Low regularity local well-posedness for the Yang-Mills equation in Lorenz gauge

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].