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

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