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Another look at regularity in transport-commutator estimates

Elias Hess-Childs, Matthew Rosenzweig, Sylvia Serfaty

TL;DR

This work analyzes how transport regularity affects Calderón-type commutator estimates for Riesz interactions in mean-field limits. It shows that replacing the Lipschitz bound $\|\nabla v\|_{L^{\infty}}$ with $\|\nabla v\|_{\mathrm{BMO}}$ fails in general except for the 1D endpoint $s=0$, and it develops a sharp regularity-trade-off principle tying interaction singularity to necessary velocity regularity. The authors introduce a defective commutator framework leveraging the Brezis–Wainger–Hansson inequality, enabling quantitative mean-field convergence in scaling-critical Sobolev spaces even when exact commutator bounds fail. They also construct explicit counterexamples and provide conditions under which global convergence holds in the sub-Coulomb regime, highlighting fundamental limits of current commutator methods and outlining open problems for localization and higher-order extensions.

Abstract

We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions, which we start by reviewing. Our first new result shows that the usual $L^\infty$ assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities $-2< s<d$, except for the one-dimensional logarithmic endpoint $s=0$. At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman-Rochberg-Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that a defective commutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis-Wainger-Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space.

Another look at regularity in transport-commutator estimates

TL;DR

This work analyzes how transport regularity affects Calderón-type commutator estimates for Riesz interactions in mean-field limits. It shows that replacing the Lipschitz bound with fails in general except for the 1D endpoint , and it develops a sharp regularity-trade-off principle tying interaction singularity to necessary velocity regularity. The authors introduce a defective commutator framework leveraging the Brezis–Wainger–Hansson inequality, enabling quantitative mean-field convergence in scaling-critical Sobolev spaces even when exact commutator bounds fail. They also construct explicit counterexamples and provide conditions under which global convergence holds in the sub-Coulomb regime, highlighting fundamental limits of current commutator methods and outlining open problems for localization and higher-order extensions.

Abstract

We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions, which we start by reviewing. Our first new result shows that the usual assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities , except for the one-dimensional logarithmic endpoint . At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman-Rochberg-Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that a defective commutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis-Wainger-Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space.
Paper Structure (25 sections, 13 theorems, 181 equations)

This paper contains 25 sections, 13 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Mean-field limits
  3. Modulated energy and commutators
  4. Transport regularity
  5. Our contributions
  6. Notation
  7. Acknowledgments
  8. Main results
  9. BMO inequalities
  10. Regularity trade-off
  11. Defective commutator estimates and mean-field convergence
  12. Counterexamples
  13. Proof of \ref{['thm:CEF1']}\ref{['item:CEF11']}
  14. Proof of \ref{['thm:CEF1']}\ref{['item:CEF12']}
  15. Proof of \ref{['thm:CEF2']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $-2<\mathsf{s}<\mathsf{d}$. Let $\mu \in L^1\cap L^p$ for $\frac{\mathsf{d}}{\mathsf{d}-\mathsf{s}}<p\le \infty$ with $\int_{{\mathbb{R}}^\mathsf{d}}d\mu = 1$, and associated to $\mu$, define the length scales When $-2<\mathsf{s}\le 0$, assume that $\int_{({\mathbb{R}}^\mathsf{d})^2}|{\mathsf{g}}|(x-y)d|\mu|^{\otimes 2}<\infty$.This condition is to ensure that the modulated energy is finite.

Theorems & Definitions (24)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 2.1
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 14 more