On Kolmogorov-type inequalities for fractional derivatives of functions of two variables
We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order.
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| Дата: | 2008 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3200 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509248333545472 |
|---|---|
| author | Babenko, V. F. Pichugov, S. A. Бабенко, В. Ф. Пичугов, С. А. Бабенко, В. Ф. Пичугов, С. А. |
| author_facet | Babenko, V. F. Pichugov, S. A. Бабенко, В. Ф. Пичугов, С. А. Бабенко, В. Ф. Пичугов, С. А. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:06Z |
| description | We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order. |
| first_indexed | 2026-03-24T02:38:05Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.5
V. F. Babenko
(Dnepropetr. nac. un-t; Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
S. A. Pyçuhov (Dnepropetr. nac. un-t ynΩenerov Ω.-d. transp.; Dnepropetr. nac. un-t)
O NERAVENSTVAX TYPA KOLMOHOROVA
DLQ DROBNÁX PROYZVODNÁX FUNKCYJ
DVUX PEREMENNÁX
A new sharp inequality of the Kolmogorov type is proved that estimates the norm of a mixed derivative
of fractional order (in the Marchaud sense) of a function of two variables with the help of the norm of
function itself and norms of its first-order partial derivatives.
Dovedeno novu toçnu nerivnist\ typu Kolmohorova, qka ocing[ normu mißano] poxidno] drobovo-
ho porqdku (v sensi Marßo) funkci] dvox zminnyx çerez normu samo] funkci] i normy ]] çastyn-
nyx poxidnyx perßoho porqdku.
Vo mnohyx voprosax analyza voznykaet neobxodymost\ rassmatryvat\ proyzvod-
n¥e y yntehral¥ drobnoho porqdka (sm., naprymer, [1]). Klassyçeskoe oprede-
lenye Lyuvyllq proyzvodn¥x drobnoho porqdka α ∈( , )0 1 funkcyy x u( ) ,
u ∈R , zadannoj na vsej dejstvytel\noj osy, takovo [1, c. 43]:
D x u+( )α ( ) : = 1
1Γ( – )
( )
( – )
–
α α
d
du
x t
u t
dt
u
∞
∫ ,
D x u– ( )α( ) : = –
( – )
( )
( – )
1
1Γ α α
d
du
x t
u t
dt
u
∞
∫ .
Ne menee vaΩn¥, a v rqde voprosov bolee udobn¥, proyzvodn¥e v sm¥sle Marßo
[2] (sm. takΩe [1, c. 95 – 97])
D x u±( )α ( ) : =
α
α αΓ( – )
( ) – ( )
1 1
0
x u x u t
t
dt
∓
+
∞
∫ . (1)
NyΩe dlq sokrawenyq zapysej budem polahat\ Aα = α
αΓ( – )1
.
Yzvestno (sm., naprymer, [1, c. 96]), çto dlq dostatoçno xoroßyx funkcyj
znaçenyq πtyx proyzvodn¥x sovpadagt:
D ±( )α x u( ) = D x u±( )α ( ) . (2)
Dlq v¥polnenyq πtoho ravenstva dostatoçno, naprymer, çtob¥ funkcyq x u( )
b¥la lokal\no absolgtno neprer¥vnoj na R (x ∈ ACloc( )R ).
Dlq δ > 0 rassmotrym zadaçu
D ± ∞
α x → sup; x ∈ ACloc( )R , x ∞ ≤ δ , ′ ≤∞x 1, (3)
hde
x ∞ : = ess sup ( ) :x u u ∈{ }R .
Zadaça (3) qvlqetsq çastn¥m sluçaem obwej zadaçy o toçnoj ocenke norm¥
© V. F. BABENKO, S. A. PYÇUHOV, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 837
838 V. F. BABENKO, S. A. PYÇUHOV
„promeΩutoçnoj” proyzvodnoj pry yzvestn¥x ocenkax norm funkcyy y ee pro-
yzvodnoj bolee v¥sokoho porqdka (sm., naprymer, [3], § 1.7), kotoraq aktyvno
yzuçalas\ mnohymy matematykamy. Obzor¥ yzvestn¥x toçn¥x rezul\tatov po ee
reßenyg v sluçae proyzvodn¥x celoho porqdka y dal\nejßye ss¥lky moΩno
najty, naprymer, v [3 – 5]. Sluçaj proyzvodn¥x drobnoho porqdka menee yssle-
dovan.
Zametym, çto dlq rassmatryvaemoho klassa funkcyj v sylu uslovyq x ∈
∈ ACloc( )R v¥polnqgtsq ravenstva (2).
Reßenye zadaçy (3) soderΩytsq v sledugwem toçnom neravenstve typa Kol-
mohorova (sm. [6]):
D ± ∞
α x ≤ 1
1
2
1
1
1
Γ( – ) –
–
–
α α
α
α αx x∞ ∞′ ,
pryçem dannoe neravenstvo obrawaetsq v ravenstvo dlq funkcyy x u( ) , kotoraq
opredelqetsq sledugwym obrazom: x u( ) = u – 1 / 2, esly u ≤ 1, y x u( ) =
= 1 / 2, esly u ≥ 1. Yz pryvedennoho neravenstva poluçaem
sup
( )
,
x AC
x x
x
∈
≤ ′ ≤
± ∞
∞ ∞
R
D
δ
α
1
= 1
1
2
1
1
Γ( – )
( )
–
–
α
δ
α
α
.
Druhye yzvestn¥e toçn¥e neravenstva typa Kolmohorova dlq drobn¥x pro-
yzvodn¥x sm. v rabotax [7 – 10].
Otmetym, çto v sluçae funkcyj dvux y bolee peremenn¥x toçn¥x neravenstv
typa Kolmohorova yzvestno nemnoho (sm. [11 – 15]).
V dannoj stat\e rassmotrym analoh zadaçy (3) dlq funkcyj dvux peremen-
n¥x x u( ) , u = ( , )u u1 2 , zadann¥x na R
2
.
Pust\ α = ( , )α α1 2 , α1, α2 ∈ (0, 1);
∆t x u u
1 1 2( , ) : = x u u( , )1 2 – x u t u( , )1 1 2+ ,
∆t x u u
2 1 2( , ) : = x u u( , )1 2 – x u u t( , )1 2 2+ ,
∆ ∆t t x u u
1 2 1 2( , ) : = x u u( , )1 2 – x u t u( , )1 1 2+ – x u u t( , )1 2 2+ + x u t u t( , )1 1 2 2+ + ,
ε = ( , )ε ε1 2 , εi = ±,
x ∞ : = esssup ( ) :x u u ∈{ }R
2 .
Smeßann¥e proyzvodn¥e Marßo D ε
α x porqdka α opredelqgtsq ravenstvom
(sm. [1, c. 347] )
D ε
α( ) x u( ) : = A
x u u
t t
dt dt
t t
α
ε ε
α α
∆ ∆– – ( , )
1 1 2 2
1 2
1 2
1
1
2
1
00
1 2+ +
∞∞
∫∫ ,
hde Aα = A Aα α1 2
.
Pust\ ACloc( )R
2
— klass funkcyj x u( ) , u = ( , )u u1 2 ∈ R2
, takyx, çto pry
lgbom fyksyrovannom znaçenyy odnoj peremennoj poluçaemaq funkcyq
druhoj peremennoj qvlqetsq lokal\no absolgtno neprer¥vnoj. Dlq funkcyj
x ∈ ACloc( )R
2
pry poçty vsex ( , )u u1 2 ∈ R2
suwestvugt çastn¥e proyzvodn¥e
D x u u( , ) ( , )1 0
1 2 = ∂
∂
x
u
u u
1
1 2( , ) y D x u u( , ) ( , )0 1
1 2 = ∂
∂
x
u
u u
2
1 2( , ).
Toçnaq formulyrovka zadaçy takova:
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
O NERAVENSTVAX TYPA KOLMOHOROVA DLQ DROBNÁX PROYZVODNÁX … 839
D ε
α x
∞
→ sup, x ∈ ACloc( )R
2 , x ∞ ≤ δ , D x( , )1 0
∞
≤ 1, D x( , )0 1
∞
≤ 1.
(4)
Dlq reßenyq πtoj zadaçy dokaΩem toçnoe neravenstvo typa Kolmohorova
dlq norm D ε
α x
∞
, x ∞ , D x( , )1 0
∞
, D x( , )0 1
∞
.
Teorema. Pust\ x u( ) ∈ ACloc( )R
2
, x ∞ < ∞, D x( , )1 0
∞
< ∞ , D x( , )0 1
∞
<
< ∞, D x( , )1 0
∞
D x( , )0 1
∞
≠ 0, α = ( , )α α1 2 , α1, α2 ∈ (0, 1), α 1 + α2 < 1. Tohda
ymeet mesto toçnoe neravenstvo
D ε
α x
∞
≤ 2
1 1
2
11 2
1
1 2
1 2
Γ Γ( – ) ( – ) – ( )
– ( )
α α α α
α α+
+
x D D∞
+
∞ ∞
1 1 0 0 11 2 1 2– ( ) ( , ) ( , )α α α α
.
(5)
Dokazatel\stvo. Ymeem
D ε
α x
∞
≤ A
x
t t
dt dt
t t
α
ε ε
α α
∆ ∆– – ( , )
1 1 2 2
1 2
1
1
2
1
00
1 2
⋅ ⋅
∞
+ +
∞∞
∫∫ . (6)
Dlq ocenky norm¥ smeßannoj raznosty yspol\zuem neravenstva (t1, t2 > 0)
∆ ∆– – ( , )ε ε1 1 2 2t t x ⋅ ⋅
∞
≤ 22 x ∞ ,
∆ ∆– – ( , )ε ε1 1 2 2t t x ⋅ ⋅
∞
≤ 2
1 1
∆– ( , )ε t x ⋅ ⋅
∞
≤ 2 1
1 0t D x( , )
∞
,
∆ ∆– – ( , )ε ε1 1 2 2t t x ⋅ ⋅
∞
≤ 2
2 2
∆– ( , )ε t x ⋅ ⋅
∞
≤ 2 2
0 1t D x( , )
∞
.
Obæedynqq πty ocenky, poluçaem
∆ ∆– – ( , )ε ε1 1 2 2t t x ⋅ ⋅
∞
≤ 2 2 1
1 0
2
0 1min , ,( , ) ( , )x t D x t D x∞ ∞ ∞{ }. (7)
Prymenqq (7) v (6), poluçaem
D ε
α x u( )
∞
≤ 2
2 1
1 0
2
0 1
1
1
2
1
00
1 21 2
A
x t D x t D x
t t
dt dtα α α
min , ,( , ) ( , )
∞ ∞ ∞
+ +
∞∞ { }
∫∫ = : 2A Iα .
(8)
Dlq proyzvol\n¥x h1, h2 > 0 razob\em oblast\ yntehryrovanyq R+
2
v (8) na try
çasty:
R+
2 = U1 ∪ U2 ∪ U3,
U1 = t h t h t t h∈ ≤ ≤{ }+R
2
2 1 1 2 1 1: , ,
U2 = t h t h t t h∈ ≤ ≤{ }+R
2
1 2 2 1 2 2: , ,
U3 = t t h t h∈ ≥ ≥{ }+R
2
1 1 2 2: , .
Poparn¥e pereseçenyq πtyx çastej ymegt nulevug ploskug meru Lebeha, tak
çto
I = I1 + I2 + I3,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
840 V. F. BABENKO, S. A. PYÇUHOV
hde Ik — yntehral v (8) po oblasty Uk, k = 1, 2, 3.
Ocenym yntehral I2:
I2 ≤
t D x
t t
dt dt
U
2
0 1
1
1
2
1 1 21 2
2
( , )
∞
+ +∫∫ α α = D x t t dt dt
h
h
h
t
( , ) – – –0 1
2
0
1
1
1 2
2
2
1
1
2
2
∞
∞
∫ ∫α α =
=
D x h
h
t dt
h( , ) –
– –
0 1
1
1
2
2
0
2
1
1 2
2
∞
∫α
α
α α
=
D x h
h
( , ) –
– ( )
0 1
1 1 2
2
1
11
2
1
∞
+( )α α α
α
α .
Analohyçno
I1 ≤
D x h
h
( , ) –
– ( )
1 0
2 1 2
1
1
21
1
2
∞
+( )α α α
α
α .
Yntehral I3 ocenyvaetsq tak:
I3 ≤
2
1
1
2
1 1 21 2
21
x
t t
dt dt
hh
∞
+ +
∞∞
∫∫ α α =
2 1
1 2 1 2
1 2
x
h h
∞
α α α α .
V rezul\tate dlq proyzvol\n¥x h1, h2 > 0 poluçaem addytyvnoe neravenstvo
D ε
α x u( )
∞
≤
≤ 2
2
1 2
1 2
1 2A
x
h hα
α α
α α
∞
– –
+
D x h
h
D x h
h
( , ) – ( , ) –
– ( ) – ( )
0 1
1 1 2
2
1
1
1 0
2 1 2
1
1
21 1
2
1
1
2
∞ ∞
+( )
+
+( )
α α α α α α
α
α
α
α .
(9)
Polahaq v pravoj çasty neravenstva (9)
h1 = 2 1 0
x
D x
∞
∞
( , ) , h2 = 2 0 1
x
D x
∞
∞
( , ) ,
ymeem
D ε
α x u( )
∞
≤ 2 1 1 0 0 11 2 1 2A x D x D xα
α α α α
∞
+
∞ ∞
– ( ) ( , ) ( , ) ×
× 2
2
2
1
2
1
1 2
1
2 1 2
1
1 1 2
1 2
1 2 1 2
α α α α α α α αα α
α α α α
+ +
+( )
+
+( )
– – – –
– ( ) – ( )
,
çto sovpadaet s (5).
Teper\ postroym funkcyg f u u( , )1 2 , kotoraq obrawaet (5) v ravenstvo.
Snaçala opredelym f u u( , )1 2 dlq ( , )u u1 2 ∈ R+
2
. PoloΩym
f u u( , )1 2 = u2 – u1 + 1
2
, esly u2 ≤ u1 ≤ 1,
f u u( , )1 2 = u2 – 1
2
, esly u2 ≤ 1 ≤ u1,
f u u( , )1 2 = u1 – u2 + 1
2
, esly u1 ≤ u2 ≤ 1,
f u u( , )1 2 = u1 – 1
2
, esly u1 ≤ 1 ≤ u2,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
O NERAVENSTVAX TYPA KOLMOHOROVA DLQ DROBNÁX PROYZVODNÁX … 841
f u u( , )1 2 = 1
2
, esly u1 > 1, u2 > 1.
ProdolΩym f u u( , )1 2 na vsg ploskost\ R2
çetn¥m obrazom po kaΩdoj pere-
mennoj.
Oçevydno, çto
f ∞ = 1
2
, D f( , )1 0
∞
= D f( , )0 1
∞
= 1. (10)
Krome toho, dlq smeßannoj raznosty v nule pry t1, t2 > 0 spravedlyv¥ raven-
stva
∆ ∆t t f
1 2
0 0( , ) = 2 2t , esly t t2 1 1≤ { }min , ,
∆ ∆t t f
1 2
0 0( , ) = 2 1t , esly t t1 2 1≤ { }min , ,
∆ ∆t t f
1 2
0 0( , ) = 2, esly t1 ≥ 1, t2 ≥ 1.
Dlq ε = ( – , – ) ocenym D ε
α f u( )
∞
snyzu:
D ε
α f u( )
∞
≥ D ε
α( ) f ( , )0 0 =
= A
f
t t
dt dt
t tt tt t
t t
α α α+ +
≥ ≥≤ { }≤ { }
+ +∫∫∫∫∫∫
1 21 22 1
1 2
1 2
1 111 1
1
2
1 1 2
0 0
,min ,min ,
( , )∆ ∆
=
= 2 1 1
1
1
11 2 2 1 2 1 1 2
Aα α α α α α α α α
+
+( )
+
+( )
– ( ) – ( )
=
2
11 2 1 2
Aα
α α α α– ( )+( )
.
Uçyt¥vaq (10) y pryvedennoe v¥ße v¥raΩenye dlq D ε
α f ( , )0 0 , poluçaem
D ε
α
α α α α
f
f D f D f
∞
∞ ∞ ∞
1 1 0 0 11 2 1 2– ( – ) ( , ) ( , )
≥ 2
1 1
2
11 2
1
1 2
1 2
Γ Γ( – ) ( – ) – ( )
– ( )
α α α α
α α+
+
,
otkuda y sleduet toçnost\ neravenstva (5).
Analohyçn¥m obrazom stroqtsq πkstremal\n¥e funkcyy y dlq druhyx zna-
çenyj vektora ε = ( , )ε ε1 2 .
Teorema dokazana.
Yz teorem¥ v¥vodym takoe sledstvye.
Sledstvye. V uslovyqx teorem¥Q1
sup ( )
loc
( , ) ( , )
( ), ,
,
x AC x
D x D x
x
∈ ≤
≤ ≤
∞
∞
∞ ∞
R
D
2
1 0 0 11 1
δ
ε
α =
2 2
1 1 1
1
1 2 1 2
1 2⋅
+( )
+
( )
( – ) ( – ) – ( )
– ( )δ
α α α α
α α
Γ Γ
.
Zameçanyq. 1. M¥ poluçyly mul\typlykatyvnoe neravenstvo (5) yz addy-
tyvnoho neravenstva (9). Lehko vydet\, çto verno y obratnoe — yz neraven-
stvaQ(5) sleduet neravenstvo (9) pry vsex znaçenyqx h1, h2 > 0.
2. Dlq v¥polnenyq neravenstva (5) dlq vsex funkcyj yz dannoho klassa po-
kazately stepenej pry normax x ∞ , D x( , )1 0
∞
, D x( , )0 1
∞
edynstvenno voz-
moΩn¥e. V πtom lehko ubedyt\sq, rassmatryvaq narqdu s funkcyej x u u( , )1 2
semejstvo funkcyj vyda Cx u u( , )δ δ1 1 2 2 , C ∈R , δ1, δ2 ∈ +R .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
842 V. F. BABENKO, S. A. PYÇUHOV
3. Uslovye α1 + α2 < 1 teorem¥ opustyt\ nel\zq, tak kak dlq postroennoj
funkcyy f u u( , )1 2 pry α1 + α2 = 1 poluçaem
D ε
α f ( , )0 0 = ∞.
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Poluçeno 26.06.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
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| id | umjimathkievua-article-3200 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:05Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/57/24034ed3e0249751918cf5ee267f9c57.pdf |
| spelling | umjimathkievua-article-32002020-03-18T19:48:06Z On Kolmogorov-type inequalities for fractional derivatives of functions of two variables О неравенствах типа Колмогорова для дробных производных функций двух переменных Babenko, V. F. Pichugov, S. A. Бабенко, В. Ф. Пичугов, С. А. Бабенко, В. Ф. Пичугов, С. А. We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order. Доведено нову точну нерівність типу Колмогорова, яка оцінює норму мішаної похідної дробового порядку (в сенсі Маршо) функції двох змінних через норму самої функції і норми її частинних похідних першого порядку. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3200 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 837–842 Український математичний журнал; Том 60 № 6 (2008); 837–842 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3200/3146 https://umj.imath.kiev.ua/index.php/umj/article/view/3200/3147 Copyright (c) 2008 Babenko V. F.; Pichugov S. A. |
| spellingShingle | Babenko, V. F. Pichugov, S. A. Бабенко, В. Ф. Пичугов, С. А. Бабенко, В. Ф. Пичугов, С. А. On Kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| title | On Kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| title_alt | О неравенствах типа Колмогорова для дробных производных функций двух переменных |
| title_full | On Kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| title_fullStr | On Kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| title_full_unstemmed | On Kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| title_short | On Kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| title_sort | on kolmogorov-type inequalities for fractional derivatives of functions of two variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3200 |
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