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

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 $H^{-3\mathrm{/}4}(\mathbb{R})$. 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 $H^s(\mathbb{R})$ when $s>-3\mathrm{/}4$ and ill-posed when $s<-3\mathrm{/}4$. By using some modified Bourgain spaces, we prove that the problem is locally well-posed in $H^{-3\mathrm{/}4}(\mathbb{R})$ with $\beta <0$and$\gamma>0.$ The new ingredient that we introduce in this paper is Lemmas 2.1-2.6.