About the influence of oscillations 381
D
EFINITION
2. We define the pseudodifferential zone Z
pd
β, N by
Z
pd
β, N :=
n t, ξ ∈ [0, ∞ ×
n
\ {0} : e
4
+ 3t |ξ| ≤ N ln e
4
+ 3t
β
o ,
the hyperbolic zone Z
hyp
β, N by
Z
hyp
β, N :=
n t, ξ ∈ [0, ∞ ×
n
\ {0} : e
4
+ 3t |ξ| ≥ N ln e
4
+ 3t
β
o .
The positive constant N will be chosen later. For |ξ | ∈ 0, p
], p = 4
β
N e
4
, we define the function t
ξ
= t |ξ | as the solution of e
4
+ 3t
ξ
|ξ| = N ln e
4
+ 3t
ξ β
. L
EMMA
3. The derivatives ∂
k |
ξ |
t
ξ
can be estimated in the following way: ∂
k |
ξ |
t
| ξ |
≤ C
k
|ξ |
− k
e
4
+ 3t
ξ
λ t
ξ
for all ξ ∈
n
, |ξ | ∈ 0, p ] .
3.3. The fundamental solution in Z
pd
β, N
Denoting At, |ξ | :=
λ t |ξ |
λ t bt
2
|ξ | +
D
t
λ λ
1 the fundamental solution X t, 0, ξ can be written explicitly in the form
X t, 0, |ξ | = I +
∞
X
k=1
Z
t
At
1
, |ξ | · · · Z
t
k−1
At
k
, |ξ | dt
k
· · · dt
1
25 for |ξ | ∈ 0, p
]. For a given positive number T let us distinguish two cases. a t
ξ
≤ T : in this case we have Z
t
kAs, |ξ |k ds ≤ CT for all
t ≤ t
ξ
; b T ≤ t
ξ
: in this case we have Z
t
kAs, |ξ |k ds ≤ CT + C
b
Z
t T
λ s|ξ | ds +
Z
t T
λ
′
s λ
s ds
≤ CT + C
b
3 t |ξ | + ln
λ t
λ T
≤ CT + C
b
N ln
e
4
+ 3t
β
+ ln λ
t λ
T ≤ CT + C
b
N ln
e
4
+ 3T
β− 1
ln e
4
+ 3t + ln
λ t
3 T
for all t ≤ t
ξ
. Consequently, exp
Z
t
kAs, |ξ |k ds ≤ CT λt e
4
+ 3t
Cb N ln e4+3T
1−β
. This gives the next statement:
382 M. Reissig – K. Yagdjian
L
EMMA
4. To each small positive ε there exists a constant C
ε
N such that for all t, ξ ∈ Z
pd
β, N it holds
kX t, 0, ξ k ≤ C
ε
N λt e
4
+ 3t
ε
, kX t, 0, ξ k ≤ C
ε
N λt |ξ |
− ε
, respectively.
To continue the solution from Z
pd
β, N to Z
hyp
β, N for |ξ | ∈ 0, p
] and to study its properties in Z
hyp
β, N we need the behaviour of ∂
k t
∂
α ξ
X t, 0, ξ , too. It is obtained among other things from 25 and 14.
T
HEOREM
3. To each small positive ε and each k and α there exists a constant C
ε, k,α
N such that
∂
k t
∂
α ξ
X t, 0, ξ ≤
C
ε, k,α
N λt λ
t |ξ | + λ
t e
4
+ 3t
k
|ξ |
−| α|−ε
for all t, ξ ∈ Z
pd
β, N .
3.4. The fundamental solution in Z
hyp
β, N
The hyperbolic zone Z
hyp
β, N can be represented as the union of the two sets
t, ξ : |ξ| ∈ 0, p
] : e
4
+ 3t |ξ| ≥ Nlne + 3t
β
and {t, ξ ∈ [0, ∞ × {|ξ| ≥ p }}. We restrict
ourselves to the first set and sketch at the end of this section the approach in the second set. In Z
hyp
β, N we apply a diagonalization procedure to the first order system 19. To carry out
this procedure we need the following classes of symbols. D
EFINITION
3. For given real numbers m
1
, m
2
, m
3
, β ∈ [0, 1 and for positive N we denote by S
β, N
{m
1
, m
2
, m
3
} the set of all symbols a = at, ξ ∈ C
∞
Z
hyp
β, N : |ξ | ∈
0, p ]
satisfying there ∂
k t
∂
α ξ
at, ξ ≤
C
k,α
|ξ |
m
1
−| α|
λ t
m
2
λ t
e
4
+ 3t ln e
4
+ 3t
β m
3
+ k
. These classes of symbols are related to the Definitions 1 and 2. To understand that the
diagonalization procedure improves properties of the remainder as usually one takes into con- sideration the following rules of the symbolic calculus:
• S
β, N
{m
1
, m
2
, m
3
} ⊂ S
β, N
{m
1
+ k, m
2
+ k, m
3
− k} , k ≥ 0 ;
• a ∈ S
β, N
{m
1
, m
2
, m
3
} , b ∈ S
β, N
{n
1
, n
2
, n
3
} , then
ab ∈ S
β, N
{m
1
+ n
1
, m
2
+ n
2
, m
3
+ n
3
} ; •
a ∈ S
β, N
{m
1
, m
2
, m
3
} , then ∂
t
a ∈ S
β, N
{m
1
, m
2
, m
3
+ 1} ; •
a ∈ S
β, N
{m
1
, m
2
, m
3
} , then ∂
α ξ
a ∈ S
β, N
{m
1
− |α|, m
2
, m
3
} . Let us define the matrices
M
− 1
t := 1
√ λ
t bt 1
1 −bt
bt ,
Mt := 1
2 s
λ t
bt bt
−1 bt
1 .
About the influence of oscillations 383
Substituting X = M
− 1
Y some calculations transform 20 into the first order system D
t
Y − D Y + B Y = 0 , 26
where Dt, ξ :=
τ
1
t, ξ τ
2
t, ξ ,
Bt, ξ := − D
t
λ t bt
2λt bt 1
1 ,
τ
1
t, ξ := −λtbt|ξ | + D
t
λ t
λ t
, τ
2
t, ξ := λtbt|ξ | + D
t
λ t
λ t
. Without difficulties one can prove D ∈ S
β, N
{1, 1, 0}, B ∈ S
β, N
{0, 0, 1}. To prove L
p
− L
q
decay estimates for the solution of 12 we need further steps of the diagonalization of 26. This is carried out in the next lemma.
L
EMMA
5. For a given nonnegative integer M there exist matrix-valued functions N
M
= N
M
t, ξ ∈ S
β, N
{0, 0, 0}, F
M
= F
M
t, ξ ∈ S
β, N
{−1, −1, 2} and R
M
= R
M
t, ξ ∈ S
β, N
{−M, −M, M + 1} such that the following operator-valued identity holds: D
t
− D + B N
M
= N
M
D
t
− D + F
M
− R
M
, where F
M
is diagonal while N
M
is invertible and its inverse N
− 1
M
belongs as N
M
to S
β, N
{0, 0, 0}. R
EMARK
5. The invertibility of the diagonalizer N
M
= N
M
t, ξ mod S
β, N
{−M, −M, M + 1} is essential. This property follows by a special choice of the positive constant N in
Definition 2. We need only a finite number of steps of diagonalization cf. proof of Theorem 2, thus N can be fixed after carrying out these steps.
Now let us devote to the system D
t
− D + F
M
− R
M
Z = 0 , Z = Zt, r, ξ ,
27 where t
ξ
≤ r ≤ t. Let E
2
= E
2
t, r, ξ ; t, r ≥ t
ξ
, is defined by E
2
t, r, ξ := λ
t λ
r
exp
−i R
t r
λ sbs ds|ξ | − i
R
t r
F
1,1 M
s, ξ ds exp
i R
t r
λ sbs ds|ξ | − i
R
t r
F
2,2 M
s, ξ ds
be the solution of the Cauchy problem D
t
− D + F
M
Z = 0, Zr, r, ξ = I . Let us denote P
M
t, r, ξ := E
2
r, t, ξ R
M
t, ξ E
2
t, r, ξ . By the aid of P
M
we define the matrix-valued function Q
M
t, r, ξ :=
∞
X
k=1
i
k
Z
t r
P
M
t
1
, r, ξ
Z
t
1
r
P
M
t
2
, r, ξ · · ·
· · · Z
t
k−1
r
P
M
t
k
, r, ξ dt
k
. . . dt
1
. 28
The function Q
M
= Q
M
t, r, ξ solves the Cauchy problem D
t
Q − P
M
Q − P
M
= 0 , Qr, r, ξ = 0 for t, r ≥ t
ξ
. Using these auxiliary functions it is easy to prove the next result.
384 M. Reissig – K. Yagdjian
L
EMMA
6. The matrix-valued function Z t, r, ξ = E
2
t, r, ξ I + Q
M
t, r, ξ solves the Cauchy problem 27 for t, r ≥ t
ξ
. Now we can go back to 20, 21 and obtain as its solution
X t, 0, ξ =M
− 1
t N
M
t, ξ E
2
t, t
ξ
, ξ I + Q
M
t, t
ξ
, ξ ·
· N
− 1
M
t
ξ
, ξ M t
ξ
X t
ξ
, 0, ξ
. 29
We write exp −i
R
t t
ξ
λ sbs ds
= exp −i
R
t
λ sbs ds − i
R
t
ξ
λ sbs ds
in correspon- dence with our goal 22 and include the second factor in the amplitudes. The matrices M and
M
− 1
are given in an explicit form. The properties of N
M
and N
− 1
M
are described by Lemma 5 using Definition 3. To estimate X t
ξ
, 0, ξ we use Theorem 3. Consequently, it remains to
estimate E
2
0, t
ξ
, ξ and Q
M
t, t
ξ
, ξ .
L
EMMA
7. For every positive small ε and every α the following estimate in Z
pd
β, N
holds: ∂
α ξ
exp i
Z
t t
ξ
λ sbs ds
≤ C
ε,α
|ξ |
−| α|−ε
, where C
ε,α
= C
ε,α
β, N .
L
EMMA
8. For every positive small ε and every α the following estimate in Z
hyp
β, N
holds, |ξ | ∈ 0, p ]:
∂
α ξ
exp −i
Z
t t
ξ
F
k,k M
s, ξ ds ≤ C
ε,α
|ξ |
−| α|−ε
, k = 1, 2 ,
where C
ε,α
= C
ε,α
β, N .
Proof. The statement for |α| = 0 follows from Z
t t
ξ
F
k,k M
s, ξ ds ≤
Z
t t
ξ
F
k,k M
s, ξ ds
≤ C
k
Z
t t
ξ
λ s ln e
4
+ 3s
2β
|ξ | e
4
+ 3s
2
ds , Z
t t
ξ
λ s ln e
4
+ 3s
2β
e
4
+ 3s
2
ds ≤
ln e
4
+ 3 t
ξ 2β
e
4
+ 3 t
ξ
+ Z
t t
ξ
2β ln e
4
+ 3s λ
s ln e
4
+ 3s
2β
e
4
+ 3s
2
ds , Definition 2 and
ln e
4
+ 3 t
ξ β
N = ln e
4
+ 3 t
ξ
1 N ln e4+3
tξ 1−β
. By induction we prove the statement for |α| 0 by using F
M
∈ S
β, N
{−1, −1, 2} and Lemma 3.
About the influence of oscillations 385
More problems appear if we derive an estimate for Q
M
. Here we refer the reader to [8]. L
EMMA
9. The matrix-valued function P
M
= P
M
t, t
ξ
, ξ satisfies for every l and α in
Z
hyp
β, N ∩ {|ξ | ∈ 0, p
]} the estimates ∂
l t
∂
α ξ
P
M
t, t
ξ
, ξ ≤
C
M,l,α
λ t |ξ |
l
e
4
+ 3t
| α|
λ t
e
4
+ 3t ·
· ln e
4
+ 3t
β
ln e
4
+ 3t
β
e
4
+ 3t |ξ|
M
. L
EMMA
10. For every positive small ε and every α, |α| ≤ M − 1, it holds the following estimate in Z
hyp
β, N , |ξ | ∈ 0, p
]: ∂
α ξ
Q
M
t, t
ξ
, ξ ≤
C
ε,α
1 + 3t
ε
|ξ |
−| α|−ε
, where C
ε,α
= C
ε,α
β, N .
Proof. We use the representation 28 with r = t
ξ
and form the derivatives ∂
α ξ
Q
M
t, t
ξ
, ξ . For
|α| = 0 the statement from Lemma 9 and similar calculations as in the proof of Lemma 8 imply the estimate for kQ
M
t, t
ξ
, ξ k. If we differentiate for |α| = 1 inside of the inte- grals, then the estimate follows immediately. If we differentiate the lower integral bound in
R
t
k−1
t
ξ
P
M
t
k
, t
ξ
, ξ dt
k
, then there appears a term of the form P
M
t
ξ
, t
ξ
, ξ ∂ t
ξ
∂ξ
l
Z
t t
ξ
P
M
t
1
, t
ξ
, ξ · · ·
Z
t
k−2
t
ξ
P
M
t
k−1
, t
ξ
, ξ dt
k−1
. . . dt
1
. Using Lemma 9 and Lemma 3 gives the desired estimate in this case, too. But we can only get
estimates for |α| ≤ M − 1. In this case we can have an integrand of the form λ
t e
4
+ 3t
2
|ξ | ln e
4
+ 3t
β M+1
. The term ln e
4
+ 3t
β M
can be estimated by C1 + 3t
ε
, the other factor is integrable and can be estimated by C. An induction procedure yields the statement for |α| ≤ M − 1 see
[8]. Now we have all tools to get an estimate for 29.
T
HEOREM
4. The fundamental solution X = X t, 0, ξ can be represented in Z
hyp
β, N
∩ {|ξ | ∈ 0, p ]} as follows:
X t, 0, ξ = X
+
t, 0, ξ exp i
Z
t
λ sbsds|ξ |
+ X
−
t, 0, ξ exp −i
Z
t
λ sbsds|ξ |
, where the matrix-valued amplitudes X
−
, X
+
satisfy for all |α| ≤ M − 1 and all positive small ε
the estimates ∂
α ξ
X
±
t, 0, ξ ≤
C
M,ε
q λ
t λ t
ξ
|ξ|
−| α|−ε
. 30
386 M. Reissig – K. Yagdjian
R
EMARK
6. There are no new difficulties to derive a corresponding estimate to 30 in {t, ξ ∈ [0, ∞ × {|ξ | ≥ p
}} which belongs to Z
hyp
β, N completely. We obtain for all
|α| ≤ M − 1 the estimates ∂
α ξ
X
±
t, 0, ξ ≤
C
M
p λ
t |ξ |
−| α|
. 31
3.5. Fourier multipliers
