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In the present article, we prove the sharp local well-posedness andill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; theinitial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ andill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtainedfrom reduction of the problem into a quadratic nonlinear Schr\"odinger equationand the contraction argument in suitably modified $X^{s,b}$ spaces. The proofof the crucial bilinear estimates in these spaces, especially in the lowestregularity, rely on some bilinear estimates for one dimensional periodicfunctions in $X^{s,b}$ spaces, which are generalization of the bilinearrefinement of the $L^4$ Strichartz estimate on $\mathbb{R}$. Our resultimproves 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 localwell-posedness in $H^{-1/2}(\mathbb{R})$ and ill-posedness below $H^{-1/2}$ forthe 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.

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