  Recently C. Bardos et al. presented in their fine paper \cite{Bardos} a proof
of an Onsager type conjecture on renormalization property and the entropy
conservation laws for the relativistic Vlasov-Maxwell system. Particularly,
authors proved that if the distribution function $u \in
L^{\infty}(0,T;W^{\alpha,p}(\mathbb{R}^6))$ and the electromagnetic field $E,B
\in L^{\infty}(0,T;W^{\beta,q}(\mathbb{R}^3))$, with $\alpha, \beta \in (0,1)$
such that $\alpha\beta + \beta + 3\alpha - 1>0$ and $1/p+1/q\le 1$, then the
renormalization property and entropy conservation laws hold. To determine a
complete proof of this work, in the present paper we improve their results
under a weaker regularity assumptions for weak solution to the relativistic
Vlasov-Maxwell equations. More precisely, we show that under the similar
hypotheses, the renormalization property and entropy conservation laws for the
weak solution to the relativistic Vlasov-Maxwell's system even hold for the end
point case $\alpha\beta + \beta + 3\alpha - 1 = 0$. Our proof is based on the
better estimations on regularization operators.
