On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces
The theorems on the exact estimates of norms of intermediate derivatives in some Sobolev type abstract spaces are obtained. The formulas for calculating the norms are given.
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irk-123456789-1066342016-10-02T03:02:49Z On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces Mirzoev, S.S. Veliev, S.G. The theorems on the exact estimates of norms of intermediate derivatives in some Sobolev type abstract spaces are obtained. The formulas for calculating the norms are given. 2010 Article On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces / S.S. Mirzoev, S.G. Veliev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 73-83. — Бібліогр.: 7 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106634 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The theorems on the exact estimates of norms of intermediate derivatives in some Sobolev type abstract spaces are obtained. The formulas for calculating the norms are given. |
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Mirzoev, S.S. Veliev, S.G. |
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Mirzoev, S.S. Veliev, S.G. On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces Журнал математической физики, анализа, геометрии |
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Mirzoev, S.S. Veliev, S.G. |
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Mirzoev, S.S. |
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On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces |
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On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces |
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On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces |
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On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces |
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On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces |
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on the estimation of the norms of intermediate derivatives in some abstract spaces |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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On the Estimation of the Norms of Intermediate Derivatives in Some Abstract Spaces / S.S. Mirzoev, S.G. Veliev // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 1. — С. 73-83. — Бібліогр.: 7 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT mirzoevss ontheestimationofthenormsofintermediatederivativesinsomeabstractspaces AT velievsg ontheestimationofthenormsofintermediatederivativesinsomeabstractspaces |
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2025-07-07T18:47:50Z |
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1837015049099542528 |
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Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 1, pp. 73–83
On the Estimation of the Norms of Intermediate
Derivatives in Some Abstract Spaces
S.S. Mirzoev
Institute of Mathematics and Mechanics of NAS of Azerbaijan
9 F. Agayev Str., Baku, AZ1141, Azerbaijan
E-mail:mirzoyev@mail.ru
S.G. Veliev
Nakhchivan Teachers Institute
1 G. Aliev Ave., Nakhchivan, AZ7003, Azerbaijan
Received January 8, 2009
The theorems on the exact estimates of norms of intermediate deriva-
tives in some Sobolev type abstract spaces are obtained. The formulas for
calculating the norms are given.
Key words: Hilbert space, intermediate derivatives, norm eigenvector.
Mathematics Subject Classification 2000: 46G05, 46E20.
Let H be a separable Hilbert space, A be a positive-definite selfadjoint ope-
rator in H. The domain of definition of the operator Aγ , γ ≥ 0, becomes a
Hilbert space Hγ with respect to the scalar product (x, y)γ = (Aγx,Aγy), x, y ∈
Hγ(H0 = H).
By L2 (R+;Hγ) we denote a Hilbert space of the vector functions f (t) with
values in Hγ , determined almost everywhere in R+ = (0,∞), measurable by
Bochner, for which
‖f‖L2(R+;Hγ) =
∞∫
0
‖f (t)‖2
γ dt
1/2
< ∞.
Further, by L (X,Y ) denote a space of linear bounded operators acting from
the space X to the space Y, σ (·) is a spectrum of the operator (·) , ρ (·) is a regular
set of the operator (·) , E is a unique operator in H.
In the sequel, everywhere du
dt = u′, d2u
dt2
= u′′ are derivatives of the vector
function u (t) in the sense of distribution theory [1].
c© S.S. Mirzoev and S.G. Veliev, 2010
S.S. Mirzoev and S.G. Veliev
Let us introduce the following spaces:
W 2
2 (R+;H) =
{
u : u ∈ L2 (R+; H2) , u′′ ∈ L2 (R+; H)
}
,
◦
W 2
2 (R+; H; 0, 1) =
{
u : u ∈ W 2
2 (R+;H) , u (0) = u′ (0) = 0
}
,
W 2
2 (R+; H; T ) =
{
u : u ∈ W 2
2 (R+; H) , u (0) = Tu′ (0) , T ∈ L
(
H1/2; H3/2
)}
,
W 2
2 (R+;H; K) =
{
u : u ∈ W 2
2 (R+; H) , u′ (0) = Ku (0) , K ∈ L
(
H3/2; H1/2
)}
(in these denotation the spaces W 2
2 (R+; H; T ) and W 2
2 (R+; H; K) depend on the
choice of the letters T and K, but it does not lead to misunderstandings in the
text).
Each of these linear sets becomes a Gilbert space with respect of the norm
[1, p. 23–29]
‖u‖W 2
2 (R+;H) =
(
‖u‖L2(R+;H) +
∥∥u′′
∥∥
L2(R+;H)
)1/2
.
For T = 0 we get the space
◦
W 2
2 (R+; H; 0) =
{
u : u ∈ W 2
2 (R+; H) , u (0) = 0
}
,
and for K = 0 we have
◦
W 2
2 (R+; H; 1) =
{
u : u ∈ W 2
2 (R+;H) , u′ (0) = 0
}
.
Notice that it follows from the theorem on traces [1, Sect. 1, Th. 3.2] that
u (0) ∈ H3/2, u′ (0) ∈ H1/2.
The space W 2
2 (R; H), where R = (−∞,∞) [1], is defined in the similar way.
By the theorem on intermediate derivatives [1, Sect. 1, Th. 2.3], the operator
A
d
dt
: W 2
2 (R+; H) → L2 (R+;H)
is bounded.
In this paper we will find the exact values of the norm of intermediate deriva-
tive operators acting from the indicated spaces to the space L2 (R+; H). Notice
that for the scalar functions (H = R, A = E) the exact values of the operator
d
dt
: W 2
2 (R+) → L2 (R+)
were found in [2–5]. Similar problems were considered in [6, 7] for some abstract
spaces.
74 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
On the Estimation of the Norms of Intermediate Derivatives...
Denote
N0,0 = sup
06=u∈
◦
W 2
2 (R+;H;0,1)
∥∥Au′
∥∥
L2(R+;H)
‖u‖−1
W 2
2 (R+;H)
, (1)
N = sup
06=u∈W 2
2 (R+;H)
∥∥Au′
∥∥
L2(R+;H)
‖u‖−1
W 2
2 (R+;H)
, (2)
NT = sup
06=u∈W 2
2 (R+;H;T )
∥∥Au′
∥∥
L2(R+;H)
‖u‖−1
W 2
2 (R+;H)
, (3)
NK = sup
06=u∈W 2
2 (R+;H;K)
∥∥Au′
∥∥
L2(R+;H)
‖u‖−1
W 2
2 (R+;H)
. (4)
In particular, for T = 0 and K = 0 we denote the norms by N0 and N1,
respectively. Find the exact values of these norms.
First, we prove the following statement.
Lemma 1. For any u ∈ W 2
2 (R+; H) and β ∈ (0, 2) there exists the identity
‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
= ‖Φ(d/dt : β : A) u‖2
L2(R+;H)
+
(
R̃ (β) ϕ̃, ϕ̃
)
H2
,
(5)
where
Φ(d/dt : β : A) u =
d2u
dt2
+
√
2− βA
du
dt
+ Au2, (6)
R̃ (β) =
√
2− βE E
E
√
2− βE
= R (β)⊗ Ẽ,
R (β) =
√
2− β 1
1
√
2− β
, Ẽ =
E 0
0 E
.
P r o o f. By D (R+; H2) we denote a set of all infinitely differentiable in
H vector functions with values in H2 that have compact supports in R+. Then
by the theorem on density [1, Sect. 1, Th. 2.1] this set is everywhere dense in
W 2
2 (R+; H). Since the operators Aj d2−j
dt2−j , j = 0, 2, are bounded from W 2
2 (R+; H)
to L2 (R+;H), then it follows from the theorem on traces that it suffices to prove
validity of equality (5) for the functions from the class D (R+; H2). Obviously,
for u ∈ D (R+; H2) there holds the equality
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 75
S.S. Mirzoev and S.G. Veliev
‖Φ(d/dt : β : A) u‖2
L2(R+;H)
=
∥∥u′′ +
√
2− βAu′ + Au2
∥∥2
L2(R+;H)
= ‖u′′‖2
L2(R+;H)
+(2− β) ‖Au′‖2
L2(R+;H)
+
∥∥A2u
∥∥2
L2(R+;H)
+ 2Re
(
u′′, A2u
)
L2(R+;H)
+ 2
√
2− βRe (u′′, Au′)L2(R+;H) +2
√
2− βRe
(
Au′, A2u
)
L2(R+;H)
.
(7)
Integrating by parts, we get the validity of the following equalities:
Re
(
u′′, A2u
)
L2(R+;H)
=
∞∫
0
(
u′′, A2u
)
H
dt = Re
− (ϕ1, ϕ0)−
∞∫
0
(
Au′, Au′
)
H
dt
= −Re (ϕ1, ϕ0)−
∥∥Au′
∥∥2
L2(R+;H)
. (8)
In a similar way we obtain
(
u′′, Au′
)
L2(R+;H)
=
∞∫
0
(
u′′, Au′
)
H
dt = − (ϕ1, ϕ1)−
∞∫
0
(
Au′, u′′
)
H
dt
= −‖ϕ1‖2 − (
Au′, u′′
)
L2(R+;H)
, ϕ1 = A1/2u′ (0),
i.e.,
2Re
(
u′′, Au′
)
L2(R+;H)
= −‖ϕ1‖2 . (9)
Similarly, we get
2Re
(
Au′, A2u
)
L2(R+;H)
= −‖ϕ0‖2 , ϕ0 = A3/2u (0) . (10)
Taking into account (8)–(10) in equality (7), we get
‖Φ(d/dt : β : A) u‖2
L2(R+;H) = ‖u′′‖2
L2(R+;H) − β ‖Au′‖2
L2(R+;H) − [2Re (ϕ0, ϕ1)
+
√
2− β ‖ϕ0‖2 +
√
2− β ‖ϕ1‖2
]
+ ‖A2u‖2
L2(R+;H).
(11)
76 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
On the Estimation of the Norms of Intermediate Derivatives...
On the other hand, there hold the equalities:
2Re (ϕ0, ϕ1) =
E 0
0 E
ϕ0
ϕ1
,
ϕ0
ϕ1
H2
,
‖ϕ0‖2 =
0 0
E 0
ϕ0
ϕ1
,
ϕ0
ϕ1
H2
,
‖ϕ1‖2 =
0 E
0 0
ϕ0
ϕ1
,
ϕ0
ϕ1
H2
.
Thus, the equality
‖Φ(d/dt : β : A) u‖2
L2(R+;H) =
∥∥u′′
∥∥2
W 2
2 (R+;H)
−β
∥∥Au′
∥∥2
L2(R+;H)
−
(
R̃ (β) ϕ̃, ϕ̃
)
H2
holds. The lemma is proved.
Hence we get the following corollaries.
Corollary 1. For u ∈
◦
W 2
2 (R+; H; 0, 1) and β ∈ (0, 2) there holds the equality
‖Φ(d/dt : β : A) u‖2
L2(R+;H)
= ‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
. (12)
Corollary 2. For u ∈ W 2
2 (R+; H; T ) and for β ∈ (0, 2) there holds the
equality
‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
= ‖Φ(d/dt : β : A) u‖2
L2(R+;H) + (RT (β) ϕ,ϕ) ,
(13)
where
(RT (β)ϕ, ϕ) = 2Re (Cϕ, ϕ) +
√
2− β
(
‖Cϕ‖2 + ‖ϕ‖2
)
,
C = A3/2TA−1/2, ϕ = A1/2u′ (0) ∈ H.
(14)
In particular, when T = 0 (C = 0), for u ∈
◦
W 2
2 (R+; H; 0) and for β ∈ (0, 2)
we have
‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
= ‖Φ(d/dt : β : A) u‖2
L2(R+;H) +
√
2− β ‖ϕ‖2 .
(15)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 77
S.S. Mirzoev and S.G. Veliev
Corollary 3. For u ∈ W 2
2 (R+; H; K) and for β ∈ (0, 2) there holds the
equality
‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
= ‖Φ(d/dt : β : A) u‖2
L2(R+;H) + (RK (β)ϕ, ϕ) ,
(16)
where
(RK (β) ϕ,ϕ) = 2Re (Sϕ,ϕ) +
√
2− β
(
‖Sϕ‖2 + ‖ϕ‖2
)
,
S = A1/2KA−3/2, ϕ = A3/2u (0) ∈ H.
(17)
In particular, when K = 0 (S = 0), for u ∈
◦
W 2
2 (R+; H; 1) and for β ∈ (0, 2)
we have
‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
= ‖Φ(d/dt : β : A) u‖2
L2(R+;H) +
√
2− β ‖ϕ‖2 .
(18)
Obviously, the lemma below holds
Lemma 2. σ
(
R̃ (β)
)
= σ (R (β)) as a geometrical set, where R̃ (β) and
R (β) are determined in Lemma 1.
Hence it follows that R̃ (β) may have only eigenvalues that coincide with
R (β).
Now we find the exact values of the norms of intermediate derivative operators
N0,0, NT , NK , N0, N1 and N , defined by formulae (1)–(4).
Theorem 1. The norm N0,0 = 1√
2
.
P r o o f. For u ∈
◦
W 2
2 (R+; H; 0, 1) and β ∈ (0, 2) equality (12) holds.
In this equality passing to the limit as β → 2 we can find that for any
u ∈
◦
W 2
2 (R+; H; 0, 1) the inequality
∥∥Au′
∥∥
L2(R+;H)
≤ 1√
2
‖u‖W 2
2 (R+;H)
holds, i.e., N0,0 ≤ 1√
2
. Prove that N0,0 = 1√
2
. Show that for any ε > 0 there
exists such a vector function uε (t) that
E (uε (t)) ≡ ‖uε‖2
W 2
2 (R;H) − (2 + ε)
∥∥Au′ε
∥∥2
L2(R;H)
< 0. (19)
Find uε (t) in the form uε (t) = g (t) ψε, where ψε ∈ H4 (‖ψε‖0 = 1), but g (t)
78 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
On the Estimation of the Norms of Intermediate Derivatives...
is a scalar function from W 2
2 (R). Then by the Plancharel theorem
E (g (t) ψε) = ‖g′′ (t) ψε‖2
L2(R;H) +
∥∥g (t)A2ψε
∥∥2
L2(R;H)
− (2 + ε) ‖g′ (t) Aψε‖2
L2(R;H)
=
+∞∫
−∞
((
ξ4E + A4 − (2 + ε) ξ2A2
)
ψε, ψε
) |ĝ (ξ)|2 dξ ≡
+∞∫
−∞
q (ξ, ψε) |ĝ (ξ)|2 dξ,
where q (ξ, ψε) = ξ4+
∥∥A2ψε
∥∥2−(2 + ε) ξ2 ‖Aψε‖2, and ĝ (ξ) is a Fourier transform
of the function g (t).
It is obvious that the function q (ξ, ψε) takes its minimal value at the points
ξ = ± (2 + ε) equal to h (ε, ψε) =
∥∥A2ψε
∥∥2 − 1
4 (2 + ε)2 ‖Aψε‖4.
If the operator A has at least one eigenvector responding to eigenvalue µ,
we can take this normed eigenvector as ψε.
Thus in this case h (ε, ψε) = µ4 − 1
4 (2 + ε)2 µ4 < 0. If µ is a point of a con-
tinuous spectrum, we can find such a vector ψε (‖ψε‖ = 1) that Alψε = λlψε +
o (δ) , l = 1, 2, . . . , for δ → 0. Obviously, for small δ the function h (ε, ψε) < 0.
Now let us fix the vector ψε, for which h (ε, ψε) < 0, and find the function g (t).
Since the function q (ξ, ψε) is continuous with respect to the argument ξ, there
can be found (η0 (ε) , η1 (ε)), where q (ξ, ψε) < 0, i.e.,
ε (g (t) ψε) =
η1(ε)∫
η0(ε)
q (ξ, ψε)
∣∣∣∣
∧
g (ξ)
∣∣∣∣
2
dξ < 0.
Further, from the continuity of the functional E (·) in the space W 2
2 (R; H)
by the theorem on density of finite infinitely differentiable vector function [1,
p. 29] there exists a vector function uN,ε (t) ∈ W 2
2 (R; H) with the support
(−N, N) ⊂ R, for which E (uN,ε (t)) < 0. Assuming uε (t) = uN,ε (t + 2N),
we get uε (t) ∈
◦
W 2
2 (R+;H; 0, 1) and E (uε (t)) = E (uN,ε (t + 2N)) < 0. Thus,
N0,0 = 1√
2
. The theorem is proved.
Since
◦
W 2
2 (R+; H; 0, 1) ⊂ W 2
2 (R+; H; T ), then NT ≥ 1√
2
. Analogously,
N ≥ NK ≥ N0,0 = 1√
2
. Explain when NT = 1√
2
or NK = 1√
2
. The following
holds.
Theorem 2. The norm NT = 1√
2
(
NK = 1√
2
)
iff for all β ∈ (0, 2) and
ϕ ∈ H (RT (β)ϕ, ϕ) > 0 ((RK (β) ϕ,ϕ) > 0).
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 79
S.S. Mirzoev and S.G. Veliev
P r o o f. Let NT = 1√
2
. Then for any u ∈ W 2
2 (R+; H; T ) and β ∈ (0, 2)
we have
‖u‖2
W 2
2 (R+;H) − β ‖Au′‖2
L2(R+;H)
= ‖u‖2
W 2
2 (R+;H)
(
1− β ‖Au′‖2
L2(R+;H) ‖u‖−2
W 2
2 (R+;H)
)
≥ ‖u‖2
W 2
2 (R+;H)
(
1− β sup
u∈W 2
2 (R+;H;T )
‖Au′‖2
L2(R+;H) ‖u‖−2
W 2
2 (R+;H)
)
= ‖u‖2
W 2
2 (R+;H)
(
1− β 1
2
)
> 0.
Then it follows from equality (13) that for any u ∈ W 2
2 (R+;H; T ) and β ∈ (0, 2)
‖Φ (d/dt : β : A) u‖2
L2(R+;H)
+ (RT (β) ϕ,ϕ) > 0, ∀ϕ ∈ H
(
ϕ = A1/2u′ (0) ∈ H
)
.
(20)
Since the characteristically polynomial Φ (λ : β : A) = λ2E +
√
2− βλA + A2
is represented in the form
Φ (λ : β : A) = (λE − ω1 (β) A) (λE − ω2 (β) A) ,
where ω1 (β) = ω2 (β) =
(−√2− β − i
√
2 + β
)
/2, (Reω1 (β) < 0, Reω2 (β) < 0),
we get that the Cauchy problem
Φ (d/dt : β : A)u = 0, u (0) = Tu′ (0) , u′ (0) = A−1/2ϕ, ∀ϕ ∈ H, (21)
has a unique solution from the space W 2
2 (R+; H)
u (t, β) =
1
ω2 − ω1
{
eω1(β)tA
(
ω2 (β) TA−1/2ϕ−A−3/2ϕ
)
+eω2(β)tA
(
A−3/2ϕ− ω1 (β) TA−1/2ϕ
)}
.
Obviously, ‖u (t, β; ϕ)‖ ≤ d1 (β) ‖ϕ‖ , d1 (β) > 0. Using the uniqueness of the
solution of the Cauchy problem and also using Banach’s theorem on invertible
operator, we get ‖u (t, β; ϕ)‖ ≥ d2 (β) ‖ϕ‖. Thus, it follows from equality (20)
that (RT (β) ϕ,ϕ) > 0 for β ∈ (0, 2) and ∀ϕ ∈ H.
Inversely, if (RT (β) ϕ,ϕ) > 0, then from equality (13) it follows that
‖u‖2
W 2
2 (R+;H) − β
∥∥Au′
∥∥2
L2(R+;H)
> 0
(∀β ∈ (0, 2) , ∀u ∈ W 2
2 (R+; H; T )
)
.
80 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
On the Estimation of the Norms of Intermediate Derivatives...
By passing to the limit as β → 2, we get NT ≤ 1
2 . Consequently, NT = 1√
2
.
We prove in a similar way that NK = 1√
2
iff (RK (β) ϕ,ϕ) > 0 for β ∈ (0, 2) and
∀ϕ ∈ H.
Using this theorem we get the following statement.
Theorem 3. The norm NT = 1√
2
iff ReC ≥ 0 (see (14)).
In fact, if NT = 1√
2
, then (RT (β) ϕ,ϕ) > 0, β ∈ (0, 2) , ϕ ∈ H. By passing to
the limit as β → 2, we get ReC ≥ 0. Inversely, if ReC ≥ 0, then (RT (β) ϕ,ϕ) > 0,
for β ∈ (0, 2), i.e., NT = 1√
2
.
Similarly is proved
Theorem 4. The norm NK = 1√
2
iff ReS ≥ 0 (see (17)).
Notice that if ReC is not a non negative operator, then the following theorem
holds.
Theorem 5. Let inf
ϕ∈H
Re (Cϕ, ϕ) < 0,
(
inf
ϕ∈H
Re(Sϕ, ϕ) < 0
)
. Then the norm
NT =
1√
2
(
1− 2
∣∣∣∣ inf
‖ϕ‖=1
Re (Cϕ, ϕ)
1 + ‖Cϕ‖2
∣∣∣∣
2
)−1/2
(22)
NK =
1√
2
(
1− 2
∣∣∣∣ inf
‖ϕ‖=1
Re (Sϕ,ϕ)
1 + ‖Sϕ‖2
∣∣∣∣
2
)−1/2
(23)
(see (14), (17)).
P r o o f. Let inf
ϕ∈H
ReC < 0. Then by Theorem 3 NT > 1√
2
. Therefore
N−2
T ∈ (0, 2). Then if in equality (13) as u (t) we take the solution of the Cauchy
problem (see (21)), for β ∈ (
0, N−2
T
)
and ‖ϕ‖ = 1 we get
(RT (β) ϕ,ϕ) = ‖u (t, β; ϕ)‖2
W 2
2 (R+;H) −
∥∥Au′ (t, β;ϕ)
∥∥2
L2(R+;H)
≥ ‖u (t, β; ϕ)‖2
W 2
2 (R+;H)
(
1− βN−2
T
)
> 0.
Thus, for β ∈ (
0, N−2
T
)
the function
m (β) = inf
‖ϕ‖=1
(R (β) ϕ,ϕ) > 0.
And for β ∈ (
N−2
T , 2
)
, by definition of NT , there can be found a vector
function υ (t, β) ∈ W 2
2 (R+;H;T ) such that
‖υ (t, β)‖2
W 2
2 (R+;H) −
∥∥Aυ′ (t, β)
∥∥2
L2(R+;H)
< 0.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1 81
S.S. Mirzoev and S.G. Veliev
Consequently, for β ∈ (
N−2
T , 2
)
it follows from equality (13) that
(RT (β)ϕβ, ϕβ) + ‖Φ(d/dt : β : A) v(t, β)‖2
L2(R+;H) < 0
(
ϕβ = A−1/2v(0, β)
)
, i.e., m(β) < 0 for β ∈ (
N−2
T , 2
)
. Thus, the continuous
function m (β), determined for β ∈ (0, 2), changes its sign at the point N−2
T , i.e.,
m
(
N−2
T
)
= 0. Hence, it follows easily that
√
2−N−2
T = −2 inf
‖ϕ‖=1
Re (Cϕ,ϕ)
/[
1 + ‖Cϕ‖2
]
,
i.e.,
NT =
1√
2
(
1− 2
∣∣∣∣ inf
‖ϕ‖=1
Re (Cϕ, ϕ)
1 + ‖Cϕ‖2
∣∣∣∣
2
)−1/2
.
Formula (23) is proved in a similar way. The theorem is proved.
It follows from Theorems 3–5 that N0 = N1 = 1√
2
(C = S = 0).
Now we find the norm N .
There holds the following.
Theorem 6. The norm N = 1, where N is determined by formula (2).
P r o o f. It is obvious that N ≥ 1√
2
. Show that N 6= 1√
2
. In fact, if N = 1√
2
,
then it follows from equality (5) that
‖Φ (d/dt : β : A)u‖2
L2(R+;H)
+
(
R̃ (β) ϕ̃, ϕ̃
)
H2
≥ ‖u‖2
W 2
2 (R+;H)
×
(
1− β sup
u∈W 2
2 (R+;H)
‖Au′‖2 ‖u‖−2
W 2
2 (R+;H)
)
≥ ‖u‖2
W 2
2 (R+;H)
(
1− β 1
2
)
> 0.
(24)
Then for β ∈ (0, 2) the Cauchy problem
Φ (d/dt : β : A) u = 0, u (0) = A−3/2ϕ0, u
′ (0) = A−1/2ϕ1, ∀ϕ0, ϕ1 ∈ H, (25)
has a unique solution from W 2
2 (R+; H), therefore for β ∈ (0, 2) (R̃ (β) ϕ̃, ϕ̃)H2
> 0. By Lemma 2 all eigenvalues of the matrix R (β) are positive. But it is seen
from the form R (β) (see Lem. 1) that for β ∈ (1, 2), R (β) has also the negative
eigenvalue λ1 (R (β)) = 1 − β < 0. Thus, N > 1
2 , i.e., N−2 ∈ (0, 2). Then for
β ∈ (
0, N−2
)
we have
‖Φ(d/dt : β : A) u‖2
L2(R+;H)
+
(
R̃ (β) ϕ̃, ϕ̃
)
≥ ‖u‖2
W 2
2
(
1− βN−2
)
> 0.
Hence it follows that if in this inequality we replace u by the solution of
the Cauchy problem (25) for β ∈
(
0, N
−2
T
)
, then we obtain
(
R̃ (β) ϕ̃, ϕ̃
)
> 0.
82 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 1
On the Estimation of the Norms of Intermediate Derivatives...
Consequently, all eigenvalues of the matrix R (β) are positive for β ∈ (
0, N−2
)
.
In particular, λ1 (β) > 0 (λ1 (β) is the first eigenvalue of the matrix R (β)).
And for β ∈
(
N
−2
T , 2
)
it follows from the definition of N that there exists such
υ (t, β) ∈ W 2
2 (R+; H) that
‖υ‖2
W 2
2 (R+;H) −
∥∥Aυ′
∥∥2
L2(R+;H)
< 0.
Consequently,
(
R̃ (β) ϕ̃β, ϕ̃β
)
< 0
(
ϕβ,0 = A−3/2υβ (0) , ϕβ,1 = A−1/2υ′β (0)
)
.
We again obtain that λ1 (β) is the first eigenvalue of the matrix R (β), is
negative for β ∈ (
N−2, 2
)
. Consequently, λ1
(
N−2
)
= 0.
I.e., ∣∣∣∣∣∣
√
2−N−2 1
1
√
2−N−2
∣∣∣∣∣∣
= 0.
Hence we find that N−2 = 1, i.e., N = 1. The theorem is proved.
References
[1] J.-L. Lions and E. Magenes, Nonhomogenous Boundary Value Problems and
Applications. Mir, Moscow, 1971.
[2] G.T Hardy, D.E Littlewood, and G. Polia, Inequalities. IL, Moscow, 1948.
[3] S.B. Stechkin, Inequalities Between the Norms of the Derivatives of an Arbitrary
Function. — Acta Sci. Math. 26 (1965), 225–230.
[4] V.V. Arestov, Precise Inequalities Between the Norms of Functions and their Deriva-
tives. — Acta Sci. Math. 33 (1972), No. 3–4, 249–267.
[5] V.N. Gabushin, On the Best Approximation of the Differentiation Operator on the
Half-Line. — Math. Zametki 6 (1969), No. 5, 573–582.
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