18 1. Simple Examples of Propagation
one in (t, x) and satisfy
1
|∇ξφ|2
∂φ
∂ξj
≤ C(t +
|x|)−1
for (t, x, ξ) ∈ Γμ × supp a .
The identity
eiφ
=
eiφ
implies
a(ξ)
eiφ
dξ = a(ξ)
N eiφ
dξ .
Denote by
†
the transpose of and integrate by parts to find
a(ξ)
eiφ
dξ = (
†)N
a(ξ)
eiφ
dξ .
The operator
(
†)N
=
|α|≤N
cα(t, x, ξ)
∂ξα
with coefficients cα smooth on a neighborhood of Γμ and homogeneous of
degree −N in t, x. Therefore,
|cα(t, x)| ≤ C(α)(1 + t +
|x|)−N
for (t, x, ξ) ∈ Γμ × supp a .
It follows that
a(ξ)
eiφ
dξ ≤ C (1 + t +
|x|)−N
.
Since the t, x derivatives of this integral are again integrals of the same
form, this suffices to prove the proposition.
Example 1.3.1. Introduce for 0 μ 1,
˜
V
μ
:=
Rd
\ Kμ an open set
slightly larger than V. For t → ∞ virtually all the energy of a solution is
contained in the cone (t, x) : x/t ∈
˜
V . This is particularly interesting
when a± are supported in a small neighborhood of a fixed ξ. For large times
virtually all the energy is localized in a small conic neighborhood of the pair
of lines x = −t ∇ξτ±(ξ) that travel with the group velocities associated to
ξ.
The integration by parts method introduced in this proof is very impor-
tant. The next estimate for nonstationary oscillatory integrals is a straight-
forward application. The fact that the estimate is uniform in the phases is
useful.
Lemma of Nonstationary Phase 1.3.2. Suppose that Ω is a bounded
open subset of
Rd,
m ∈ N, and C1 1. Then there is a constant C2 0 so
that for all f ∈ C0
m(Ω)
and φ ∈
Cm(Ω
; R) satisfying
∀|α| ≤ m ,
∂αφ
L∞
≤ C1 , and ∀x ∈ Ω , C1
−1
≤ |∇xφ| ≤ C1 ,
