On the order of growth of ring $Q$-homeomorphisms at infinity
For ring homeomorphisms $f : ℝn → ℝn$ , we establish the order of growth at infinity.
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| author | Salimov, R. R. Smolovaya, E. S. Салимов, Р. Р. Смоловая, E. С. Салимов, Р. Р. Смоловая, E. С. |
| author_facet | Salimov, R. R. Smolovaya, E. S. Салимов, Р. Р. Смоловая, E. С. Салимов, Р. Р. Смоловая, E. С. |
| author_sort | Salimov, R. R. |
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| description | For ring homeomorphisms $f : ℝn → ℝn$ , we establish the order of growth at infinity. |
| first_indexed | 2026-03-24T02:32:43Z |
| format | Article |
| fulltext |
UDK 517.5
R. R. Salymov, E. S. Smolovaq
(Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
O PORQDKE ROSTA KOL|CEVÁX Q-HOMEOMORFYZMOV
NA BESKONEÇNOSTY
For ring Q-homeomorphisms f n n: R R→ , an order of the growth at infinity is established.
Dlq kil\cevyx Q-homeomorfizmiv f n n: R R→ vstanovleno porqdok zrostannq na neskinçen-
nosti.
1. Vvedenye. Kak yzvestno, v osnovu heometryçeskoho opredelenyq kvazykon-
formn¥x otobraΩenyj po Vqjsqlq (sm. [1]), zadann¥x v oblasty G yz Rn
, po-
loΩeno uslovye
M f KM( ) ( )Γ Γ≤
dlq proyzvol\noho semejstva Γ kryv¥x γ v oblasty G, hde M — konform-
n¥j modul\ semejstva kryv¥x v Rn
, a K ≥ 1 — nekotoraq postoqnnaq. Druhy-
my slovamy, modul\ lgboho semejstva kryv¥x yskaΩaetsq ne bolee çem v K
raz.
Emkosty y moduly qvlqgtsq osnovn¥m heometryçeskym metodom v sovremen-
noj teoryy otobraΩenyj. V svqzy s πtym v monohrafyy [2] yzuçalsq sledug-
wyj klass otobraΩenyj.
Pust\ G — oblast\ v Rn
, n ≥ 2, y Q D: ,→ ∞[ ]1 — yzmerymaq funkcyq.
Homeomorfyzm f G n: → R = Rn
∪ ∞{ } naz¥vaetsq Q -homeomorfyzmom,
esly
M f Q x x dm xn
G
( ) ( ) ( ) ( )Γ ≤ ⋅∫ ρ (1)
dlq lgboho semejstva Γ putej v G y lgboj dopustymoj funkcyy ρ dlq Γ.
Zdes\ m oboznaçaet meru Lebeha v Rn
. Cel\g teoryy Q-homeomorfyzmov qv-
lqetsq yzuçenye vzaymosvqzej meΩdu razlyçn¥my svojstvamy maΩorant¥ Q y
samoho otobraΩenyq f.
Vperv¥e neravenstvo vyda (1) b¥lo ustanovleno na ploskosty dlq kvazykon-
formn¥x otobraΩenyj (sm. [3]). V dal\nejßem v rabote [4] dlq prostranstven-
n¥x kvazykonformn¥x otobraΩenyj to Ωe neravenstvo poluçeno s Q x( ) =
= K x fI ( , ) , hde K x fI ( , ) — vnutrennqq dylatacyq f,
K x f
J x f
l f x
I n
( , )
( , )
( )
=
′( )
,
esly J x f( , ) ≠ 0; K x fI ( , ) = 1, esly ′f x( ) = 0, y K x fI ( , ) = ∞ v ostal\n¥x
toçkax. Zdes\
l f x
f x h
hh n
′( ) = ′
∈ { }
( ) : min
( )
\R 0
.
© R. R. SALYMOV, E. S. SMOLOVAQ, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 829
830 R. R. SALYMOV, E. S. SMOLOVAQ
Otmetym, çto neravenstvo vyda (1) b¥lo ustanovleno v rabote [5] dlq tak naz¥-
vaem¥x otobraΩenyj, kvazykonformn¥x v srednem. Podobn¥e klass¥ otobra-
Ωenyj v termynax emkostej yssledovalys\ takΩe v rabotax V.?M.?Myklgkova
(sm., naprymer, [6]).
Napomnym, çto boreleva funkcyq ρ : ,Rn → ∞[ ]0 naz¥vaetsq dopustymoj
dlq semejstva kryv¥x Γ v Rn
(pyßut ρ ∈ adm Γ ), esly
ρ
γ
ds∫ ≥ 1
dlq vsex γ ∈Γ . Modul\ (konformn¥j) semejstva kryv¥x Γ opredelqetsq ra-
venstvom
M x dm xn
G
( ) inf ( ) ( )Γ
Γ
=
∈ ∫ρ
ρ
adm
,
hde m — mera Lebeha v Rn
.
Problem¥ lokal\noho y hranyçnoho povedenyq Q-homeomorfyzmov v Rn
yzuçalys\ v sluçae Q BMO∈ (ohranyçennoho sredneho kolebanyq) v rabote [7]
(sm. takΩe [2]), Q FMO∈ (koneçnoho sredneho kolebanyq) y v druhyx sluçaqx
v rabotax [8 – 11]. V rabote [12] ustanovleno svojstvo ACL dlq Q-homeomor-
fyzmov v Rn
, n ≥ 2, pry Q L G∈ loc
1 ( ) . Pokazana takΩe prynadleΩnost\ takyx
Q-homeomorfyzmov sobolevskomu klassu Wloc
1 1,
y dyfferencyruemost\ poçty
vsgdu. ∏ty rezul\tat¥ b¥ly perenesen¥ na kol\cev¥e Q-homeomorfyzm¥ v ra-
bote [13]. V dannoj rabote dlq kol\cev¥x Q-homeomorfyzmov f n n: R R→
ustanavlyvaetsq porqdok rosta na beskoneçnosty.
Pust\ G y ′G — oblast\ v Rn
, n ≥ 2. Sohlasno rabote [11] (sm. takΩe [14,
15]), budem hovoryt\, çto homeomorfyzm f G G: → ′ naz¥vaetsq kol\cev¥m
Q-homeomorfyzmom otnosytel\no toçky x G0 ∈ , esly
M f S f S G∆( , , )1 2 ′( ) ≤ Q x x x dm xn
A
( ) ( )⋅ −( )∫ η 0 (2)
v¥polnqetsq dlq lgboho kol\ca A = A r r x( , , )1 2 0 = x n∈{ R : r1 < x x− 0 <
< r2} , Si = S x ri( , )0 = x n∈{ R : x x− 0 = ri} , i = 1, 2, 0 < r1 < r2 < d0 =
= dist ( , )x G0 ∂ , y dlq lgboj yzmerymoj funkcyy η : ( , )r r1 2 → 0, ∞[ ] takoj,
çto
η( )r dr
r
r
≥∫ 1
1
2
.
Budem takΩe hovoryt\, çto homeomorfyzm f G G: → ′ qvlqetsq kol\ce-
v¥m Q-homeomorfyzmom v oblasty G , esly uslovye (2) v¥polneno dlq vsex
toçek x G0 ∈ .
Pry n = 2 ponqtye kol\cevoho homeomorfyzma b¥lo vperv¥e vvedeno y plo-
dotvorno yspol\zovalas\ dlq yzuçenyq v¥roΩdenn¥x uravnenyj Bel\tramy
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
O PORQDKE ROSTA KOL|CEVÁX Q-HOMEOMORFYZMOV … 831
f z fz z= µ( ) (3)
v rabotax [14, 15]. V ukazann¥x rabotax ustanovlen rqd teorem suwestvovanyq
reßenyj (3), qvlqgwyxsq kol\cev¥my Q -homeomorfyzmamy otnosytel\no
kaΩdoj toçky z0 ∈C s
Q z
z z
z z
z
z
( )
( )
( )
=
− −
−
−
1
1
0
0
2
2
µ
µ
(sr. s [16]). Takym obrazom, v sluçae kol\cev¥x Q-homeomorfyzmov moΩet
b¥t\ Q z( ) < 1 na mnoΩestve poloΩytel\noj mer¥, çto suwestvenno otlyçaet
yx ot Q-homeomorfyzmov.
Sleduq rabote [17], paru E = (A, C), hde A n⊂ R — otkr¥toe mnoΩestvo y
C — nepustoe kompaktnoe mnoΩestvo, soderΩaweesq v A, naz¥vaem kondensa-
torom. E naz¥vaem kol\cev¥m kondensatorom, esly B = A C\ — kol\co, t.?e.
esly B — oblast\, dopolnenye kotoroj Rn B\ sostoyt v toçnosty yz dvux
komponent. E naz¥vaem ohranyçenn¥m kondensatorom, esly mnoΩestvo A qv-
lqetsq ohranyçenn¥m. Hovorqt, çto kondensator E = (A, C) leΩyt v oblasty
G, esly A G⊂ . Oçevydno, çto esly f G n: → R — otkr¥toe otobraΩenye y
E = (A, C) — kondensator v G , to ( , )fA fC takΩe kondensator v fG . Dalee
f E = ( , )fA fC .
Pust\ E = (A, C) — kondensator. W E0( ) = W A C0( , ) — semejstvo neotryca-
tel\n¥x funkcyj u A R: → 1
takyx, çto: 1) u prynadleΩyt C A0( ) ,
2)? u x( ) ≥ 1 dlq vsex x C∈ , 3) u prynadleΩyt klassu A CL, y pust\
∇ = ∂
=
∑u ui
i
n
( )
/
2
1
1 2
.
Velyçynu
cap E = cap ( , )A C = inf
( )u W E
u
A
u dm
∈
∇∫
0
naz¥vagt emkost\g kondensatora E.
Yzvestno, çto
cap E
m S
m A C
n
n
n
≥
[ ]
−
−
(inf )
( \ )
1
1 . (4)
Zdes\ m Sn−1 — (n – 1)-mernaq mera Lebeha C∞
-mnohoobrazyq S, qvlqgweho-
sq hranycej S U= ∂ ohranyçennoho otkr¥toho mnoΩestva U, soderΩaweho C
y soderΩawehosq vmeste so svoym zam¥kanyem U v A, hde toçnaq nyΩnqq
hran\ beretsq po vsem takym S (sm. predloΩenye 5 yz [18]). Yz (4), prymenqq
yzoperymetryçeskoe neravenstvo, poluçaem
cap E n
m C
m A C
n
n
n
≥
−
Ω
( )
( \ )
1
, (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
832 R. R. SALYMOV, E. S. SMOLOVAQ
hde Ωn — n-mern¥j obæem edynyçnoho ßara v Rn
. V dal\nejßem budem ys-
pol\zovat\ ravenstvo
cap E M A C A C= ∂ ∂( )∆( , ; \ ) , (6)
hde dlq mnoΩestv S1 , S2 y S3 v Rn
, n ≥ 2, ∆( , ; )S S S1 2 3 oboznaçaet se-
mejstvo vsex neprer¥vn¥x kryv¥x, soedynqgwyx S1 y S2 v S3 (sm. [19 – 21]).
2. Osnovnaq lemma. V dal\nejßem B x r( , )0 = x n∈{ R : x x− 0 < r} .
Dlq dokazatel\stva osnovnoho rezul\tata nam potrebuetsq sledugwaq lemma.
Lemma 1. Pust\ f n n: R R→ — kol\cevoj Q -homeomorfyzm otnosy-
tel\no toçky x n
0 ∈R , 0 < r < R < ∞. Tohda
m f B x r( , )0( ) ≤ m f B x R n
dt
tq tx
n
r
R
( , ) exp
( )/( )0 1 1
0
( ) −
−∫
, (7)
hde q tx0
( ) — srednee znaçenye funkcyy Q x( ) po sfere S x t( , )0 = x n∈{ R :
x x− 0 = t} .
Dokazatel\stvo. Rassmotrym sferyçeskoe kol\co R xt ( )0 = x t:{ <
< x x− 0 < t + ∆ t} . Pust\ Ct = B x t( , )0 , At t+∆ = B x t t( , )0 + ∆ , ymeem kon-
densator ( , )A Ct t t+∆ , tohda ( , )f A f Ct t t+∆ — kol\cevoj kondensator v Rn
.
Sohlasno (6) ymeem
cap ( , )f A f Ct t t+∆ = M f A f C f R xt t t t∆ ∆∂ ∂( )( )+ , ; ( )0 .
Opredelym funkcyg Φ( )t dlq dannoho homeomorfyzma f sledugwym ob-
razom: Φ( )t = m f B x t( , )0( ) . V sylu neravenstva (5)
cap ( , )f A f Ct t t+∆ ≥ n
m f C
m f A f C
n
n
t
t t t
n
Ω
∆
( )
\+
−
( )
1
. (8)
Sohlasno kryteryg kol\cevoho Q-homeomorfyzma (sm. naprymer, [2, s. 137])
cap ( , )f A f Ct t t+∆ ≤
w
ds
sq s
n
x
nt
t t
n
−
−
+
−
∫
1
1 1
1
0
/ ( ) ( )
∆
. (9)
Sledovatel\no, yz (8), (9) poluçaem
n
m f C
m f A f C
n
n
t
t t t
n
Ω
∆
( )
\+
−
( )
1
≤
w
ds
sq s
n
x
nt
t t
n
−
−
+
−
∫
1
1 1
1
0
/ ( ) ( )
∆
,
hde wn−1 — (n – 1)-mernaq plowad\ edynyçnoj sfer¥ v Rn
.
Dalee ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
O PORQDKE ROSTA KOL|CEVÁX Q-HOMEOMORFYZMOV … 833
n
t
t t t
Φ
Φ ∆ Φ
( )
( ) ( )+ −
≤
1
0
1 1
ds
sq sx
nt
t t
/( )( )−
+
∫
∆
.
Razdelyv obe çasty na ∆ t , poluçym
n
t
ds
sq sx
n
t
t t
1
0
1 1∆
∆
/( )( )−
+
∫ ≤
1
Φ
Φ ∆ Φ
∆( )
( ) ( )
t
t t t
t
+ −
.
Ustremlqq ∆ t k nulg y uçyt¥vaq monotonnoe vozrastanye funkcyy Φ po t,
dlq poçty vsex t ymeem
n
tq t
t
t
x
n
1
0
1 1/( )( )
( )
( )− ≤ ′Φ
Φ
. (10)
Yntehryruq obe çasty neravenstva (10) po t r R∈[ ], y uçyt¥vaq, çto
′ ≤∫
Φ
Φ
Φ
Φ
( )
( )
ln
( )
( )
t
t
dt
R
rr
R
(sm., naprymer, teoremu 7.4. hl. IV v [22]), poluçaem
n
dt
tq t
R
r
x
n
r
R
0
1 1/( )( )
ln
( )
( )−∫ ≤
Φ
Φ
.
Sledovatel\no,
exp
( )
( )
( )/( )n
dt
tq t
R
r
x
n
r
R
0
1 1−∫
≤
Φ
Φ
.
Nakonec, poskol\ku Φ( )r = m f B x r( , )0( ) y Φ( )R = m f B x R( , )0( ) , poluçaem
ocenku (7), çto y trebovalos\ dokazat\.
3. O povedenyy na beskoneçnosty kol\cev¥x Q-homeomorfyzmov. For-
mulyruemaq nyΩe teorema predstavlqet soboj analoh yzvestnoj teorem¥ Lyu-
vyllq o nesuwestvovanyy otlyçnoj ot konstant¥, ohranyçennoj vo vsej plos-
kosty analytyçeskoj funkcyy. Dlq kvazyrehulqrn¥x otobraΩenyj dannoe ut-
verΩdenye poluçeno v rabote [23]. Analohyçn¥e vopros¥ dlq kol\cev¥x Q-
otobraΩenyj yssledovalys\ v rabotax [24, 25].
Teorema 1. Pust\ f n n: R R→ — kol\cevoj Q-homeomorfyzm otnosy-
tel\no toçky x n
0 ∈R . Tohda
lim inf ( , , ) exp
( )/ ( )
R x
n
r
R
L x f R
dt
tq t→∞
−−
>∫0 1 1
00
0 , (11)
hde
L x f R( , , )0 = sup ( ) ( )
x x R
f x f x
− ≤
−
0
0
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
834 R. R. SALYMOV, E. S. SMOLOVAQ
y r0 — proyzvol\noe poloΩytel\noe çyslo.
Dokazatel\stvo. Zameçaq, çto m f B x R( , )0( ) ≤ Ωn
nL x f R⋅ ( , , )0 , yz lem-
m¥?1 poluçaem
m fB x r( , )0 0( ) ≤ Ωn
n
x
n
r
R
L x f R n
dt
tq t
( , , ) exp
( )/ ( )0 1 1
00
−
−∫ . (12)
Oçevydno, çto m fB x r( , )0 0( ) > 0 y ot R ne zavysyt. Perexodq k nyΩnemu pre-
delu v (12) pry R → ∞, poluçaem (11).
Zameçanye. Ponqtye kol\cevoho Q-homeomorfyzma moΩet\ b¥t\ oprede-
leno otnosytel\no ∞. V πtom sluçae sformulyrovann¥j v¥ße rezul\tat moΩ-
no rasprostranyt\ na otobraΩenye f, zadannoe v oblastqx G n⊂ R , soderΩa-
wyx ∞ (sm., naprymer, [26]).
Sledstvye 1. Pust\ f n n: R R→ — kol\cevoj Q-homeomorfyzm otno-
sytel\no toçky x n
0 ∈R s q tx0
( ) ≡ q. Tohda
lim inf
( , , )
R
L x f R
R→∞
>0 0γ ,
hde γ = q n1 1/( )−
.
Sledstvye 2. Pust\ f n n: R R→ — kol\cevoj Q-homeomorfyzm otno-
sytel\no toçky x n
0 ∈R y q tx0
( ) ≤ c tnln −1
pry t > t0 ≥ 1. Tohda
lim inf
( , , )
(ln )R
L x f R
R→∞
>0 0γ ,
hde γ = c n1 1/( )−
.
4. O povedenyy na beskoneçnosty homeomorfyzmov klassa W n n
loc
1, ( )R .
Pust\ dlq otobraΩenyq f G n: → R , ymegweho v G çastn¥e proyzvodn¥e po-
çty vsgdu, ′f x( ) — qkobyeva matryca otobraΩenyq f v toçke x, J(x, f ) — qko-
byan otobraΩenyq f v toçke x, t. e. determynant ′f x( ) . V dal\nejßem (sm. [2])
′ = ′
∈ { }
f x
f x h
hh n
( ) max
( )
\R 0
— matryçnaq norma ′f x( ) . Pust\, krome toho,
l f x
f x h
hh n
′( ) = ′
∈ { }
( ) min
( )
\R 0
.
Napomnym, çto vneßnqq dylatacyq otobraΩenyq f v toçke x est\ velyçyna
K x f
f x
J x f
O
n
( , )
( )
( , )
= ′
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
O PORQDKE ROSTA KOL|CEVÁX Q-HOMEOMORFYZMOV … 835
esly J x f( , ) ≠ 0; K x fO ( , ) = 1, esly ′f x( ) = 0, y K x fO ( , ) = ∞ v ostal\n¥x
toçkax.
Vnutrennqq dylatacyq otobraΩenyq f v toçke x est\ velyçyna
K x f
J x f
l f x
I n
( , )
( , )
( )
=
′( )
,
esly J x f( , ) ≠ 0; K x fI ( , ) = 1, esly ′f x( ) = 0, y K x fI ( , ) = ∞ v ostal\n¥x
toçkax.
Teorema 2. Pust\ f n n: R R→ — homeomorfyzm klassa W n n
loc
1, ( )R , dlq
kotoroho K x f0( , ) ∈ Ln n
loc
−1( )R , y r0 — proyzvol\noe poloΩytel\no çyslo.
Tohda
lim inf ( , , ) exp
( )/ ( )
R x
n
r
R
L x f R
dt
tk t→∞
−−
∫0 1 1
00
= M > 0,
hde k tx0
( ) — srednee znaçenye funkcyy K x fI ( , ) po sfere S x t( , )0 =
= x n∈{ R : x x− 0 = t} .
Sledstvye 3. Esly k tx0
( ) ≤ c tnln −1
pry t > t0 ≥ 1, to
lim inf
( , , )
(ln )R
L x f R
R→∞
>0 0γ ,
hde γ = c n1 1/( )−
.
1. Väisälä J. Lectures on n-dimensional quasiconformal mappings // Lect. Notes Math. – 1971. –
229.
2. Martio O., Ryazanov V., Srebro U., Yakubov E. Moduli in modern mapping theory. – New York:
Springer, 2009.
3. Lehto O., Virtanen K. Quasiconformal mappings in the plane. – New York etc.: Springer, 1973.
4. Bishop C. J., Gutlyanskii V. Ya., Martio O., Vuorinen M. On conformal dilatation in space // Int. J.
Math. and Math. Sci. – 2003. – 22. – P. 1397 – 1420.
5. Struhov G. F. Kompaktnost\ klassov otobraΩenyj, kvazykonformn¥x v srednem // Dokl.
AN SSSR. – 1978. – 243, # 4. – S. 859 – 861.
6. Myklgkov V. M. Konformnoe otobraΩenye nerehulqrnoj poverxnosty y eho prymenenyq. –
Volhohrad: Yzd-vo Volhohrad. un-ta, 2005.
7. Martio O., Ryazanov V., Srebro U., Yakubov E. On Q-homeomorphisms // Ann. Acad. Sci. Fenn.
Ser. A1. Math. – 2005. – 30. – P. 49 – 69.
8. Yhnat\ev F., Rqzanov V. Koneçnoe srednee kolebanye v teoryy otobraΩenyj // Ukr. mat.
vestn. – 2005. – 2, # 3. – S. 395 – 417.
9. Yhnat\ev A., Rqzanov V. K teoryy hranyçnoho povedenyq prostranstvenn¥x otobraΩenyj //
Tam Ωe. – 2006. – 3, # 2. – S. 199 – 211.
10. Rqzanov V., Salymov R. Slabo ploskye prostranstva y hranyc¥ v teoryy otobraΩenyj //
Tam Ωe. – 2007. – 4, # 2. – S. 199 – 234.
11. Rqzanov V. Y., Sevost\qnov E. A. Ravnostepenno neprer¥vn¥e klass¥ kol\cev¥x Q-homeo-
morfyzmov // Syb. mat. Ωurn. – 2007. – 48, # 6. – S. 1361 – 1376.
12. Salymov R. Absolgtnaq neprer¥vnost\ na lynyqx y dyfferencyruemost\ odnoho obobwe-
nyq kvazykonformn¥x otobraΩenyj // Yzv. RAN. Ser. mat. – 2008. – 72, # 5. – S. 141 – 148.
13. Salymov R., Sevost\qnov E. ACL y dyfferencyruemost\ poçty vsgdu kol\cev¥x homeo-
morfyzmov // Tr. Yn-ta prykl. matematyky y mexanyky NAN Ukrayn¥. – 2008. – 16. – S. 171
– 178.
14. Ryazanov V., Srebro U., Yakubov E. On ring solution of Beltrami equations // J. d’Anal. Math. –
2005. – 96. – P. 117 – 150.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
836 R. R. SALYMOV, E. S. SMOLOVAQ
15. Ryazanov V., Srebro U., Yakubov E. The Beltrami equation and ring homeomorphisms // Ukr.
Math. Bull. – 2007. – 4, # 1. – P. 79 – 115.
16. Gutlyanski V., Martio O., Sugava T., Vuorinen M. On the degenerate Beltrami equation // Trans.
Amer. Math. Soc. – 2005. – 357, # 3. – P. 875 – 900.
17. Martio O., Rickman S., Vaisala J. Definitions for quasiregular mappings // Ann. Acad. Sci. Fenn.
Ser. A1. Math. – 1969. – 448. – 40 p.
18. Kruhlykov V. Y. Emkosty kondensatorov y prostranstvenn¥e otobraΩenyq, kvazykonform-
n¥e v srednem // Mat. sb. – 1986. – 130, # 2. – S. 185 – 206.
19. Gering F. W. Quasiconformal mappings in complex analysis and its applications. – Vienna: Int.
Atomic Energy Agency, 1976. – Vol. 2.
20. Hesse J. A p-extremal length and p-capacity equality // Ark. mat. – 1975. – 13. – P. 131 – 144.
21. Shlyk V. A. On the equality between p-capacity and p-modulus // Sibirsk. Mat. Zh. – 1993. – 34,
# 6. – P. 216 – 221.
22. Saks S. Teoryq yntehrala. – M.: Yzd-vo ynostr. lyt., 1949.
23. Martio O., Rickman S., Vaisala J. Distortion and singularities of quasiregular mappings // Ann.
Acad. Sci. Fenn. Ser. A. Math. – 1970. – 465. – 13 p.
24. Sevost\qnov E. Teorem¥ Lyuvyllq, Pykara y Soxockoho dlq kol\cev¥x otobraΩenyj //
Ukr. mat. visn. – 2008. – 5, # 3. – S. 366 – 381.
25. Sevost\qnov E. A. Ustranenye osobennostej y analohy teorem¥ Soxockoho – Vejerßtrassa
dlq Q-otobraΩenyj // Ukr. mat. Ωurn. – 2009. – 61, # 1. – S. 116 – 126.
26. Sevost\qnov E. K teoryy ustranenyq osobennostej otobraΩenyj s neohranyçennoj xarak-
terystykoj kvazykonformnosty // Yzv. RAN. Ser. mat. – 2010. – 74, # 1. – S. 159 – 174.
Poluçeno 07.12.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
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| id | umjimathkievua-article-2913 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:32:43Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/be/0d36ca7302933e1fe084547cddefcbbe.pdf |
| spelling | umjimathkievua-article-29132020-03-18T19:40:12Z On the order of growth of ring $Q$-homeomorphisms at infinity О порядке роста кольцевых $Q$-гомеоморфизмов на бесконечности Salimov, R. R. Smolovaya, E. S. Салимов, Р. Р. Смоловая, E. С. Салимов, Р. Р. Смоловая, E. С. For ring homeomorphisms $f : ℝn → ℝn$ , we establish the order of growth at infinity. Для кільцевих $Q$-гомеоморфізмів $f : ℝn → ℝn$ встановлено порядок зростання на нескінченності. Institute of Mathematics, NAS of Ukraine 2010-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2913 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 6 (2010); 829 – 836 Український математичний журнал; Том 62 № 6 (2010); 829 – 836 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2913/2577 https://umj.imath.kiev.ua/index.php/umj/article/view/2913/2578 Copyright (c) 2010 Salimov R. R.; Smolovaya E. S. |
| spellingShingle | Salimov, R. R. Smolovaya, E. S. Салимов, Р. Р. Смоловая, E. С. Салимов, Р. Р. Смоловая, E. С. On the order of growth of ring $Q$-homeomorphisms at infinity |
| title | On the order of growth of ring $Q$-homeomorphisms at infinity |
| title_alt | О порядке роста кольцевых $Q$-гомеоморфизмов на
бесконечности |
| title_full | On the order of growth of ring $Q$-homeomorphisms at infinity |
| title_fullStr | On the order of growth of ring $Q$-homeomorphisms at infinity |
| title_full_unstemmed | On the order of growth of ring $Q$-homeomorphisms at infinity |
| title_short | On the order of growth of ring $Q$-homeomorphisms at infinity |
| title_sort | on the order of growth of ring $q$-homeomorphisms at infinity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2913 |
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