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