A sharp commutator estimate for all Riesz modulated energies

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.