Point-wise time-space estimates for a class of oscillatory integrals and their applications

Abstract

This paper investigates the point-wise time-space estimates for a class of oscillatory integrals given by \(\int _{\mathbb R^{n} }e^{i<x,\; \xi >\pm itP^{\frac{1}{2} } (\xi )} P^{-\frac{\alpha }{2} } (\xi )d\xi \), where P is a real non-degenerate elliptic polynomial of order \(m\ge 4\) on \(\mathbb R^{n} \). These estimates are applied to obtain time-space integrability estimates with regularity for solutions to higher order wave-type equations.