in L^1({\mathbb{R}}^\mathsf{d})\cap L^\infty({\mathbb{R}}^\mathsf{d})$ with $\int_{{\mathbb{R}}^\mathsf{d}}\mu=1$. If $\mathsf{s}\le 0$, suppose further that $\int_{({\mathbb{R}}^\mathsf{d}) Furthermore, for any integers $m\ge 0$ and $n\ge 1$, there exists a constant $C>0$ depending only $

Theorems & Definitions (59)

  • Theorem 1.1: Global estimates
  • Theorem 1.2: Localized estimates
  • Corollary 1.3
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • ...and 49 more