Vlasov-Poisson equation in weighted Sobolev space \(W^{m, p}(w)\)

In this paper, we are concerned about the well-posedness of Vlasov-Poisson equation near vaccum in weighted Sobolev space \(W^{m, p}(w)\). The most difficult part comes from estimates electronic term \(\nabla_{x}\phi\). To overcome difficulty, establish \(L^p\)-\(L^q\) \(\nabla_{x}\phi\); some weight is introduced as well to obtain off-diagonal estimate. also useful when it control higher-order derivative term.