Lp–Lq estimates for dispersive equations and related applications

This paper is mainly concerned with the Lp–Lq estimates of solutions for a class of general dispersive equations under the condition (Hb) with an index b∈(0,1]. To the end, the new pointwise decay estimates of oscillatory integrals related to the fundamental solutions are proved. If b=1, then the pointwise estimates are particularly consistent with the sharp results of the nondegenerate cases. Moreover, as an application of the Lp–Lq estimates, we also show that higher-order differential operator iP(D)+V(x,D) generates a fractionally integrated group on some Lp(Rn), from which certain Lp-estimates for the solutions of generalized Schrödinger equations are obtained.