Majorants of functions with vanishing integrals over balls
We prove the existence of nontrivial functions in Rn , n > 2, with vanishing integrals over balls of fixed radius and given majoranta of growth.
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2008
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| author | Ochakovskaya, O. A. Очаковская, О. А. Очаковская, О. А. |
| author_facet | Ochakovskaya, O. A. Очаковская, О. А. Очаковская, О. А. |
| author_sort | Ochakovskaya, O. A. |
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| datestamp_date | 2020-03-18T19:48:06Z |
| description | We prove the existence of nontrivial functions in Rn , n > 2, with vanishing integrals over balls of fixed radius and given majoranta of growth.
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| first_indexed | 2026-03-24T02:38:09Z |
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UDK 517.5
O. A. Oçakovskaq (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
MAÛORANTÁ FUNKCYJ
S NULEVÁMY YNTEHRALAMY PO ÍARAM
We prove the existence of nontrivial functions in R
n
, n ≥ 2 , with vanishing integrals over balls of
fixed radius and given majoranta of growth.
Dovedeno isnuvannq nenul\ovyx funkcij u R
n
, n ≥ 2 , z nul\ovymy intehralamy po kulqx fik-
sovanoho radiusa ta zadanog maΩorantog rostu.
Vvedenye. Pust\ R
n
— vewestvennoe evklydovo prostranstvo razmernosty
n ≥ 2 s evklydovoj normoj ⋅ . PredpoloΩym, çto f L n∈ loc
1 ( )R y dlq nekoto-
roho fyksyrovannoho r > 0 y vsex y n∈R ymeet mesto ravenstvo
f x y dx
x r
( )+
≤
∫ = 0. (1)
Pust\ takΩe dlq poçty vsex x n∈R v¥polneno neravenstvo
f x( ) ≤ F x( ), (2)
hde F — zadannaq poloΩytel\naq funkcyq na R
n. Dlq kakyx F yz uslovyj
(1), (2) sleduet, çto f — nulevaq funkcyq?
Otmetym, çto klass funkcyj, udovletvorqgwyx uslovyg (1), dostatoçno
ßyrok. Polnoe eho opysanye poluçeno v [1] (sm. takΩe [2]). Postavlenn¥j v¥-
ße vopros yzuçalsq ranee mnohymy avtoramy (sm., naprymer, rabotu [1] y pryve-
dennug v nej byblyohrafyg). V çastnosty, D.5Smyt ustanovyl [3], çto esly
f C n∈ ( )R , udovletvorqet (1) y
F x( ) = o x n( )/1 2−( ) pry x → +∞, (3)
to f = 0. ∏tot rezul\tat stanovytsq nevern¥m, esly uslovye (3) zamenyt\ uslo-
vyem
F x( ) = O x n( )/1 2−( ) pry x → +∞.
Yzvesten takΩe rqd analohyçn¥x rezul\tatov, v kotor¥x vmesto uslovyq (2)
rassmatryvagtsq ocenky sverxu dlq razlyçn¥x yntehral\n¥x srednyx funkcyy
f [4 – 8]. Po povodu analohyçn¥x zadaç dlq druhyx klassov funkcyj (v çastno-
sty, reßenyj uravnenyq svertky) sm. [1]. V rabotax [9, 10] rassmatryvalas\ po-
dobnaq zadaça dlq funkcyj, zadann¥x na poluprostranstve.
Xarakternoj osobennost\g uslovyq (3) qvlqetsq eho ynvaryantnost\ otno-
sytel\no hrupp¥ vrawenyj R
n. ∏to oznaçaet, çto f dolΩna ub¥vat\ s polyno-
myal\noj skorost\g vdol\ lgboho napravlenyq. Yz rezul\tatov rabot¥ [10]
sleduet, çto uslovye (3) moΩno zamenyt\ uslovyem
F x( ) = exp
ln
− +…+
+ +…+( )
+
−
−
α βx x
x x
xn
n
n
1 1
1 12
, x n∈R ,
hde α , β — proyzvol\n¥e poloΩytel\n¥e postoqnn¥e. Takym obrazom, esly
daΩe dopustyt\ πksponencyal\n¥j rost f vdol\ peremennoj xn , to vsledstvye
© O. A. OÇAKOVSKAQ, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 857
858 O. A. OÇAKOVSKAQ
dostatoçno b¥stroho ub¥vanyq vdol\ x xn1 1, ,… − y uslovyq (1) poluçaem f = 0.
V dannoj rabote dokazano suwestvovanye nenulev¥x funkcyj f C n∈ ∞( )R ,
udovletvorqgwyx (1) y (2), pryçem funkcyq F ymeet vyd
F x( ) = exp − +…+
+…+( )
+
−
−
x x
x x
xn
n
n
1 1
1 1κ
ε .
Zdes\ ε > 0 moΩet b¥t\ v¥brano skol\ uhodno mal¥m, a v kaçestve funkcyy κ
moΩno vzqt\ lgbug neprer¥vnug poloΩytel\nug y vozrastagwug na [ 0, + ∞ )
funkcyg, udovletvorqgwug uslovyg
dt
t tκ( )
1
∞
∫ < + ∞ . (4)
Otmetym takΩe, çto razlyçn¥e svojstva funkcyj s nulev¥my yntehralamy po
druhym podmnoΩestvam R
n
yzuçalys\ v rabotax [11 – 15].
1. Formulyrovka osnovnoho rezul\tata. Perexodq k formulyrovke os-
novnoho rezul\tata dannoj rabot¥, napomnym, çto vsgdu v dal\nejßem predpo-
lahaetsq, çto n ≥ 2 y r > 0 fyksyrovano.
Teorema. Dlq lgboho ε > 0 y lgboj poloΩytel\noj vozrastagwej funk-
cyy κ ∈ C [ 0, + ∞ ) , udovletvorqgwej (4), suwestvuet nenulevaq funkcyq
f C n∈ ∞( )R , udovletvorqgwaq (1) pry vsex y n∈R , dlq kotoroj v¥polneno
neravenstvo
f x( ) ≤ exp − +…+
+…+( )
+
−
−
x x
x x
xn
n
n
1 1
1 1κ
ε (5)
dlq vsex x ∈ R
n.
2. Dokazatel\stvo osnovnoho rezul\tata. PreΩde vseho otmetym, çto
sformulyrovannug teoremu dostatoçno dokazat\ dlq sluçaq r = 1. Obwyj
sluçaj budet sledovat\ otsgda s pomow\g prostoj zamen¥ peremenn¥x v ynteh-
rale (1). Pust\ t ∈ R
1
, ν > 0 y Jn / ( )2 ν = 0 (zdes\ y dalee yspol\zuetsq stan-
dartnoe oboznaçenye dlq funkcyy Besselq Jk pervoho roda s yndeksom k ).
Rassmotrym funkcyg
g x( ) = I t x x ch t xn n n( )/− −+…+( ) −( )3 2 1
2
1
2 2 2ν , x n∈R ,
hde
I zn( )/ ( )−3 2 = I z zn
n
( )/
( )/( )−
−
3 2
3 2 .
Yspol\zuq formul¥ dlq dyfferencyrovanyq funkcyy Besselq [16] (formul¥
(6.1), (6.2)), poluçaem, çto g udovletvorqet uravnenyg Hel\mhol\ca ∆g g+ ν2 =
= 0. Po teoreme o srednyx dlq reßenyj uravnenyq Hel\mhol\ca (sm. [2], § 4)
ymeem
g x dx
x y
( )
− ≤
∫
1
= ( ) ( ) ( )/
/2 2
2π νn
nI g y = 0 (6)
dlq lgboho y n∈R . Pust\ ϕ ∈ ∞C ( )R
1
— nenulevaq neotrycatel\naq funkcyq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
MAÛORANTÁ FUNKCYJ S NULEVÁMY YNTEHRALAMY PO ÍARAM 859
s nosytelem na otrezke [ a, b ] ⊂ ( ν, + ∞ ) . UmnoΩaq ravenstvo (6) na ϕ ( t ) y yn-
tehryruq po [ a, b ] , naxodym
f x dx
x y
( )
− ≤
∫
1
= 0
dlq vsex y n∈R , hde
f x( ) =
a
b
n n nI t x x t x t dt∫ − −+…+( ) −( )( )/ ( )3 2 1
2
1
2 2 2ch ν ϕ . (7)
Takym obrazom, f C n∈ ∞( )R y udovletvorqet (1) pry r = 1 y vsex y n∈R . Yz
(7) sleduet, çto f ( )0 0> . Krome toho, pry x xn1
2
1
2+…+ − ≤ 1 ymeem
f x( ) ≤ e t dt I tb x
a
b
t b
n
n
2 2
0
3 2
−
≤ ≤
−∫ν ϕ( ) max ( )( )/ . (8)
Dalee, pust\ ε > 0 y κ ∈ C [ 0, + ∞ ) — poloΩytel\naq vozrastagwaq funkcyq,
udovletvorqgwaq (4). Pust\ takΩe ζ > 0 y
b > a > ζ , ( )( )ν ζ ν ζ+ + − +b b < ε
2
. (9)
DokaΩem, çto funkcyg ϕ moΩno v¥brat\ tak, çtob¥ f udovletvorqla uslo-
vyg (5). Yz svojstv funkcyy κ sleduet, çto
k k k=
∞
∑
1
1
κ( )
< + ∞ .
Tohda suwestvuet posledovatel\nost\ { }ηk k =
∞
1 poloΩytel\n¥x çysel takaq,
çto
lim
k
k
→∞
η < + ∞ ,
k
k
k k=
∞
∑
1
η
κ( )
< + ∞ , (10)
y posledovatel\nost\ η κk k k/ ( ( )) ub¥vaet.
PoloΩym
µk =
η
κ
k
k
k k( )
−
, Eα = x x xn
n∈ +…+ ≥{ }−R : 1
2
1
2 α , α ≥ 0. (11)
Yz (10), (11) y [17] (teorema51.3.8) v¥tekaet, çto suwestvuet nenulevaq neotry-
catel\naq funkcyq ϕ0
1∈ ∞C ( )R s nosytelem na [ a, b ] takaq, çto
ϕ0
( )
[ , ]
j
C a b
≤ µ j (12)
pry vsex j ∈N.
PoloΩym
f x0( ) =
a
b
n n nI t x x u t x dt∫ − −+…+( )( )/ ( , )3 2 1
2
1
2
0 , x n∈R , (13)
hde u t xn0( , ) = ch t x tn
2 2
0−( )ν ϕ ( ). Prymenyv formulu (7.3) yz [16], pry n =
= 2 ravenstvo (13) moΩno perepysat\ v vyde
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
860 O. A. OÇAKOVSKAQ
f x0( ) = 2 1
0 2π
Re ( , )
a
b
itxe u t x dt∫ . (14)
Pry n ≥ 3 yz yntehral\noho predstavlenyq Puassona (sm. [16], formulu (14.6))
y (13) sleduet
f x x0 1 2( , ) = λ ξ ξρξ
n
n
a
b
it
ne u t x dt d
−
−
−∫ ∫ ∫+
−
1
1 2
1 2
1
2 4 2
01
/
/
( )/( ) ( , ) –
– λ
γ
ρ
n
n
a
b
it z
nz e u t x dt dz∫ ∫−
−( )( )/ ( , )1 2 4 2
0 . (15)
Zdes\
λn = 2 2
2
3 2
1
( )/−
−−
n nπ Γ , ρ = x xn1
2
1
2+…+ − ,
γ — poluokruΩnost\ radyusa 1 / 2 s centrom v nule, leΩawaq v verxnej polu-
ploskosty; yntehryrovanye vdol\ γ vedetsq protyv çasovoj strelky. Ynteh-
ryruq v (14), (15) po çastqm y yspol\zuq (11), (12) y (9), pry ρ ≥ 1, q ∈ N polu-
çaem
f x0( ) ≤ e
cx
q n
q n
nε µ
ρ+ +
+ +
1
1
1
, (16)
hde postoqnnaq c1 > 0 ne zavysyt ot x y q. Dalee, ocenky (8) y (9) pokaz¥va-
gt, çto dlq lgboho x n∈R takoho, çto ρ < 1, v¥polneno neravenstvo
f x0( ) ≤ c e xn
2
ε , (17)
hde c2 > 0 ne zavysyt ot x . Pust\ α > 1 takoe, çto
κ ( )q n+ +1 ≤ κ x xn1 1+…+( )−
dlq vsex x E∈ α . V neravenstve (16) poloΩym
q n+ + 1 =
x x
x x
n
n
1 1
1 1
+…+
+…+( )
−
−κ
,
hde ⋅[ ] — celaq çast\ çysla. Yspol\zuq (10) y (11), yz ocenok (16) y (17) v¥vo-
dym, çto
f x0( ) ≤ c
x x
x x
xn
n
n3
1 1
1 1
exp − +…+
+…+( )
+
−
−κ
ε
dlq vsex x n∈R , hde c3 > 0 ne zavysyt ot x . Tohda funkcyq f x( ) = f x c0 3( ) /
udovletvorqet vsem trebuem¥m uslovyqm.
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MAÛORANTÁ FUNKCYJ S NULEVÁMY YNTEHRALAMY PO ÍARAM 861
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| id | umjimathkievua-article-3203 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:09Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a8/63306c5dacf97d3ce25e9d5848080fa8.pdf |
| spelling | umjimathkievua-article-32032020-03-18T19:48:06Z Majorants of functions with vanishing integrals over balls Мажоранты функций с нулевыми интегралами по шарам Ochakovskaya, O. A. Очаковская, О. А. Очаковская, О. А. We prove the existence of nontrivial functions in Rn , n > 2, with vanishing integrals over balls of fixed radius and given majoranta of growth. Доведено існування ненульових функцій у Rn , n > 2, з нульовими інтегралами по кулях фіксованого радіуса та заданою мажорантою росту. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3203 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 857–861 Український математичний журнал; Том 60 № 6 (2008); 857–861 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3203/3152 https://umj.imath.kiev.ua/index.php/umj/article/view/3203/3153 Copyright (c) 2008 Ochakovskaya O. A. |
| spellingShingle | Ochakovskaya, O. A. Очаковская, О. А. Очаковская, О. А. Majorants of functions with vanishing integrals over balls |
| title | Majorants of functions with vanishing integrals over balls |
| title_alt | Мажоранты функций с нулевыми интегралами по шарам |
| title_full | Majorants of functions with vanishing integrals over balls |
| title_fullStr | Majorants of functions with vanishing integrals over balls |
| title_full_unstemmed | Majorants of functions with vanishing integrals over balls |
| title_short | Majorants of functions with vanishing integrals over balls |
| title_sort | majorants of functions with vanishing integrals over balls |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3203 |
| work_keys_str_mv | AT ochakovskayaoa majorantsoffunctionswithvanishingintegralsoverballs AT očakovskaâoa majorantsoffunctionswithvanishingintegralsoverballs AT očakovskaâoa majorantsoffunctionswithvanishingintegralsoverballs AT ochakovskayaoa mažorantyfunkcijsnulevymiintegralamipošaram AT očakovskaâoa mažorantyfunkcijsnulevymiintegralamipošaram AT očakovskaâoa mažorantyfunkcijsnulevymiintegralamipošaram |