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 ∂ t t u + P ( D x ) u = 0 , where the symbol P ( ξ ) is a real, non-degenerate elliptic polynomial of the order m ≥ 4 on 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.