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In this paper, we investigate the Cauchy problem for the Ostrovsky equation\begin{eqnarray*}  \partial_{x}\left(u_{t}-\beta \partial_{x}^{3}u  +\frac{1}{2}\partial_{x}(u^{2})\right)  -\gamma u=0, \end{eqnarray*} in the Sobolev space $H^{-3/4}(\R)$. Here$\beta>0(<0)$ corresponds to the positive (negative) dispersion of the media,respectively. P. Isaza and J. Mej\'{\i}a (J. Diff. Eqns. 230(2006), 601-681;Nonli. Anal. 70(2009), 2306-2316), K. Tsugawa (J. Diff. Eqns. 247(2009),3163-3180) proved that the problem is locally well-posed in $H^s(\R)$ when$s>-3/4$ and ill-posed when $s<-3/4$. By using some modified Bourgain spaces,we prove that the problem is locally well-posed in $H^{-3/4}(\R)$ with $\beta<0$ and $\gamma>0.$ The new ingredient that we introduce in this paper isLemmas 2.1-2.6.

The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity