  In this paper, we establish the well-posedness for the Cauchy problem of the
fifth order KdV equation with low regularity data. The nonlinear term has more
derivatives than can be recovered by the smoothing effect, which implies that
the iteration argument is not available when initial data is given in $H^s$ for
any $s \in \mathbb{R}$. So we give initial data in $H^{s,a}=H^s \cap \dot{H}^a$
when $a \leq s$ and $a \leq 0$. Then we can use the Fourier restriction norm
method to obtain the local well-posedness in $H^{s,a}$ when $s \geq \max\{-1/4,
-2a-2 \}$, $-3/2<a \leq -1/4$ and $(s,a) \neq (-1/4,-7/8)$. This result is
optimal in some sense.
