find for $x \in \bfR$ $$\vert E_1(t,x) \vert \leq \iint f(t,y,v) \ dv \ dy + \Vert \phi \Vert_1 \leq C.$$ The second conclusion of the theorem then follows by adding the field estimates.

We begin by noting that $v$ is an eigenvector of $D$ since $$\label{D2}
Dv = v_0^{-1} [ \mathbb{I} + v \times v] = v_0^{-1} [ v + (v \cdot v) v ] = v_0^{-1} (1 + \vert v \vert^2) v = v_0 v.$$ Now, let $\gamma \geq 0$ be given. Multiplying the Vlasov equation by $v_0^\gamma$, we find $$\label{Vlasovmoment}
\partial_t(v_0^\gamma f) + \partial_x(\hat{v}_1 v_0^\gamma f) + \nabla_v \cdot [ K v_0^\gamma f] - \nabla_v(v_0^\gamma) \cdot K f = v_0^\gamma \nabla_v \cdot [D \nabla_v f].$$ We first compute the right side of this equation. Using (\[D1\]) and (\[D2\]), we find $$\begin{aligned}
v_0^\gamma \nabla_v \cdot [D \nabla_v f] & = & \nabla_v \cdot [v_0^\gamma D\nabla_v f ] - \nabla_v (v_0^\gamma) \cdot D\nabla_v f\\
& = & \nabla_v \cdot [D\nabla_v (v_0^\gamma f) ]  - \nabla_v \cdot [D\nabla_v (v_0^\gamma) f ] - \nabla_v (v_0^\gamma) \cdot D\nabla_v f\\
& = & \nabla_v \cdot [D\nabla_v (v_0^\gamma f) ]  - \gamma \nabla_v \cdot [ v_0^{\gamma-2}f D v  ] - \gamma v_0^{\gamma-2} v \cdot 