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 of the electronic term \(\nabla_{x}\phi\). To overcome this difficulty, we establish the \(L^p\)-\(L^q\) estimates of the electronic term \(\nabla_{x}\phi\); some weight is introduced as well  to obtain the off-diagonal estimate. The weight is also useful when it comes to control the higher-order derivative term.