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

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