Lp–Lqestimates for dispersive equations and related applications

Abstract

This paper is mainly concerned with the L p – L q estimates of solutions for a class of general dispersive equations under the condition ( H b ) 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 L p – L q estimates, we also show that higher-order differential operator i P ( D ) + V ( x , D ) generates a fractionally integrated group on some L p ( R n ) , from which certain L p -estimates for the solutions of generalized Schrödinger equations are obtained.