Prime ends in the mapping theory on the Riemann surfaces
It is proved criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1693152020-06-11T01:26:25Z Prime ends in the mapping theory on the Riemann surfaces Ryazanov, V.I. Volkov, S.V. It is proved criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory. 2017 Article Prime ends in the mapping theory on the Riemann surfaces / V.I. Ryazanov, S.V. Volkov // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 103-125. — Бібліогр.: 37 назв. — англ. 1810-3200 2010 MSC. Primary 31A05, 31A20, 31A25, 31B25, 35Q15; Secondary 30E25, 31C05, 34M50, 35F45. http://dspace.nbuv.gov.ua/handle/123456789/169315 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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It is proved criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory. |
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Ryazanov, V.I. Volkov, S.V. |
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Ryazanov, V.I. Volkov, S.V. Prime ends in the mapping theory on the Riemann surfaces Український математичний вісник |
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Ryazanov, V.I. Volkov, S.V. |
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Ryazanov, V.I. |
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Prime ends in the mapping theory on the Riemann surfaces |
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Prime ends in the mapping theory on the Riemann surfaces |
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Prime ends in the mapping theory on the Riemann surfaces |
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Prime ends in the mapping theory on the Riemann surfaces |
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Prime ends in the mapping theory on the Riemann surfaces |
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prime ends in the mapping theory on the riemann surfaces |
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Інститут прикладної математики і механіки НАН України |
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Prime ends in the mapping theory on the Riemann surfaces / V.I. Ryazanov, S.V. Volkov // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 103-125. — Бібліогр.: 37 назв. — англ. |
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Український математичний вiсник
Том 14 (2017), № 1, 103 – 125
Prime ends in the mapping theory
on the Riemann surfaces
Vladimir Ryazanov, Sergei Volkov
(Presented by V. Ya. Gutlyanskii)
Abstract. It is proved criteria for continuous and homeomorphic
extension to the boundary of mappings with finite distortion between
domains on the Riemann surfaces by prime ends of Caratheodory.
2010 MSC. Primary 31A05, 31A20, 31A25, 31B25, 35Q15; Secondary
30E25, 31C05, 34M50, 35F45.
Key words and phrases. Prime ends, Riemann sufaces, mappings of
finite distortion, boundary behavior, Sobolev classes.
1. Introduction
The theory of the boundary behavior in the prime ends for the map-
pings with finite distortion has been developed in [12] for the plane do-
mains and in [15] for the spatial domains. The pointwise boundary behav-
ior of the mappings with finite distortion in regular domains on Riemann
surfaces was recently studied by us in [30] and [31]. Moreover, the prob-
lem was investigated in regular domains on the Riemann manifolds for
n ≥ 3 as well as in metric spaces, see e.g. [1] and [34]. It is necessary to
mention also that the theory of the boundary behavior of Sobolev’s map-
pings has significant applications to the boundary value problems for the
Beltrami equations and for analogs of the Laplace equation in anisotropic
and inhomogeneous media, see e.g. [3, 8, 10, 11, 13, 14, 20, 23, 26] and rele-
vant references therein.
For basic definitions and notations, discussions and historic comments
in the mapping theory on the Riemann surfaces, see our previous papers
[29–32].
Received 26.03.2017
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
104 Prime ends in the mapping theory on Riemann surfaces
2. Definition of the prime ends and preliminary remarks
We act similarly to Caratheodory [5] under the definition of the prime
ends of domains on a Riemann surface S, see Chapter 9 in [6]. First of
all, recall that a continuous mapping σ : I → S, I = (0, 1), is called a
Jordan arc in S if σ(t1) ̸= σ(t2) for t1 ̸= t2. We also use the notations
σ, σ and ∂σ for σ(I), σ(I) and σ(I) \σ(I), correspondingly. A Jordan arc
σ in a domain D ⊂ S is called a cross–cut of the domain D if σ splits
D, i.e. D \ σ has more than one (connected) component, ∂σ ⊆ ∂D and
σ is a compact set in S.
A sequence σ1, . . . , σm, . . . of cross-cuts of D is called a chain in D
if:
(i) σi ∩ σj = ∅ for every i ̸= j, i, j = 1, 2, . . .;
(ii) σm splits D into 2 domains one of which contains σm+1 and an-
other one σm−1 for every m > 1;
(iii) δ(σm) → 0 as m→ ∞.
Here δ(E) = sup
p1,p2∈S
δ(p1, p2) denotes the diameter of a set E in S with
respect to an arbitrary metric δ in S agreed with its topology, see [29]–
[31].
Correspondingly to the definition, a chain of cross-cuts σm generates
a sequence of domains dm ⊂ D such that d1 ⊃ d2 ⊃ . . . ⊃ dm ⊃ . . .
and D ∩ ∂ dm = σm. Two chains of cross-cuts {σm} and {σ′k} are called
equivalent if, for everym = 1, 2, . . ., the domain dm contains all domains
d′k except a finite number and, for every k = 1, 2, . . ., the domain d′k
contains all domains dm except a finite number, too. A prime end P
of the domain D is an equivalence class of chains of cross-cuts of D.
Later on, ED denote the collection of all prime ends of a domain D and
DP = D ∪ED is its completion by prime ends.
Next, we say that a sequence of points pl ∈ D is convergent to a
prime end P of D if, for a chain of cross–cuts {σm} in P , for every
m = 1, 2, . . ., the domain dm contains all points pl except their finite
collection. Further, we say that a sequence of prime ends Pl converge to a
prime end P if, for a chain of cross–cuts {σm} in P , for everym = 1, 2, . . .,
the domain dm contains chains of cross–cuts {σ′k} in all prime ends Pl
except their finite collection.
Now, let D be a domain in the compactification S of a Riemann
surface S by Kerekjarto–Stoilow, see a discussion in [29]– [31]. Denote
by ED the union of D and all its prime ends. Open neighborhoods of
points in D is induced by the topology of S. A basis of neighborhoods
of a prime end P of D can be defined in the following way. Let d be an
arbitrary domain from a chain in P . Denote by d∗ the union of d and all
V. Ryazanov, S. Volkov 105
prime ends of D having some chains in d. Just all such d∗ form a basis
of open neighborhoods of the prime end P . The corresponding topology
on DP is called the topology of prime ends.
Let P be a prime end of D on a Riemann surface S, {σm} and {σ′m}
be two chains in P , dm and d′m be domains corresponding to σm and σ′m.
Then
∞∩
m=1
dm ⊆
∞∩
m=1
d′m ⊂
∞∩
m=1
dm ,
and, thus,
∞∩
m=1
dm =
∞∩
m=1
d′m ,
i.e. the set named by a body of the prime end P
I(P ) :=
∞∩
m=1
dm (2.1)
depends only on P but not on a choice of a chain of cross–cuts {σm} in
P .
It is necessary to note also that, for any chain {σm} in the prime end
P ,
Ω :=
∞∩
m=1
dm = ∅ . (2.2)
Indeed, every point p in Ω belongs to D. Moreover, some open neighbor-
hood of p in D should belong to Ω. In the contrary case each neighbor-
hood of p should have a point in some σm. However, in view of condition
(iii) then p ∈ ∂D that should contradict the inclusion p ∈ D. Thus, Ω is
an open set and if Ω would be not empty, then the connectedness of D
would be broken because D = Ω ∪Ω∗ with the open set Ω∗ := D \ I(P ).
In view of conditions (i) and (ii), we have by (2.2) that
I(P ) =
∞∩
m=1
(∂dm ∩ ∂D) = ∂D ∩
∞∩
m=1
∂dm .
Thus, we obtain the following statement.
Proposition 2.1. For each prime end P of a domain D on a Riemann
surface,
I(P ) ⊆ ∂D. (2.3)
106 Prime ends in the mapping theory on Riemann surfaces
Remark 2.1. If ∂D is a compact set in S, then I(P ) is a continuum,
i.e. it is a connected compact set, see e.g. I(9.12) in [37], see also I.9.3
in [4], and I(P ) belongs to only one (connected) component Γ of ∂D. In
the case, we say that the component Γ is associated with the prime
end P .
Moreover, in the case of a compact boundary of D, every prime end
of D contains a convergent chain {σm}, i.e., that is contracted to a
point p0 ∈ ∂D. Furthermore, each prime end P contains a spherical
chain {σm} lying on circles S(p0, rm) = {p ∈ S : δ(p, p0) = rm} with
p0 ∈ ∂D and rm → 0 as m → ∞. The proof is perfectly similar to
Lemma 1 in [15] after the replacement of metrics, see also Theorem 7.1
in [22], and hence we omit it. Note by the way that condition (iii) does
not depend on the choice of the metric δ agreed with the topology of S
because ∂D has a compact neighborhood.
3. The main lemma
Lemma 3.1. Let D be a domain in a Riemann surface S and let Γ be
a compact isolated component of ∂D in S that is not degenerated to a
point. Then Γ has a neighborhood U with a conformal mapping h of
U∗ := U ∩D onto a ring R = {z ∈ C : 0 < r < |z| < 1} where one may
assume that γ := ∂U∗ ∩D is a closed Jordan curve and
C(γ, h) = {z ∈ C : |z| = 1}, C(Γ, h) = {z ∈ C : |z| = r} .
Furthermore, the map h can be extended to a homeomorphism of EU∗
onto R.
Here we use the notation of the cluster set of the mapping h for
B ⊆ ∂D,
C(B, h) :=
{
z ∈ C : z = lim
k→∞
h(pk), pk → p ∈ B, pk ∈ D
}
Note that the first statement is obvious in the case of isolated boundary
points of ∂D with r = 0 and the punctured unit disk R = D0 := {z ∈ C :
0 < |z| < 1}.
Proof. By the Kerekjarto–Stoilow representation of S, see a discussion
in [29]– [31], Γ has an open neighborhood V in S of a finite genus. With-
out loss of generality, we may assume that V is connected and does not
intersect ∂D \Γ because Γ is an isolated component of ∂D. Thus, V ∩D
is a Riemann surface of finite genus with an isolated boundary element
V. Ryazanov, S. Volkov 107
g corresponding to Γ. However, a Riemann surface of finite genus has
only boundary elements of the first kind, see, e.g., IV.II.6 in [35]. Con-
sequently, Γ has a neighborhood U∗ from the side of D of genus zero
with a closed Jordan curve γ = ∂U∗ ∩D. Set U = U∗ ∪ (V \D). Cor-
respondingly to the Kerekjarto–Stoilow representation, the latter means
that U∗ is homeomorphic to a plane domain and, consequently, by the
general principle of Koebe, see e.g. Section II.3 in [17], U∗ is confor-
mally equivalent to a plane domain D∗. Note that by the construction
U∗ had two boundary components. Hence there is a conformal map-
ping h of U∗ onto a ring D∗ = R = {z ∈ C : 0 < r < |z| < 1} with
C(γ, h) = {z ∈ C : |z| = 1} and C(Γ, h) = {z ∈ C : |z| = r}, see e.g.
Proposition 2.5 in [25] or Proposition 13.5 in [20].
Now, U∗ and R are Riemann surfaces of hyperbolic type and the
modulus M of curve families are invariant under the conformal mapping
h, see a discussion in [29]– [31]. By condition (i) we have, for a chain
{σm} in a prime end P associated with the component Γ and localized
in U∗, that
M(∆(σm, σm+1, U
∗)) < ∞ ∀ m = 1, 2, . . . (3.1)
where ∆(E,F,G) denotes a family of all curves joining the sets E and F
through the set G. Moreover, by Remark 2.1 the prime end P contains
a convergent chain {σm} for which and any continuum C in U∗
lim
m→∞
M(∆(σm, C, U
∗)) = 0 . (3.2)
Similarly, prime ends associated with γ satisfy conditions (3.1) and (3.2).
Thus, the prime ends of U∗ in the sense (i)–(iii) and their images in R are
the prime ends in the sense of Section 4 in [21]. The Näkki prime ends in
R has a natural one-to-one correspondence with the points of ∂R whose
extension to the correspondence between R and RP by the identity in
R is a homeomorphism with respect to the topologies of R and RP or
with respect to convergence of points and prime ends, correspondingly,
see Theorems 4.1 and 4.2 in [21].
Remark 3.1. So, the space of U∗
P with the topology of prime ends is
metrizable by ρ(p1, p2) := |h̃(p1) − h̃(p2)|, where h̃ is the extension of
h : U∗ → R to the homeomorphism h̃ : U∗
P → R from Lemma 3.1, and
the space (U∗
P , ρ) is compact.
Furthermore, if D be a domain in the Kerekjarto–Stoilow compactifi-
cation S of a Riemann surface S and ∂D is a set in S with a finite collection
of components, then the whole space DP can be metrized through the
theory of pseudometric spaces, see e.g. Section 2.21.XV in [18], and it
108 Prime ends in the mapping theory on Riemann surfaces
is compact. Namely, let ρ0 be one of the metrics on S and let ρ1, . . . , ρn
be the above metrics on U∗
1 P , . . . , U
∗
nP for the corresponding components
Γ1, . . . ,Γn of ∂D. Then ρ∗j := ρj/(1 + ρj) ≤ 1, j = 0, 1, . . . , n, be also
metrics generated the same topologies on S, U∗
1 P , . . . , U
∗
nP , correspond-
ingly, see e.g. Section 2.21.V in [18]. Then the topology of prime ends
on DP is generated by the metric ρ =
n∑
j=0
2−(j+1)ρ̃j < 1 where the pseu-
dometrics ρ̃j are extensions of ρ∗j onto DP by 1, see e.g. Remark 2 in
point 2.21.XV of [18].
4. Some general topological lemmas
Let us give definitions of topological notions and facts of a general
character that will be useful in what follows. Let T be an arbitrary
topological space. Then a path in T is a continuous map γ : [a, b] → T.
Given A,B, C ⊆ T, ∆(A,B,C) denotes a collection of all paths γ joining
A and B in C, i.e., γ(a) ∈ A, γ(b) ∈ B and γ(t) ∈ C for all t ∈ (a, b). In
what follows, |γ| denotes the locus of γ, i.e. the image γ([a, b]).
Proposition 4.1. Let T be a topological space. Suppose that E1 and E2
are sets in T with E1 ∩ E2 = ∅. Then
∆(E1 , E2 , T ) > ∆( ∂E1 , ∂E2 , T \ (E1 ∪ E2) ) . (4.1)
Proof. Indeed, let γ ∈ ∆(E1, E2, T ), i.e. the path γ : [a, b] → T is such
that γ(a) ∈ E1 and γ(b) ∈ E2. Note that the set α := γ−1(E1) is a closed
subset of the segment [a, b] because γ is continuous, see e.g. Theorem
1 in Section I.2.1 of [4]. Consequently, α is compact because [a, b] is a
compact space, see e.g. I.9.3 in [4]. Then there is a∗ := maxt∈α t < b
because γ(b) ∈ E2 and by the hypothesis of the proposition E1∩E2 = ∅.
Thus, γ′ := γ|[a∗,b] belongs to ∆(∂E1, E2, T \E1) because γ is continuous
and hence γ′(a∗) cannot be an inner point of E1.
Arguing similarly in the space T ′ = T \E1 with E′
1 := E2 and E′
2 :=
∂E1, we obtain that there is b∗ := minγ′(t)∈E2
t > a∗. Thus, by the given
construction γ∗ := γ|[a∗,b∗] just belongs to ∆(∂E1, ∂E2, T \(E1∪E2)).
Lemma 4.1. In addition to the hypothesis of Proposition 4.1, let T be
a subspace of a metric space (M,ρ). Suppose that
∂E1 ⊆ C1 := {p ∈M : ρ(p, p0) = R1},
∂E2 ⊆ C2 := {p ∈M : ρ(p, p0) = R2}
V. Ryazanov, S. Volkov 109
with p0 ∈M \ T and R1 < R2. Then
∆(E1 , E2 , T ) > ∆(C1 , C2 , A ) (4.2)
where
A = A(p0, R1, R2) := {p ∈M : R1 < ρ(p, p0) < R2} .
Note that here, generally speaking, C1 ∩ T ̸= E1 and C2 ∩ T ̸= E2 as
well as γ∗ in the proof of Proposition 4.1 is not in R.
Proof. First of all, note that by the continuity of γ∗ the set ω := γ−1
∗ (R)
is open in [a∗, b∗] and ω is the union of a countable collection of disjoint
intervals (a1, b1), (a2, b2), . . . with ends in Γ := γ−1
∗ (∂R). If there is a
pair ak and bk in the different sets Γi := γ−1
∗ (Ci), i = 1, 2, Γ = Γ1 ∪ Γ2,
Γ1 ∩ Γ2 = ∅, then the proof is complete.
Let us assume that such a pair is absent. Then the given collection
is split into 2 collections of disjoint intervals (a′l, b
′
l) and (a′′l , b
′′
l ) with
ends a′l, b
′
l ∈ Γ1 and a′′l , b
′′
l ∈ Γ2, l = 1, 2, . . .. Set α1 =
∪
l
(a′l, b
′
l) and
α2 =
∪
l
(a′′l , b
′′
l ).
Arguing by contradiction, it is easy to show that γ∗ : [a∗, b∗] → (M,ρ)
is uniformly continuous because [a∗, b∗] is a compact space. Indeed, let us
assume that there is ε > 0 and a sequence of pairs a∗n and b∗n ∈ [a∗, b∗],
n = 1, 2, . . ., such that |b∗n − a∗n| → 0 as n → ∞ and simultaneously
ρ(γ∗(a
∗
n), γ∗(b
∗
n)) ≥ ε. However, by compactness of [a∗, b∗] there is a
subsequence a∗nk → a0 ∈ [a∗, b∗] and then also b∗nk → a0 as k → ∞.
Hence by the continuity of γ∗ it should be ρ(γ∗(a
∗
nk
), γ∗(a0)) → 0 as
well as ρ(γ∗(b
∗
nk
), γ∗(a0)) → 0 and then by the triangle inequality also
ρ(γ∗(a
∗
nk
), γ∗(b
∗
nk
)) → 0 as k → ∞. The contradiction disproves the
assumption.
Note that b′l − a′l → 0 as l → ∞ and by the uniform continuity of γ∗
on [a∗, b∗] we have that |γ′l| → C1 in the sense that
sup
p∈|γ′l |
inf
q∈C1
ρ(p, q) → 0 as l → ∞
where γ′l := γ∗|[a′l,b′l], l = 1, 2, . . .. Thus, there is R′
2 ∈ (R1, R2) such that
the set L1 :=
∪
l
|γ′l| lies outside of B2 := {p ∈M : ρ(p, p0) > R′
2}.
Arguing similarly, we obtain that there is R′
1 ∈ (R1, R
′
2) such that
the set L2 :=
∪
l
|γ′′l | lies outside of B1 := {p ∈ M : ρ(p, p0) < R′
1}.
Remark that the sets β1 := γ−1
∗ (B1) and β2 := γ−1
∗ (B2) are open in
[a∗, b∗] because γ∗ is continuous and by the construction δ1 := α1 ∪ β1
110 Prime ends in the mapping theory on Riemann surfaces
and δ2 := α2 ∪ β2 are open, mutually disjoint and together cover the
segment [a∗, b∗]. The latter contradicts to connectedness of the segment
and, thus, disproves the above assumption.
5. On boundary behavior in prime ends of inverse maps
The main base for extending inverse mappings is the following fact.
Lemma 5.1. Let S and S′ be Riemann surfaces, D and D′ be domains in
S and S′, ∂D ⊂ S and ∂D′ ⊂ S′ have finite collections of components, and
let f : D → D∗ be a homeomorphism of finite distortion with Kf ∈ L1
loc.
Then
C(P1, f) ∩ C(P2, f) = ∅ (5.1)
for all prime ends P1 ̸= P2 in the domain D.
Here we use the notation of the cluster set of the mapping f at
P ∈ ED,
C(P, f) :=
{
P ′ ∈ ED′ : P ′ = lim
k→∞
f(pk), pk → P, pk ∈ D
}
.
As usual, we also assume here that the dilatation Kf of the mapping
f is extended by zero outside of the domain D.
Proof. First of all note that S and S′ are metrizable spaces. Hence their
compactness is equivalent to their sequential compactness, see e.g. Re-
mark 41.I.3 in [19], and, consequently, ∂D and ∂D′ are compact subsets
of S and S′, correspondingly, see e.g. Proposition I.9.3 in [4]. Thus,
in view of Remarks 2.1 and 3.1 and Lemma 3.1, we may assume that
S is hyperbolic, D is a compact set in S, Kf ∈ L1(D), P1 and P2
are associated with the same component Γ of ∂D and D′ is a ring
R = {z ∈ C : 0 < r < |z| < 1} and
Ak := C(Pk, f) , k = 1, 2
are sets of points in the circle Cr := {z ∈ C : |z| = r}, ∂D consists of
2 components: Γ and a closed Jordan curve γ, C(γ, f) = C∗ := {z ∈
C : |z| = 1}, C(C∗, f
−1) = γ, C(Cr, f
−1) = Γ, see also Proposition 2.5
in [25] or Proposition 13.5 in [20]. Furthermore, then the sets Ak are
continua, i.e. closed arcs of the circle Cr, because
Ak =
∞∩
m=1
f
(
d
(k)
m
)
, k = 1, 2 ,
V. Ryazanov, S. Volkov 111
where d
(k)
m are domains corresponding to chains of cross–cuts {σ(k)m } in
the prime ends Pk, k = 1, 2, see e.g. I(9.12) in [37] and also I.9.3 in [4].
In addition, by Remark 2.1 we may assume also that σ
(k)
m are open arcs
of the hyperbolic circles C
(k)
m := {p ∈ S : h(p, pk) = r
(k)
m } on S with
pk ∈ ∂D and r
(k)
m → 0 as m→ ∞, k = 1, 2.
Set p0 = p1. By the definition of the topology of the prime ends in
the space DP , we have that d
(1)
m ∩d(2)m = ∅ for all large enough m because
P1 ̸= P2. For a such m, set R1 = r
(1)
m+1 < R2 = r
(1)
m and
Uk = d(k)m , Σk = σ(k)m , Ck = {p ∈ S : h(p, p0) = Rk}, k = 1, 2 .
Let K1 and K2 be arbitrary continua in U1 and U2, correspondingly.
Applying Proposition 4.1 and Lemma 4.1 with T = D, E1 = d
(1)
m+1 and
E2 = D \ d(1)m , and taking into account the inclusion ∆(K1,K2, D) ⊂
∆(E1, E2, D), we obtain that
∆(K1,K2, D) > ∆(C1, C2, A) , A := {p ∈ S : R1 < h(p, p0) < R2} ,
(5.2)
which means that any path α : [a, b] → S joining K1 and K2 in D,
α(a) ∈ K1, α(b) ∈ K2 and α(t) ∈ D, t ∈ (a, b), has a subpath joining C1
and C2 in A. Thus, since f is a homeomorphism, we have also that
∆(fK1, fK2, fD) > ∆(fC1, fC2, fA) (5.3)
and by the minorization principle, see e.g. [7, p. 178], we obtain that
M(∆(fK1, fK2, fD)) ≤ M(∆(fC1, fC2, fA)) . (5.4)
So, by Lemma 3.1 in [30] and [31] we conclude that
M(∆(fK1, fK2, fD)) 6
∫
A
Kf (p) · ξ2(h(p, p0)) dh(p) (5.5)
for all measurable functions ξ : (R1, R2) → [0,∞] such that
R2∫
R1
ξ(R) dR > 1 . (5.6)
In particular, for ξ(R) ≡ 1/δ, δ = R2 −R1 > 0, we get from here that
M(∆(fK1, fK2, fD)) 6 M0 :=
1
δ
∫
D
Kf (p) dh(p) < ∞ . (5.7)
112 Prime ends in the mapping theory on Riemann surfaces
Since f is a homeomorphism, (5.7) means that
M(∆(K1,K2, D
′)) 6 M0 < ∞ (5.8)
for all continua K1 and K2 in the domains V1 = fU1 and V2 = fU2,
correspondingly.
Let us assume that A1 ∩ A2 ̸= ∅. Then by the construction there is
p0 ∈ ∂R ∩ ∂V1 ∩ ∂V2. However, the latter contradicts (5.8) because the
ring R is a QED (quasiextremal distance) domains, see e.g. Theorem 3.2
in [20], see also Theorem 10.12 in [36].
Theorem 5.1. Let S and S′ be Riemann surfaces, D and D′ be do-
mains in S and S′, correspondingly, ∂D ⊂ S and ∂D′ ⊂ S′ have finite
collections of nondegenerate components, and let f : D → D′ be a homeo-
morphism of finite distortion with Kf ∈ L1
loc. Then the inverse mapping
g = f−1 : D′ → D can be extended to a continuous mapping g̃ of D′
P
onto DP .
Proof. Recall that by Remark 3.1 the spaces DP and D′
P are compact
and metrizable with metrics ρ and ρ′. Let a sequence pn ∈ D′ converges
as n→ ∞ to a prime end P ′ ∈ ED′ . Then any subsequence of p∗n := g(pn)
has a convergent subsequence by compactness of DP . By Lemma 5.1
any such convergent subsequence should have the same limit. Thus, the
sequence p∗n is convergent, see e.g. Theorem 2 of Section 2.20.II in [18].
Note that p∗n cannot converge to an inner point of D because I(P ) ⊆ ∂D
by Proposition 2.1 and, consequently, pn is convergent to ∂D′, see e.g.
Proposition 2.5 in [25] or Proposition 13.5 in [20]. Thus, ED′ is mapped
into ED under this extension g̃ of g. In fact, g̃ maps ED′ onto ED because
pn = f(p∗n) has a convergent subsequence for every sequence p∗n ∈ D
that is convergent to a prime end P of the domain D because D′
P is
compact. The map g̃ is continuous. Indeed, let a sequence P ′
n ∈ D′
P be
convergent to P ′ ∈ D′
P . Then there is a sequence pn ∈ D′ such that
ρ′(P ′
n, pn) < 2−n and ρ(p∗n, P
∗
n) < 2−n where p∗n := g(pn), P
∗
n := g̃(Pn)
and P ∗ = g̃(P ′). Then pn → P ′ and by the above p∗n → P ∗ as well as
P ∗
n → P ∗ as n→ ∞.
6. Lemma on extension to boundary of direct mappings
In contrast with the case of the inverse mappings, as it was already
established in the plane, no degree of integrability of the dilatation leads
to the extension to the boundary of direct mappings with finite distortion,
see the example in the proof of Proposition 6.3 in [20]. The nature of
the corresponding conditions has a much more refined character as the
following lemma demonstrates.
V. Ryazanov, S. Volkov 113
Lemma 6.1. Under the hypothesis of Theorem 5.1, let in addition∫
R(p0,ε,ε0)
Kf (p) · ψ2
p0,ε,ε0(h(p, p0)) dh(p) = o
(
I2p0,ε0(ε)
)
∀ p0 ∈ ∂D
(6.1)
as ε → 0 for all ε0 < δ(p0) where R(p0, ε, ε0) = {p ∈ S : ε < h(p, p0) <
ε0} and ψp0,ε,ε0(t) : (0,∞) → [0,∞], ε ∈ (0, ε0), is a family of measurable
functions such that
0 < Ip0,ε0(ε) :=
ε0∫
ε
ψp0,ε,ε0(t) dt < ∞ ∀ ε ∈ (0, ε0) .
Then f can be extended to a continuous mapping f̃ of DP onto D′
P .
We assume here that the function Kf is extended by zero outside of
D.
Proof. By Remarks 2.1 and 3.1 and Lemma 3.1, arguing as in the begin-
ning of the proof of Lemma 5.1, we may assume that D is a compact set
in S, ∂D consists of 2 components: a closed Jordan curve γ and one more
nondegenerate component Γ, D′ is a ring R = {z ∈ C : 0 < r < |z| < 1},
D′
P = R,
C(Γ, f) = Cr := {z ∈ C : |z| = r}, C(γ, f) = C∗ := {z ∈ C : |z| = 1}
and that f is extended to a homeomorphism of D ∪ γ onto D′ ∪ C∗.
Let us first prove that the set L := C(P, f) consists of a single point
of Cr for a prime end P of the domain D associated with Γ. Note that
L ̸= ∅ by compactness of the set R and, moreover, L ⊆ Cr by Proposition
2.1.
Let us assume that there is at least two points ζ0 and ζ∗ ∈ L. Set
U = {ζ ∈ C : |ζ − ζ0| < ρ0} where 0 < ρ0 < |ζ∗ − ζ0|.
Let σk, k = 1, 2, . . . , be a chain in the prime end P from Remark
2.1 lying on the circles Sk := {p ∈ S : h(p, p0) = rk} where p0 ∈ Γ and
rk → 0 as k → ∞. Let dk be the domains associated with σk. Then
there exist points ζk and ζ∗k in the domains d′k = f(dk) ⊂ R such that
|ζ0 − ζk| < ρ0 and |ζ0 − ζ∗k | > ρ0 and, moreover, ζk → ζ0 and ζ∗k → ζ∗
as k → ∞. Let γk be paths joining ζk and ζ∗k in d′k. Note that by the
construction ∂U ∩ γk ̸= ∅, k = 1, 2, . . ..
By the condition of strong accessibility of the point ζ0 in the ring R,
there is a continuum E ⊂ R and a number δ > 0 such that
M(∆(E, γk;R)) > δ (6.2)
114 Prime ends in the mapping theory on Riemann surfaces
for all large enough k. Note that C = f−1(E) is a compact subset of D
and hence h(p0, C)) > 0. Let ε0 ∈ (0, δ0) where δ0 := min (δ(p0), h(p0, C)).
Without loss of generality, we may assume that rk < ε0 and that (6.2)
holds for all k = 1, 2, . . ..
Let Γm be the family of paths joining the circle S0 := {p ∈ S :
h(p, p0) = ε0} and σm, m = 1, 2, . . ., in the intersection of D\dm and the
ring Rm := {p ∈ S : rm < h(p, p0) < ε0}. Applying Proposition 4.1 and
Lemma 4.1 with T = D, E1 = dm and E2 = B0 := {p ∈ S : h(p, p0) >
ε0}, and taking into account the inclusion ∆(C,Ck, D) ⊂ ∆(E1, E2, D) =
∆(B0, dm, D) where Ck = f−1(γk), we have that ∆(C,Ck, D) > Γm for
all k > m because by the construction Ck ⊂ dk ⊂ dm. Thus, since f is
a homeomorphism, we have also that ∆(E, γk, D) > fΓm for all k > m,
and by the principle of minorization, see e.g. [7], p. 178, we obtain that
M(f(Γm)) > δ for all m = 1, 2, . . ..
On the other hand, every function
ξ(t) = ξm(t) := ψp0,rm,ε0(t)/Ip0,ε0(rm), m = 1, 2, . . . ,
satisfies the condition (5.6) and by Lemma 3.1 in [30] and [31]
M(fΓm) 6
∫
Rm
Kf (p) · ξ2m(h(p, p0)) dh(p) ,
i.e., M(fΓm) → 0 as m→ ∞ in view of (6.1).
The obtained contradiction disproves the assumption that the cluster
set C(P, f) consists of more than one point.
Thus, we have the extension f̃ of f to DP such that f̃(ED) ⊆ ED′ . In
fact, f̃(ED) = ED′ . Indeed, if ζ0 ∈ D′, then there is a sequence ζn in D′
that is convergent to ζ0. We may assume with no loss of generality that
f−1(ζn) → P0 ∈ DP because DP is compact, see Remark 3.1. Hence
ζ0 ∈ ED because ζ0 /∈ D, see e.g. Proposition 2.5 in [25] or Proposition
13.5 in [20].
Finally, let us show that the extended mapping f̃ : DP → D′
P is
continuous. Indeed, let Pn → P0 in DP . The statement is obvious for
P0 ∈ D. If P0 ∈ ED, then by the last item we are able to choose P ∗
n ∈ D
such that ρ(Pn, P
∗
n) < 2−n and ρ′(f̃(Pn), f̃(P
∗
n)) < 2−n where ρ and
ρ′ are some metrics on DP and D′
P , correspondingly, see Remark 3.1.
Note that by the first part of the proof f(P ∗
n) → f(P0) because P
∗
n → P0.
Consequently, f̃(Pn) → f̃(P0), too.
Remark 6.1. Note that condition (6.1) holds, in particular, if∫
D(p0,ε0)
Kf (p) · ψ2(h(p, p0)) dh(p) < ∞ ∀ p0 ∈ ∂D (6.3)
V. Ryazanov, S. Volkov 115
where D(p0, ε0) = {p ∈ S : h(p, p0) < ε0} and where ψ(t) : (0,∞) →
[0,∞] is a locally integrable function such that Ip0,ε0(ε) → ∞ as ε → 0.
In other words, for the extendability of f to a continuous mapping of DP
onto D′
P , it suffices for the integrals in (6.3) to be convergent for some
nonnegative function ψ(t) that is locally integrable on (0,∞) but that
has a non-integrable singularity at zero.
7. On the homeomorphic extension to the boundary
Combining Lemma 6.1 and Theorem 5.1, we obtain the significant
conclusion:
Lemma 7.1. Under the hypothesis of Lemma 6.1, the homeomorphism
f : D → D′ can be extended to a homeomorphism f̃ : DP → D′
P .
Proof. Indeed, by Lemma 5.1 the mapping f̃ : DP → D′
P from Lemma
6.1 is injective and hence it has the well defined inverse mapping f̃−1 :
D′
P → DP and the latter coincides with the mapping g̃ : D′
P → DP
from Theorem 5.1 because a limit under a metric convergence is unique.
The continuity of the mappings g̃ and f̃ follows from Theorem 5.1 and
Lemma 6.1, respectively.
We assume everywhere in this section that the functionKf is extended
by zero outside of D.
Theorem 7.1. Under the hypothesis of Theorem 5.1, let in addition
ε0∫
0
dr
||Kf ||(p0, r)
= ∞ ∀ p0 ∈ ∂D, ε0 < δ(p0) (7.1)
where
||Kf ||(p0, r) :=
∫
S(p0,r)
Kf (p) dsh(p) . (7.2)
Then f can be extended to a homeomorphism of DP onto D′
P .
Here S(p0, r) denotes the circle {p ∈ S : h(p, p0) = r}.
Proof. Indeed, for the functions
ψp0,ε0(t) :=
{
1/||Kf ||(p0, t), t ∈ (0, ε0),
0, t ∈ [ε0,∞),
(7.3)
116 Prime ends in the mapping theory on Riemann surfaces
we have by the Fubini theorem that
∫
R(p0,ε,ε0)
Kf (p) · ψ2
p0,ε0(h(p, p0) dh(p) =
ε0∫
ε
dr
||Kf ||(p0, r)
(7.4)
where R(p0, ε, ε0) denotes the ring {p ∈ S : ε < h(p, p0) < ε0} and,
consequently, condition (6.1) holds by (7.1) for all p0 ∈ ∂D and ε0 ∈
(0, ε(p0)).
Here we have used the standard conventions in the integral theory
that a/∞ = 0 for a ̸= ∞ and 0 · ∞ = 0, see, e.g., Section I.3 in [33].
Thus, Theorem 7.1 follows immediately from Lemma 7.1.
Corollary 7.1. In particular, the conclusion of Theorem 7.1 holds if
kp0(r) = O
(
log
1
r
)
∀ p0 ∈ ∂D (7.5)
as r → 0 where kp0(r) is the average of Kf over the infinitesimal circle
S(p0, r).
Choosing in (6.1) ψ(t) := 1
t log 1/t , we obtain by Lemma 7.1 the next
result, see also Lemma 4.1 in [25] or Lemma 13.2 in [20].
Theorem 7.2. Under the hypothesis of Theorem 5.1, let Kf have a
dominant Qp0 in a neighborhood of each point p0 ∈ ∂D with finite mean
oscillation at p0. Then f can be extended to a homeomorphism f̃ : DP →
D′
P .
By Corollary 4.1 in [25] or Corollary 13.3 in [20] we obtain the follow-
ing.
Corollary 7.2. In particular, the conclusion of Theorem 7.2 holds if
lim
ε→0
−
∫
D(p0,ε)
Kf (p) dh(p) < ∞ ∀ p0 ∈ ∂D (7.6)
where D(p0, ε) is the infinitesimal disk {p ∈ S : h(p, p0) < ε}.
Corollary 7.3. The conslusion of Theorem 7.2 holds if every point p0 ∈
∂D is a Lebesgue point of the function Kf or its dominant Qp0.
The next statement also follows from Lemma 7.1 under the choice
ψ(t) = 1/t.
V. Ryazanov, S. Volkov 117
Theorem 7.3. Under the hypothesis of Theorem 5.1, let, for some ε0 >
0, ∫
ε<h(p,p0)<ε0
Kf (p)
dh(p)
h2(p, p0)
= o
([
log
1
ε
]2)
as ε→ 0 ∀ p0 ∈ ∂D
(7.7)
Then f can be extended to a homeomorphism of DP onto D′
P .
Remark 7.1. Choosing in Lemma 7.1 the function ψ(t) = 1/(t log 1/t)
instead of ψ(t) = 1/t, (7.7) can be replaced by the more weak condition∫
ε<h(p,p0)<ε0
Kf (p) dh(p)(
h(p, p0) log 1
h(p,p0)
)2 = o
([
log log
1
ε
]2)
(7.8)
and (7.5) by the condition
kp0(r) = o
(
log
1
ε
log log
1
ε
)
. (7.9)
Of course, we could give here the whole scale of the corresponding con-
dition of the logarithmic type using suitable functions ψ(t).
8. On interconnections between integral conditions
For every non-decreasing function Φ : [0,∞] → [0,∞], the inverse
function Φ−1 can be well defined by setting
Φ−1(τ) = inf
Φ(t)≥τ
t . (8.1)
As usual, here inf is equal to ∞ if the set of t ∈ [0,∞] such that Φ(t) ≥ τ
is empty. Note that the function Φ−1 is non-decreasing, too.
Remark 8.1. Immediately by the definition it is evident that
Φ−1(Φ(t)) ≤ t ∀ t ∈ [0,∞] (8.2)
with the equality in (8.2) except intervals of constancy of the function
Φ(t).
Recall that a function Φ : [0,∞] → [0,∞] is called convex if
Φ(λt1 + (1− λ)t2) ≤ λ Φ(t1) + (1− λ) Φ(t2)
for all t1, t2 ∈ [0,∞] and λ ∈ [0, 1].
118 Prime ends in the mapping theory on Riemann surfaces
In what follows, H(R) denotes the hyperbolic disk centered at the
origin with the hyperbolic radius R = log (1+ r)/(1− r), r ∈ (0, 1) is its
Euclidean radius:
H(R) = { z ∈ C : h(z, 0) < R } , R ∈ (0,∞) . (8.3)
Further we also use the notation of the hyperbolic sine: sinh t :=
(et − e−t)/2 .
The following statement is an analog of Lemma 3.1 in [28] adopted to
the hyperbolic geometry in the unit disk D := {z ∈ C : |z| < 1}.
Lemma 8.1. Let Q : H(ε) → [0,∞], ε ∈ (0, 1), be a measurable function
and Φ : [0,∞] → (0,∞] be a non-decreasing convex function with a finite
mean integral value M(ε) of the function Φ ◦Q on H(ε). Then
ε∫
0
dρ
ρq(ρ)
≥ 1
2
∞∫
δ(ε)
dτ
τ [Φ−1(τ)]
(8.4)
where q(ρ) is the average of Q on the circle S(ρ) = {z ∈ D : h(z, 0) = ρ}
and
δ(ε) = exp
(
4 sinh2
ε
2
)
· M(ε)
ε2
> τ0 := Φ(0) > 0 . (8.5)
Proof. Since M(ε) < ∞ we may assume with no loss of generality that
Φ(t) < ∞ for all t ∈ [0,∞) because in the contrary case Q ∈ L∞ and
then the left-hand side in (8.4) is equal to ∞. Moreover, we may assume
that Φ(t) is not constant because in the contrary case Φ−1(τ) ≡ ∞ for
all τ > τ0 and hence the right-hand side in (8.4) is equal to 0. Note also
that Φ(τ) is (strictly) increasing, convex and continuous in the segment
[t∗,∞] and
Φ(t) ≡ τ0 ∀ t ∈ [0, t∗] where t∗ := sup
Φ(t)=τ0
t . (8.6)
Setting H(t) : = log Φ(t), we see that H−1(η) = Φ−1(eη), Φ−1(τ) =
H−1(log τ). Thus, we obtain that
q(ρ) = H−1
(
log
h(ρ)
ρ2
)
= H−1
(
2 log
1
ρ
+ log h(ρ)
)
∀ ρ ∈ R∗
(8.7)
where h(ρ) := ρ2Φ(q(ρ)) and R∗ = { ρ ∈ (0, ε) : q(ρ) > t∗}. Then also
q(e−s) = H−1
(
2s + log h(e−s)
)
∀ s ∈ S∗ (8.8)
V. Ryazanov, S. Volkov 119
where S∗ = {s ∈ ( log 1
ε , ∞ ) : q(e−s) > t∗}.
Now, by the Jensen inequality, see e.g. Theorem 2.6.2 in [24], we
have that
∞∫
log 1
ε
h(e−s) ds =
ε∫
0
h(ρ)
dρ
ρ
=
ε∫
0
Φ(q(ρ)) ρ dρ (8.9)
≤
ε∫
0
(
−
∫
S(ρ)
Φ(Q(z)) dsh(z)
)
ρ dρ ≤ 2 sinh2
ε
2
·M(ε)
because H(ε) has the hyperbolic area A(ε) = 4π sinh2 ε2 and S(ρ) has
the hyperbolic length L(ρ) = 2π sinh ρ, see e.g. Theorem 7.2.2 in [2],
and, moreover, sinh ρ ≥ ρ by the Taylor expansion. Then arguing by
contradiction it is easy to see for the set T := { s ∈ ( log 1
ε , ∞ ) :
h(e−s) > M(ε) } that its length
|T | =
∫
T
ds ≤ 2 sinh2
ε
2
. (8.10)
Next, let us show for T∗ : = T ∩ S∗ that
q
(
e−s
)
≤ H−1 (2s + log M(ε)) ∀ s ∈
(
log
1
ε
,∞
)
\T∗ . (8.11)
Indeed, note that
(
log 1
ε ,∞
)
\T∗ =
[(
log 1
ε ,∞
)
\ S∗
]
∪
[(
log 1
ε ,∞
)
\ T
]
=[(
log 1
ε ,∞
)
\ S∗
]
∪ [S∗ \ T ]. The inequality (8.11) holds for s ∈ S∗ \ T
by (8.8) because H−1 is a non-decreasing function. Note also that
e2sM(ε) > Φ(0) = τ0 ∀ s ∈
(
log
1
ε
, ∞
)
(8.12)
and then
t∗ < Φ−1
(
e2sM(ε)
)
= H−1 (2s + log M(ε)) ∀ s ∈
(
log
1
ε
, ∞
)
(8.13)
Consequently, (8.11) holds for all s ∈ ( log 1
ε , ∞ ) \ S∗, too.
Since H−1 is non-decreasing, we have by (8.10)–(8.11) that, for ∆ :=
logM(ε),
ε∫
0
dρ
ρq(ρ)
=
∞∫
log 1
ε
ds
q(e−s)
≥
∫
(log 1
ε
,∞)\T∗
ds
H−1(2s+∆)
(8.14)
120 Prime ends in the mapping theory on Riemann surfaces
≥
∞∫
|T∗|+log 1
ε
ds
H−1(2s+∆)
≥
∞∫
2 sinh2 ε
2
+log 1
ε
ds
H−1(2s+∆)
=
1
2
∞∫
4 sinh2 ε
2
+log
M(ε)
ε2
dη
H−1(η)
and after the replacement of variables η = log τ , τ = eη, we come to
(8.4).
Theorem 8.1. Let Q : H(ε) → [0,∞], ε ∈ (0, 1), be a measurable func-
tion such that ∫
H(ε)
Φ(Q(z)) dh(z) < ∞ (8.15)
where Φ : [0,∞] → [0,∞] is a non-decreasing convex function with
∞∫
δ0
dτ
τΦ−1(τ)
= ∞ (8.16)
for some δ0 > τ0 := Φ(0). Then
ε∫
0
dρ
ρq(ρ)
= ∞ , (8.17)
where q(ρ) is the average of Q on the hyperbolic circle h(z, 0) = ρ.
Proof. If Φ(0) ̸= 0, then Theorem 8.1 directly follows from Lemma 8.1
because Φ−1 is strictly increasing on the interval (τ0,∞) and Φ−1(δ0) > 0.
In the case Φ(0) = 0, let us fix a number δ ∈ (0, δ0) and set Φ∗(t) = Φ(t),
if Φ(t) > δ, and Φ∗(t) = δ, if Φ(t) ≤ δ. Then by (8.15) we have that∫
H(ε)
Φ∗(Q(z)) dh(z) < ∞ because |Φ∗(t)− Φ(t)| ≤ δ and the measure of
H(ε) is finite. Moreover, Φ−1
∗ (τ) = Φ−1(τ) for τ ≥ δ and then by (8.16)
∞∫
δ0
dτ
τΦ−1
∗ (τ)
= ∞. Thus, (8.17) holds again by Lemma 8.1.
Remark 8.2. Note that condition (8.16) implies that
∞∫
δ
dτ
τΦ−1(τ)
= ∞ ∀ δ ∈ [0,∞) . (8.18)
V. Ryazanov, S. Volkov 121
but relation (8.18) for some δ ∈ [0,∞), generally speaking, does not
imply (8.16). Indeed, (8.16) evidently implies (8.18) for δ ∈ [0, δ0), and,
for δ ∈ (δ0,∞), we have that
0 ≤
δ∫
δ0
dτ
τΦ−1(τ)
≤ 1
Φ−1(δ0)
log
δ
δ0
< ∞ (8.19)
because the function Φ−1 is non-decreasing and Φ−1(δ0) > 0. Moreover,
by the definition of the inverse function Φ−1(τ) ≡ 0 for all τ ∈ [0, τ0],
τ0 = Φ(0), and hence (8.18) for δ ∈ [0, τ0), generally speaking, does not
imply (8.16). If τ0 > 0, then
τ0∫
δ
dτ
τΦ−1(τ)
= ∞ ∀ δ ∈ [0, τ0) (8.20)
However, relation (8.20) gives no information on the function Q itself
and, consequently, (8.18) for δ < Φ(0) cannot imply (8.17) at all.
9. Other criteria for homeomorphic extension in prime
ends
Theorem 7.1 has a magnitude of other consequences thanking to Theo-
rem 8.1.
Theorem 9.1. Under the hypothesis of Theorem 5.1, let∫
D(p0,ε0)
Φp0 (Kf (p)) dh(p) < ∞ ∀ p0 ∈ ∂D (9.1)
for ε0 = ε(p0) and a nondecreasing convex function Φp0 : [0,∞) → [0,∞)
with
∞∫
δ(p0)
dτ
τΦ−1
p0 (τ)
= ∞ (9.2)
for δ(p0) > Φp0(0). Then f is extended to a homeomorphism of DP onto
D′
P .
Proof. Indeed, in the case of the hyperbolic Riemann surfaces, (9.1) and
(9.2) imply (7.1) by Theorem 8.1 and, after this, Theorem 9.1 becomes
a direct consequence of Theorem 7.1. In the more simple case of the
elliptic and parabolic Riemann surfaces, we similarly can apply Theorem
3.1 in [28] for the Euclidean plane instead of Theorem 8.1.
122 Prime ends in the mapping theory on Riemann surfaces
Corollary 9.1. In particular, the conclusion of Theorem 9.1 holds if∫
D(p0,ε0)
eα0Kf (p) dh(p) < ∞ ∀ p0 ∈ ∂D (9.3)
for some ε0 = ε(p0) > 0 and α0 = α(p0) > 0.
Remark 9.1. Note that by Theorem 5.1 and Remark 5.1 in [16] con-
dition (9.2) is not only sufficient but also necessary for a continuous ex-
tendibility to the boundary of all mappings f with the integral restriction
(9.1).
Note also that by Theorem 2.1 in [28], see also Proposition 2.3 in [27],
(9.2) is equivalent to every of the conditions from the following series:
∞∫
δ(p0)
H ′
p0(t)
dt
t
= ∞ , δ(p0) > 0 , (9.4)
∞∫
δ(p0)
dHp0(t)
t
= ∞ , δ(p0) > 0 , (9.5)
∞∫
δ(p0)
Hp0(t)
dt
t2
= ∞ , δ(p0) > 0 , (9.6)
∆(p0)∫
0
Hp0
(
1
t
)
dt = ∞ , ∆(p0) > 0 , (9.7)
∞∫
δ∗(p0)
dη
H−1
p0 (η)
= ∞ , δ∗(p0) > Hp0(0) , (9.8)
where
Hp0(t) = log Φp0(t) . (9.9)
Here the integral in (9.5) is understood as the Lebesgue–Stieltjes in-
tegral and the integrals in (9.4) and (9.6)–(9.8) as the ordinary Lebesgue
integrals.
It is necessary to give one more explanation. From the right hand
sides in the conditions (9.4)–(9.8) we have in mind +∞. If Φp0(t) = 0 for
t ∈ [0, t∗(p0)], then Hp0(t) = −∞ for t ∈ [0, t∗(p0)] and we complete the
definition H ′
p0(t) = 0 for t ∈ [0, t∗(p0)]. Note, the conditions (9.5) and
(9.6) exclude that t∗(p0) belongs to the interval of integrability because
V. Ryazanov, S. Volkov 123
in the contrary case the left hand sides in (9.5) and (9.6) are either
equal to −∞ or indeterminate. Hence we may assume in (9.4)–(9.7) that
δ(p0) > t0, correspondingly, ∆(p0) < 1/t(p0) where t(p0) := sup
Φp0 (t)=0
t,
set t(p0) = 0 if Φp0(0) > 0.
The most interesting among the above conditions is (9.6), i.e. the
condition:
∞∫
δ(p0)
log Φp0(t)
dt
t2
= +∞ for some δ(p0) > 0 . (9.10)
Finally, it is necessary to note the restriction on nondegeneracy of
boundary components of domains in Theorem 5.1 as well as in all other
theorems is not essential because this simplest case is included in our
previous papers [30,31].
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Contact information
Vladimir Ryazanov Institute of Applied Mathematics
and Mechanics of the NAS of Ukraine,
Slavyansk, Ukraine
E-Mail: vl.ryazanov1@gmail.com,
vlryazanov1@rambler.ru
Sergei Volkov Institute of Applied Mathematics
and Mechanics of the NAS of Ukraine,
Slavyansk, Ukraine
E-Mail: serhii.volkov@donntu.edu.ua
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