Sharp local well-posedness for the "good" Boussinesq equation

In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schr\"odinger equation and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for one dimensional periodic functions in $X^{s,b}$ spaces, which are generalization of the bilinear refinement of the $L^4$ Strichartz estimate on $\mathbb{R}$. Our result improves the known local well-posedness in $H^s(\mathbb{T})$ with $s>-3/8$ given by Oh and Stefanov (2012) to the regularity threshold $H^{-1/2}(\mathbb{T})$. Similar ideas also establish the sharp local well-posedness in $H^{-1/2}(\mathbb{R})$ and ill-posedness below $H^{-1/2}$ for the nonperiodic case, which improves the result of Tsugawa and the author (2010) in $H^s(\mathbb{R})$ with $s>-1/2$ to the limiting regularity.