On dispersive decay for the generalized Korteweg--de Vries equation

We prove pointwise-in-time dispersive estimates for solutions to the generalized Korteweg–de Vries (gKdV) equation. In particular, for solutions to the mass-critical model, we assume only that initial data lie in ${\dot{H}}^{\frac{1}{4}}\cap {\dot{H}}^{-\frac{1}{12}}$ and show that solutions decay in $L^{\mathrm{\infty }}$ like $\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{3}}$. To accomplish this, we develop a persistence of negative regularity for solutions to gKdV and extend Lorentz–Strichartz estimates to the mixed norm case.