Mappings with finite length distortion and Riemann surfaces
We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to...
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irk-123456789-1706182020-07-21T01:26:02Z Mappings with finite length distortion and Riemann surfaces Ryazanov, V.I. Volkov, S.V. Математика We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be a very simple condition on the integrability of the dilatations in the first power. Moreover, the domain of the mapping is assumed to be locally connected on the boundary and its range has a weakly flat boundary. The criteria for the continuous extension of the direct map pings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at the corresponding boundary point. A stronger (but simpler) one is that the mean value of the dilatation over infinitesimal disks centered at the corresponding boundary point is finite. The domain is again assumed to be locally connected on the boundary and its range has a strongly accessible boundary. We give also many other criteria for the continuous extension of the direct mappings to the boundary. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains are obtained. У термінах дилатацій доведено ряд критеріїв для неперервного та гомеоморфного продовження на границю відображень зі скінченним спотворенням довжини між областями на ріманових поверхнях. Критерієм для неперервного продовження обернених відображень на границю є дуже проста умова про інтегрованість дилатації в першому степені. При цьому область визначення відображення передбачається локально зв'язною на границі, а область значень — зі слабо плоскою границею. Критерії для неперервного продовження на границю прямих відображень мають набагато тоншу природу. Один із критеріїв полягає в існуванні мажоранти дилатації в класі функцій зі скінченним середнім коливанням, тобто таких, що мають кінцеве середнє відхилення від свого середнього значення над інфінітезимальними (нескінченно малими) колами з центром у відповідній граничній точці. Більш жорстка, але більш проста вимога полягає в тому, що середнє значення дилатації над інфінітезимальними колами з центром у відповідній граничній точці скінченне. Область визначення знову передбачається локально зв'язною на границі, а область значень — із сильно досяжною границею. Також наведено багато інших критеріїв неперервного продовження на границю прямих відображень. Як наслідки отримуємо відповідні критерії для гомеоморфного продовження на границю областей відображень зі скінченним спотворенням довжини. В терминах дилатаций доказан ряд критериев для непрерывного и гомеоморфного продолжения на границу отображений с конечным искажением длины между областями на римановых поверхностях. Критерием для непрерывного продолжения обратных отображений на границу оказывается очень простое условие об интегрируемости дилатаций в первой степени. При этом область определения отображения предполагается локально связной на границе, а область значений — со слабо плоской границей. Критерии для непрерывного продолжения на границу прямых отображений имеют гораздо более тонкую природу. Один из критериев состоит в существовании мажоранты дилатации в классе функций с конечным средним колебанием, т. е. имеющих конечное среднее отклонение от своего среднего значения над инфинитезимальными (бесконечно малыми) кругами с центром в соответствующей граничной точке. Более сильное, но более простое требование состоит в том, что среднее значение дилатации над инфинитезимальными кругами с центром в соответствующей граничной точке конечно. Область определения снова предполагается локально связной на границе, а область значений — с сильно достижимой границей. Также приведены многие другие критерии непрерывного продолжения на границу прямых отображений. В качестве следствий получаются соответствующие критерии для гомеоморфного продолжения в замыкание областей отображений с конечным искажением длины. 2020 Article Mappings with finite length distortion and Riemann surfaces. / V.I. Ryazanov, S.V. Volkov // Доповіді Національної академії наук України. — 2020. — № 6. — С. 7-14. — Бібліогр.: 14 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2020.06.007 http://dspace.nbuv.gov.ua/handle/123456789/170618 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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English |
| topic |
Математика Математика |
| spellingShingle |
Математика Математика Ryazanov, V.I. Volkov, S.V. Mappings with finite length distortion and Riemann surfaces Доповіді НАН України |
| description |
We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the
map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for
the continuous extension of the inverse mapping to the boundary is turned out to be a very simple condition on the
integrability of the dilatations in the first power. Moreover, the domain of the mapping is assumed to be locally
connected on the boundary and its range has a weakly flat boundary. The criteria for the continuous extension of
the direct map pings to the boundary have a much more refined nature. One of such criteria is the existence of a
majorant for the dilation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation
from its mean value over infinitesimal disks centered at the corresponding boundary point. A stronger (but simpler)
one is that the mean value of the dilatation over infinitesimal disks centered at the corresponding boundary point is
finite. The domain is again assumed to be locally connected on the boundary and its range has a strongly accessible
boundary. We give also many other criteria for the continuous extension of the direct mappings to the boundary.
As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion
to the closures of domains are obtained. |
| format |
Article |
| author |
Ryazanov, V.I. Volkov, S.V. |
| author_facet |
Ryazanov, V.I. Volkov, S.V. |
| author_sort |
Ryazanov, V.I. |
| title |
Mappings with finite length distortion and Riemann surfaces |
| title_short |
Mappings with finite length distortion and Riemann surfaces |
| title_full |
Mappings with finite length distortion and Riemann surfaces |
| title_fullStr |
Mappings with finite length distortion and Riemann surfaces |
| title_full_unstemmed |
Mappings with finite length distortion and Riemann surfaces |
| title_sort |
mappings with finite length distortion and riemann surfaces |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2020 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/170618 |
| citation_txt |
Mappings with finite length distortion and Riemann surfaces. / V.I. Ryazanov, S.V. Volkov // Доповіді Національної академії наук України. — 2020. — № 6. — С. 7-14. — Бібліогр.: 14 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT ryazanovvi mappingswithfinitelengthdistortionandriemannsurfaces AT volkovsv mappingswithfinitelengthdistortionandriemannsurfaces |
| first_indexed |
2025-07-15T05:50:57Z |
| last_indexed |
2025-07-15T05:50:57Z |
| _version_ |
1837690947814555648 |
| fulltext |
7ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 6: 7—14
Ц и т у в а н н я: Ryazanov V.I., Volkov S.V. Mappings with finite length distortion and Riemann surfaces.
Допов. Нац. акад. наук Укр. 2020. № 6. С. 7—14. https://doi.org/10.15407/dopovidi2020.06.007
1. Introduction. The present paper is a natural continuation of our previous papers [1-3], where
the reader can find the corresponding historic comments and a discussion of many definitions
and relevant results. The given papers were devoted to the theory of the boundary behavior of
mappings with finite distortion by Iwaniec.
Here, we will develop the theory of the boundary behavior of the so-called mappings with
finite length distortion first introduced in [4] for n , 2n� , see also Chapter 8 in [5]. As was
shown in [6], such mappings, generally speaking, are not mappings with finite distortion by
Iwaniec, because their first partial derivatives can be not locally integrable. At the same time,
this class is a natural generalization of the well-known classes of bi-Lipschitz mappings, as well as
isometries and quasi-isometries.
https://doi.org/10.15407/dopovidi2020.06.007
UDC 517.5
V.I. Ryazanov 1, 2, S.V. Volkov 3
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
2 Bogdan Khmelnytsky National University of Cherkasy
3 Donetsk National Technical University, Pokrovsk
E-mail: vl.ryazanov1@gmail.com, serhii.volkov@donntu.edu.ua
Mappings with finite length distortion and Riemann surfaces
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ
We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the
map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for
the continuous extension of the inverse mapping to the boundary is turned out to be a very simple condition on the
integrability of the dilatations in the first power. Moreover, the domain of the mapping is assumed to be locally
connected on the boundary and its range has a weakly flat boundary. The criteria for the continuous extension of
the direct map pings to the boundary have a much more refined nature. One of such criteria is the existence of a
majorant for the dilation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation
from its mean value over infinitesimal disks centered at the corresponding boundary point. A stronger (but simpler)
one is that the mean value of the dilatation over infinitesimal disks centered at the corresponding boundary point is
finite. The domain is again assumed to be locally connected on the boundary and its range has a strongly accessible
boundary. We give also many other criteria for the continuous extension of the direct mappings to the boundary.
As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion
to the closures of domains are obtained.
Keywords: Riemann surfaces, boundary behavior, mappings with finite length distortion, strongly accessible and
weakly flat boundaries.
8 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 6
V.I. Ryazanov, S.V. Volkov
2. Definitions and preliminary remarks. We assume that all mappings under consideration
are continuous. The previous definitions can be found in [1-3]. Here, we restrict ourselves in the
main by new conceptions.
Let us start from the main definitions in [4] adopted to the case of domains D in the complex
plane , see also Chapter 8 in [5]. It is said that a mapping :f D → is of finite metric distortion,
written f ∈ FMD, if f has the ( )N -property by Luzin with respect to the area in and
0 ( , ) ( , )l z f L z f< < ∞� a.e. (1)
where
, ,
| ( ) ( ) | | ( ) ( ) |
( , ) : , ( , ) : lim suplim inf
| | | |z D z D
f f z f f z
l z f L z f
z zζ→ ζ∈ ζ→ ζ∈
ζ − ζ −= =
ζ − ζ −
. (2)
Now, we say that a mapping :f D → has ( )L -property, if, for a.e. path γ in D , the path
fγ = γ is locally rectifiable, and |f γ has the (N)-property by Luzin with respect to the length
measure. Recall that a path γ in D is a mapping :y DΔ → , where Δ is an interval in . More-
over, it is said that a property holds for almost every (a.e.) path of a family, if the property fails
only for its subfamily of paths of conformal modulus zero, see the definition of the conformal
modulus on Riemann surfaces in [1–3].
We say also that a homeomorphism f between domains D and D∗ in is of finite length
distortion, written f ∈ FLD, if f ∈FMD and, moreover, f and 1f − have (L)-property. A special
case is bi-Lipschitz homeomorphisms for which the quantities in (1) are uniformly in the do-
main D separated from zero, as well as from infinity. Thus, homeomorphisms of finite length
distortion are a far reaching generalization of isometries and quasiisometries.
Remark 1. By Theorem 6.10 in [4] or Theorem 8.6 in [5], a homeomorphism f ∈FLD between
domains D and D∗ in satisfies the inequality
2( ) ( ) ( ) ( )
D
M f Q z z dm zΓ ⋅ρ∫� (3)
with fQ K= for any family Γ of paths γ in D and admρ∈ Γ, see [1—3] for definitions of the dila-
tation fK , the conformal modulus M of families of paths, and admissible functions : [0, ]Dρ → ∞ .
Homeomorphisms f between domains D and D∗ in the complex plane satisfying con-
ditions of the type (3) are called Q-homeomorphisms, see [7], and also Chapters 4—6 in [5].
Correspondingly to Remark 1, such homeomorphisms form a wider class of mappings than home-
omorphisms with finite length distortion.
Let us pass to the corresponding definitions on Riemann surfaces. So, let f be a homeo-
morphism between domains D and D∗ on Riemann surfaces S and ∗S . First of all, we say that f
is a mapping with finite length distortion, written f ∈ FLD, if f is so in charts of S and ∗S . In
view of properties of conformal mappings, namely, the (N)-properties by Luzin with respect to
area, as well as to length, and invariance of local rectifiable paths, see e.g. Theorem 5.6 in [8],
the definition is independent of the choice of charts. We also say that f is a local Q-homeomor-
phism for a measurable function : (0, )Q → ∞S , if (3) holds for any family Γ of paths γ in D lay-
ing inside an arbitrary prescribed chart U of the Riemann surface S .
9ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 6
Mappings with finite length distortion and Riemann surfaces
Remark 2. As is known, if a function : [0, ]Vρ → ∞ is admissible for a family A of paths α in
an open set V of the complex plane , then the function * 1 1( ) ( ( ))/ | ( ( )) |− −′ρ ζ = ρ ϕ ζ ϕ ϕ ζ is
admissible for the family :B A= ϕ of paths :β = ϕ α under every conformal mapping :Vϕ → ,
see again Theorem 5.6 in [8]. Thus, the right-hand side in inequality (3) is a conformal invariant,
because the Jacobian of ( )zϕ is equal to 2| ( ) |z′ϕ .
Proposition 1. Every homeomorphism f with finite length distortion between domains D and
D∗ on Riemann surfaces S and ∗S , correspondingly, is a local Q-homeomorphism with fQ K= .
Here and below, we assume that fK is extended by zero outside of D .
Proof. Let :g U → be a chart of the Riemann surface S . Since the space S is separable,
the open set D U∩ consists of a countable collection of its components kU every of which is
homeomorphic to the plane domain : ( )k kV g U= . Thus, every domain * : ( )k kU f U= is also ho-
meomorphic to the plane domains kV and, consequently, by the general Koebe principle, see
Section II.3 in [9], *
kU is a chart of the Riemann surface ∗S .
Note also that the path family Γ is split into a countable collection of mutually disjoint path
families kΓ lying in the domains kU . Hence, the path family * : fΓ = Γ is split also into a countable
collection of mutually disjoint path families :k kf∗Γ = Γ lying in the domains *
kU , i.e., in the cor-
responding charts of the Riemann surface ∗S . Thus, by Remark 1 in [1] and by Remarks 1 and 2
of the present paper, we obtain the desired conclusion.
3. The main lemma. Recall that the factor /GD of the unit disk D with a discrete group G
of fractional mappings of D onto itself without fixed points is a Riemann surface with charts from
the natural (locally homeomorphic) projection : /Gπ →D D , see Theorem 6.2.1 in [10].
Lemma 1. Let G be a discrete group of fractional maps of D onto itself with no fixed points,
and :f D D∗→ be a homeomorphism of finite length distortion between domains D and D∗ on
Riemann surfaces : /G=S D and ∗S , 0p D∈ .
Then there is 0( )pε such that the natural projection : /Gπ →D D is injective on a hyperbolic
disk 0 0 0: { : ( , ) ( )}B z h z z p= ∈ < εD , where 1
0 0( )z p−∈π , and
2( ( )) ( ) ( ) ( )f
D
M f K p p dh pΓ ξ∫� (4)
for families Γ of paths in 0( )D B∩π and measurable functions : [0, ]Dξ → ∞ , such that
( ) ( ) 1hp ds p
γ
ξ ∀γ∈Γ∫ � . (5)
Remark 3. By the Klein–Poincaré theorem on the uniformization, see II.3 in [9], an arbitrary
Riemann space S is conformally equivalent to the unit disk D factored by a discrete group G of
fractional mappings of D onto itself without fixed points, excepting the simplest cases of S that
are conformally equivalent to , , a ring, or a torus.
In the case of a torus, S is conformally equivalent to a factor /G with respect to a group G
of shifts in with 2 generators 1z z→ +ω and 2z z→ +ω , where 1ω and 2 \ {0}ω ∈ and
1 2Im / 0ω ω > . In this case, a fundamental domain F is a parallelogram whose sides are parallel
to 1ω and 2ω , and gluing its opposite sides just gives a torus. The metric and the area on the
surface /G in small coincide with the Euclidean ones, because the Euclidean metric and area
are invariant under shifts.
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 6
V.I. Ryazanov, S.V. Volkov
By the scheme of the proof below relations (4) are also valid for all the given special cases
with the Euclidean metric and area instead of hyperbolic ones. Later on, for the universality, we
keep the same notations in these cases, too.
Proof. By Section 2 in either [1] or [3], we may identify /GD with a fundamental set F in D
for G with the metric d defined by (2.10) in [1] that contains a fundamental Poincaré polygon
0
Pz for G centered at 1
0 0( )z p−∈π . Let us choose 0( ) 0pε > such that 0 0( , ) ( , )d z z h z z= for
0 0( , ) ( )d z z pε� and
0
0 0 0 0( ) min inf ( , ),sup ( , )
zP z D
p d z d z z
ς∈∂ ∈
⎡ ⎤
ε < δ := ζ⎢ ⎥
⎣ ⎦
.
Since 2( ) 2 | | /(1 | | )hds z dz z= − , we see that, for every ξ satisfying (5), ( ) | | 1z dz
γ
η∫ � ∀γ∈Γ ,
where
2
2 ( )
( ) :
1 | |
z
z
z
ξη =
−
, i.e., the function η is admissible for the family Γ of paths γ in 0( )D B∩π .
Moreover, since 2 2( ) 4 / (1 | | )dh z dxdy z= − , z x iy= + , we obtain that
2 2( ) ( ) ( ) ( ) ( ) ( )f f
D D
K z z dh z K z z dm zξ = η∫ ∫ , (6)
where ( ) :dm z dxdy= corresponds to the Lebesgue area in the plane . Thus, the conclusion of
Lemma 1 follows from Proposition 1.
Remark 4. In other words, the statement of Lemma 1 means that every homeomorphism f of
finite length distortion between domains on Riemann surfaces is a local fK -homeomorphism
with respect to the hyperbolic metric and the hyperbolic area. Note also that Riemann surfaces
are locally the so-called Ahlfors 2-regular spaces with the given metric and measure h , see
Theorem 7.2.2 in [10]. Hence, we may apply results in [11] on the boundary behavior of Q -home-
omorphisms in metric spaces with measures to homeomorphisms with finite length distortion
between domains on Riemann surfaces. It makes possible us, in comparison with [12], to formu-
late new results in terms of the metric and measure h but not in terms of local coordinates on
arbitrary Riemann surfaces, see [1] or [3] and the end of Remark 3 on notations.
4. On the extending of the inverse mapping to the boundary. By contrast with the direct
mappings, see the next section, we have the following simple criterion for the inverse mappings,
see definitions in [1-3].
Theorem 1. Let S and ∗S be Riemann surfaces, D and D∗ be domains in S and ∗S , cor-
respondingly, D∂ ⊂ S and D∗ ∗∂ ⊂ S , D be locally connected on its boundary, and let D∗∂ be
weakly flat. Suppose that :f D D∗→ is a homeomorphism of finite length distortion with 1
locfK L∈ .
Then the mapping 1 :g f D D− ∗= → can be extended by continuity to a mapping :g D D∗ → .
Proof. By the Uryson theorem, S is a metrizable space. Hence, the compactness of S is
equivalent to its sequential compactness, and the closure D is a compact subset of S . Thus, the
conclusion of Theorem 1 is true by Theorem 6.1 in [11] and by Lemma 1 and Remarks 3–4.
5. On the extending of the direct mappings to the boundary. As was already established
in the plane, no degree of integrability of Q leads to the extension of direct mappings of Q -home-
omorphisms to the boundary, see Proposition 6.3 in [5]. The corresponding criterion for FLD
given below is much more refined. Namely, by Lemma 5.1 in [11], as well as Lemma 1 and Re-
marks 3-4 above, we obtain the following, see definitions in [1-3].
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 6
Mappings with finite length distortion and Riemann surfaces
Lemma 2. Let S and ∗S be Riemann surfaces, D and D∗ be domains in S and ∗S , D∂ ⊂ S ,
D∗ ∗∂ ⊂ S , D be locally connected at a point 0p D∈∂ . Suppose that :f D D∗→ is a homeomor-
phism of finite length distortion, D∗∂ is strongly accessible at least at one point in the cluster set
0( , )C p f , and
2 2
, 0 ,0 0 0
( , )0 0
( ) ( ( , )) ( ) ( ( ))f p p
h p p
K p h p p dh p o Iε ε
ε < <ε
⋅ψ = ε∫ as 0ε → (7)
for some 0 0ε > , where ,0
( )p tεψ is a family of nonnegative measurable functions on (0, )∞ such that
0
, , 00 0 0
0 ( ) : ( ) (0, )p pI t dt
ε
ε ε
ε
< ε = ψ < ∞ ∀ε ∈ ε∫ . (8)
Then f is extended by continuity to the point 0p , and 0( )f p D∗∈∂ .
Note that conditions (7), (8) imply that ,0 0
( )pI ε ε →∞ as 0ε → , and 0ε can be chosen ar-
bitrarily small with keeping (7), (8).
Lemma 2 makes it possible to derive a series of criteria on the continuous extension of
mappings with finite length distortion to the boundary, for instance:
Theorem 2. Let S, ∗S be Riemann surfaces, D , D∗ be domains on S, ∗S , D∂ ⊂ S , D∗ ∗∂ ⊂ S , D
be locally connected on D∂ , D∗∂ be strongly accessible. Suppose that :f D D∗→ is a homeo-
morphism in FLD and, for all 0p D∈∂ ,
( )0
0
00 ( , )0
, || || ( , ) : ( ) ( )
|| || ( , )
p
f f h
f h p p r
dr
K p r K p ds p
K p r
ε
=
= ∞ =∫ ∫ . (9)
Then the mapping f is extended by continuity to D and *( )f D D∂ = ∂ .
Proof. Indeed, setting 00
( ) 1/ || || ( , )p ft K p tψ = for all 0(0, )t ∈ ε , 0 0: ( )pε = ε , and
0
( ) 1p tψ =
for all 0( , )t ∈ ε ∞ , we obtain from condition (9) that
2 2
0 , ,0 0 0 0 0
( , )0 0
( ) ( ( , )) ( ) ( ) ( ( )) 0f p p p
h p p
K p h p p d h p I o I asε ε
ε< <ε
⋅ ψ = ε = ε ε →∫ ,
where, in view of the condition ( ) [1, )fK p ∈ ∞ a.e. in D ,
0
,0 0 0
0 ( ) : ( )p pI t dt
ε
ε
ε
< ε = ψ < ∞∫ .
Thus, the first conclusions of Theorem 2 follow from Lemma 2. The second conclusion of
Theorem 2 follows, for instance, from Proposition 2.5 in [11], see also Proposition 13.5 in [5].
Corollary 1. In particular, the conclusion of Theorem 2 holds, if
0
1
( ) log
( , )fK p O
h p p
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
as 0 0p p p D→ ∀ ∈∂ (10)
or, more generally,
0
1
( ) logpk O⎛ ⎞ε = ⎜ ⎟ε⎝ ⎠
as 00 p Dε → ∀ ∈∂ , (11)
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 6
V.I. Ryazanov, S.V. Volkov
where
0
( )pk ε is the mean value of the function fK over the circle 0( , )h p p = ε .
By Theorem 3.1 in [13], we have the following consequence of Theorem 2.
Theorem 3. Under hypotheses of Theorem 2, suppose, instead of (9), that
( ( )) ( )f
U
K p dh pΦ < ∞∫ (12)
in a neighborhood U of D∂ , where : R R
+ +
Φ → is a nondecreasing convex function with the con-
dition
1
, (0)
( )
d∞
−
δ
τ = ∞ δ > Φ
τΦ τ∫ . (13)
Then the mapping f is extended by continuity to D and *( )f D D∂ = ∂ .
Remark 5. Note that, by Theorem 5.1 and Remark 5.1 in [14], condition (13) is not only
sufficient, but also necessary for the continuous extension of all mappings f of finite length
distortion with integral restrictions of the form (12) to the boundary. Note also that, by Theo-
rem 2.1 in [13], condition (13) is equivalent to a series of other conditions, and the most interes-
ting of them is
2
log ( ) 0
dt
t for some
t
∞
Δ
Φ = +∞ Δ >∫ . (14)
Corollary 2. In particular, the conclusion of Theorem 3 holds, if, for some 0α > ,
( )
( )
K pf
U
e dh p
α
< ∞∫ . (15)
The next statement holds by Remarks 3—4 and Lemma 2 with ( ) 1/t tψ = .
Theorem 4. Under the hypotheses of Theorem 2, if, instead of (9),
2
2
0( , )0 0
( ) 1
( ) log
( , )
f
h p p
dh p
K p o
h p pε< <ε
⎛ ⎞⎡ ⎤⎜ ⎟= ⎢ ⎥⎜ ⎟ε⎣ ⎦⎝ ⎠
∫ as 00 p Dε → ∀ ∈∂ , (16)
then the mapping f is extended by continuity to D and *( )f D D∂ = ∂ .
Following [11], see also Section 13.4 in [5], we say that a function : Rϕ →S has finite mean
oscillation at a point 0p ∈S , written 0F ( )MO pϕ∈ , if
0( , )0 0
1
| ( ) | ( )lim sup
| ( , ) | B p
p dh p
B p εεε→
ϕ −ϕ < ∞
ε ∫ , (17)
where εϕ is the mean value of ϕ over the disk 0 0( , ) { : ( , ) }B p p h p pε = ∈ < εS .
By Remarks 3-4 and Lemma 2 with the choice ,0
1
( ) 1/ logp t t
tεψ ≡ , in view of Lemma 4.1 and
Remark 4.1 in [11], see also Lemma 13.2 and Remark 13.3 in [5], we come to the next
Theorem 5. Under the hypotheses of Theorem 2, if, instead of (9),
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 6
Mappings with finite length distortion and Riemann surfaces
0 0( ) ( ) ( ) , :fK p Q p FMO p p D for some Q R+∈ ∀ ∈∂ →� S . (18)
Then the mapping f is extended by continuity to D and *( )f D D∂ = ∂ .
By Corollary 4.1 in [11], see also Corollary 13.3 in [5], we have the following proposition.
Corollary 3. In particular, the conclusion of Theorem 5 holds, if
0
0( , )0 0
1
( ) ( )lim sup
( , ) fB p
K p dh p p D
B p εε→
< ∞ ∀ ∈∂
ε ∫ . (19)
Remark 6. Note that Lemma 2 allows the pointwise analysis. Note also that, combining the
above results on the continuous extension with Theorem 1, we come to the corresponding results
on the homeomorphic extension of mappings with finite length distortion to the boundary.
However, we do not formulate them in the explicit form here because of the restrictions on the
volume of the paper.
This work was partially supported by grants of Ministry of Education and Science of Ukraine,
project number is 0119U100421.
REFERENCES
1. Volkov, S. V. & Ryazanov, V. I. (2015). On the boundary behavior of mappings in the class 1, 1
locW on Rieman-
nian surfaces. Trudy Instituta Prikladnoi Matematiki i Mehaniki NAN Ukrainy, 29, pp. 34-53 (in Russian).
2. Volkov, S. V. & Ryazanov, V. I. (2016). Toward a theory of the boundary behavior of mappings of Sobolev
class on Riemann surfaces. Dopov. Nac. akad. nauk Ukr., No. 10, pp. 5-9. https://doi.org/10.15407/dopovidi
2016.10.005
3. Ryazanov, V. & Volkov, S. (2017). On the boundary behavior of mappings in the class 1, 1
locW on Riemann sur-
faces. Complex Anal. Oper. Theory, 11, No. 7, pp. 1503-1520. https://doi.org/10.1007/s11785-016-0618-4
4. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2004). Mappings with finite length distortion. J. Anal.
Math., 93, pp. 215-236. https://doi.org/10.1007/BF02789308
5. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2009). Moduli in modern mapping theory. Springer
Monographs in Mathematics. New York: Springer. https://doi.org/10.1007/978-0-387-85588-2
6. Kovtonyuk, D., Petkov, I. & Ryazanov, V. (2017). Prime ends in theory of mappings with finite distortion in
the plane. Filomat, 31, No. 5, pp. 1349-1366. https://doi.org/10.2298/FIL1705349K
7. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2005). On Q-homeomorphisms. Ann. Acad. Sci. Fenn.
Math., 30, No. 1, pp. 49-69.
8. Väisälä, J. (1971). Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics,
Vol. 229. Berlin, New York: Springer.
9. Krushkal’, S. L., Apanasov, B. N. & Gusevskii, N. A. (1986). Kleinian groups and uniformization in examples
and problems. Translations of Mathematical Monographs, Vol. 62. Providence, RI: AMS.
10. Beardon, A. F. (1983). The geometry of discrete groups. Graduate Texts in Matheamatics, Vol. 91. New York:
Springer.
11. Ryazanov, V. & Salimov, R. (2007). Weakly flat spaces and boundaries in the theory of mappings. Ukrainian
Math. Bull., 4, No. 2, pp. 199-234.
12. Volkov, S. V. & Ryazanov, V. I. (2019). On mappings of finite length distortion on Riemannian surfaces. Trudy
Instituta Prikladnoi Matematiki i Mehaniki NAN Ukrainy, 33, pp. 1-16 (in Ukrainian).
13. Ryazanov, V., Srebro, U. & Yakubov, E. (2010). On integral conditions in the mapping theory. Math. Sci. J.,
173, No. 4, pp. 397-407. https://doi.org/10.1007/s10958-011-0257-2
14. Kovtonyuk, D. & Ryazanov, V. (2008). On the theory of mappings with finite area distortion. J. Anal. Math.,
104, pp. 291-306. https://doi.org/10.1007/s11854-008-0025-5
Received 01.03.2020
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 6
V.I. Ryazanov, S.V. Volkov
В.I. Рязанов 1, 2, С.В. Волков 3
1 Інститут прикладної математики і механіки НАН України, Слов’янськ
2 Черкаський національний університет ім. Богдана Хмельницького
3 Донецький національний технічний університет, Покровськ
E-mail: vl.ryazanov1@gmail.com, serhii.volkov@donntu.edu.ua
ВІДОБРАЖЕННЯ ЗІ СКІНЧЕННИМ СПОТВОРЕННЯМ
ДОВЖИНИ ТА РІМАНОВІ ПОВЕРХНІ
У термінах дилатацій доведено ряд критеріїв для неперервного та гомеоморфного продовження на грани-
цю відображень зі скінченним спотворенням довжини між областями на ріманових поверхнях. Критерієм
для неперервного продовження обернених відображень на границю є дуже проста умова про інтегрова-
ність дилатації в першому степені. При цьому область визначення відображення передбачається локально
зв’язною на границі, а область значень — зі слабо плоскою границею. Критерії для неперервного продо-
вження на границю прямих відображень мають набагато тоншу природу. Один із критеріїв полягає в іс-
нуванні мажоранти дилатації в класі функцій зі скінченним середнім коливанням, тобто таких, що мають
кінцеве середнє відхилення від свого середнього значення над інфінітезимальними (нескінченно малими)
колами з центром у відповідній граничній точці. Більш жорстка, але більш проста вимога полягає в тому,
що середнє значення дилатації над інфінітезимальними колами з центром у відповідній граничній точці
скінченне. Область визначення знову передбачається локально зв’язною на границі, а область значень — із
сильно досяжною границею. Також наведено багато інших критеріїв неперервного продовження на грани-
цю прямих відображень. Як наслідки отримуємо відповідні критерії для гомеоморфного продовження на
границю областей відображень зі скінченним спотворенням довжини.
Ключові слова: ріманові поверхні, відображення зі скінченним спотворенням довжини, сильно досяжні й
слабо плоскі границі.
В.И. Рязанов 1, 2, С.В. Волков 3
1 Институт прикладной математики и механики НАН Украины, Славянск
2 Черкаcский национальный университет им. Богдана Хмельницкого
3 Донецкий национальный технический университет, Покровск
E-mail: vl.ryazanov1@gmail.com, serhii.volkov@donntu.edu.ua
ОТОБРАЖЕНИЯ С КОНЕЧНЫМ ИСКАЖЕНИЕМ
ДЛИНЫ И РИМАНОВЫ ПОВЕРХНОСТИ
В терминах дилатаций доказан ряд критериев для непрерывного и гомеоморфного продолжения на гра-
ницу отображений с конечным искажением длины между областями на римановых поверхностях. Кри-
терием для непрерывного продолжения обратных отображений на границу оказывается очень простое
условие об интегрируемости дилатаций в первой степени. При этом область определения отображения
предполагается локально связной на границе, а область значений — со слабо плоской границей. Критерии
для непрерывного продолжения на границу прямых отображений имеют гораздо более тонкую природу.
Один из критериев состоит в существовании мажоранты дилатации в классе функций с конечным сред-
ним колебанием, т. е. имеющих конечное среднее отклонение от своего среднего значения над инфини-
тезимальными (бесконечно малыми) кругами с центром в соответствующей граничной точке. Более
сильное, но более простое требование состоит в том, что среднее значение дилатации над инфинитези-
мальными кругами с центром в соответствующей граничной точке конечно. Область определения снова
предполагается локально связной на границе, а область значений — с сильно достижимой границей. Так-
же приведены многие другие критерии непрерывного продолжения на границу прямых отображений.
В качестве следствий получаются соответствующие критерии для гомеоморфного продолжения в замы-
кание областей отображений с конечным искажением длины.
Ключевые слова: римановы поверхности, отображения с конечным искажением длины, сильно дости-
жимые и слабо плоские границы.
|