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

This paper investigates higher order wave-type equations of the form ∂ttu+P(Dx)u=0, where the symbol P(ξ) is a real, non-degenerate elliptic polynomial of the order m≥4 on Rn. 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) Lp−Lq estimates on the wave solution in terms of the initial conditions.