  In this paper, we investigate the Cauchy problem for the shallow water type
equation \[ u_{t}+\partial_{x}^{3}u
  + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x}
  (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in
{\mathbf T}=\R/2\pi
  \lambda \] with low regularity data in the periodic settings and
$\lambda\geq1$. We prove that the bilinear estimate in $X_{s,b}$ with
$s<\frac{1}{2}$ is invalid.
  We also prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$
with $\frac{1}{6}<s<\frac{1}{2}$ for small initial data. The result of this
paper improves the result of case $j=1$ of Himonas and Misiolek (Communications
in Partial Differential Equations, 23(1998), 123-139.). The new ingredients are
some new function spaces and some new Strichartz estimates.
