Uniform decay estimates for a class of oscillatory integrals and applications

Abstract

One dimensional oscillatory integrals of the type $\int^\infty _0 \xi ^\alpha \rm\exp \big[it(p(\xi )-\xi x)\big]\rm {d}\xi $ are considered, where $p(\xi )$ is a real polynomial of degree $m\geq 3$. Long-time decay and global smoothing estimates are established, as well as short-time behavior as $t\to 0$. The results are applied to the fundamental solutions of a class of linearized Kadomtsev-Petviashvili equations with higher dispersion