In this paper we study the Hp–Hq estimates of the solutions for a class of dispersive equations
$$\left\{ {\matrix{{i{\partial _t}u(t,x) = - P(|\nabla |)u(t,x),} \hfill & {(t,x) \in \mathbb{R} \times {\mathbb{R}^n},} \hfill \cr {u(0,x) = {u_0}(x),} \hfill & {x \in {\mathbb{R}^n},} \hfill \cr } } \right.$$
where P: ℝ+ → ℝ is smooth away from the origin and enjoy different scalings. As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrödinger equations.
