26
5 Stable Structures, Hilbert Sectors, Phases
Proof. From the identity
G
(χ
(2)
) − G
(χ
(1)
) = U
(ϕ 0 + χ
(2)
) − U
(ϕ 0 + χ
(1)
)
=
1
0
dσ
d
dσ
U
(ϕ 0 + χ
(1)
+ σ(χ
(2)
− χ
(1)
))
=
1
0
dσU
(ϕ 0 + χ
(1)
+ σ(χ
(2)
− χ
(1)
))(χ
(2)
− χ
(1)
),
equation (5.10) will follow if, for any ρ > 0, there exists a constant C(ρ) such that
sup
k=1,...n
n
j=1
∂
2 U
∂z j ∂z k
(ϕ 0 + χ
)χ j L 2 ≤ C(ρ)χ H 1 ,
(5.14)
for all χ
, χ ∈ H
1 with χ
≤ ρ, χ ≤ ρ.
For the class of potentials under consideration, the proof of (5.14) reduces to the
estimate of terms of the type (ϕ + χ
(1)
)
α
χ
(2) with χ
(i)
∈ H
1
, i = 1, 2, α ∈ N
n for
s = 1, 2 and |α| ≤ 2 for s = 3. Now, since |a + b|
p
≤ 2
p
(|a|
p
+ |b|
p
), ∀a, b ∈ R,
p ≥ 1, one has
(ϕ 0 + χ
(1)
)
α
χ
(2)
L 2 ≤ 2
|α|
{{|ϕ 0 |
|α|
|χ
(2)
|| L 2 + +|χ
(1)
|
|α|
|χ
(2)
|| L 2 }
(5.15)
and the first term on the r.h.s. is immediately estimated by
2
|α|
|ϕ 0 |
|α|
|χ
(2)
|| L 2 ≤ A
|α|
(ϕ 0 L ∞ )
|α|
χ
(2)
H 1 .
(5.16)
The second term can be estimated by using the usual Hölder and the Sobolev inequalities
21
2
|α|
|χ
(1)
|
|α|
|χ
(2)
|| L 2 ≤ 2
|α|
|χ
(1)
||
|α|
L 2(|α|+1) |χ
(2)
|| L 2(|α|+1)
≤ B
|α| C s (2|α| + 2)
|α|+1
χ
(1)
|α|
H 1 χ
(2)
H 1 .
(5.17)
21 See, e.g. L.R. Volevic and B.P. Paneyakh, Russian Math. Surveys 20, 1 (1965). We list them for
the convenience of the reader
s = 1, f ; L
p (R
1 ) ≤ C 1 ( p) f ; H
1 (R
1 ), 2 ≤ p ≤ ∞, C 1 ( p) = O(1) ,
s = 2, f ; L
p (R
2 ) ≤ C 2 ( p) f ; H
1 (R
2 ), 2 ≤ p < ∞, C 2 ( p) = O( p
1
2 ) ,
s = 3, f ; L
p (R
3 ) ≤ C 3 ( p) f ; H
1 (R
3 ), 2 ≤ p ≤ 6, C 3 ( p) = O(1) .
The same kind of estimates hold locally. In particular, for any cube K ⊂ R s of size R, they take
the form
f ; L
p (K ) ≤ C s,R ( p) f ; H
1 (K ),
with p ∈ [2, +∞] for s = 1, p ∈ [2, +∞[ for s = 2 and p ∈ [2, 6] for s = 3. The constants
C s,R ( p) depend only on the size R and exhibit the same dependence on p as in the global case.
5 Stable Structures, Hilbert Sectors, Phases
Proof. From the identity
G
(χ
(2)
) − G
(χ
(1)
) = U
(ϕ 0 + χ
(2)
) − U
(ϕ 0 + χ
(1)
)
=
1
0
dσ
d
dσ
U
(ϕ 0 + χ
(1)
+ σ(χ
(2)
− χ
(1)
))
=
1
0
dσU
(ϕ 0 + χ
(1)
+ σ(χ
(2)
− χ
(1)
))(χ
(2)
− χ
(1)
),
equation (5.10) will follow if, for any ρ > 0, there exists a constant C(ρ) such that
sup
k=1,...n
n
j=1
∂
2 U
∂z j ∂z k
(ϕ 0 + χ
)χ j L 2 ≤ C(ρ)χ H 1 ,
(5.14)
for all χ
, χ ∈ H
1 with χ
≤ ρ, χ ≤ ρ.
For the class of potentials under consideration, the proof of (5.14) reduces to the
estimate of terms of the type (ϕ + χ
(1)
)
α
χ
(2) with χ
(i)
∈ H
1
, i = 1, 2, α ∈ N
n for
s = 1, 2 and |α| ≤ 2 for s = 3. Now, since |a + b|
p
≤ 2
p
(|a|
p
+ |b|
p
), ∀a, b ∈ R,
p ≥ 1, one has
(ϕ 0 + χ
(1)
)
α
χ
(2)
L 2 ≤ 2
|α|
{{|ϕ 0 |
|α|
|χ
(2)
|| L 2 + +|χ
(1)
|
|α|
|χ
(2)
|| L 2 }
(5.15)
and the first term on the r.h.s. is immediately estimated by
2
|α|
|ϕ 0 |
|α|
|χ
(2)
|| L 2 ≤ A
|α|
(ϕ 0 L ∞ )
|α|
χ
(2)
H 1 .
(5.16)
The second term can be estimated by using the usual Hölder and the Sobolev inequalities
21
2
|α|
|χ
(1)
|
|α|
|χ
(2)
|| L 2 ≤ 2
|α|
|χ
(1)
||
|α|
L 2(|α|+1) |χ
(2)
|| L 2(|α|+1)
≤ B
|α| C s (2|α| + 2)
|α|+1
χ
(1)
|α|
H 1 χ
(2)
H 1 .
(5.17)
21 See, e.g. L.R. Volevic and B.P. Paneyakh, Russian Math. Surveys 20, 1 (1965). We list them for
the convenience of the reader
s = 1, f ; L
p (R
1 ) ≤ C 1 ( p) f ; H
1 (R
1 ), 2 ≤ p ≤ ∞, C 1 ( p) = O(1) ,
s = 2, f ; L
p (R
2 ) ≤ C 2 ( p) f ; H
1 (R
2 ), 2 ≤ p < ∞, C 2 ( p) = O( p
1
2 ) ,
s = 3, f ; L
p (R
3 ) ≤ C 3 ( p) f ; H
1 (R
3 ), 2 ≤ p ≤ 6, C 3 ( p) = O(1) .
The same kind of estimates hold locally. In particular, for any cube K ⊂ R s of size R, they take
the form
f ; L
p (K ) ≤ C s,R ( p) f ; H
1 (K ),
with p ∈ [2, +∞] for s = 1, p ∈ [2, +∞[ for s = 2 and p ∈ [2, 6] for s = 3. The constants
C s,R ( p) depend only on the size R and exhibit the same dependence on p as in the global case.