Stability estimates for the Vlasov-Poisson system in $p$-kinetic Wasserstein distances

We extend Loeper’s $L^2$-estimate relating the electric fields to the densities for the Vlasov-Poisson system to $L^p$, with $1<p<+\mathrm{\infty }$, based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical Loeper’s $2$-Wasserstein stability estimate and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance to kinetic Wasserstein distances of order $1<p<+\mathrm{\infty }$.