On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical $\hat{L}^r$-space

Abstract

The purpose of this paper is to study local and global well-posedness of the initial value problem for the generalized Korteweg–de Vries (gKdV) equation in ${\widehat{L}}^r=\left\{f\in {\mathcal{S}}^{\prime }\left(ℝ\right):\parallel f{\parallel }_{{\widehat{L}}^r}=\parallel \widehat{f}{\parallel }_{L^{r^{\prime }}}<\infty \right\}$. We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical ${\widehat{L}}^r$-space. A key ingredient is a Stein–Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for ${\widehat{L}}^r$-framework.