On the boundary behavior of imbeddings of metric spaces into a Euclidean space
We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a bo...
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| Дата: | 2007 |
|---|---|
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| Мова: | Українська Англійська |
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2007
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509449008971776 |
|---|---|
| author | Salimov, R. R. Салімов, Р. Р. |
| author_facet | Salimov, R. R. Салімов, Р. Р. |
| author_sort | Salimov, R. R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:34Z |
| description | We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a boundary point. |
| first_indexed | 2026-03-24T02:41:16Z |
| format | Article |
| fulltext |
UDK 517.5
R. R. Salymov (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
O HRANYÇNOM POVEDENYY VLOÛENYJ
METRYÇESKYX PROSTRANSTV V EVKLYDOVO
The boundary behavior of the so-called Q-homeomorphisms with respect to a measure in some metric
spaces is investigated. A series of conditions on the function Q ( x ) and on the boundary of a domain are
formulated under which every Q-homeomorphism with respect to a measure admits a continuous
extension to a boundary point.
DoslidΩu[t\sq hranyçna povedinka tak zvanyx Q-homeomorfizmiv vidnosno miry v deqkyx met-
ryçnyx prostorax. Sformul\ovano nyz\ku umov na funkcig Q ( x ) i meΩu oblasti, pry qkyx
bud\-qkyj Q-homeomorfizm vidnosno miry dopuska[ neperervne prodovΩennq v toçku meΩi.
1. Vvedenye. V poslednee desqtyletye v teoryy otobraΩenyj yntensyvno yzu-
çagtsq razlyçn¥e klass¥ otobraΩenyj s koneçn¥m yskaΩenyem (sm., naprymer,
[1 – 8]). OtobraΩenyq s koneçn¥m yskaΩenyem dlyn¥ b¥ly vveden¥ V. Y. Rq-
zanov¥m y yssledovalys\ ym sovmestno s O. Martyo, U. Srebro y ∏. Qkubov¥m v
rabote [9]. Ony predstavlqgt soboj znaçytel\no bolee ßyrokyj klass otobra-
Ωenyj, çem nepostoqnn¥e otobraΩenyq s ohranyçenn¥m yskaΩenyem po Reßet-
nqku. Naprymer, lgboj homeomorfyzm f ∈ W n
loc
,1
s f –
1 ∈ W n
loc
,1
qvlqetsq oto-
braΩenyem s koneçn¥m yskaΩenyem dlyn¥. V teoryy kvazykonformn¥x oto-
braΩenyj y yx obobwenyj bol\ßug rol\ yhragt razlyçn¥e modul\n¥e nera-
venstva.
Sledugwaq koncepcyq b¥la predloΩena O. Martyo (sm., naprymer, [10]).
Pust\ G — oblast\ v R
n
, n ≥ 2, y Q : G → [ 1, ∞ ] — yzmerymaq funkcyq. Ho-
meomorfyzm f : G → Rn = Rn ∪ { ∞ } naz¥vaetsq Q-homeomorfyzmom, esly
M f Q x x dm xn
G
( ) ≤ ( ) ( ) ( )∫Γ ρ
dlq lgboho semejstva Γ putej v G y lgboj dopustymoj funkcyy ρ dlq Γ.
Napomnym, çto boreleva funkcyq ρ : R
n → [ 0, ∞ ] naz¥vaetsq dopustymoj
dlq semejstva kryv¥x Γ v R
n
(pyßut ρ ∈ adm Γ ), esly
ρ ρ
γ γ
ds x dx∫ ∫= ( ) ≥ 1 (1)
dlq vsex γ ∈ Γ. Modul\ semejstva kryv¥x Γ opredelqetsq ravenstvom
M x dm xn
G
( ) = ( ) ( )
∈ ∫Γ
Γ
inf
admρ
ρ ,
hde m — mera Lebeha v R
n
.
Problema hranyçnoho povedenyq Q-homeomorfyzmov yzuçalas\ v sluçae
Q ∈ B M O (ohranyçennoho sredneho kolebanyq) v rabote [10], a v sluçae Q ∈
∈ F M O (koneçnoho sredneho kolebanyq) y v druhyx sluçaqx v rabote [11].
Zdes\ problema yzuçaetsq v metryçeskyx prostranstvax dlq nov¥x klassov oto-
braΩenyj y funkcyj. Ranee modul\naq texnyka dlq metryçeskyx prostranstv
razvyvalas\ v rabotax [12 – 14].
Pust\ ( X, d, µ ) — prostranstvo X s metrykoj d y borelevoj meroj µ. Na-
pomnym, çto prostranstvo ( X, d, µ ) naz¥vaetsq n-rehulqrn¥m po Al\forsu,
esly suwestvuet postoqnnaq C ≥ 1 takaq, çto
© R. R. SALYMOV, 2007
1068 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
O HRANYÇNOM POVEDENYY VLOÛENYJ METRYÇESKYX … 1069
C R B CRn
R
n− ≤ ( ) ≤1 µ
dlq vsex ßarov BR v X radyusa R < diam X. Budem hovoryt\, çto prostranstvo
( X, d, µ ) — n-rehulqrno sverxu, esly
µ( ) ≤B CRR
n
(2)
dlq vsex ßarov BR v X radyusa R < diam X. Oblast\g v X budem naz¥vat\ ot-
kr¥toe svqznoe mnoΩestvo.
Pust\ G — oblast\ v prostranstve ( X, d, µ ), G′ — oblast\ v R
n
y Q : G →
→ [ 1, ∞ ] — yzmerymaq funkcyq. Budem hovoryt\, çto homeomorfyzm f : G →
→ G′ qvlqetsq Q-homeomorfyzmom otnosytel\no mer¥ µ, esly
M f Q x x d xn
G
( ) ≤ ( ) ( ) ( )∫Γ ρ µ (3)
dlq lgboho semejstva Γ putej v G y lgboj dopustymoj funkcyy ρ dlq Γ.
Mera dlyn¥ y dopustym¥e funkcyy dlq semejstv kryv¥x v metryçeskyx pros-
transtvax opredelqgtsq analohyçno (1) (sm., naprymer, [14 – 16]).
2. O koneçnom srednem kolebanyy otnosytel\no mer¥. Pust\ G — ob-
last\ v prostranstve ( X, d, µ ). Budem hovoryt\, çto funkcyq ϕ : G → R ymeet
koneçnoe srednee kolebanye otnosytel\no mer¥ µ v toçke x0 ∈ G ( sokrawen-
no ϕ ∈ F M O
µ
( x0 ) ), esly
lim
,
ε
ε
ε
ϕ ϕ µ
→
( )
( ) − ( )−∫
0
0
x d x
G x
< ∞, (4)
hde
ϕ ϕ µε
ε
= ( ) ( )−
( )
∫ x d x
G x0,
=
1
0
0
µ ε
ϕ µ
ε
( )( )
( ) ( )
( )
∫G x
x d x
G x
,
,
— srednee znaçenye funkcyy ϕ ( x ) po G ( x0 , ε ) = { x ∈ G : d ( x, x0 ) < ε } otnosy-
tel\no mer¥ µ. Zdes\ uslovye (4) vklgçaet predpoloΩenye, çto ϕ yntehryru-
ema otnosytel\no mer¥ µ v okrestnosty toçky x0 .
PredloΩenye. Esly dlq nekotoroho nabora çysel ϕε ∈ R, ε ∈ ( 0, ε0 ],
lim
,
ε
ε
ε
ϕ ϕ µ
→
( )
( ) − ( )−∫
0
0
x d x
G x
< ∞,
to ϕ ∈ F M O
µ
( x0 ).
Dokazatel\stvo. Dejstvytel\no, po neravenstvu treuhol\nyka
ϕ ϕ µε
ε
( ) − ( )−
( )
∫ x d x
G x0 ,
≤ − ( ) − ( ) + − ( )
( )
∫ ϕ ϕ µ ϕ ϕε ε ε
ε
x d x x
G x
0
0,
≤
≤ 2
0
ϕ ϕ µε
ε
( ) − ( )−
( )
∫ x d x
G x ,
.
Sledstvye 1. V çastnosty, esly
lim
,
ε ε
ϕ µ
→ ( )
( ) ( )−∫
0
0
x d x
G x
< ∞,
to ϕ ∈ F M O
µ
( x0 ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1070 R. R. SALYMOV
Lemma. Pust\ G — oblast\ v n-rehulqrnom sverxu prostranstve ( X, d,
µ ), n ≥ 2, v toçke x0 ∈ ∂G v¥polneno uslovye
µ γ µ( ) ( )( ) ≤ ( )−G B x r
r
G B x rn∩ ∩0 2
2
02
1
, log , (5)
y ϕ : G → R — neotrycatel\naq funkcyq klassa F M O
µ
( x0 ). Tohda
ϕ µ
ε
ε ε
( ) ( )
( )
( )
=
( )
∫ x d x
d x x
d x x
On
G A , log
,
log log
,
0
0
1
1
0∩
pry ε → 0 y nekotorom ε0 ∈ ( 0, δ0 ), hde δ0 = min ( e–
e, d0 ), d0 = sup ,
x G
d x x
∈
( )0 ,
A x X d x x( ) = ∈ < ( ) <{ }ε ε ε ε, : ,0 0 0 .
Dokazatel\stvo. V¥berem ε0 ∈ ( 0, δ0 ) takoe, çto funkcyq ϕ yntehryru-
ema v G0 = G ∩ B0 otnosytel\no mer¥ µ, hde B0 = B ( x0 , ε0 ),
δ = sup
,r
r
G r
x d x
∈( ) ( )
( ) − ( )−∫
0 0ε
ϕ ϕ µ < ∞,
G ( r ) = G ∩ B ( r ), B ( r ) = B ( x0 , r ) = { }∈ ( ) <x X d x x r: , 0 . Dalee, pust\ ε < 2–
1
ε0 ,
εk < 2–
k
ε0 , Ak = { }∈ ≤ ( ) <+x X d x xk k: ,ε ε1 0 , Bk = B ( εk ) y ϕk — srednee znaçe-
nye funkcyy ϕ ( x ) v Gk = G ∩ Bk , k = 0, 1, 2, … , otnosytel\no mer¥ µ. V¥be-
rem natural\noe çyslo N takoe, çto ε ∈ [ εN + 1 , εN ) , y oboznaçym α ( t ) =
= ( t log2 1 / t )
–
n
. Tohda G ∩ A ( ε, ε0 ) ⊂ ∆ ( ε ) : =
∆kk
N
= 0∪ , hde ∆k = G ∩ Ak
, y
η ε ϕ α µ
ε
( ) = ( ) ( ) ( ) ≤ +( )
( )
∫ x d x x d x S S, 0 1 2
∆
,
S x d x x d xk
k
N
k
1 0
1
( ) = ( ) − ( ) ( )( ) ( )∫∑
=
ε ϕ ϕ α µ,
∆
,
S d x x d xk
k
N
k
2 0
1
( ) = ( ) ( )( )∫∑
=
ε ϕ ς µ,
∆
.
Poskol\ku Gk ⊂ G ( 2d ( x, x0 ) ) dlq x ∈ ∆k , po uslovyg (2)
µ ( Gk ) ≤ µ ( G ( 2d ( x, x0 ) ) ) ≤ C ⋅ 2n
⋅ d ( x, x0 )
n, t. e.
1
2
1
0d x x
C
Gn
n
k( )
≤
( )
⋅
, µ
.
Krome toho,
1
1
1
2
0
log
,d x x
kn n
( )
≤ dlq x ∈ ∆k y, takym obrazom,
S C
k
Cn
n
k
N
n
1
1
2
1
2 2≤ ≤⋅ ⋅
=
∑δ δ ,
poskol\ku pry n ≥ 2
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
O HRANYÇNOM POVEDENYY VLOÛENYJ METRYÇESKYX … 1071
1 1
1
2 1
k
dt
t nn
k
n
=
∞ ∞
∑ ∫< =
−
≤ 1.
Dalee,
α µ µ µ µ
µ
( ) ⋅ ⋅( ) ( ) ≤ ( )
( )
≤
( ) − ( )
( )
≤∫ ∫ +
d x x d x
k
d x
d x x
C
k
G G
G
C
k
k k
n n
A
n
n
k k
k
n
n,
,0
0
11 2 2
∆
.
Krome toho, sohlasno uslovyg (4)
µ µ ε γ
ε
µ( ) = ( ) ≤ ( )−
−( )G B G Gk k
n
k
k1 2
22
1∩ log ,
a potomu
ϕ ϕ
µ
ϕ ϕ µk k
k
k
G
G
x d x
k
− =
( )
( ( ) − ) ( )− −∫1 1
1
≤
≤
γ
ε
µ
ϕ ϕ µ δγ
ε
log
log
2
2
1
1 2
2
1
1
1
n
k
k
k
G
n
kG
x d x
k
−
−
−
−
( )
( ( ) − ) ( ) ≤
−
∫
y vsledstvye ub¥vanyq εk
ϕ ϕ ϕ ϕ ϕ ϕ δγ
εk k l l
l
k
n
k
k= ≤ + − ≤ +−
=
−∑1 1
1
1 2
2 1
log .
Sledovatel\no, pry n ≥ 2
S S C
k
C
k
k
n k
n
k
N
n
n
k
n
k
N
2 2
1
1 2
2
1
2 2
1
= ≤ ≤
+
⋅ ⋅
=
−
=
∑ ∑ϕ
ϕ δγ
ε
log
≤
≤ C
k
k
n
n
n
k
N
⋅ + ( + )
− −
−
=
∑2 2 1
2 0
1 2
1
1
ϕ δγ εlog
=
= C
k
k
k
n
n
n
k
N
⋅ + ( + )
− −
−
=
∑2 2
1
1
2 0
1 2
2
1
ϕ δγ εlog
≤
≤ C
k
n n
k
n
⋅ + ( + )
− −
=
∑2 2 1
1
1 2 0
1 2
1
ϕ δγ εlog
y
η ε δ ϕ δγ ε( ) ≤ ( + ) + ( + )+ − −
=
∑2 2 1
11
1 2 0
1 2
1
n n n
k
n
C C
k
log .
Poskol\ku
1
2 1
2k
dt
t
N N
k
N N
=
∑ ∫< = <log log
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1072 R. R. SALYMOV
y dlq ε0 ∈ ( 0, 2–
1
) y ε < εN
N N
N
< +
= =log log log2
0
2 2
1 1 1
ε ε ε
,
pry ε0 ∈ ( 0, δ0 ), δ0 = min ( e–
e, d0 ) y ε → 0
η ε δ ϕ δγ ε
ε
( ) ≤ ( + ) + ( + ) +
+ − −2 2 1 1
11
1 2 0
1 2
2 2
n n nC C log log log =
= O log log
1
ε
.
3. O hranyçnom povedenyy. V dal\nejßem R
n = R n ∪ { ∞ } budem ras-
smatryvat\ kak metryçeskoe prostranstvo so sferyçeskoj (xordal\noj) metry-
koj h ( x, y ) = | π ( x ) – π ( y ) |, hde π qvlqetsq stereohrafyçeskoj proekcyej R
n
na sferu S en
n
1
2
1
21+
, v R
n
+
1
:
h ( x, ∞ ) =
1
1 2+ x
, h ( x, y ) =
x y
x y
−
+ +1 12 2
, x ≠ ∞ ≠ y.
Takym obrazom, po opredelenyg h ( x, y ) ≤ 1 dlq vsex x y y ∈ R
n
.
Pust\ D ⊂ Rn
, n ≥ 2, — oblast\. ∂D naz¥vaetsq syl\no dostyΩymoj, esly
dlq nev¥roΩdenn¥x kontynuumov E y F v D
M E F D( )( )∆ , ; > 0,
y slabo ploskoj, esly dlq nev¥roΩdenn¥x kontynuumov E y F v D s E ∩
∩ F ≠ ∅
M E F D( )( )∆ , ; = ∞,
hde ∆( )E F D, ; — semejstvo vsex putej, soedynqgwyx E y F v D. Yzvestno,
çto lgbaq slabo ploskaq hranyca qvlqetsq syl\no dostyΩymoj (sm. lemmu 5.6 v
[10]).
Oblast\ G ⊂ X naz¥vaetsq lokal\no svqznoj v toçke x0 ∈ ∂G, esly x0
ymeet proyzvol\no mal¥e okrestnosty U v X takye, çto mnoΩestva U ∩ G
qvlqgtsq svqzn¥my.
Teorema. Pust\ G — oblast\ v n-rehulqrnom sverxu prostranstve ( X, d,
µ ), n ≥ 2, G ′ — oblast\ v R
n
y f : G → G ′ — Q -homeomorfyzm otnosy-
tel\no mer¥ µ. Esly oblast\ G lokal\no svqzna v toçke x0 ∈ ∂G y pry r <
< diam G udovletvorqet uslovyg (5), Q ∈ F M O
µ
( x0 ), a oblast\ ′G ymeet
syl\no dostyΩymug hranycu, to homeomorfyzm f prodolΩym v toçku x0 po
neprer¥vnosty v R
n
.
Dokazatel\stvo. PokaΩem, çto predel\noe mnoΩestvo E = C( x0
, f ) =
= { y ∈ R
n
: y = lim
k
kf x
→∞
( ), xk → x0
, xk ∈ G } sostoyt yz edynstvennoj toçky.
Zametym, çto E — kontynuum, tak kak oblast\ G lokal\no svqzna v toçke x0.
Dejstvytel\no,
E = lim sup
m
mf G
→∞
( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
O HRANYÇNOM POVEDENYY VLOÛENYJ METRYÇESKYX … 1073
hde Gm = G ∩ Um — nekotoraq monotonno ub¥vagwaq posledovatel\nost\ ob-
lastej s okrestnostqmy Um toçky x0 y d ( Gm ) → 0 pry m → ∞ (sm., napry-
mer, utverΩdenye I (9.12) v [17, s. 15]).
PredpoloΩym, çto kontynuum E — nev¥roΩdenn¥j. Pust\ x1 y x 2 ∈ G,
x1 ≠ x2
, d ( x1
, x0 ) < e– e
, y γ0 : [ 0, 1 ] → G — neprer¥vnaq kryvaq, soedynqgwaq
x1 y x2 v G. Zametym, çto K = γ0 ( [ 0, 1 ] ) — kompakt v G, kak neprer¥vn¥j
obraz kompakta [ 0, 1 ]. Takym obrazom, ε0 = dist ( x0
, K ) > 0 y ε0 < e– e
. Pust\ Γε
— semejstvo vsex putej, soedynqgwyx ßar Bε = { x ∈ X : d ( x, x0 ) < ε } y K v G,
ε ∈ ( 0, ε0 ). Tohda funkcyq
ρε ( x ) =
1
1
0
0
0 0
d x x
d x x
x G
x X G
( )
( )
∈
∈
, log
,
log
log
log
, ,
, \ ,
ε
ε
ε
ε
hde Gε = { x ∈ G : ε < d ( x, x0 ) < ε0 }, dopustyma dlq Γε y, sledovatel\no, v sylu
(3) y dokazannoj lemm¥
M f
c
n
( ) ≤
Γε
ε
ε
ε
log log
log
log
log
1
0
,
t. e. M f( )Γε → 0 pry ε → 0. S druhoj storon¥, M f( )Γε ≥ M0 = M ( ∆ ( f K, E;
G′ ) ), a sohlasno syl\noj dostyΩymosty hranyc¥ ∂G′ ymeem M0 > 0. Poluçen-
noe protyvoreçye oproverhaet predpoloΩenye.
Kombynyruq teoremu y sledstvye 1, poluçaem sledugwee utverΩdenye.
Sledstvye 2. Pust\ G — oblast\ v n-rehulqrnom sverxu prostranstve
( X, d, µ ) y f : G → G′ ⊂ Rn
— Q-homeomorfyzm otnosytel\no mer¥ µ s
lim
,
ε ε
µ
→ ( )
( ) ( )−∫
0
0
Q x d x
G x
< ∞.
Esly oblast\ G v toçke x0 ∈ ∂G lokal\no svqzna y udovletvorqet uslovyg
(5), a oblast\ ′G ymeet syl\no dostyΩymug hranycu, to homeomorfyzm f
prodolΩym v toçku x0 po neprer¥vnosty v R
n
.
Prymer. Pust\ M — rymanovo n-mernoe mnohoobrazye, n ≥ 2, s metrykoj
d y µ — n-mernaq xausdorfova mera na M. Po klassyçeskoj teoreme sravne-
nyq Byßopa polnoe mnohoobrazye M s neotrycatel\noj kryvyznoj Ryççy qv-
lqetsq n-rehulqrn¥m sverxu otnosytel\no rymanovoj mer¥ µ (sm., naprymer,
[14, s. 75; 18, 19, s. 123]).
Sledstvye 3. Pust\ G — oblast\ v polnom rymanovom mnohoobrazyy M
s neotrycatel\noj kryvyznoj Ryççy y f : G → G ′ ⊂ R
n
, n ≥ 2, — Q-homeomor-
fyzm otnosytel\no rymanovoj mer¥ µ . Esly oblast\ G v toçke x0 ∈ ∂G
lokal\no svqzna y udovletvorqet uslovyg (5), Q ∈ F M O
µ
( x0 ), a oblast\ ′G
ymeet syl\no dostyΩymug hranycu, to homeomorfyzm f prodolΩym v toçku
x0 po neprer¥vnosty v R
n
.
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| id | umjimathkievua-article-3370 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:16Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/72/df5001faf59bab49d3a29327c6a40b72.pdf |
| spelling | umjimathkievua-article-33702020-03-18T19:52:34Z On the boundary behavior of imbeddings of metric spaces into a Euclidean space O граничном поведении вложений метрических пространств в евклидово Salimov, R. R. Салімов, Р. Р. We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a boundary point. Досліджується гранична поведінка так званих Q-гомеоморфізмів відносно міри в деяких метричних просторах. Сформульовано низьку умов на функцію Q ( x ) і межу області, при яких будь-який Q-гомеоморфізм відносно міри допускає неперервне продовження в точку межі. Institute of Mathematics, NAS of Ukraine 2007-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3370 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 8 (2007); 1068–1074 Український математичний журнал; Том 59 № 8 (2007); 1068–1074 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3370/3483 https://umj.imath.kiev.ua/index.php/umj/article/view/3370/3484 Copyright (c) 2007 Salimov R. R. |
| spellingShingle | Salimov, R. R. Салімов, Р. Р. On the boundary behavior of imbeddings of metric spaces into a Euclidean space |
| title | On the boundary behavior of imbeddings of metric spaces into a Euclidean space |
| title_alt | O граничном поведении вложений метрических пространств в евклидово |
| title_full | On the boundary behavior of imbeddings of metric spaces into a Euclidean space |
| title_fullStr | On the boundary behavior of imbeddings of metric spaces into a Euclidean space |
| title_full_unstemmed | On the boundary behavior of imbeddings of metric spaces into a Euclidean space |
| title_short | On the boundary behavior of imbeddings of metric spaces into a Euclidean space |
| title_sort | on the boundary behavior of imbeddings of metric spaces into a euclidean space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3370 |
| work_keys_str_mv | AT salimovrr ontheboundarybehaviorofimbeddingsofmetricspacesintoaeuclideanspace AT salímovrr ontheboundarybehaviorofimbeddingsofmetricspacesintoaeuclideanspace AT salimovrr ograničnompovedeniivloženijmetričeskihprostranstvvevklidovo AT salímovrr ograničnompovedeniivloženijmetričeskihprostranstvvevklidovo |