v_1)f(s, x \pm (t-s),v) \ dv + \frac{1}{2} \vert E_1(s, x \pm (t-s)) \vert^2 \right. \\ \qquad + \left. \frac{1}{2}\vert E_2 \pm B \vert^2 (s, x \pm (t-s))   \right ) \ ds \leq C(1+t),
\end{gathered}$$ $$\label{j2}
\int_0^t \vert j_2(s, x \pm (t-s)) \vert  ds \leq C(1 + t),$$ and $$\label{fieldbounds}
\Vert E(t) \Vert_\infty +\Vert B(t) \Vert_\infty \leq C(1 + t).$$

Once control of the fields is obtained, higher moments of the particle distribution function can be controlled as well.

\[L3\] Let the assumptions of Lemma \[L2\] hold. Then, for any $\gamma \in [0,\a]$ and $t \in [0,T]$ $$\label{densitybounds}
\Vert v_0^\gamma f(t) \Vert_\infty \leq C(1 + t)^{2\gamma}$$ and for any $\gamma \in [0,\a-2)$ and $t \in [0,T]$ $$\label{rhobounds}
{\ensuremath{\left\|  \int v_0^{\gamma} f(t) \ dv  \right\|}}_\infty \leq C(1 + t)^{2\a}.$$

With control on moments of the density, we may bound derivatives of the field by adapting a well-known argument [@GlaSch90; @GlStr] that projects these derivatives onto the backward light cone.

\[L4\] Let the assumptions of Lemma \[L2\] hold, and assume additionally that $E_2^0, B^0 \in H^2(\bfR)$. Then, for any $t \in [0,T]$, we have $$\label{fieldderivbounds}
\Vert \partial_t E(t) \Vert_\infty + \Vert \partial_x E(t) \Vert_\infty + \Vert \partial_t B(t) \Vert_\infty + \Vert \partial_xB(t) 