Abstract We consider the generalized Korteweg–de Vries (gKdV) equation ∂tu+∂x3u+μ∂x(uk+1)=0, where k>4 is an integer number and μ=±1. We give an alternative proof of the Kenig, Ponce and Vega result in Kenig, Ponce and Vega (1993) , which asserts local and global well-posedness in H˙sk(R), with sk=(k−4)/2k. A blow-up alternative in suitable Strichartz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space H˙sk(R), extending the results of Côte (2006) .