  We prove that the Yang-Mills equation in Lorenz gauge in the
(n+1)-dimensional case is locally well-posed for data of the gauge potential in
$H^s$ and the curvature in $H^r$ , where $s >\frac{n}{2}-\frac{7}{8}$ , $r >
\frac{n}{2}-\frac{7}{4}$ , if $n \ge 4$, and $ s > \frac{3}{4}$ , $ r > -
\frac{1}{8}$ , if $n=3$. The proof is based on the fundamental results of
Klainerman-Selberg [KS] and on the null structure of most of the nonlinear
terms detected by Selberg-Tesfahun [ST] and Tesfahun [Te].
