Finite mean oscillation on Finsler manifolds
We study functions of the finite mean oscillation in Finsler spaces with respect to the boundary behavior of ring Q-homeomorphisms.
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irk-123456789-1265382017-11-27T03:02:38Z Finite mean oscillation on Finsler manifolds Afanas’eva, E.S. Математика We study functions of the finite mean oscillation in Finsler spaces with respect to the boundary behavior of ring Q-homeomorphisms. Вивчаються функції скінченного середнього коливання у фінслерових просторах відносно граничної поведінки кільцевих Q-гомеоморфізмів. Изучаются функции конечного среднего колебания в финслеровых пространствах относительно граничного поведения кольцевых Q-гомеоморфизмов. 2017 Article Finite mean oscillation on Finsler manifolds / E.S. Afanas’eva // Доповіді Національної академії наук України. — 2017. — № 3. — С. 14-17. — Бібліогр.: 11 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2017.03.014 http://dspace.nbuv.gov.ua/handle/123456789/126538 517.5 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Математика Математика Afanas’eva, E.S. Finite mean oscillation on Finsler manifolds Доповіді НАН України |
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We study functions of the finite mean oscillation in Finsler spaces with respect to the boundary behavior of ring Q-homeomorphisms. |
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Article |
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Afanas’eva, E.S. |
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Afanas’eva, E.S. |
| author_sort |
Afanas’eva, E.S. |
| title |
Finite mean oscillation on Finsler manifolds |
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Finite mean oscillation on Finsler manifolds |
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Finite mean oscillation on Finsler manifolds |
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Finite mean oscillation on Finsler manifolds |
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Finite mean oscillation on Finsler manifolds |
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finite mean oscillation on finsler manifolds |
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Видавничий дім "Академперіодика" НАН України |
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2017 |
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Математика |
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http://dspace.nbuv.gov.ua/handle/123456789/126538 |
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Finite mean oscillation on Finsler manifolds / E.S. Afanas’eva // Доповіді Національної академії наук України. — 2017. — № 3. — С. 14-17. — Бібліогр.: 11 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT afanasevaes finitemeanoscillationonfinslermanifolds |
| first_indexed |
2025-07-09T05:13:30Z |
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2025-07-09T05:13:30Z |
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1837145012573306880 |
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14 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 3
doi: https://doi.org/10.15407/dopovidi2017.03.014
UDC 517.5
E.S. Afanas’eva
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk
E-mail: es.afanasjeva@yandex.ru
Finite mean oscillation on Finsler manifolds
Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskii
We study functions of the finite mean oscillation in Finsler spaces with respect to the boundary behavior of ring Q-ho-
meo morphisms.
Keywords: Finsler manifolds, FMO class functions, ring Q-homeomorphisms.
In this article, we continue our study of mappings on Finsler manifolds ( , )nM Ф started in [1].
For historical remarks, we refer to [2]. Recall some needed definitions. By a domain in the topo-
logical space T, we mean an open linearly connected set. A domain D is called locally connected
at a point 0x D∈∂ , if, for any neighborhood U of 0x , there is a neighborhood V U⊆ of 0x such
that V D∩ is connected (cf. [3]). Similarly, we say that a domain D is locally linearly connected at
a point 0x D∈∂ if, for any neighborhood U of 0x , there exists a neighborhood V U⊆ of 0x such
that V D∩ is linearly connected. Recall that the n-dimensional topological manifold nM means
a Hausdorff topological space with countable base such that every point has a neighborhood ho-
meomorphic to nR . The manifold of the class rC with 1r � is called smooth.
Let D denote a domain in the Finsler space ( , )nM Ф , 2,n � and let n n
xTM T M= ∪ be a
tangent bundle of ( , )nM Ф , nx M∀ ∈ . By a Finsler manifold ( , )nM Ф , 2,n � we mean a smooth
manifold of the class C∞ with defined Finsler structure ( , )Ф x ξ , where ( , )Ф x ξ : nTM R+→ is a
function satisfying the following conditions:
1) Ф∈C∞ ( \ {0})nTM ;
2) 0a∀ > hold ( , )Ф x a aξ = ( , )Ф x ξ and ( , ) 0Ф x ξ > for 0ξ ≠ ;
3) the n n× Hessian matrix ijg
2 21 ( , )
( , )
2 i j
Ф x
x
∂ ξξ =
∂ξ ∂ξ
is positive definite at every point of
\ {0}nTM (cf. [4]).
By the geodesic distance ( , )Фd x y , we mean the infimum of lengths of piecewise-smooth
curves joining x and y in ( , )nM Ф , 2n � . It is well known that, for such metric, only two axioms
of metric spaces hold, namely the identity and triangle inequality axioms. Therefore, the Finsler
manifold provides a quasimetric space, for which the symmetry axiom fails.
© E.S. Afanas'eva, 2017
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 3
Finite mean oscillation on Finsler manifolds
Remark 1. Later, we consider a Finsler structure of the type
1
( , ) ( ( , ) ( , ))
2
Ф x Ф x Ф xξ = ξ + −ξ� ,
thereby obtaining a Finsler manifold ( , )nM Ф� with symmetrized (reversible) function Ф� . Clear-
ly, if Ф� is reversible, then the induced distance function Фd � is reversible, i.e., ( , ) ( , )Ф Фd x y d y x=� � ,
for all pairs of points ,x y∈ nM , see [5]. It is also known that the reversible Finsler metric coin-
cides with the Riemannian one, see, e.g., [6]. Therefore, in our further discussion, we can rely on
the results of [2].
Later, : [ , ] na b Mγ → is a piecewise-smooth curve, and ( )x t is its parametrization. An element
of length in ( , )nM Ф� , 2,n � is defined as a differential of the path for an infinitesimal measured
part of a curve Dγ ∈ by
2
, 1
n
Ф
i j
ds
=
= ∑� ( , )ij i jg x d dξ η η ;
see, e.g., [7]. So, the distance Фds � in the Finsler space, as in the case of a Riemannian space, is
determined by a metric tensor. Using the quadratic form Фds � , we determine the length of Dγ ⊂
by
( )Ф Фs ds
γ
γ = ∫� �
2
1
( , )
t
t
Ф x dx dt= ∫ � ,
see, e.g., [8, 9]. The invariance of this integral requires above-given restrictions 2—3 on the Lagran-
gian Ф� ( , )x dx .
Following [10], in view of Remark 1, an element of volume on the Finsler manifold is defined
by 1( ) det ( , ) .n
ijФd x g x dx dxσ = ξ� … It is known that the volume in the Finsler space coincides
with its Hausdorff measure induced by the metric ( , )Фd x y , if ( , )Ф x ξ is an invertible function,
see, e.g., [5].
Let Γ be a family of curves in a domain D. By the family of curves Γ, we mean a fixed set of
curves ,γ and, for an arbitrary mapping : n nf M M ∗→ , ( ) : { | }f fΓ = γ γ∈Γ� . The modulus of the
family Γ is defined by
( )
adm
: inf ( ) ( )n
Ф
D
M x d x
ρ∈ Γ
Γ = ρ σ∫ � ,
where the infimum is taken over all nonnegative Borel functions ρ such that the condition
Ф
γ
ρ∫ � ( , )x dx 1Фds
γ
= ρ∫ � �
holds for any curve γ ∈Γ . The functions ρ satisfying this condition are called admissible for Γ, cf. [4].
Later, for sets A, B, and C from ( , )nM Ф� , 2,n � by ( , ; )A B CΔ , we denote a set of all curves
: [ , ] na b Mγ → , which join A and B in C, i.e. ( )a Aγ ∈ , ( )b Bγ ∈ , and ( )t Cγ ∈ for all ( , )t a b∈ .
By Remark 1, one can apply the following well-known facts: Proposition 1 and Remark 1 in
[2]. Thus, we assume that the geodesic spheres 0( , )S x r , geodesic balls 0( , )B x r , and geodesic
rings 0 1 2( , , )A A x r r= lie in a normal neighborhood of the point 0x .
Let D and D' be domains on the Finsler manifolds ( , )nM Ф� and ( , )nM Ф∗ ∗
� , 2,n � respective-
ly, and let : (0, )nQ M → ∞ be a measurable function. We say that a homeomorphism : 'f D D→ is
16 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 3
E.S. Afanas’eva
the ring Q-homeomorphism at a point 0x D∈ , if
0 1 0( ( ( ), ( ); ) ( ) ( ( , )) ( )
A D
M f C f C D Q x d x x d xα
∩
Δ ⋅η μ′ ∫� (1)
holds for any geodesic ring 0 0( , , )A A x= ε ε , 00 ,< ε < ε any two continua (compact connected
sets) 0 0 1( , )C B x r D⊂ ∩ and 1 0 2\ ( , )C D B x r⊂ , and each Borel function 1 2: ( , ) [0, ]r rη → ∞ such
that
2
1
( ) 1
r
r
r drη∫ � .
We say that f is a ring Q-homeomorphism in D, if (1) holds for all points 0x D∈ .
We say that the boundary of the domain D is weakly flat at a point 0x D∈∂ , if, for any
number 0P > and any neighborhood U of 0x , there exists a neighborhood V U⊂ such that
( ( , ; ))M E F D PΔ � for any continua E and F in D intersecting U∂ and .V∂ We also say that
the boundary D is strongly accessible at a point 0x D∈∂ , if, for any neighborhood U of 0x ,
there are a compactum U of E D⊂ , a neighborhood V U⊂ of 0x , and a number 0δ > such that
( ( , ; ))M E F DΔ δ� for any continuum F in D intersecting U∂ and .V∂ The boundary of D is
called strongly accessible and weakly flat, if it has the corresponding property at every its point,
cf. [11].
Similarly to [11], we say that a function : nMφ → R has the finite mean oscillation at a point
0x ∈ nM , abbr. 0FMO( )xφ∈ , if
0
0 0 ( , )
1
| ( ) | ( ) < ,lim
( ( , )) Ф
Ф B x
x d x
B x ε
ε→ ε
φ − φ σ ∞
σ ε ∫ �
�
�
where
0
0 ( , )
1
= ( ) ( )
( ( , )) Ф
Ф B x
x d x
B xε
ε
φ φ σ
σ ε ∫ �
�
�
is the mean value of the function ( )xφ over the 0( , )B x ε with respect to the measure Фσ � .
Theorem 1. Let D be locally connected at a point 0x D∈∂ , let D∂ ′ be strongly accessible, and
let the closure D′ be compact. If 0FMO( )Q x∈ , then any ring Q-homeomorphism :f D D→ ′ can
be continued to the point 0x by continuity on ( , )nM Ф∗ ∗
� .
Corollary 1. Let D be locally connected at the point 0x D∈∂ , let D∂ ′ be strongly accessible,
and let D′ be compact. If
0
0 0 ( , )
1
( ) < ,lim
( ( , )) Ф
Ф B x
Qd x
B xε→ ε
σ ∞
σ ε ∫ �
�
any ring Q-homeomorphism :f D D→ ′ can be continued to the point 0x by continuity on ( , )nM Ф∗ ∗
� .
Theorem 2. Let D be locally connected on the boundary, let D∂ ′ be strongly accessible, and let
D′ be compact. If Q belongs to FMO, then any ring Q-homeomorphism :f D D→ ′ admits a con-
tinuous continuation :f D D→ ′ .
Theorem 3. Let D be locally connected on the boundary, let D∂ ′ be weakly flat, and let D
and D′ be compact. If Q belongs to FMO, then any ring Q-homeomorphism :f D D→ ′ admits the
continuation to the homeomorphism :f D D→ ′.
17ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 3
Finite mean oscillation on Finsler manifolds
REFERENCES
1. Afanas’eva, E. S. (2016). Boundary behavior of Q-homeomorphisms on Finsler spaces. Ukr. Math. Bull., 214,
No. 2, pp. 161-171.
2. Afanas’eva, E. S. (2012). Boundary behavior of ring Q-homeomorphisms on Riemannian manifolds. Ukr.
Math. J., 63, No. 10, pp. 1479-1493. doi: https://doi.org/10.1007/s11253-012-0594-4.
3. Kuratowski, K. (1968). Topology. Vol. II. New York, London: Acad. Press.
4. Dymchenko, Yu. V. (2014). A Relation Between the Condenser Capacity and the Module of Separating Sur-
faces in Finsler Spaces. J. Math. Sci., 200, Iss. 5, pp. 559-567.
5. Cheng, X., & Shen, Z. (2012). Finsler geometry. An approach via Randers spaces. Heidelberg: Springer.
6. Bogoslovsky, G. Yu. (2009). Finsler geometry and the theory of relativity. Sb. nauch. trudov RNOTS “Logos”,
Iss. 4, pp. 169-177 (in Russian).
7. Rutz, S. F., & Paiva, F. M. (2000). Gravity in Finsler spaces. Finslerian geometries. Fundamental Theories of
Physics. Vol. 109, pp. 223-244. Dordrecht: Kluwer Acad. Publ.
8. Dymchenko, Yu. V. (2009). Equality of the capacity and the modulus of a condenser in Finsler spaces, 85,
Iss. 3, pp. 566-573. doi: https://doi.org/10.1134/S0001434609030274.
9. Zhotikov, V. G. (2009). Finsler geometry (according to Wagner) and the equations of the motion in the rela-
tivistic dynamics. Proceedings XV Int. Sci. Meeting PIRT—2009, pp. 133-144. Moscow.
10. Rund, H. (1959). The differential geometry of Finsler spaces. Die Grundlehren der Mathematischen Wis-
senschaften. Bd. 101. Berlin, Göttingen, Heidelberg: Springer.
11. Martio, O., Ryazanov, V., Srebro, U., & Yakubov, E. (2009). Moduli in Modern Mapping Theory. Springer
Mono graphs in Mathematics. New York: Springer.
Received 19.11.2016
О.С. Афанасьєва
Інститут прикладної математики і механіки НАН України, Слов’янськ
E-mail: es.afanasjeva@yandex.ru
СКІНЧЕННЕ СЕРЕДНЄ КОЛИВАННЯ У ФІНСЛЕРОВИХ МНОГОВИДАХ
Вивчаються функції скінченного середнього коливання у фінслерових просторах відносно граничної по-
ведінки кільцевих Q-гомеоморфізмів.
Ключові слова: фінслерові многовиди, функції класу FMO, кільцеві Q-гомеоморфізми.
Е.С. Афанасьева
Институт прикладной математики и механики НАН Украины, Славянск
E-mail: es.afanasjeva@yandex.ru
КОНЕЧНОЕ СРЕДНЕЕ КОЛЕБАНИЕ НА ФИНСЛЕРОВЫХ МНОГООБРАЗИЯХ
Изучаются функции конечного среднего колебания в финслеровых пространствах относительно гранич-
ного поведения кольцевых Q-гомеоморфизмов.
Ключевые слова: финслеровы многообразия, функции класса FMO, кольцевые Q-гомеоморфизмы.
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