On convergence theory for Beltrami equations
This paper is devoted to convergence theorems which play an important role in our scheme for deriving theorems on the existence of solutions of the Beltrami equations.
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irk-123456789-109502014-02-27T03:00:18Z On convergence theory for Beltrami equations Ryazanov, V. Srebro, U. Yakubov, E. This paper is devoted to convergence theorems which play an important role in our scheme for deriving theorems on the existence of solutions of the Beltrami equations. 2008 Article On convergence theory for Beltrami equations / V. Ryazanov, U. Srebro, E. Yakubov // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 517-528 — Бібліогр.: 14 назв. — англ. 1810-3200 http://dspace.nbuv.gov.ua/handle/123456789/10950 en Інститут прикладної математики і механіки НАН України |
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This paper is devoted to convergence theorems which play an important role in our scheme for deriving theorems on the existence of solutions of the Beltrami equations. |
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Ryazanov, V. Srebro, U. Yakubov, E. |
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Ryazanov, V. Srebro, U. Yakubov, E. On convergence theory for Beltrami equations |
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Ryazanov, V. Srebro, U. Yakubov, E. |
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Ryazanov, V. |
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On convergence theory for Beltrami equations |
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On convergence theory for Beltrami equations |
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On convergence theory for Beltrami equations |
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On convergence theory for Beltrami equations |
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On convergence theory for Beltrami equations |
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on convergence theory for beltrami equations |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/10950 |
| citation_txt |
On convergence theory for Beltrami equations / V. Ryazanov, U. Srebro, E. Yakubov // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 517-528 — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT ryazanovv onconvergencetheoryforbeltramiequations AT srebrou onconvergencetheoryforbeltramiequations AT yakubove onconvergencetheoryforbeltramiequations |
| first_indexed |
2025-07-02T13:15:49Z |
| last_indexed |
2025-07-02T13:15:49Z |
| _version_ |
1836541175615455232 |
| fulltext |
Ukrainian Mathematical Bulletin
Volume 5 (2008), № 4, 517 – 528 UMB
On convergence theory for Beltrami equations
Vladimir Ryazanov, Uri Srebro,
Eduard Yakubov
(Presented by V. Ya. Gutlyanskii)
Abstract. This paper is devoted to convergence theorems which play
an important role in our scheme for deriving theorems on the existence
of solutions of the Beltrami equations.
2000 MSC. 30C65, 30C75.
Key words and phrases. Weakly complete space, weakly fundamental
sequences, weak convergence, ring Q-homeomorphism, ACL, Sobolev
classes.
1. Introduction
Let D be a domain in the complex plane C, i.e., a connected and open
subset of C, and let µ : D → C be a measurable function with |µ(z)| < 1
a.e. The Beltrami equation is an equation of the form
fz = µ(z) · fz, (1.1)
where fz = ∂f = (fx + ify)/2, fz = ∂f = (fx− ify)/2, z = x+ iy, and fx
and fy are partial derivatives of f with respect to x and y, respectively.
The function µ is called the complex coefficient and
Kµ(z) =
1 + |µ(z)|
1 − |µ(z)|
(1.2)
the maximal dilatation or, in short, the dilatation of Eq. (1.1). The
Beltrami equation (1.1) is said to be degenerate if ess sup Kµ(z) = ∞.
Received 18.03.2008
ISSN 1812 – 3309. c© Institute of Mathematics of NAS of Ukraine
518 On convergence theory...
Recall that a function f : D → C is absolutely continuous on lines,
abbr. f ∈ ACL, if, for every closed rectangle R in D, whose sides are
parallel to the coordinate axes, f |R is absolutely continuous on almost
all line segments in R which are parallel to the sides of R. In particular,
f is ACL (possibly modified on a set of Lebesgue measure zero) if it
belongs to the Sobolev class W 1,1
loc of locally integrable functions with
locally integrable first generalized derivatives and, conversely, if f ∈ ACL
has locally integrable first partial derivatives, then f ∈ W 1,1
loc , see, e.g.,
1.2.4 in [9]. For a sense-preserving ACL homeomorphism f : D → C,
the Jacobian Jf (z) = |fz|
2 − |fz|
2 is nonnegative a.e. In this case, the
complex dilatation µf of f is the ratio µ(z) = fz/fz, if fz 6= 0 and µ(z) = 0
otherwise, and the dilatation Kf (z) of f is Kµ(z), see (1.2). Note that
|µ(z)| ≤ 1 a.e. and Kµ(z) ≥ 1 a.e. Given a function Q : D → [1,∞],
a sense-preserving ACL homeomorphism f : D → C is called a Q(z)-
quasiconformal mapping if Kf (z) ≤ Q(z) a.e., see [11].
Recall also that, given a family of paths Γ in C, a Borel function
ρ : C → [0,∞] is called admissible for Γ, abbr. ρ ∈ adm Γ, if
∫
γ
ρ(z) |dz| ≥ 1 (1.3)
for each γ ∈ Γ. The modulus of Γ is defined by
M(Γ) = inf
ρ∈adm Γ
∫
C
ρ2(z) dx dy. (1.4)
Motivated by the ring definition of quasiconformality in [6], we in-
troduce the following notion that extends and localizes the notion of a
quasiconformal mapping. Let D be a domain in C, z0 ∈ D, and Q : D →
[0,∞] a measurable function. We say that a homeomorphism f : D → C
is a ring Q-homeomorphism at the point z0 if
M(∆(fC0, fC1, fD)) ≤
∫
A∩D
Q(z) · η2(|z − z0|) dx dy (1.5)
for every ring
A = A(z0, r1, r2) = {z ∈ C : r1 < |z − z0| < r2}, 0 < r1 < r2 < ∞,
and for every continua C0 and C1 in D which belong to the different
components of the complement to the ring A in C, containing z0 and ∞,
V. Ryazanov, U. Srebro, E. Yakubov 519
respectively, and for every measurable function η : (r1, r2) → [0,∞] such
that
r2
∫
r1
η(r) dr = 1. (1.6)
2. On convergence of Sobolev’s functions
First of all, let us recall the necessary definitions and basic facts on
the Sobolev spaces W l,p and Lp, p ∈ [1,∞]. Given an open set U in
R
n and a natural number l, C l
0
(U) denotes a collection of all functions
ϕ : U → R with compact support having all partial continuous derivatives
of order at most l in U. ϕ ∈ C∞
0
(U) if ϕ ∈ C l
0
(U) for all l = 1, 2, . . .. A
vector α = (α1, . . . , αn) with natural coordinates is called a multiindex.
Every multiindex α is associated with the differential operator Dα =
∂|α|/∂xα1
1
· · · ∂xαn
n where |α| = α1 + · · · + αn.
Now, let u and v : U → R be locally integrable functions. The
function v is called the generalized derivative Dαu of u if
∫
Ω
u Dαϕ dx = (−1)|α|
∫
Ω
v ϕ dx ∀ϕ ∈ C∞
0 . (2.1)
The concept of the generalized derivative was introduced by Sobolev
in [13]. The Sobolev class W l,p(Ω) consists of all functions u : U → R in
Lp(U), p ≥ 1, with generalized derivatives of order l summable of order
p. A function u : U → R belongs to W l,p
loc(U) if u ∈ W l,p(U∗) for every
open set U∗ with compact closure U∗ ⊂ U. A similar notion introduced
for vector-functions f : U → R
m in the component-wise sense.
A function ω : R
n → R with a compact support in B is called a
Sobolev averaging kernel if ω is nonnegative, belongs to C∞
0
(Rn), and
∫
Rn
ω(x) dx = 1. (2.2)
The well-known example of such a function is ω(x) = γϕ(|x|2− 1
4
), where
ϕ(t) = e1/t for t < 0 and ϕ(t) ≡ 0 for t ≥ 0, and the constant γ is chosen
so that (2.2) holds. Later on, we use only ω depending on |x|.
520 On convergence theory...
Let U be a nonempty bounded open subset of R
n and f ∈ L1(U).
Extending f by zero outside of U, we set
fh = ωh ∗ f =
∫
|y|≤1
f(x + hy)ω(y) dy =
1
hn
∫
U
f(z)ω
(
z − x
h
)
dz, (2.3)
where fh = ωh ∗ f, ωh(y) = ω (y/h) , h > 0, is called the Sobolev mean
function for f. It is known that fh ∈ C∞
0
(Rn), ‖fh‖p ≤ ‖f‖p for every
f ∈ Lp(U), p ∈ [1,∞], and fh → f in Lp(U) for every f ∈ Lp(U),
p ∈ [1,∞) (see, e.g., 1.2.1 in [9]). It is clear that if f has a compact
support in U, then fh also has a compact support in U for small enough h.
A sequence ϕk ∈ L1(U) is called weakly fundamental if
lim
k1,k2→∞
∫
U
Φ(x) (ϕk1
(x) − ϕk2
(x)) dx = 0 ∀Φ ∈ L∞(U)
It is well known that the space L1(U) is weakly complete, i.e., every weakly
fundamental sequence ϕk ∈ L1(U) converges weakly in L1(U) (see, e.g.,
Theorem IV.8.6 in [3]). Give also the following useful statement (see,
e.g., Theorem 1.2.5 in [7]).
Proposition 2.1. Let f and g ∈ L1
loc(U). If
∫
f ϕ dx =
∫
g ϕ dx ∀ϕ ∈ C∞
0 (U), (2.4)
then f = g a.e.
Later on, in comparison with [11], we apply the following lemma in-
stead of Lemma III.3.5 in [10] which is not valid for p = 1.
Lemma 2.1. Let U be a bounded open set in R
n, and let fk : U → R
be a sequence of functions of the class W 1,1(U). Suppose that fk → f
as k → ∞ weakly in L1(U), ∂fk/∂xj, k = 1, 2, . . ., j = 1, 2, . . . , n are
uniformly bounded in L1(U) and their indefinite integrals are absolutely
equicontinuous. Then f ∈ W 1,1(U) and ∂fk/∂xj → ∂f/∂xj as k → ∞
weakly in L1(U).
Remark 2.1. The weak convergence fk → f in L1(U) implies that
sup
k
‖fk‖1 < ∞
(see, e.g., IV.8.7 in [3]). The latter together with
V. Ryazanov, U. Srebro, E. Yakubov 521
sup
k
‖∂fk/∂xj‖1 < ∞,
j = 1, 2, . . . , n, implies that fk → f by the norm in Lq for every 1 < q <
n/(n− 1), the limit function f belongs to BV (U), the class of functions
of bounded variation, but, generally speaking, not to the class W 1,1(U)
(see, e.g., Remark in 4.6 and Theorem 5.2.1 in [4]). Thus, the additional
condition of Lemma 2.1 on absolute equicontinuity of the indefinite inte-
grals of ∂fk/∂xj is essential (cf. also Remark to Theorem I.2.4 in [10]).
Proof of Lemma 2.1. It is known that the space L1 is weakly complete
(see Theorem IV.8.6 in [3]). Thus, it suffices to prove that the sequences
∂fk
∂xj
are weakly fundamental in L1.
Indeed, by the definition of generalized derivatives, we have
∫
U
ϕ(x)
∂fk
∂xj
dx = −
∫
U
fk(x)
∂ϕ
∂xj
dx ∀ϕ ∈ C∞
0 (U). (2.5)
Note that the integrals on the right-hand side in (2.5) are bounded linear
functionals in L1(U), and the sequence fk is weakly fundamental in L1(U)
because fk → f weakly in L1(U). Hence, in particular,
∫
U
ϕ(x)
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx → 0 ∀ϕ ∈ C∞
0 (U)
as k1 and k2 → ∞.
Now, let Φ ∈ L∞(U). Then ‖Φh‖∞ ≤ ‖Φ‖∞ and Φh → Φ in the
norm of L1(U) for its Sobolev mean functions Φh, and, hence, Φh → Φ
in measure as h → 0. Set ϕm = Φhm
, where Φhm
→ Φ a.e. as m → ∞.
Considering the restrictions of Φ to compacta in U, we may assume that
ϕm ∈ C ∞
0
(U). By the Egoroff theorem, ϕm → Φ uniformly on a set
S ⊂ U such that |U \ S| < δ, where δ > 0 can be arbitrary small (see,
e.g., III.6.12 in [3]). Given ε > 0, we have
∣
∣
∣
∣
∣
∫
S
(Φ(x) − ϕm(x))
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
≤ 2 · max
x∈S
| Φ(x) − ϕm(x)| · sup
k=1,2,...
∫
U
∣
∣
∣
∂fk
∂xj
∣
∣
∣
dx ≤
ε
3
for all large enough m. Choosing one such m, we have
∣
∣
∣
∣
∣
∫
U
ϕm(x)
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
≤
ε
3
522 On convergence theory...
for k1 and k2 large enough. By the absolute equicontinuity of the indef-
inite integrals of ∂fk/∂xj , there is δ > 0 such that
∫
E
∣
∣
∣
∂fk
∂xj
∣
∣
∣
dx ≤
1
12
ε
‖Φ‖∞
for all k = 1, 2, . . . and every measurable set E ⊂ U with |E| < δ (see
IV.8.10 and IV.8.11 in [3]). Setting E = U \ S, we obtain
∣
∣
∣
∣
∣
∫
U
Φ(x)
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
≤ I1 + I2 + I3,
where
I1 =
∣
∣
∣
∣
∣
∫
E
(Φ(x) − ϕm(x))
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
,
I2 =
∣
∣
∣
∣
∣
∫
S
(Φ(x) − ϕm(x))
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
,
I3 =
∣
∣
∣
∣
∣
∫
U
ϕm(x)
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
,
and, hence by the above arguments,
∣
∣
∣
∣
∣
∫
U
Φ(x)
(
∂fk1
∂xj
−
∂fk2
∂xj
)
dx
∣
∣
∣
∣
∣
≤ ε
for large enough k1 and k2. Thus, ∂fk
∂xj
is weakly fundamental in L1(U),
and hence ∂fk
∂xj
converges weakly in L1(U) just to ∂f
∂xj
by (2.5), see Propo-
sition 2.1.
3. On convergence of ACL homeomorphisms
Theorem 3.1. Let D be a domain in C, and let fn : D → C be a sequence
of sense-preserving ACL homeomorphisms with complex dilatations µn
such that
1 + |µn(z)|
1 − |µn(z)|
≤ Q(z) ∈ L1
loc ∀n = 1, 2, . . . (3.1)
If fn → f uniformly on each compact set in D, where f is a homeomor-
phism, then f ∈ ACL and ∂fn and ∂fn converge weakly in L1
loc
to ∂f and
∂f , respectively. Moreover, if, in addition, µn → µ a.e., then ∂f = µ∂f
a.e.
V. Ryazanov, U. Srebro, E. Yakubov 523
Remark 3.1. In fact, it is easy to show that, under condition (3.1), fn
and f belong to W 1,1
loc
(see, e.g., (3.2) below and II.3.27 in [3]). Moreover,
if, in addition, Q ∈ Lp
loc
, then fn and f belong to W 1,s
loc
, ∂fn → ∂f and
∂fn → ∂f weakly in Ls
loc
, where s = 2p/(1 + p) (see, e.g., Lemma 2.2
in [1]). Finally, f is a Q(z)-quasiconformal mapping, see [11].
Proof of Theorem 3.1. To prove the first part of the theorem, it suffices
by Lemma 2.1 to show that ∂fn and ∂fn are uniformly bounded in L1
loc
and have locally absolute equicontinuous indefinite integrals. So, let C
be a compact set in D, and let V be an open set with their compact
closure V in D such that C ⊂ V, say V = {z ∈ D : dist(z, C) < r},
where r < dist(C, ∂D). Note that
|∂fn| ≤ |∂fn| ≤ |∂fn| + |∂fn| ≤ Q1/2(z) · J1/2
n (z) a.e.,
where Jn is the Jacobian of fn. Consequently, by the Hölder inequality
and Lemma III.3.3 in [8],
∫
E
|∂fn| dx dy ≤
∣
∣
∣
∣
∣
∫
E
Q(z) dx dy
∣
∣
∣
∣
∣
1/2
|fn(C)|1/2
for every measurable set E ⊆ C. Hence, by the uniform convergence of
fn to f on C,
∫
E
|∂fn| dx dy ≤
∣
∣
∣
∣
∣
∫
E
Q(z) dx dy
∣
∣
∣
∣
∣
1/2
|f(V )|1/2 (3.2)
for large enough n and, thus, the first part of the proof is completed.
We now assume that µn(z) → µ(z) a.e. Set ζ(z) = ∂f(z)−µ(z) ∂f(z)
and show that ζ(z) = 0 a.e. Indeed, for every disk B with B ⊂ D, by
the triangle inequality
∣
∣
∣
∣
∣
∫
B
ζ(z) dx dy
∣
∣
∣
∣
∣
≤ I1(n) + I2(n),
where
I1(n) =
∣
∣
∣
∣
∣
∫
B
(
∂f(z) − ∂fn(z)
)
dx dy
∣
∣
∣
∣
∣
and
524 On convergence theory...
I2(n) =
∣
∣
∣
∣
∣
∫
B
(µ(z) ∂f(z) − µn(z) ∂fn(z)) dx dy
∣
∣
∣
∣
∣
Note that I1(n) → 0 because ∂fn → ∂f weakly in L1
loc
by the first part
of the proof. Next, I2(n) = I ′
2
(n) + I ′′
2
(n), where
I ′2(n) =
∣
∣
∣
∣
∣
∫
B
µ(z)(∂f(z) − ∂fn(z)) dx dy
∣
∣
∣
∣
∣
and
I ′′2 (n) =
∣
∣
∣
∣
∣
∫
B
(µ(z) − µn(z))∂fn(z) dx dy
∣
∣
∣
∣
∣
.
Again, by the weak convergence ∂fn → ∂f in L1
loc
, we have that I ′
2
(n) →
0 because µ ∈ L∞. Moreover, given ε > 0, by (3.2)
∫
E
|∂fn(z)| dx dy < ε, n = 1, 2, . . . , (3.3)
whenever E is every measurable set in B with |E| < δ for small enough
δ > 0.
Further, by the Egoroff theorem (see, e.g., III.6.12 in [3]), µn(z) →
µ(z) uniformly on some set S ⊂ B such that |E| < δ, where E = B \ S.
Hence, |µn(z) − µ(z)| < ε on S and, by (3.3),
I ′′2 (n) ≤ ε
∫
S
|∂fn(z)| dx dy + 2
∫
E
|∂fn(z)| dx dy
≤ ε
{(
∫
B
Q(z) dx dy
)1/2
· |f(λB)|1/2 + 2
}
for some λ > 1 and for all large enough n, i.e. I ′′
2
(n) → 0, because ε > 0 is
arbitrary. Thus,
∫
B ζ(z) dx dy = 0 for all disks B with B ⊂ D. Finally,
by the Lebesgue theorem on the differentiability of indefinite integrals
(see, e.g., IV(6.3) in [12]), ζ(z) = 0 a.e. in D.
Proposition 3.1. Let D be a domain in C and f
V. Ryazanov, U. Srebro, E. Yakubov 525
Proof. Indeed, suppose that f(z1) = f(z2) for some z1 6= z2 in D. For
small t > 0, let Dt be a disk of the spherical radius t centered at z1 such
that Dt ⊂ D and z2 /∈ Dt. Then, for all n, fn(∂Dt) separates fn(z1)
from fn(z2) and, hence, s(fn(z1), fn(∂Dt)) < s(fn(z1), fn(z2)). Thus,
for every such t, there is ζn(t) ∈ ∂Dt such that s(fn(z1), fn(ζn(t)) <
s(fn(z1), fn(z2)). Moreover, there is a subsequence ζnk
(t) → ζ0(t) ∈ ∂Dt,
because the circle ∂Dt is a compact set. However, the locally uniform
convergence fnk
→ f implies that fnk
(ζnk
(t)) → f(ζ0(t)) (see, e.g., [2,
p. 268]). Consequently, s(f(z1), f(ζ0(t)) ≤ s(f(z1), f(z2)). Then, since
f(z1) = f(z2), there is a point zt = ζ0(t) on ∂Dt such that f(z1) = f(zt)
for every small t contradicting the discreteness of f .
Corollary 3.1. Let D be a domain in C and fn : D → C, n = 1, 2, . . . ,
a sequence of quasiconformal mappings which satisfy (3.1). If fn → f
locally uniformly, then either f is constant or f is an ACL homeomor-
phism, and ∂fn and ∂̄fn converge weakly in L1
loc(D \ {f−1(∞)}) to ∂f
and ∂̄f , respectively. If, in addition, µn → µ a.e., then ∂̄f = µ∂f a.e.
Proof. Consider the case where f is not constant in D. Let us show that
then no point in D has a neighborhood of the constancy for f . Indeed,
assume that there is at least one point z0 ∈ D such that f(z) ≡ c for
some c ∈ C in a neighborhood of z0. Note that the set Ω0 of such points
z0 is open. The set Ec = {z ∈ D : s(f(z), c) > 0}, where s is the
spherical (chordal) distance in C, is also open in view of continuity of f
and not empty in the considered case. Thus, there is a point z0 ∈ ∂Ω0∩D
because D is connected. By continuity of f, we have f(z0) = c. However,
by construction, there is a point z1 ∈ Ec = D \ Ω0 such that |z0 −
z1| < r0 = dist (z0, ∂D) and, thus, by the lower estimate of the distance
s(f(z0), f(z)) in Lemma 3.12 from [11], we obtain a contradiction for
z ∈ Ω0. Then, again by Lemma 3.12 in [11], we obtain that f is discrete,
and f is a homeomorphism by Proposition 3.1. All other assertions follow
from Theorem 3.1.
4. On convergence of ring Q-homeomorphisms
Theorem 4.1. Let fn : D → C, n = 1, 2, . . . , be a sequence of ring Q-
homeomorphisms at a point z0 ∈ D. If fn converges locally uniformly to
a homeomorphism f : D → C, then f is also a ring Q-homeomorphism
at z0.
Proof. Note first that every point w0 ∈ D′ = fD belongs to D′
n =
fnD for all n ≥ N together with D(w0, ε), where D(w0, ε) = {w ∈
C : s(w, w0) < ε} for some ε > 0. Indeed, set δ = 1
2
s(z0, ∂D), where
526 On convergence theory...
z0 = f−1(w0) and εn = s(w0, ∂fnD(z0, δ)). Note that the sets fnD(z0, δ)
are open, and εn > 0 is the radius of the maximal closed disk centered at
w0 which is inside of fnD(z0, δ). Assume that εn → 0 as n → ∞. Since
∂D(z0, δ) and ∂fnD(z0, δ) = fn∂D(z0, δ) are compact, there exist zn ∈
∂D(z0, δ), s(zn, z0) = δ, such that εn = s(w0, fn(zn)), and we may assume
that zn → z∗ ∈ ∂D(z0, δ) as n → ∞ and then fn(zn) → f(z∗) as n → ∞
(see, e.g., [2, p. 268]). However, by construction, s(w0, fn(zn)) = εn → 0
as n → ∞, and, hence, f(z∗) = f(z0), i.e., z = z∗. This contradiction
disproves the above assumption. Thus, we obtain also that every compact
set C ⊂ D′ belongs to D′
n for all n ≥ N for some N.
Now remark that D′ =
⋃∞
m=1
Cm, where Cm = D∗
m, D∗
m is a con-
nected component of the open set Ωm = {w ∈ D′ : s(w, ∂D′) > 1/m},
m = 1, 2, . . . , including a fixed point w0 ∈ D′. Indeed, every point w ∈ D′
can be joined with w0 by a path γ in D′. Because |γ| is compact, we have
s(|γ|, ∂D′) > 0 and, consequently, w ∈ D∗
m for large enough m = 1, 2, . . . .
Next, take an arbitrary pair of continua E and F in D which belong
to the different connected components of the complement of a ring A =
A(z0, r1, r2) = {z ∈ C : r1 < |z − z0| < r2}, z0 ∈ D, 0 < r1 < r2 < r0 ≤
supz∈D |z − z0|. For m ≥ m0, the continua fE and fF belong to D∗
m.
Fix one of such m. Then the continua fnE and fnF also belong to D∗
m
for large enough n. As well known,
M(∆(fnE, fnF ; D∗
m)) → M(∆(fE, fF ; D∗
m))
as n → ∞, see [14, Theorem 1]. However, D∗
m ⊂ fnD for large enough
n, and hence
M(∆(fnE, fnF ; D∗
m)) ≤ M(∆(fnE, fnF ; fnD))
and, thus, by (1.5),
M(∆(fE, fF ; D∗
m)) ≤
∫
A∩D
Q(z) · η2(|z − z0|) dx dy
for every measurable function η : (r1, r2) → [0,∞] such that
r2
∫
r1
η(r) dr = 1.
V. Ryazanov, U. Srebro, E. Yakubov 527
Finally, since Γ =
⋃∞
m=m0
Γm where Γ = ∆(fE, fF ; fD) and Γm : =
∆(fE, fF ; D∗
m) is increasing in m = 1, 2, . . . , we obtain that M(Γ) =
limm→∞ M(Γm) (see, e.g., [5, Theorem 7]), and, thus,
M(∆(fE, fF ; fD)) ≤
∫
A∩D
Q(z) · η2(|z − z0|) dx dy,
i.e., f is a ring Q-homeomorphism at z0.
Acknowledgements. The research of the first author was partially
supported by grants from the University of Helsinki, from Technion —
Israel Institute of Technology, Haifa, and Holon Institute of Technology,
Israel, from Institute of Mathematics of PAN, Warsaw, Poland, and by
Grant F25.1/055 of the State Foundation of Fundamental Researches of
Ukraine; the research of the second author was partially supported by
a grant from the Israel Science Foundation (grant no. 198/00) and by
Technion Fund for the Promotion of Research, and the third author was
partially supported by a grant from the Israel Science Foundation (grant
no. 198/00).
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Contact information
Vladimir Ryazanov Institute of Applied Mathematics
and Mechanics, NAS of Ukraine,
R. Luxemburg Str. 74,
83114, Donetsk,
Ukraine
E-Mail: vlryazanov1@rambler.ru
Uri Srebro Technion — Israel Institute of Technology,
Haifa 32000,
Israel
E-Mail: srebro@math.technion.ac.il
Eduard Yakubov Holon Institute of Technology,
52 Golomb St., P.O.Box 305,
Holon 58102,
Israel
E-Mail: yakubov@hit.ac.il
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