In this paper we study the Hp–Hq estimates of the solutions for a class of dispersive equations $$\{ {{{i{∂ _t}u(t,x) = - P(|∇ |)u(t,x),} & {(t,x) ∈ R × {R^n},} {u(0,x) = {u_0}(x),} & {x ∈ {R^n},} } } .$$ 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.
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Deng et al. (2024) studied this question.