  The 1D Cauchy problem for the Zakharov system is shown to be locally
well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave
data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain
assumptions on the parameters k,l and 1<p\le 2, where \|u_0\|_{\hat{H^{k,p}}}
:= \| < \xi >^k \hat{u_0}\|_{L^{p'}}, generalizing the results for p=2 by
Ginibre, Tsutsumi, and Velo. Especially we are able to improve the results from
the scaling point of view, and also allow suitable k<0, l<-1/2, i.e. data u_0
\not\in L^2 and (n_0,n_1)\not\in H^{-1/2}\times H^{-3/2}, which was excluded in
the case p=2.
