  In this paper we study a class of variable coefficient third order partial
differential operators on $\mathbb{R}^{n+1}$, containing, as a subclass, some
variable coefficient operators of KdV-type in any space dimension. For such a
class, as well as for the adjoint class, we obtain a Carleman estimate and the
local solvability at any point of $\mathbb{R}^{n+1}$. A discussion of possible
applications in the context of dispersive equations is provided.
