8 2. FUNDAMENTAL SOLUTION AND COVARIANCE FUNCTION
The change of variables y
t
s
y yields
Gn t,
t
s
x =
R3
G(t, dy)ψn t,
t
s
x y
=
R3
G(s, dy)ψn t,
t
s
(x y)
t
s
.
Since
ψn t,
t
s
(x y) =
n
t
3
ψ
n
s
(x y) = ψn (s, x y)
s
t
3
,
it follows that
Gn t,
t
s
x =
R3
G(s, dy)ψn(s, x y)
s
t
2
=
s
t
2
Gn(s, x).
This proves the lemma.
We recall the following integrability condition of the fundamental solution of
the wave equation, valid for any β ]0, 2[:
(2.5) sup
t∈[0,T ] Rd
|FG(t)(ξ)|2
|ξ|d−β
C(1 + T
2).
Indeed,
(2.6) FG(t)(ξ) =
|ξ|−1
sin(t|ξ|)
(see [37]), and therefore
Rd
|FG(t)(ξ)|2
|ξ|d−β
I1(t) + I2(t),
where
I1(t) =
|ξ|≤1
t2
|ξ|d−β
C1T
2,
I2(t) =
|ξ| 1

|ξ|d+2−β
C2.
A similar property holds for Gn. In fact, since |Fψn(t)(ξ)|≤ 1,
(2.7) |FGn(t)(ξ)| = |Fψn(t)(ξ)||FG(t)(ξ)|≤ |FG(t)(ξ)|
and therefore
(2.8) sup
n≥1
sup
t∈[0,T ] Rd
|FGn(t)(ξ)|2
|ξ|d−β
C(1 + T
2).
The next statement gives a more precise result than (2.5).
Lemma 2.2. For any t [0, T ] and β ]0, 2[,
(2.9)
t
0
ds
Rd
|FG(s)(ξ)|2
|ξ|d−β

Ct3−β
.
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