1.3. Group velocity and the method of nonstationary phase 17
Consider the behavior for large times. The phases φ±(t, x, ξ) := ±ξ t +
xξ have gradients
∇ξφ±(t, x, ξ) := ∇ξ ± ξ t + xξ =
±tξ
ξ
+ x = t
±ξ
ξ
+
x
t
.
At space-time points (t, x) with t 1 and
±ξ
ξ
+
x
t
= 0 ,
the phase oscillates rapidly and the contribution to the integral is expected
to be small. The contribution to the a± integral from ξ ∼ ξ is felt predomi-
nantly at points where x/t ∼ ∓ξ/ ξ . Setting τ±(ξ) := ±ξ , one has
∓ξ
ξ
= −∇ξτ±(ξ) .
This agrees with the formula for the group velocity (re)introduced on purely
geometric grounds in §2.4.
For t → ∞ the contributions of the plane waves
a±(ξ)ei(τ±(ξ)t+xξ)
with
ξ ∼ ξ are expected to be felt at points with x/t ∼ −∇ξτ±(ξ). A precise
version is proved using the method of nonstationary phase.
Proposition 1.3.1. Suppose that a±(ξ) ∈
S(Rd),
and define
V :=
±
v : v = −∇ξτ±(ξ) for some ξ ∈ supp a±
to be the closed set of group velocities that appear in the plane wave decom-
position of u. For μ 0, let Kμ ⊂
Rd
denote the set of points at distance
≥ μ from V. Denote by Γμ the cone
Γμ := (t, x) : t 0 and x/t ∈ Kμ .
Then for all N 0 and α,
(1 + t +
|x|)N
∂t,xu(t,
α
x) ∈
L∞(Γμ)
.
Proof. The solution u is smooth with values in S so one need only consider
{t ≥ 1}. We estimate the u+ summand. The u− summand is treated
similarly. The subscript + is suppressed in u+, φ+, a+ and τ+.
Introduce the first order differential operator
(1.3.2) (t, x, ∂) :=
1
i|∇ξφ|2
j
∂φ
∂ξj
∂
∂ξj
, so (t, x, ∂ξ)
eiφ
=
eiφ
.
The operator is only defined where ∇ξφ = 0. The coefficients are smooth
functions on a neighborhood of Γμ, and are homogeneous of degree minus
