 
Summary: Estimates for a class of oscillatory integrals and
decay rates for wavetype equations
JinMyong Kim, Anton Arnold and Xiaohua Yao
Abstract. In this paper we first establish global pointwise timespace esti
mates for a class of oscillatory integrals. Then, we use them to establish
Lp
 Lq
estimates for a class of higher order wavetype equations of the form
ttu + P(Dx)u = 0, where the symbol P() is a real nondegenerate elliptic
polynomial of Rn
.
Mathematics Subject Classification (2000). 42B25; 42A45; 35Q53; 35Q35.
Keywords. Oscillatory integral, higherorder wave equation, fundamental so
lution estimate.
1. Introduction
It is well known that the solution u(t, x) of the Cauchy problem for the wave
equation:
ttu(t, x)  u(t, x) = 0, (t, x) R × Rn
u(0, x) = u0(x), tu(0, x) = u1(x), x Rn
has the following form:
