Estimates for a class of oscillatory integrals and decay rates for wave-type equations

TL;DR

This paper derives decay estimates for solutions to higher order wave equations with elliptic polynomial symbols, using harmonic analysis to analyze oscillatory integrals and establish $L^p-L^q$ bounds.

Contribution

It introduces new global pointwise estimates for oscillatory integrals associated with higher order wave equations, leading to $L^p-L^q$ decay bounds.

Findings

Established global pointwise time-space estimates for oscillatory integrals.

Derived $L^p-L^q$ estimates for wave solutions based on initial data.

Extended decay rate analysis to higher order elliptic wave equations.

Abstract

This paper investigates higher order wave-type equations of the form $\partial_{tt}u+P(D_{x})u=0$, where the symbol $P(\xi)$ is a real, non-degenerate elliptic polynomial of the order $m\ge4$ on ${\bf R}^n$. Using methods from harmonic analysis, we first establish global pointwise time-space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) $L^p-L^q$ estimates on the wave solution in terms of the initial conditions.