Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables
We investigate the correlation between the constants $K(ℝ^n)$ and $K(T^n)$, where $$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \ri...
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| Datum: | 2006 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509576281980928 |
|---|---|
| author | Babenko, V. F. Churilova, M. S. Бабенко, В. Ф. Чурилова, М. С. Бабенко, В. Ф. Чурилова, М. С. |
| author_facet | Babenko, V. F. Churilova, M. S. Бабенко, В. Ф. Чурилова, М. С. Бабенко, В. Ф. Чурилова, М. С. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:42Z |
| description | We investigate the correlation between the constants $K(ℝ^n)$ and $K(T^n)$, where
$$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$
is the exact constant in a Kolmogorov-type inequality, $ℝ$ is the real straight line, $T = [0,2π],\; L^l_{p, p} (G^n)$ is the set of functions $ƒ ∈ L_p (G^n)$ such that the partial derivative $D_i^{l_i } f(x)$ belongs to $L_p (G^n), i = \overline {1,n}, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^{α} f$ is the mixed derivative of a function $ƒ, 0 < µi < 1, i = \overline {0,n},$
and $∑_{i=0}^n µ_i = 1$.
If $G^n = ℝ$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i),\; µ_i = α_i/l_i,\; i = \overline {1,n}$
if $G^n = T^n$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i) − ∑_{i=0}^n (λ/l_i),\; µ_i = α_i/ l_i + λ/l_i , i=
\overline {1,n},\; λ ≥ 0$.
We prove that, for $λ = 0$, the equality $K(ℝ^n) = K(T^n)$ is true. |
| first_indexed | 2026-03-24T02:43:18Z |
| format | Article |
| fulltext |
UDK 517.5
V. F. Babenko, M. S. Çurylova (Dnepropetr. nac. un-t)
SRAVNENYE TOÇNÁX KONSTANT
V NERAVENSTVAX TYPA KOLMOHOROVA
DLQ PERYODYÇESKYX Y NEPERYODYÇESKYX
FUNKCYJ MNOHYX PEREMENNÁX
We investigate the correlation between the constants K n( )R and K n( )T , where
K G
D f
f D f
n
f L G
D f
L G
L G i
n
i
l
L Gp p
l n
i
n
i
li
Lp G
n
p
n
p
n
i
p
n
i
( ) : sup
,
( )
( )
( )
( ) ( )
=
∈
∏ ≠
=
=
∏
1
0
0
1
α
µ µ
is the exact constant in the Kolmogorov-type inequality; R — is the real line, T = [ , ]0 2π ; L Gp p
l n
, ( ) is
a set of functions f L Gp
n∈ ( ) such that the partial derivative D f xi
li ( ) belongs to L Gp
n( ), i n= 1, ,
1 ≤ ≤ ∞p , l n∈N , α ∈ =N N0 0n n( { })∪ , D fα is a mixed derivative of the function f ; 0 1< <µi ,
i n= 0, , µii
n
=∑ =
0
1. If Gn n= R , then µ α0 1
1= −
=∑ ( / )i ii
n
l , µ αi i il= / , i n= 1, ; if Gn n= T ,
then µ0 1= – ( / ) ( / )α λi ii
n
ii
n
l l
= =∑ ∑−
1 1
, µ α λi i i il l= +/ / , i n= 1, , λ ≥ 0 . We prove that if λ = 0 ,
then the equality K Kn n( ) ( )R T= is true.
DoslidΩu[t\sq vza[mozv’qzok miΩ konstantamy K n( )R ta K n( )T , de
K G
D f
f D f
n
f L G
D f
L G
L G i
n
i
l
L Gp p
l n
i
n
i
li
Lp G
n
p
n
p
n
i
p
n
i
( ) : sup
,
( )
( )
( )
( ) ( )
=
∈
∏ ≠
=
=
∏
1
0
0
1
α
µ µ
— toçna konstanta v nerivnosti typu Kolmohorova; R — dijsna prqma, T = [ , ]0 2π ; L Gp p
l n
, ( )
— mnoΩyna funkcij f L Gp
n∈ ( ) takyx, wo çastynna poxidna D f x L Gi
l
p
ni ( ) ( )∈ , i n= 1, ,
1 ≤ ≤ ∞p , l n∈N , α ∈ =N N0 0n n( { })∪ , D fα
— mißana poxidna funkci] f ; 0 1< <µi ,
i n= 0, , µii
n
=∑ =
0
1. Qkwo Gn n= R , to µ α0 1
1= −
=∑ ( / )i ii
n
l , µ αi i il= / , i n= 1, ; qkwo
Gn n= T , to µ0 1= – ( / ) ( / )α λi ii
n
ii
n
l l
= =∑ ∑−
1 1
, µ α λi i i il l= +/ / , i n= 1, , λ ≥ 0 . Dovedeno, wo
pry λ = 0 spravdΩu[t\sq rivnist\ K Kn n( ) ( )R T= .
Pust\ 1 ≤ p ≤ ∞ , Lp
n( )R ( )R R
1 = — prostranstvo yzmerym¥x funkcyj f :
R
n → R s koneçnoj Lp-normoj
f Lp
n( )R
: = R
R
n
n
f x dx p
f x p
p p
x
∫( ) ≤ < ∞
= ∞
∈
( ) , ,
sup ( ) , .
/1
1
vrai
Opredelym takΩe prostranstvo Lp
n( )T ( )[ , ]T T
1 0 2= = π yzmerym¥x funk-
cyj f : R
n → R, 2π-peryodyçeskyx po kaΩdoj peremennoj, s koneçnoj normoj
© V. F. BABENKO, M. S. ÇURYLOVA, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5 597
598 V. F. BABENKO, M. S. ÇURYLOVA
f Lp
n( )T
: = T
T
n
n
f x dx p
f x p
p p
x
∫( ) ≤ < ∞
= ∞
∈
( ) , ,
sup ( ) , .
/1
1
vrai
Pust\ r n∈N , s n∈ ∞[ , ]1 , G est\ R yly T. Pry n = 1 çerez L Gs
r ( ) obo-
znaçym mnoΩestvo funkcyj f : R → R , ymegwyx poçty vsgdu lokal\no abso-
lgtno neprer¥vn¥e proyzvodn¥e f i( ) , i = 0 1, r − ( )( )f f0 = , y takyx, çto
f L Gr
s
( ) ( )∈ . Pust\ teper\ n ≥ 2. Voz\mem çastnug proyzvodnug D f xi
ki ( )
funkcyy f po i-j peremennoj y zafyksyruem vse peremenn¥e, krome i-j:
D f xi
ki ( ) = D f x yi
k
i
i ( , ), y n∈ −
R
1. Çerez L Gs
r n
i
i ( ) oboznaçym mnoΩestvo funk-
cyj f : R
n → R takyx, çto çastn¥e proyzvodn¥e D fi
ki , k i = 0 1, ri −
( )D f fi
0 = , lokal\no absolgtno neprer¥vn¥ kak funkcyy xi poçty dlq vsex
dopustym¥x y, a D f L Gi
r
s
ni
i
∈ ( ) ,
L G L Gs
r n
i
n
s
r n
i
i( ) ( )= =1∩ , y poloΩym L Gp s
r n
, ( ) =
= L G L Gp
n
s
r n( ) ( )∩ ∀ n ∈ N .
Kak ob¥çno, çerez C n∞( )R oboznaçym prostranstvo beskoneçno dyfferen-
cyruem¥x na R
n
funkcyj, a çerez C n
0
∞( )R — beskoneçno dyfferencyruem¥x
fynytn¥x na R
n.
VaΩnug rol\ vo mnohyx voprosax analyza yhragt neravenstva dlq norm
promeΩutoçn¥x proyzvodn¥x funkcyj f L Gp s
r∈ , ( ) vyda
f k
L Gq
( )
( )
≤ K f fL G
r
L Gq s
( )
( )
( )
µ µ1−
, (1)
hde k, r ∈ N0 = N ∪ { 0 } , k < r, µ ∈ ( 0, 1 ) , s neuluçßaem¥my konstantamy
K = K ( G ) = K G q p sk r, ( ; , , ; )µ = sup
,
( )
( )
( )
( )
( )
( )
( )
f L G
f
k
L G
L G
r
L Gp s
r
r
q
p s
f
f f∈
≠
−
0
1µ µ . (2)
Yssledovanyq mnohyx matematykov b¥ly posvqwen¥ v¥çyslenyg konstant
(2) pry razlyçn¥x znaçenyqx parametrov G, k, r, q, p, s, µ . Odnym yz perv¥x
y naybolee qrkyx rezul\tatov v πtom napravlenyy qvlqetsq sledugwee
neravenstvo A. N. Kolmohorova [1]:
pry vsex k, r ∈ N , k < r, dlq lgboj funkcyy f Lr∈ ∞ ∞, ( )R
f k
L
( )
( )∞ R
≤
ϕ
ϕ
r k L
r L
k r L
k r r
L
k r
f f
−
−
−∞
∞
∞ ∞
( )
( )
/ ( )
/ ( )
( )
/R
R
R R1
1 ,
hde ϕr — r-j peryodyçeskyj yntehral, ymegwyj nulevoe srednee znaçenye na
peryode, ot funkcyy ϕ0 ( t ) = sgn sin t . Poπtomu za neravenstvamy vyda (1) za-
krepylos\ nazvanye „neravenstva typa Kolmohorova”. Obzor¥ druhyx rezul\ta-
tov v πtom napravlenyy dlq G = R, R + , T sm., naprymer, v [2 – 7].
Neobxodym¥e y dostatoçn¥e uslovyq suwestvovanyq neravenstva (1) pry G =
= R sostoqt v tom, çto [8]
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
SRAVNENYE TOÇNÁX KONSTANT V NERAVENSTVAX TYPA KOLMOHOROVA … 599
r k
p
k
s
− + ≥
r
q
, (3)
y pry πtom neobxodymo dolΩno b¥t\
µ =
r k s q
r s p
− − +
− +
1 1
1 1
/ /
/ /
.
V peryodyçeskom sluçae, kak dokazano v [9], neravenstvo (1) v¥polnqetsq dlq
lgboho f Lp s
r∈ , ( )T , 1 ≤ p, s ≤ ∞ , k, r ∈ N , k < r, esly y tol\ko esly
µ ≤ µcr : = min , / /
/ /
1
1 1
1 1
− − − +
− +
k
r
r k s q
r s p
, (4)
pryçem naybol\ßyj ynteres predstavlqgt neravenstva (1) s µ = µcr .
V rabote [10] ustanovlen¥ sootnoßenyq meΩdu toçn¥my konstantamy (2) v
neravenstvax typa (1) dlq peryodyçeskyx y neperyodyçeskyx funkcyj, zadan-
n¥x na vewestvennoj osy. A ymenno, dokazano, çto
K ( R ) ≤ K ( T ) pry
r k s q
r s p
− − +
− +
1 1
1 1
/ /
/ /
< 1 − k
r
, (5)
K ( R ) = K ( T ) pry
r k s q
r s p
− − +
− +
1 1
1 1
/ /
/ /
= 1 − k
r
. (6)
UtverΩdenye (6) pozvolqet poluçyt\ nekotor¥e nov¥e rezul\tat¥ na osy,
poskol\ku k nastoqwemu vremeny vopros o toçn¥x konstantax (2) v neravenst-
vax (1) dlq peryodyçeskyx funkcyj yzuçen polnee, çem dlq neperyodyçeskyx.
V sluçae funkcyj mnohyx peremenn¥x est\ mnoho razlyçn¥x typov nera-
venstv dlq norm promeΩutoçn¥x proyzvodn¥x, y vopros¥ suwestvovanyq takyx
neravenstv rassmatryvalys\ v rabotax V. P. Yl\yna, ∏rmynha, Nyrenberha,
Hal\qrdo, V.LN. Habußyna, A.LF. Tymana, V.LA. Solonnykova, O.LV. Besova,
H.LH.LMaharyl-Yl\qeva, ∏. M. Haleeva y dr.
V mnohomernom sluçae sravnenye konstant, analohyçnoe odnomernomu, voob-
we hovorq, ne ymeet mesta (sm., naprymer, [11 – 13]). V dannoj rabote m¥ doka-
z¥vaem, çto dlq neravenstv nekotoroho specyal\noho vyda v sluçae odynakov¥x
metryk vo vsex normax konstant¥ dlq peryodyçeskyx y neperyodyçeskyx funk-
cyj sovpadagt.
Toçnee, dlq funkcyj f L Gp s
l n∈ , ( ), l n∈N , 1 ≤ p , q , s i ≤ ∞ , i = 1, n ,
yzvestn¥ neravenstva vyda
D f
L Gq
n
α
( )
≤ C f D f
L G
i
n
i
l
L Gp
n
i
s
n
i
i
( ) ( )
µ µ
0
1=
∏ , (7)
hde D fα
— smeßannaq proyzvodnaq funkcyy f, α ∈N0
n , 0 < µi < 1, i = 0, n ,
µii
n
=∑ 0
= 1.
Pry 1 < p , q , si < ∞ dostatoçn¥e uslovyq suwestvovanyq takyx nera-
venstv dlq neperyodyçeskyx funkcyj b¥ly poluçen¥ v [14], v bolee obwej sy-
tuacyy, kohda metryky vektorn¥e, — v [15], neobxodym¥e y dostatoçn¥e — v
[16], v peryodyçeskom sluçae — v [17].
Budem rassmatryvat\ sluçaj, kohda p = q = si , i = 1, n ; tohda prostranstvo
L Gp s
l n
, ( ) budem oboznaçat\ L Gp p
l n
, ( ), 1 ≤ p < ∞ . Yz rabot¥ [18] (§ 6.3) sledu-
et, çto v πtom sluçae dlq neperyodyçeskyx funkcyj v¥polnqetsq neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
600 V. F. BABENKO, M. S. ÇURYLOVA
D f
Lp
n
α
( )R
≤ C f D f
L
l
i
n
i
l
L
l
p
n
i ii
n
i
p
n
i i
( )
( / )
( )
/
R R
1
1
1− ∑
=
= ∏α α
, (8)
hde l n∈N , α ∈N0
n , ( / )αi ii
n
l=∑ 1
< 1; a dlq peryodyçeskyx —
D f
Lp
n
α
( )T
≤ C f D f
L
l l
i
n
i
l
L
l l
p
n
i ii
n
ii
n
i
p
n
i i i
( )
( / ) ( / )
( )
/ /
T T
1
1
1 1− ∑ − ∑
=
+
= = ∏α λ α λ
, (9)
hde λ ≥ 0, C = C ( λ ) , l n∈N , α ∈N0
n , (( )/ )α λi ii
n
l+=∑ 1
< 1. Naybol\ßyj yn-
teres predstavlqgt neravenstva (9) s λ = 0.
Takym obrazom, analohyçno odnomernomu sluçag, neravenstvo (8) dlq
neperyodyçeskyx funkcyj ymeet mesto s pokazatelqmy µ0cr
= 1
1
− =∑ ( )/αi ii
n
l ,
µicr
= αi il/ , i = 1, n , a neravenstvo (9) dlq peryodyçeskyx funkcyj — pry vsex
µ0 = 1
1 1
− −= =∑ ∑( ) ( )/ /α λi ii
n
ii
n
l l , µi = α λi i il l/ /+ , i = 1, n , λ ≥ 0; pokazately
µicr
, i = 0, n , naz¥vagtsq krytyçeskymy.
Neuluçßaem¥e konstant¥ v neravenstvax (8), (9) ymegt sootvetstvenno vyd
K n( )R : = sup
,
( )
( )
( )
( )
( / )
( )
/
f L
D f
L
L
l
i
n
i
l
L
l
p p
l n
i
n
i
li
Lp
n
p
n
p
n
i ii
n
i
p
n
i i
D f
f D f∈
∏ ≠
− ∑
=
=
= ∏R
R
R R
R
1
1
0
1
1
α
α α , (10)
K n( )T : = sup
,
( )
( )
( )
( )
( / ) ( / )
( )
/ /
f L
D f
L
L
l l
i
n
i
l
L
l l
p p
l n
i
n
i
li
Lp
n
p
n
p
n
i ii
n
ii
n
i
p
n
i i i
D f
f D f∈
∏ ≠
− ∑ − ∑
=
+
=
= = ∏T
T
T T
T
1
1 1
0
1
1
α
α λ α λ . (11)
Cel\ dannoj rabot¥ — pokazat\, çto K ( R
n
) = K ( T
n
) pry λ = 0. A ymenno,
spravedlyva sledugwaq teorema.
Teorema. Pust\ 1 ≤ p ≤ ∞ , l n∈N , α ∈N0
n , ( )/αi ii
n
l=∑ 1
< 1, λ = 0. Toh-
da ymeet mesto ravenstvo
K ( R
n
) = K ( T
n
) . (12)
Zametym, çto pry dokazatel\stve teorem¥ suwestvenno yspol\zuetsq odno-
mernaq sxema (sm. [10]).
Pryvedem snaçala nekotor¥e vspomohatel\n¥e utverΩdenyq.
Lemma(1 (sm., naprymer, [19, c. 16]). Dlq lgboho δ > 0 suwestvuet posle-
dovatel\nost\ funkcyj { }ζm m∈N so sledugwymy specyal\n¥my svojstvamy:
1) ∀ ∈ →m mN R R: :ζ ;
2) ζm C∈ ∞( )R ;
3) ζ δm x x m= ∀ > +( )0 ;
4) 0 1≤ ≤ ∀ ∈ζm x R ;
5) ζm = 1, esly x m≤ ;
6) ∀ ∈ ∀ ∈ + + + = −+m x m m x xm mN [ , ]: ( ) ( )1 1 11δ ζ ζ ;
7) ζ δm
k
k
kx C( )( ) ≤ − , k ∈ N .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
SRAVNENYE TOÇNÁX KONSTANT V NERAVENSTVAX TYPA KOLMOHOROVA … 601
Lemma(2. Dlq lgboj funkcyy f Lp p
l n∈ , ( )R , 1 ≤ p < ∞ , najdetsq posle-
dovatel\nost\ { }gm m∈N fynytn¥x beskoneçno dyfferencyruem¥x funkcyj
yz Lp p
l n
, ( )R takaq, çto pry m → ∞
f gm Lp
n− ( )R
→ 0,
D f D gi
l
i
l
m L
i i
p
n−
( )R
→ 0, i = 1, n .
Otmetym, çto takoho roda approksymacyy çasto yspol\zugtsq v teoryy
funkcyj mnohyx peremenn¥x, osobenno v teoremax vloΩenyq (sm., naprymer,
[20], § 14); dokazatel\stvo dannoho utverΩdenyq m¥ pryvodym dlq polnot¥ yz-
loΩenyq.
Dokazatel\stvo. Voz\mem proyzvol\no f Lp p
l n∈ , ( )R y ε > 0. Yzvestno
(sm., naprymer, [20], § 5, 6), çto najdetsq funkcyq f C Ln
p p
l n
ε ∈ ∞( ) ( ),R R∩ ta-
kaq, çto
f f Lp
n− ε ( )R
< ε, D f D fi
l
i
l
L
i i
p
n− ε ( )R
< ε, i = 1, n .
PoloΩym g x f x xm m( ) ( ) ( )= ε η , η ζm i
n
m ix x( ) ( )= =∏ 1
, m ∈ N , hde funkcyy
ζm ix( ) v¥bran¥ sohlasno lemme 1 po nekotoromu δ > 0. Tohda g xm( ) ∈
∈ C n
0
∞( )R ∩ Lp p
l n
, ( )R y
f gm Lp
nε − ( )R
→ 0 pry m → ∞ ,
D f D gi
l
i
l
m L
i i
p
nε −
( )R
≤ D f D f C D f Di
l
i
l
m L
k
l
l
k
i
k
i
l k
m L
i i
p
n
i
i
i
p
nε ε εη η− +
=
−
−∑( ) ( )
( )( )
R R
0
1
.
Oçevydno, pry m → ∞
D f D fi
l
i
l
m L
i i
p
nε ε η−
( )R
→ 0.
Dalee, v sylu svojstv funkcyj ζ m çastnaq proyzvodnaq D xi
l k
m
i − η ( ) otlyçna
ot nulq tol\ko na mnoΩestve [ ; ] [ ; ]\− − + −m m m mn n1 1 ; takym obrazom, so-
hlasno svojstvu 7 suwestvugt takye konstanta C y δ > 1, çto
k
l
l
k
i
k
i
l k
m L
i
i
i
p
nC D f D
=
−
−∑
0
1
( )( )
( )ε η
R
≤
C
C D f
k
l
l
k
i
k
L m m m m
i
i p
n nδ ε
=
−
− − + −∑
0
1
1 1
( )
([ ; ] \ [ ; ] )
.
Poskol\ku dlq funkcyy fε velyçyna ( )
([ ; ] \ [ ; ] )
D fi
k
L m m m mp
n nε − − + −1 1
koneçna,
to δ v lemme 1 moΩno v¥brat\ nastol\ko bol\ßym, çtob¥ pravaq çast\ posled-
neho neravenstva b¥la men\ße ε.
Takym obrazom, dlq lgboho ε > 0 najdetsq M takoe, çto dlq lgboho m > M
f gm Lp
n− ( )R
≤ f f Lp
n− ε ( )R
+ f gm Lp
nε − ( )R
< 2 ε ,
D f D gi
l
i
l
m L
i i
p
n−
( )R
≤
≤ D f D fi
l
i
l
L
i i
p
n− ε ( )R
+ D f D gi
l
i
l
m L
i i
p
nε −
( )R
< 3 ε , i = 1, n .
Lemma dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
602 V. F. BABENKO, M. S. ÇURYLOVA
Dokazatel\stvo teorem¥. Otmetym, çto pry λ = 0 pokazately v nera-
venstvax (8) y (9) ( y, sootvetstvenno, v opredelenyqx konstant K ( R
n
) y K ( T
n
))
sovpadagt y ravn¥ krytyçeskym pokazatelqm v neravenstve (9): µ0cr
, µicr
, i =
= 1, n . Vsgdu v dokazatel\stve m¥ budem oboznaçat\ pokazately çerez µ0 , µi ,
i = 1, n , ukaz¥vaq, hde πto neobxodymo, çto ony qvlqgtsq krytyçeskymy.
Suwestvenn¥m obstoqtel\stvom qvlqetsq tot fakt, çto dlq krytyçeskyx
pokazatelej konstanta K ( T
n ) v neravenstve dlq peryodyçeskyx funkcyj na
samom dele ne zavysyt ot dlyn¥ peryoda. Dejstvytel\no, oboznaçym çerez b T ,
b > 0, otrezok [ 0, 2 π b ] . Dlq funkcyy f L bp p
l n∈ , ( )( )T poloΩym g ( x ) =
= f ( bx ) . Qsno, çto g Lp p
l n∈ , ( )T y
K ( T
n
) = sup
,
( )
( )
( )
( ) ( )
g L
D g
L
L i
n
i
l
Lp p
l n
i
n
i
li
Lp
n
p
n
p
n
i
p
n
i
D g
g D g∈
∏ ≠
=
=
∏T
T
T T
T
1
0
0
1
α
µ µ =
= sup
,
(( ) )
(( ) )
/
(( ) )
( / ) ( ( / ))
(( ) ) (( ) )
f L b
D f
n p
L b
n p l n p
L b i
n
i
l
L b
p p
l n
i
n
i
li
Lp b n
ii
n
p
n
i ii
n
p
n
i
p
n
i
b D f
b f D f∈
∏ ≠
∑ −
− + −
=
=
=
=∑ ∏T
T
T T
T
1
1
0 1 0
0
1
α α
µ µ µ µ =
= b K bii
n
i i ii
n
i
nn p n p l n p nα µ µ µ= ==∑ − + − +∑ ∑1 0 11/ ( / ) ( / )) ( )( )T = K ( ( b T )
n
) ,
poskol\ku dlq krytyçeskyx pokazatelej
i
n
i i i i
i
n
l
n
p= =
∑ ∑− + −
1 0
1( )α µ µ = 0
(pry p = ∞ sçytaem 1 / p = 0 ).
Perejdem k dokazatel\stvu ravenstva (12).
Pust\ snaçala 1 ≤ p < ∞ . V¥berem proyzvol\no f Lp p
l n∈ , ( )R y ε > 0. Ys-
pol\zovav lemmuL2, najdem funkcyg g C n∈ ∞
0 ( )R takug, çto
f Lp
n( )R
≥
1
1 + ε
g Lp
n( )R
y
D fi
l
L
i
p
n( )R
≥
1
1 + ε
D gi
l
L
i
p
n( )R
, i = 1, n .
Prymenyv neravenstvo (8) dlq funkcyy f g Lp p
l n− ∈ , ( )R , yz lemm¥ 2 poluçym
takΩe
D f
Lp
n
α
( )R
≤ ( )
( )
1 + C D g
Lp
nε α
R
s nekotoroj konstantoj C.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
SRAVNENYE TOÇNÁX KONSTANT V NERAVENSTVAX TYPA KOLMOHOROVA … 603
Pust\ teper\ f Ll n∈ ∞ ∞, ( )R . Poskol\ku v sylu neravenstva (8) D f xα ( ) su-
westvenno ohranyçena na R
n, to po opredelenyg suwestvennoho supremuma dlq
lgboho ε > 0 najdetsq mnoΩestvo poloΩytel\noj mer¥ ∆ε , na kotorom
D f xα ( ) > D f
L n
α ε
∞
−
( )R
. V takom sluçae najdetsq y çyslo m takoe, çto
pereseçenye ∆ m = [ ; ]−m m n ∩ ∆ε ymeet poloΩytel\nug meru. Rassmotrym te-
per\ funkcyg g ( x ) = f x xm( ) ( )η , ηm x( ) = ζm ii
n
x( )=∏ 1
, ζ m opredelen¥ v lem-
meL1. Tohda
D g
L n
α
∞ ( )R
= sup ( ) ( )vrai
x
m
m
f x x
∈supp η
η ≥ sup ( ) ( )vrai
x
m
m
f x x
∈∆
η =
= sup ( )vrai
x m
f x
∈∆
= D f
L n
α
∞ ( )R
,
g L n
∞ ( )R
= sup ( ) ( )vrai
x
m
n
f x x
∈R
η ≤ f L n
∞ ( )R
,
D gi
l
L
i
n
∞ ( )R
≤ ( ) ( )( )
( ) ( )
D f C D f Di
l
m L
k
l
l
k
i
k
i
l k
m L
i
n
i
i
i
nη η
∞ ∞
+
=
−
−∑
R R
0
1
.
V sylu svojstv funkcyj ζ m (sm. lemmuL1) najdutsq takye C > 0 y δ > 1, çto
k
l
l
k
i
k
i
l k
m L
i
i
i
nC D f D
=
−
−∑
∞0
1
( )( )
( )
η
R
≤
C
C D f
k
l
l
k
i
k
L m m m m
i
i n nδ =
−
− − + −∑
∞0
1
1 1
( )
([ ; ] \ [ ; ] )
.
Poskol\ku velyçyna ( )
([ ; ] \ [ ; ] )
D fi
k
L m m m mn n
∞ − − + −1 1
koneçna (v sylu odnomerno-
ho neravenstva Kolmohorova), to δ v lemme 1 moΩno v¥brat\ takym obrazom,
çtob¥ pravaq çast\ posledneho neravenstva b¥la men\ße ε D fi
l
L
i
n
∞ ( )R
. Okon-
çatel\no poluçym
D gi
l
L
i
n
∞ ( )R
≤ D fi
l
L
i
n
∞
+
( )
( )
R
1 ε .
Dlq lgboho 1 ≤ p ≤ ∞ v¥berem poloΩytel\noe çyslo b, bol\ßee dlyn
proekcyj nosytelq sootvetstvugwej funkcyy g na koordynatn¥e osy, y polo-
Ωym
˜ ( )g x = g x b
n
( )+
∈
∑ 2π ν
ν Z
.
Qsno, çto ˜ ( ), ( )g L bp p
l n∈ T , y dlq krytyçeskyx pokazatelej ymeem
D f
f D f
L
L i
l
Li
n
p
n
p
n
i
p
n
i
α
µ µ
( )
( ) ( )
R
R R
0
1=∏
≤
≤
( )
( ) ( )
( )
( ) ( )
1
1 11
1
10
+
+( ) +
−
=
−∏
C D g
g D g
L
L i
n
i
l
L
p
n
p
n
i
p
n
i
ε
ε ε
α
µ µ
R
R R
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
604 V. F. BABENKO, M. S. ÇURYLOVA
=
1
1
1
1
0 0
+
+ ∑− ⋅
=
= ∏
C
D g
g D gii
n
p
n
p
n
i
p
n
i
L b
L b i
n
i
l
L b
ε
ε µ
α
µ µ
( )
˜
˜ ˜
(( ) )
(( ) ) (( ) )
T
T T
≤
≤ ( )( ) ( )( )1 1+ +C K b nε ε T = ( )( ) ( )1 1+ +C K nε ε T
pry 1 ≤ p < ∞ y
D f
f D f
L
L i
l
Li
n
n
n
i
n
i
α
µ µ
∞
∞ ∞=∏
( )
( ) ( )
R
R R
0
1
≤
D g
g D g
L
L i
l
Li
n
n
n
i
n
i
α
µ
µ
ε
∞
∞ ∞
+
−
=∏
( )
( ) ( )
( )
R
R R
0 1 1
1
=
=
1
1
1
1
1 0( )
˜
˜ ˜
(( ) )
(( ) ) (( ) )
+ ∑− ⋅
=
=
∞
∏ε µ
α
µ µ
ii
n
p
n
p
n
i
n
i
D g
g D g
L b
L b i
n
i
l
L b
T
T T
≤
≤ ( ) ( )( )1 1+ ∑ =ε µii
n
K b n
T = ( ) ( )1 1+ ∑ =ε µii
n
K n
T
pry p = ∞ .
Vsledstvye proyzvol\nosty ε > 0 otsgda sleduet, çto
K n( )R ≤ K n( )T .
DokaΩem teper\ protyvopoloΩnoe neravenstvo. Zametym, çto Ll n
∞ ∞, ( )T ⊂
⊂ Ll n
∞ ∞, ( )R , poπtomu K n( )T ≤ K n( )R pry p = ∞ .
Pust\ 1 ≤ p < ∞ , T1 — otrezok [ 0, 1 ] . V¥berem proyzvol\no fL∈
∈ Lp p
l n
, ( )T1 . PoloΩym ηm x( ) = ζm ii
n
x( )=∏ 1
(ζm v¥bran¥ pry δ = 1 ), g xm( ) =
= f x xm( ) ( )η , x ∈ R
n
, m ∈ N . Uçyt¥vaq svojstva funkcyj ζm (sm. lemmuL1),
ymeem
D gm Lp
n
α
( )R
≥ ( ) /
( )
2
1
m D fn p
Lp
n
α
T
,
gm Lp
n( )R
≤ ( ) /
( )2 2
1
m fn p
Lp
n+
T
,
D gi
l
m L
i
p
n( )R
= ( ) ( )( )
( )
D f C D f Di
l
m l
k
i
k
i
l k
m
k
l
L
i
i
i
i
p
n
η η+ −
=
−
∑
0
1
R
≤
≤ ( ) ( )( )
( ) ( )
D f C D f Di
l
m L l
k
i
k
i
l k
m L
k
l
i
p
n i
i
p
n
i
η η
R R
+ −
=
−
∑
0
1
≤
≤ ( ) /
( )
/
( )
2 2 2
1 11
1
0
1
m D f C m M C D fn p
i
l
L
n
n
j n j
j
n p
l
k
i
k
L
k
l
i
p
n i p
n
i
+ +
−
= =
−
∑ ∑
T T
,
hde M = max
( )0 1≤ ≤ −
−
∞k l
i
l k
m L
i
i
nD η
R
ne zavysyt ot m. Zdes\ pry ocenke
( )( )
( )
D f Di
k
i
l k
m L
i
p
n
− η
R
yspol\zovan tot fakt, çto Di
l k
m
i − η ≠ 0 na mnoΩestve
mer¥ 2
1
n
n
j n j
j
n
C m −
=∑ = ( ) ( )2 2 2m mn n+ − .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
SRAVNENYE TOÇNÁX KONSTANT V NERAVENSTVAX TYPA KOLMOHOROVA … 605
Yz poluçenn¥x ocenok sleduet, çto
K n( )R ≥
D g
g D g
m L
m L i
l
m Li
n
p
n
p
n
i
p
n
i
α
µ µ
( )
( ) ( )
R
R R
0
1=∏
≥
≥
( )
( ) ( )
/
( )
/
( )
/
( )
2
2 2 2 2
1
1
0
11
m D f
m f m D f A
n p
L
n p
L
i
n
n p
i
l
L
p
n
p
n
i
p
n
i
α
µ µ
T
T T
+{ } + +
=
∏
= :
= : F f m( , ) ∀ m ∈ N,
hde
A = 2
1
1
0
1
1
n
n
j n j
j
n p
l
k
i
k
L
k
l
C m M C D f
i p
n
i
−
= =
−
∑ ∑
/
( )T
,
t. e.
K n( )R ≥ F f m( , ).
Ustremlqq m → ∞ , poluçaem
K n( )R ≥
D f
f D f
L
L i
l
Li
n
p
n
p
n
i
p
n
i
α
µ µ
( )
( ) ( )
T
T T
1
1
0
11=∏
,
poskol\ku
lim
( )
( ) ( )
/
( / ) ( / )m
n p
n p n p
m
m m ii
n
→∞ + + ∑ =
2
2 2 2 20 1µ µ
= 1,
a v slahaemom A znamenatelq maksymal\naq stepen\ m ravna ( n – 1 ) / p , y, zna-
çyt, πto slahaemoe na znaçenye predela ne povlyqet.
No tohda sohlasno svojstvu toçnoj verxnej hrany
K n( )R ≥ K n( )T1 = K n( )T .
Teorema dokazana.
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2. Tyxomyrov V. M., Maharyl-Yl\qev H. H. Neravenstva dlq proyzvodn¥x // Yzbrann¥e trud¥.
Matematyka, mexanyka / A. N. Kolmohorov. – M.: Nauka, 1985. – S.L387 – 390.
3. Arestov V. V., Habußyn V. N. Nayluçßee pryblyΩenye neohranyçenn¥x operatorov
ohranyçenn¥my // Yzv. vuzov. Matematyka. – 1995. – # 11. – S. 44 – 66.
4. Arestov V. V. PryblyΩenye neohranyçenn¥x operatorov ohranyçenn¥my y rodstvenn¥e
πkstremal\n¥e zadaçy // Uspexy mat. nauk. – 1996. – 51, # 6. – S.L88 – 124.
5. Babenko V. F., Kofanov V. A., Pichugov S. A. Inequalities of Kolmogorov type and some their ap-
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P. 223 – 237.
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some of their applications // New Approaches in Nonlinear Analysis. – Palm Harbor (USA):
Hadronic Press, 1999. – P. 9 – 50.
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8. Habußyn V. N. Neravenstva dlq norm funkcyy y ee proyzvodn¥x v metrykax Lp // Mat.
zametky. – 1967. – 1, # 3. – S. 291 – 298.
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Poluçeno 04.10.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
|
| id | umjimathkievua-article-3478 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:43:18Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a7/c3c1b872775ed75020a4f41970d677a7.pdf |
| spelling | umjimathkievua-article-34782020-03-18T19:55:42Z Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables Сравнение точных констант в неравенствах типа Колмогорова для периодических и непериодических функций многих переменных Babenko, V. F. Churilova, M. S. Бабенко, В. Ф. Чурилова, М. С. Бабенко, В. Ф. Чурилова, М. С. We investigate the correlation between the constants $K(ℝ^n)$ and $K(T^n)$, where $$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$ is the exact constant in a Kolmogorov-type inequality, $ℝ$ is the real straight line, $T = [0,2π],\; L^l_{p, p} (G^n)$ is the set of functions $ƒ ∈ L_p (G^n)$ such that the partial derivative $D_i^{l_i } f(x)$ belongs to $L_p (G^n), i = \overline {1,n}, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^{α} f$ is the mixed derivative of a function $ƒ, 0 < µi < 1, i = \overline {0,n},$ and $∑_{i=0}^n µ_i = 1$. If $G^n = ℝ$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i),\; µ_i = α_i/l_i,\; i = \overline {1,n}$ if $G^n = T^n$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i) − ∑_{i=0}^n (λ/l_i),\; µ_i = α_i/ l_i + λ/l_i , i= \overline {1,n},\; λ ≥ 0$. We prove that, for $λ = 0$, the equality $K(ℝ^n) = K(T^n)$ is true. Досліджується взаємозв'язок між константами $K(ℝ^n)$ та $K(T^n)$, де $$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$ —точна константа в нерівності типу Колмогорова; $ℝ$ — дійсна пряма, $T = [0,2π],\; L^l_{p, p} (G^n)$— множина функцій $ƒ ∈ L_p (G^n)$ таких, що частинна похідна $D_i^{l_i} f(x),\; i = \overline {1,n}, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^{α} f$— мішана похідна функції $ƒ, 0 < µi < 1, i = \overline {0,n}$, $∑_{i=0}^n µ_i = 1$. Якщо $G^n = ℝ$, то $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i),\; µ_i = α_i/l_i,\; i = \overline {1,n}$; якщо $G^n = T^n$, то $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i) − ∑_{i=0}^n (λ/l_i),\; µ_i = α_i/ l_i + λ/l_i , i= \overline {1,n},\; λ ≥ 0$. Доведено, що при $λ = 0$ справджується рівність $K(ℝ^n) = K(T^n)$. Institute of Mathematics, NAS of Ukraine 2006-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3478 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 5 (2006); 597–606 Український математичний журнал; Том 58 № 5 (2006); 597–606 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3478/3693 https://umj.imath.kiev.ua/index.php/umj/article/view/3478/3694 Copyright (c) 2006 Babenko V. F.; Churilova M. S. |
| spellingShingle | Babenko, V. F. Churilova, M. S. Бабенко, В. Ф. Чурилова, М. С. Бабенко, В. Ф. Чурилова, М. С. Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| title | Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| title_alt | Сравнение точных констант в неравенствах типа Колмогорова для периодических и непериодических функций многих переменных |
| title_full | Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| title_fullStr | Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| title_full_unstemmed | Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| title_short | Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| title_sort | comparison of exact constants in kolmogorov-type inequalities for periodic and nonperiodic functions of many variables |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3478 |
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