On persistence properties in fractional weighted spaces

In this work we derive a point-wise formula that will allows us to study the well-posedness of initial value problem associated to nonlinear dispersive equations in fractional weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(\mathrm{\mid }x{\mathrm{\mid }}^{2r}dx)$, $s,r\in \mathbb{R}$. 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.