Another look at regularity in transport-commutator estimates

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 through modulated energy techniques, which we start by reviewing. In these applications, the transport field is generated by the limiting PDE or by its adjoint linearized flow. Relaxing the regularity demanded of the transport field to control the commutator enlarges the class of limiting densities and fluctuation observables accessible to the method, especially at scaling-critical regularities. Our first new result shows that the usual L ∞ 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.