Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $

The Zakharov system in dimension \begin{document}$ d = 2,3 $\end{document} is shown to have a local unique solution for any initial values in the space \begin{document}$ H^{s} \times H^{l} \times H^{l-1} $\end{document} , where a new range of regularity \begin{document}$ (s, l) $\end{document} is given, especially at the line \begin{document}$ s-l = -1 $\end{document} . The result is obtained mainly by the normal form reduction and the Strichartz estimates.