Sharp well-posedness for the Benjamin equation

Having the ill-posedness in the range s<−3/4s<−3/4 of the Cauchy problem for the Benjamin equation with an initial Hs(R)Hs(R) data, we prove that the already-established local well-posedness in the range s>−3/4s>−3/4 of this initial value problem is extendable to s=−3/4s=−3/4 and also that such a well-posed property is globally valid for s∈[−3/4,∞)s∈[−3/4,∞).