Analytic smoothing estimates for the Korteweg-de Vries equation with steplike data

In this paper, we prove analytic smoothing estimates for the Korteweg-de Vries equation. In the first result, we obtain explicit analytic smoothing estimates for initial data in Faddeev class which decays exponentially in the positive direction. In the second result, we go beyond Faddeev class and generalize the result to non-decaying initial data. Particularly, step functions supported in the left half line and their perturbations by Faddeev class potentials decaying exponentially in positive direction are involved by the second result. Finally, we discuss some of its applications to control problems such as observability inequalities for the KdV equation.