In this work we derive a pointwise formula that will allow us to study the well posedness of the initial value problem associated to nonlinear dispersive equations in fractional weighted Sobolev spaces $H^s(\mathbb {R})\cap L^2(|x|^{2r}dx)$, $s, r \in \mathbb {R}$. As an application of this formula we will study local and global well posedness of the $k$-generalized Korteweg-de Vries equation in these weighted Sobolev spaces.