Another look at regularity in transport-commutator estimates

TL;DR

This paper investigates the regularity requirements of transport velocity fields for controlling Riesz-type commutators, revealing limitations of BMO assumptions, a trade-off between potential singularity and regularity, and establishing convergence rates in critical Sobolev spaces.

Contribution

It demonstrates that BMO regularity cannot replace $L^ abla$ assumptions generally, identifies a potential-regularity trade-off, and introduces defective commutator estimates for almost-Lipschitz fields.

Findings

BMO assumption cannot replace $L^ abla$ in general.

Trade-off between potential singularity and velocity regularity.

Defective commutator estimate enables convergence rates in critical Sobolev spaces.

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.