Uniform existence time for propagation of infinitely many moments

Prove that the local existence time for solutions of the Vlasov–Poisson equation with initial data in the weighted Sobolev spaces mathcal{H}^n_r is independent of the derivative index n and velocity-weight index r, thereby establishing propagation of an infinite number of velocity moments.

Background

The paper uses a result from \cite{MR3842911} asserting local existence for Vlasov–Poisson solutions in \mathcal{H}^n_r, where n controls Sobolev regularity and r controls velocity decay. The time-analytic injectivity theorem requires uniform bounds on arbitrarily high velocity moments and spatial derivatives.

The authors note that the cited result does not explicitly establish that the existence time T is uniform with respect to n and r. They identify proving this uniformity as necessary for obtaining propagation of infinitely many moments, and subsequently provide a proposition and argument intended to establish it.