The authors start the work with a wide panoramic view of the results related to the initial value problem for the Korteweg-de Vries equation. The main result of the paper is to have found the lower best index \(s\), such that we have local well posedness in \(H^s(\mathbb{R})\): Existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on \(|u_0|_{H^s}\). The value \(s> -3/4\), is the optimal one provided by the used method. The result improves a previous one of the same authors. The method combines oscillatory integral estimates with bilinear estimates for \(\partial_x(u^2/2)\) in the Bourgain function spaces associated with the index \(s\). The estimates extend to the periodic case.