Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions
Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j...
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| author | Babenko, V. F. Parfinovych, N. V. Pichugov, S. A. Бабенко, В. Ф. Парфінович, Н. В. Пічугов, С. О. |
| author_facet | Babenko, V. F. Parfinovych, N. V. Pichugov, S. A. Бабенко, В. Ф. Парфінович, Н. В. Пічугов, С. О. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:19Z |
| description | Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$
and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote
$$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$
We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented. |
| first_indexed | 2026-03-24T02:31:54Z |
| format | Article |
| fulltext |
UDC 517.5
V. F. Babenko (Dnepropetrovsk Nat. Univ., Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Donetsk),
N. V. Parfinovych (Dnepropetrovsk Nat. Univ.),
S. A. Pichugov (Dnepropetrovsk State Technical Univ. Railway Transport)
SHARP KOLMOGOROV-TYPE INEQUALITIES
FOR NORMS OF FRACTIONAL DERIVATIVES
OF MULTIVARIATE FUNCTIONS
ТОЧНI НЕРIВНОСТI ТИПУ КОЛМОГОРОВА ДЛЯ НОРМ
ДРОБОВИХ ПОХIДНИХ ФУНКЦIЙ БАГАТЬОХ ЗМIННИХ
Let C(Rm) be spaces of bounded and continuous functions x : Rm → R, endowed with the norms ‖x‖C =
= ‖x‖C(Rm) := sup{|x(t)| : t ∈ Rm}. Let ej , j = 1, . . . , m, be the standard basis in Rm. Given moduli
of continuity ωj , j = 1, . . . , m, denote
Hj,ωj :=
(
x ∈ C(Rm) : ‖x‖ωj = ‖x‖
H
j,ωj = sup
tj 6=0
‖∆tjejx(·)‖C
ωj(|tj |)
<∞
)
.
In this paper, new sharp Kolmogorov-type inequalities for norms of mixed fractional derivatives ‖Dα
ε x‖C
of functions x ∈
mT
j=1
Hj,ωj are obtained. Some applications of these inequalities are presented.
Нехай C(Rm) — простори неперервних обмежених функцiй x : Rm → R з нормами ‖x‖C =
= ‖x‖C(Rm) := sup{|x(t)| : t ∈ Rm}, ej , j = 1, . . . , m, — звичайна база в Rm. Для заданих мо-
дулiв неперервностi ωj , j = 1, . . . , m, позначимо
Hj,ωj :=
(
x ∈ C(Rm) : ‖x‖ωj = ‖x‖
H
j,ωj = sup
tj 6=0
‖∆tjejx(·)‖C
ωj(|tj |)
<∞
)
.
У роботi отримано новi точнi нерiвностi типу Колмогорова для норм мiшаних частинних похiдних
‖Dα
ε x‖C функцiй x ∈
mT
j=1
Hj,ωj . Наведенi деякi застосування цих нерiвностей.
1. Introduction. Statements of the problems. Main results. Sharp Kolmogorov-type
inequalities for univariate and multivariate functions, estimating the norms of intermedi-
ate derivatives through the norms of the function itself and its derivatives of higher
order, are of great importance for many branches of mathematics and its applications.
After A.N. Kolmogorov obtained his inequality (see [1 – 3]) many inequalities of this
type for norms of integer derivatives of univariate functions were obtained (see, for
example, [4 – 8]).
In the case of functions of two or more variables very few results of this type are
known (see [9]–[15]).
Many questions in Analysis require to consider derivatives and antiderivatives of
fractional order (see, for instance, [16]). One of the natural and useful definitions of the
fractional derivative for a univariate function x(u), u ∈ R, is the following definition of
Marchaud fractional derivative [17] (see also [16, p. 95 – 97]):
(Dα
±x)(u) :=
α
Γ(1− α)
∞∫
0
x(u)− x(u∓ t)
t1+α
dt, α ∈ (0, 1).
For brevity, we denote Aα =
α
Γ(1− α)
.
c© V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3 301
302 V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV
Let C(R) be the space of all bounded and continuous functions x : R→ R endowed
with the norm
‖x‖C := sup
{
|x(t)| : t ∈ R
}
.
Let ω(t) be a modulus of continuity, i.e., continuous, nondecreasing, subadditive
function defined on [0,+∞) and such that ω(0) = 0. By Hω = Hω(R) denote the
space of functions x ∈ C(R), for which
‖x‖ω = ‖x‖Hω = sup
t1, t2∈R
t1 6=t2
|x(t1)− x(t2)|
ω(|t1 − t2|)
<∞.
If ω(t) = tβ , β ∈ (0, 1], then we write Hβ instead of Hω.
Let α ∈ (0, 1) and ω(t) be such that
1∫
0
ω(t)
t1+α
dt <∞,
or, equivalently, ∫
R+
min{1, ω(t)}
t1+α
dt <∞.
It was proved in [18] that for any h > 0 the following additive Kolmogorov-type
inequality:
‖Dα
±x‖C ≤ Aα
‖x‖ω h∫
0
ω(t)
t1+α
dt+
2‖x‖C
αhα
(1)
holds.
Moreover, this inequality becomes an equality for the function
xh(u) =
ω(|u|)− ω(h)
2
, |u| ≤ h,
ω(h)
2
, |u| > h.
Note that after minimization over h of the right-hand side of inequality (1), it can be
rewritten in the following form:
‖Dα
±x‖C ≤ Aα
∞∫
0
min{2‖x‖C , ‖x‖ωω(t)}
t1+α
dt. (2)
In the case ω(t) = tβ , β ∈ (0, 1], α < β ≤ 1, inequality (2) becomes
‖Dα
±x‖C ≤
1
Γ(1− α)
21−α/β
1− α/β
‖x‖1−α/βC ‖x‖α/β
Hβ
. (3)
Other known results on sharp Kolmogorov-type inequalities for fractional derivatives
can be found in [18 – 21].
Let Rm be the space of points t = (t1, . . . , tm) and {ei}mi=1 be the standard basis
in Rm.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES . . . 303
By C(Rm) denote the space of all bounded and continuous functions x : Rm → R
with the norm
‖x‖C = ‖x‖C(Rm) := sup
{
|x(t)| : t ∈ Rm
}
.
For a given vector t = (t1, . . . , tm) ∈ Rm, by ∆tjejx(u) denote the first difference of
the function x(u) along the variable uj with the step tj , j = 1, . . . ,m,
∆tjejx(u) := x(u)− x(u+ tjej),
and define as
∆tx(u) := ∆t1e1∆t2e2 . . .∆tmemx(u)
the mixed difference of the function x(u) with the step t.
Let ωj(tj), tj ≥ 0, j = 1, . . . ,m, be given moduli of continuity. We will consider
the following spaces:
Hj,ωj :=
{
x ∈ C(Rm) : ‖x‖ωj = ‖x‖Hj,ωj = sup
tj 6=0
‖∆tjejx(·)‖C
ωj(|tj |)
<∞
}
.
If ωj(tj) = t
βj
j , βj ∈ (0, 1], then we write Hj,βj instead of Hj,ωj , j = 1, . . . ,m.
For a given function x(u), u ∈ Rm, and a vector of smoothness α = (α1, . . . , αm),
αj ∈ (0, 1), j = 1, . . . ,m, and a vector of sign distribution ε = (ε1, . . . , εm), εj = ±,
j = 1, . . . ,m, the mixed Marchaud derivative of order α is defined in the following way
(see [17, p. 347]):
(Dα
ε x)(u) := Aα
∫
Rm+
∆εtx(u)
m∏
j=1
t
−αj−1
j dt,
where x ∈ C(Rm), Aα =
∏m
j=1
Aαj , Aαj =
αj
Γ(1− αj)
, εt := (ε1t1, . . . , εmtm).
V. F. Babenko and S. A. Pichugov [15] proved that for any function x ∈
m⋂
j=1
Hj,βj ,
the sharp inequality holds
‖Dα
ε x‖C ≤
2m−1∏m
j=1
Γ(1− αj)
21−
Pm
j=1
αj
βj
1−
∑m
j=1
αj
βj
‖x‖
1−
Pm
j=1
αj
βj
C
m∏
j=1
‖x‖αj/βj
Hj,βj
, (4)
provided that βj ∈ (0, 1] and αj ∈ (0, 1), j = 1, . . . ,m, satisfy the condition∑m
j=1
αj
βj
< 1.
The inequality (4) is the multivariate analog of (3). In this paper we obtain an
inequality, which is a generalization of (4) and represents a multivariate analog of (2).
In what follows, for αj ∈ (0, 1), j = 1, . . . ,m, and for given moduli of continuity
ω1(t1), . . . , ωm(tm), we will need the following condition:∫
Rm+
min
{
1, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt <∞. (5)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
304 V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV
Theorem 1. Let moduli of continuity ω1(t1), . . . , ωm(tm) and numbers αj ∈
∈ (0, 1), j = 1, . . . ,m, be such that condition (5) is satisfied. Then for any function
x ∈
m⋂
j=1
Hj,ωj and any vector ε of sign distribution, the following sharp inequality holds:
∥∥Dα
ε x
∥∥
C
≤
≤ 2m−1Aα
∫
Rm+
min
{
2‖x‖C , ‖x‖ω1ω1(t1), . . . , ‖x‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt. (6)
For given moduli of continuity ω1, . . . , ωm, by UHj,ωj , j = 1, . . . ,m, denote the
unit ball in the space Hj,ωj .
Consider the function
Ω
δ, m⋂
j=1
UHj,ωj
:= sup
x∈
Tm
j=1UH
j,ωj,
‖x‖C≤δ
‖Dα
ε x‖C , δ ≥ 0. (7)
The function (7) is called the modulus of continuity of the operator Dα
ε on the set
m⋂
j=1
UHj,ωj .
Theorem 1 implies the following statement.
Corollary 1. Under conditions of Theorem 1 for any δ > 0,
Ω
δ, m⋂
j=1
UHj,ωj
= 2m−1Aα
∫
Rm+
min
{
2δ, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt. (8)
In particular, if βj ∈ (0, 1] and αj ∈ (0, 1), j = 1, . . . ,m, are such that
∑m
j=1
αj
βj
< 1,
then
Ω
δ, m⋂
j=1
UHj,βj
=
2m−1∏m
j=1
Γ(1− αj)
21−
Pm
j=1
αj
βj
1−
∑m
j=1
αj
βj
δ
1−
Pm
j=1
αj
βj .
The problem of finding of the modulus of continuity for a given operator on a given
set is closely related to the problem about approximation of an unbounded operator by
bounded ones.
We now consider the general statement of this problem.
Let X and Y be the Banach spaces, let L(X,Y ) be the space of linear bounded
operators S : X → Y, and let A : X → Y be an operator (not necessarily linear) with
the domain DA ⊂ X. Let also Q ⊂ DA be some class of elements.
For N > 0, the quantity
EN (A,Q) = inf
S∈L(X,Y )
‖S‖≤N
sup
x∈Q
‖Ax− Sx‖Y (9)
is called the best approximation of the operator A on the set Q by linear operators
S : X → Y such that ‖S‖ = ‖S‖X→Y ≤ N.
The problem is to compute the quantity (9) and to find the extremal operator, i.e.,
the operator delivering the infimum on the right-hand side of (9).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES . . . 305
This problem appeared in Stechkin’s investigations in 1965. The statement of this
problem, the first important results and the solution of this problem for low order
differentiation operators were presented in [22]. For a survey of further research on this
problem see [4, 5].
The function
Ω(δ,Q) := sup
x∈Q
‖x‖X≤δ
‖Ax‖Y , δ ≥ 0,
is called the modulus of continuity of the operator A on the set Q.
Note, that this definition generalizes the above presented definition of the modulus
of continuity of the operator Dα
ε on the set
m⋂
j=1
UHj,ωj .
It is easily seen that the problem of computation of the function Ω(δ,Q) for a given
operator is the abstract version of the problem about the Kolmogorov inequality.
S. B. Stechkin [22] proved that
EN (A,Q) ≥ sup
δ≥0
{
Ω(δ,Q)−Nδ
}
. (10)
Namely, the inequality (10) shows the relation between the Stechkin problem and
Kolmogorov-type inequalities.
The following theorem gives the solution of the Stechkin problem for the operator
Dα
ε on the class
m⋂
j=1
UHj,ωj .
Theorem 2. Let the strictly increasing moduli of continuity ω1(t1), . . . , ωm(tm)
and the numbers αj ∈ (0, 1), j = 1, . . . ,m, be such that the condition (5) is satisfied.
Given N > 0, let hN = (hN1 , . . . , h
N
m) ∈ Rm+ be such that
ω1(hN1 ) = . . . = ωm(hNm) and
2mAα
α1 . . . αm
m∏
j=1
(hNj )−αj = N. (11)
Let
G(hN ) :=
{
u = (u1, . . . , um) ∈ Rm : |u1| ≥ hN1 , . . . , |um| ≥ hNm
}
.
Then
EN
Dα
ε ,
m⋂
j=1
UHj,ωj
= 2m−1Aα
∫
Rm+ \G(hN )
min
{
ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt.
In addition, the operator
BhNx(u) = Aα
∫
G(hN )
∆εtx(u)
m∏
j=1
t
−αj−1
j dt
is the extremal operator.
Note that the lower estimate for EN
(
Dα
ε ,
m⋂
j=1
UHj,ωj
)
will be obtained with the
help of Corollary 1. In order to obtain the upper bound for EN
(
Dα
ε ,
m⋂
j=1
UHj,ωj
)
we
will estimate from above the quantity
∥∥Dα
ε x−BhNx
∥∥
C
on the class
m⋂
j=1
UHj,ωj .
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
306 V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV
Note also that in the case ω(tj) = t
βj
j , βj ∈ (0, 1], j = 1, . . . ,m, applying
Theorem 2, we immediately obtain the following statement.
Corollary 2. Suppose that βj ∈ (0, 1] and αj ∈ (0, 1), j = 1, . . . ,m, satisfy the
inequality
∑m
j=1
αj
βj
< 1. Then for any N > 0,
EN
Dα
ε ,
m⋂
j=1
UHj,βj
=
2m−
Pm
j=1
αj
βj∏m
j=1
Γ(1− αj)
1Pm
j=1
αj
βj
∑m
j=1
αj
βj
1−
∑m
j=1
αj
βj
N
1− 1Pm
j=1
αj
βj .
The inequalities for intermediate derivatives are also closely related to the Kolmogorov
problem about necessary and sufficient conditions of the existence of a function, for
which given numbers are the upper bounds of absolute values of its derivatives of
corresponding order (see [2, 3]). For some known results in this direction see, for
example, [23 – 26] and [6].
We consider the Kolmogorov-type problem in the following setting. It is required to
find the necessary and sufficient conditions on the numbers M0,Mα,Mω1 , . . . ,Mωm for
existence of the function x ∈
m⋂
j=1
Hj,ωj such that
‖x‖C = M0, ‖Dα
ε x‖C = Mα, ‖x‖ω1 = Mω1 , . . . , ‖x‖ωm = Mωm .
Theorem 3. Let moduli of continuity ω1(t1), . . . , ωm(tm) and numbers αj ∈
∈ (0, 1), j = 1, . . . ,m, be such that (5) holds, and let numbersM0, Mα, Mω1 , . . . ,Mωm
be given. There exists a function x ∈
m⋂
j=1
Hj,ωj such that
‖x‖C = M0, ‖Dα
ε x‖C = Mα, ‖x‖ω1 = Mω1 , . . . , ‖x‖ωm = Mωm ,
if and only if the inequality
Mα ≤ 2m−1Aα
∫
Rm+
min
{
2M0,Mω1ω1(t1), . . . ,Mωmωm(tm)
} m∏
j=1
t
−αj−1
j dt (12)
holds.
Corollary 3. Suppose that βj ∈ (0, 1] and αj ∈ (0, 1), j = 1, . . . ,m, satisfy the
condition
∑m
j=1
αj
βj
< 1, and let numbers M0, Mα, Mβ1 , . . . ,Mβm be given. There
exists a function x ∈
m⋂
j=1
Hj,βj such that
‖x‖C = M0, ‖Dα
ε x‖C = Mα, ‖x‖H1,β1 = Mβ1 , . . . , ‖x‖Hm,βm = Mβm ,
if and only if the inequality
Mα ≤
2m−1∏m
j=1
Γ(1− αj)
21−
Pm
j=1
αj
βj
1−
∑m
j=1
αj
βj
M
1−
Pm
j=1
αj
βj
0
m∏
j=1
M
αj/βj
Hj,βj
holds.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES . . . 307
2. Proofs. Proof of Theorem 1. We prove the theorem in the case ε = (+, . . . ,+)
only, since for any other ε one can use analogous arguments. Taking into account the
definition of the fractional derivative we have
∀u ∈ Rm : |Dα
ε x(u)| ≤ Aα
∫
Rm+
‖∆tx(·)‖C
m∏
j=1
t
−αj−1
j dt. (13)
To estimate the norm ‖∆tx‖C we will use the following inequalities:
‖∆tx‖C ≤ 2m‖x‖C
and
‖∆tx‖C ≤ 2m−1‖∆tjejx‖C ≤ 2m−1‖x‖ωjωj(|tj |), j = 1, . . . ,m.
Combining these estimates we obtain
‖∆tx‖C ≤ 2m−1 min
{
2‖x‖C , ‖x‖ω1ω1(|t1|), . . . , ‖x‖ωmωm(|tm|)
}
.
Applying the last estimate to the right-hand side of (13) we have
∀u ∈ Rm :
∣∣Dα
ε x(u)
∣∣ ≤
≤ 2m−1Aα
∫
Rm+
min
{
2‖x‖C , ‖x‖ω1ω1(t1), . . . , ‖x‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt. (14)
Let us show that for every function x ∈
m⋂
j=1
Hj,ωj , its fractional derivative Dα
ε x(u)
depends on u continuously.
Let
ω(x, θ) := sup
|t|<θ
‖x(·)− x(·+ t)‖C ,
where |t| =
√
t21 + . . .+ t2m, t = (t1, . . . , tm).
Applying the inequality (14) to the difference x(u)− x(u+ δ), δ ∈ Rm, we obtain∣∣Dα
ε x(u)−Dα
ε x(u+ δ)
∣∣ ≤
≤ 2m−1Aα
∫
Rm+
min
{
2ω(x, |δ|), 2‖x‖ω1ω1(t1), . . . , 2‖x‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt.
Note, that the function
min
{
2ω(x, |δ|), 2‖x‖ω1ω1(t1), . . . , 2‖x‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j
uniformly converges to zero (as δ → 0) on any set of points (t1, . . . , tm) ∈
∏m
j=1
[σj ,∞),
σj > 0, j = 1, . . . ,m, and the integral∫
Rm+
min
{
2ω(x, |δ|), 2‖x‖ω1ω1(t1), . . . , 2‖x‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt
uniformly converges on any bounded set of values of the parameter δ.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
308 V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV
Therefore
|Dα
ε x(u)−Dα
ε x(u+ δ)| → 0, |δ| → 0,
which proves the continuity of Dα
ε x(u) for all u ∈ Rm.
Thus, we obtain from (14): ∥∥Dα
ε x
∥∥
C
≤
≤ 2m−1Aα
∫
Rm+
min
{
2‖x‖C , ‖x‖ω1ω1(t1), . . . , ‖x‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt,
and inequality (6) is proved. Construct now the function f(t), which turns (6) into
equality. To this end, we will use the methods from [15]. Define f(u) for u ∈ Rm+ ,
and then extend it to the whole space Rm evenly with respect to each variable. For
u = (u1, . . . , um) ∈ Rm+ and δ > 0, set ωδj (uj) = min{ωj(uj), 2δ}.
Consider the vector ωδ(u) =
(
ωδ1(u1), . . . , ωδm(um)
)
and denote v = v(u) : =
: =
(
v1(u), . . . , vm(u)
)
=
(
ωδ(u)
)∗
, where (ωδ(u))∗ is the rearrangement of the
numbers ωδ1(u1), . . . , ωδm(um) in nonincreasing order. Now, define the function f(u)
by setting for u ∈ Rm+ ,
f(u) = v1(u)− v2(u) + . . .+ (−1)m−1vm(u)− δ.
Since 0 ≤
∑m
j=1
(−1)j−1vj(u) ≤ 2δ, we have ‖f‖C ≤ δ. Let us verify, that f ∈
∈
m⋂
j=1
Hj,ωj , and estimate ‖f‖ωj , j = 1, . . . ,m. For this purpose, consider the di-
fference
f(u+ hej)− f(u) =
= v1(u+ hej)− v2(u+ hej) + . . .+ (−1)m−1vm(u+ hej)−
−(v1(u)− v2(u) + . . .+ (−1)m−1vm(u)), h > 0.
The vector u+hej differs from the vector u in the j-th coordinate only, which is greater
then the j-th coordinate of the vector u. Therefore, v(u+ hej) differs from v(u) in the
following way. Let the number ωδj (uj) be the ν-th coordinate of vector v(u). Then there
exists µ ≤ ν such that ωδj (uj + hej) is the µ-th coordinate of v(u + hej). Moreover,
coordinates of v(u+ hej) coincide with coordinates of the vector v(u), as soon as they
have indices less than µ or greater than ν.
Thus,
f(u+ hej)− f(u) =
= (−1)µ−1ωδj (uj + h) + (−1)µvµ(u) + . . .+ (−1)ν−1vν−1(u)−
−(−1)µ−1vµ(u)− . . .− (−1)ν−2vν−1(u)− (−1)ν−1ωδj (uj) =
= (−1)µ−1ωδj (uj + h) + 2(−1)µvµ(u) + . . .
. . .+ 2(−1)ν−1vν−1(u)− (−1)ν−1ωδj (uj).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES . . . 309
Since
ωδj (uj + h) ≥ vµ(u) ≥ . . . ≥ vν−1(u) ≥ ωδj (uj),
it is easy to verify that
ωδj (uj)− ωδj (uj + h) ≤
≤ (−1)µ−1ωδj (uj + h) + 2(−1)µvµ(u) + . . .
. . .+ 2(−1)ν−1vν−1(u)− (−1)ν−1ωδj (uj) ≤
≤ ωδj (uj + h)− ωδj (uj). (15)
Taking into account that
ωδj (uj + h)− ωδj (uj) ≤ ωj(uj + h)− ωj(uj) ≤ ωj(h),
we obtain f ∈ Hj,ωj and ‖f‖ωj ≤ 1, j = 1, . . . ,m.
We now compute |(Dα
ε f)(0)| when ε = (+, . . . ,+). In order to do this, we firstly
show that for all t ∈ Rm+ ,
∆t1 . . .∆tmf(0, . . . , 0) = −2m−1 min
{
2δ, ω1(t1), . . . , ωm(tm)
}
=
= −2m−1 min
{
ωδ1(t1), . . . , ωδm(tm)
}
.
The proof is by induction on m. For m = 2 (induction basis), this fact can be verified
directly.
Since the operators ∆ti and ∆tj commute, we can take ∆t1 , . . . ,∆tm in any conveni-
ent order, while computing ∆t1 . . .∆tmf(0, . . . , 0). For definiteness, suppose that ωδ1(t1)
is the greatest number among ωδ1(t1), . . . , ωδm(tm). We will compute the difference in
t1 in last turn. Represent the difference ∆t1 . . .∆tmf(0, . . . , 0) in the following way:
∆t1 . . .∆tmf(0, . . . , 0) =
=
1∑
j1=0
1∑
j2=0
. . .
1∑
jm=0
(−1)j1+...+jmf(j1t1, j2t2, . . . , jmtm) =
=
1∑
j2=0
. . .
1∑
jm=0
(−1)j2+...+jmf(0, j2t2, . . . , jmtm) =
−
1∑
j2=0
. . .
1∑
jm=0
(−1)j2+...+jmf(t1, j2t2, . . . , jmtm).
By the induction assumption,
1∑
j2=0
. . .
1∑
jm=0
(−1)j2+...+jmf(0, j2t2, . . . , jmtm) =
= −2m−2 min
{
ωδ2(t2), . . . , ωδm(tm)
}
. (16)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
310 V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV
Using the fact that ωδ1(t1) is the greatest number among ωδ1(t1), . . . , ωδm(tm), and the
definition of f, we obtain
f(t1, j2t2, . . . , jmtm) = ωδ1(t1)− f(0, j2t2, . . . , jmtm).
Thus,
−
1∑
j2=0
. . .
1∑
jm=0
(−1)j2+...+jm
(
ωδ1(t1)− f(0, j2t2, . . . , jmtm)
)
=
= −2m−2 min
{
ωδ2(t2), . . . , ωδm(tm)
}
(here we have used the induction hypothesis (16) again). Finaly, we have
∆t1 . . .∆tmf(0, . . . , 0) =
= −2m−2 min
{
ωδ2(t2), . . . , ωδm(tm)
}
− 2m−2 min
{
ωδ2(t2), . . . , ωδm(tm)
}
=
= −2m−1 min
{
ωδ2(t2), . . . , ωδm(tm)
}
=
= −2m−1 min
{
ωδ1(t1), ωδ2(t2), . . . , ωδm(tm)
}
.
For ε = (+, . . . ,+), we estimate ‖Dα
ε f‖C from below:∥∥Dα
ε f
∥∥
C
≥
∣∣(Dα
ε f)(0, . . . , 0)
∣∣ =
= 2m−1Aα
∫
Rm+
m∏
j=1
t−αj−1∆tf(0, . . . , 0) dt =
= 2m−1Aα
∫
Rm+
min
{
2δ, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t−αj−1 dt ≥
≥ 2m−1Aα
∫
Rm+
min
{
2‖f‖C , ‖f‖ω1ω1(t1), . . . , ‖f‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt. (17)
Combining (6) (for the function f ) with (17), we see that
‖Dα
ε f‖C = 2m−1Aα
∫
Rm+
min
{
2δ, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t−αj−1 dt =
= 2m−1Aα
∫
Rm+
min
{
2‖f‖C , ‖f‖ω1ω1(t1), . . . , ‖f‖ωmωm(tm)
} m∏
j=1
t
−αj−1
j dt, (18)
i.e., relation (6) turns into equality.
The proof is complete.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES . . . 311
Proof of Corollary 1. It follows from equality (6) that for any δ > 0,
Ω
δ, m⋂
j=1
UHj,ωj
≤ 2m−1Aα
∫
Rm+
min
{
2δ, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt.
For the function f, constructed in proof of Theorem 1,
‖f‖C ≤ δ, f ∈
m⋂
j=1
Hj,ωj .
Using (18) we obtain
Ω
δ, m⋂
j=1
UHj,ωj
≥ ‖Dα
ε f‖C =
= 2m−1Aα
∫
Rm+
min
{
2δ, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt.
The corollary is proved.
Proof of Theorem 2. As in the proof of Theorem 1 suppose that ε = (+, . . . ,+).
Remind that for a given N > 0, the vector hN = (hN1 , . . . , h
N
m) ∈ Rm+ is defined by the
following conditions:
ω1(hN1 ) = . . . = ωm(hNm),
2mAα
α1 . . . αm
m∏
j=1
(hNj )−αj = N,
and
G(hN ) :=
{
u = (u1, . . . , um) ∈ Rm : u1 ≥ hN1 , . . . , um ≥ hNm
}
.
Define the operator BhN as follows
BhNx(u) = Aα
∫
G(hN )
∆tx(u)
m∏
j=1
t
−αj−1
j dt.
Show that BhN is the bounded operator from C(Rm) to C(Rm), and moreover
‖BhN ‖ ≤ N. Indeed for all x ∈ C(Rm),
∥∥BhNx∥∥C ≤ 2mAα
∫
G(hN )
m∏
j=1
t
−αj−1
j dt‖x‖C =
=
2mAα
α1 . . . αm
m∏
j=1
(
hNj
)−αj ‖x‖C = N‖x‖C .
For any x ∈
m⋂
j=1
Hj,ωj , estimate the deviation ‖Dα
ε x−BhNx‖C . We have
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
312 V. F. BABENKO, N. V. PARFINOVYCH, S. A. PICHUGOV
∥∥Dα
ε x−BhNx
∥∥
C
≤
∥∥∥∥∥∥∥Aα
∫
Rm+ \G(hN )
∆tx(u)
m∏
j=1
t
−αj−1
j dt
∥∥∥∥∥∥∥
C
≤
≤ 2m−1Aα
∫
Rm+ \G(hN )
min
{
ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt.
We have obtained the estimation of the value EN
(
Dα
ε ,
m⋂
j=1
UHj,ωj
)
from above.
Let us estimate this value from below. From (10) we obtain
EN
Dα
ε ,
m⋂
j=1
UHj,ωj
≥ sup
δ>0
Ω
δ, m⋂
j=1
UHj,ωj
−Nδ
. (19)
Using Corollary 1 and condition (11) we have
EN
Dα
ε ,
m⋂
j=1
UHj,ωj
≥
≥ sup
δ>0
2m−1Aα
∫
Rm+
min
{
2δ, ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt −
−2mAαδ
∫
G(hN )
m∏
j=1
t
−αj−1
j dt
. (20)
Set
δN = ω1(hN1 ) = . . . = ωm(hNm).
Note that for t ∈ G(hN ),
min
{
2δN , ω1(t1), . . . , ω(tm)
}
= 2δN
and for t ∈ Rm+ \G(hN ),
min
{
2δN , ω1(t1), . . . , ω(tm)
}
= min
{
ω1(t1), . . . , ω(tm)
}
.
From (20) we derive
EN
Dα
ε ,
m⋂
j=1
UHj,ωj
≥
≥ 2m−1Aα
∫
Rm+
min
{
2δN , ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt−
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES . . . 313
−2m−1Aα
∫
G(hN )
min
{
2δN , ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt =
= 2m−1Aα
∫
Rm+ \G(hN )
min
{
ω1(t1), . . . , ωm(tm)
} m∏
j=1
t
−αj−1
j dt.
We have obtained the required estimation from below.
Theorem 2 is proved.
Proof of Theorem 3. Let us consider the case ε = (+, . . . ,+). For δ > 0
and modulus of continuity ω1, . . . , ωm, by f( · ; δ;ω1, . . . , ωm) denote the function
f constructed in the proof of Theorem 1. Suppose the inequality (12) holds true and
select 0 < L0 ≤M0 such that
Mα = 2m−1Aα
∫
Rm+
min
{
2L0,Mω1ω1(t1), . . . ,Mωmωm(tm)
} m∏
j=1
t
−αj−1
j dt.
For the function f(· ;L0;Mω1ω1, . . . ,Mωmωm), we have∥∥f(· ;L0;Mω1ω1, . . . ,Mωmωm)
∥∥
C
≤ L0 ≤M0.
In addition, it is easy to verify that∥∥f(· ;L0;Mω1ω1, . . . ,Mωmωm)
∥∥
ωj
= Mωj , j = 1, . . . ,m.
As in the proof of Theorem 1, we obtain∥∥Dα
ε f(· ;L0;Mω1ω1, . . . ,Mωmωm)
∥∥
C
=
= 2m−1Aα
∫
Rm+
min
{
2L0,Mω1ω1(t1), . . . ,Mωmωm(tm)
} m∏
j=1
t
−αj−1
j dt = Mα.
Now construct the function
x(u) = f(u;L0;Mω1ω1, . . . ,Mωmωm) +M0 −
∥∥f(· ;L0;Mω1ω1, . . . ,Mωmωm)
∥∥
C
.
It is obvious that x ∈
m⋂
j=1
Hj,ωj , and also
‖x‖C = M0, ‖Dα
ε x‖C = Mα, ‖x‖ωj = Mωj , j = 1, . . . ,m.
Theorem 3 is proved.
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Received 14.12.09
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
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| id | umjimathkievua-article-2869 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:54Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e8/8a339d1cffc6c325f8b2e7eafda59ae8.pdf |
| spelling | umjimathkievua-article-28692020-03-18T19:39:19Z Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions Точні нерівності типу колмогорова для норм дробових похідних функцій багатьох змінних Babenko, V. F. Parfinovych, N. V. Pichugov, S. A. Бабенко, В. Ф. Парфінович, Н. В. Пічугов, С. О. Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented. Нехай $C(\mathbb{R}^m)$ — простори неперервних обмежених функцій $x: \mathbb{R}^m → \mathbb{R}$ з нормами $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$, $e_j,\; j = 1,…,m$ — звичайна база в $\mathbb{R}^m$. Для заданих модулів неперервності $ω_j,\; j = 1,…, m$, позначимо $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ У роботі отримано нові точні нерівності типу Колмогорова для норм мішаних частинних похідних $∥D^{α}_{ε}x∥_C$ функцій $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Наведені деякі застосування цих нерівностей. Institute of Mathematics, NAS of Ukraine 2010-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2869 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 3 (2010); 301–314 Український математичний журнал; Том 62 № 3 (2010); 301–314 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2869/2489 https://umj.imath.kiev.ua/index.php/umj/article/view/2869/2490 Copyright (c) 2010 Babenko V. F.; Parfinovych N. V.; Pichugov S. A. |
| spellingShingle | Babenko, V. F. Parfinovych, N. V. Pichugov, S. A. Бабенко, В. Ф. Парфінович, Н. В. Пічугов, С. О. Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| title | Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| title_alt | Точні нерівності типу колмогорова для норм дробових похідних функцій багатьох змінних |
| title_full | Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| title_fullStr | Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| title_full_unstemmed | Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| title_short | Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| title_sort | sharp kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2869 |
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