Summary: Let us consider a solution \(f(x,v,t)\in L^ 1({\mathbb{R}}^{2N}\times [0,T])\) of the kinetic equation \[ \partial_ tf+v\cdot \nabla_ xf=div_ vg,\quad t\geq 0,\quad x,v\in {\mathbb{R}}^ N,\quad f(x,v,0)=f_ 0,\quad f\geq 0, \] where \(| v|^{\alpha +1}f_ 0\), \(| v|^{\alpha}g\in L^ 1({\mathbb{R}}^{2N}\times [0,T])\) for some \(\alpha >0\). We prove that f has a higher moment than what is expected. Namely, for any bounded set \(K_ x\), we have \(v^{\alpha +3/2}f\in L^ 1(K_ x\times {\mathbb{R}}^ N_{\nu}\times [0,T])\). We use this result to improve the regularity of the local density \(\rho (x,t)=\int f dv\) for the Vlasov-Poisson equation, which corresponds to \(g=Ef\), where E is the force field created by the repartition f itself. We also apply this to the Bhatnagar-Gross-Krook model with an external force, and we prove that the solution of the Fokker-Planck equation with a source term in \(L^ 2\) belongs to \(L^ 2([0,T]\); \(H^{1/2}({\mathbb{R}}^{2N}_{x,v}))\).