Vlasov–Poisson–Fokker–Planck equation in E2,1s space

In this paper, we are concerned about the local-in-time well-posedness of the Vlasov–Poisson–Fokker–Planck equation in E2,1s(R2n) which is a hybrid modulation Lebesgue space and related to the Gevery class only with respect to x variable. The difficulty lies in the estimates of the electronic term ∇xϕ. To handle this, we establish a product formula and L2-Lq estimate.