On integral conditions in the mapping theory
It is established interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane.
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1243812017-09-25T03:03:03Z On integral conditions in the mapping theory Ryazanov, V. Srebro, U. Yakubov, E. It is established interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane. 2010 Article On integral conditions in the mapping theory / V. Ryazanov, U. Srebro, E. Yakubov // Український математичний вісник. — 2010. — Т. 7, № 1. — С. 73-87. — Бібліогр.: 28 назв. — англ. 1810-3200 2000 MSC. 30C65, 30C75. http://dspace.nbuv.gov.ua/handle/123456789/124381 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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It is established interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane. |
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Ryazanov, V. Srebro, U. Yakubov, E. |
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Ryazanov, V. Srebro, U. Yakubov, E. On integral conditions in the mapping theory Український математичний вісник |
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On integral conditions in the mapping theory |
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On integral conditions in the mapping theory |
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On integral conditions in the mapping theory |
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On integral conditions in the mapping theory |
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On integral conditions in the mapping theory |
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on integral conditions in the mapping theory |
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Інститут прикладної математики і механіки НАН України |
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On integral conditions in the mapping theory / V. Ryazanov, U. Srebro, E. Yakubov // Український математичний вісник. — 2010. — Т. 7, № 1. — С. 73-87. — Бібліогр.: 28 назв. — англ. |
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Український математичний вісник |
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2025-07-09T01:20:46Z |
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2025-07-09T01:20:46Z |
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1837130368472317952 |
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Український математичний вiсник
Том 7 (2010), № 1, 73 – 87
On integral conditions in the mapping theory
Vladimir Ryazanov, Uri Srebro,
Eduard Yakubov
(Presented by V. Ya. Gutlyanskii)
Abstract. It is established interconnections between various integral
conditions that play an important role in the theory of space mappings
and in the theory of degenerate Beltrami equations in the plane.
2000 MSC. 30C65, 30C75.
Key words and phrases. Integral conditions, mapping theory, Bel-
trami equations, Sobolev classes.
1. Introduction
In the theory of space mappings and in the Beltrami equation theory
in the plane, the integral conditions of the following type
1
∫
0
dr
rqλ(r)
= ∞, λ ∈ (0, 1], (1.1)
are often met with the function Q is given say in the unit ball B
n = {x ∈
R
n : |x| < 1} and q(r) is the average of the function Q(x) over the sphere
|x| = r, see e.g. [2, 4, 6, 7, 14–17,20–23,27,28].
On the other hand, in the theory of mappings called quasiconformal
in the mean, conditions of the type
∫
Bn
Φ(Q(x)) dx <∞ (1.2)
are standard for various characteristics Q of these mappings, see e.g.
[1, 3, 8, 10–13,17–19,26]. Here dx corresponds to the Lebsgue measure in
R
n, n ≥ 2.
Received 24.12.2009
ISSN 1810 – 3200. c© Iнститут математики НАН України
74 On integral conditions...
In this connection, we establish interconnections between integral con-
ditions on the function Φ and between (1.1) and (1.2). More precisely,
we give a series of equivalent conditions for Φ under which (1.2) implies
(1.1). It makes possible to apply many known results formulated under
the condition (1.1) to the theory of the mean quasiconformal mappings
in space as well as to the theory of the degenerate Beltrami equations in
the plane.
2. On some equivalent integral conditions
In this section we establish equivalence of integral conditions, see also
Section 3 for one more related condition.
Further we use the following notion of the inverse function for mono-
tone functions. For every non-decreasing function Φ : [0,∞] → [0,∞],
the inverse function Φ−1 : [0,∞] → [0,∞] can be well defined by setting
Φ−1(τ) = inf
Φ(t)≥τ
t. (2.1)
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 2.1. Immediately by the definition it is evident that
Φ−1(Φ(t)) ≤ t ∀ t ∈ [0,∞] (2.2)
with the equality in (2.2) except intervals of constancy of the function
Φ(t).
Similarly, for every non-increasing function ϕ : [0,∞] → [0,∞], set
ϕ−1(τ) = inf
ϕ(t)≤τ
t. (2.3)
Again, 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 also non-increasing.
Lemma 2.1. Let ψ : [0,∞] → [0,∞] be a sense-reversing homeomor-
phism and ϕ : [0,∞] → [0,∞] a monotone function. Then
[ψ ◦ ϕ]−1(τ) = ϕ−1 ◦ ψ−1(τ) ∀ τ ∈ [0,∞] (2.4)
and
[ϕ ◦ ψ]−1(τ) ≤ ψ−1 ◦ ϕ−1(τ) ∀ τ ∈ [0,∞] (2.5)
with the equality in (2.5) except a countable collection of τ ∈ [0,∞].
V. Ryazanov, U. Srebro, E. Yakubov 75
Remark 2.2. If ψ is a sense-preserving homeomorphism, then (2.4) and
(2.1) are obvious for every monotone function ϕ. Similar notations and
statements also hold for other segments [a, b], where a and b ∈ [−∞,+∞],
instead of the segment [0,∞].
Proof of Lemma 2.1. We first prove (2.4). If ϕ is non-increasing, then
[ψ ◦ ϕ]−1 (τ) = inf
ψ(ϕ(t))≥τ
t = inf
ϕ(t)≤ψ−1(τ)
t = ϕ−1 ◦ ψ−1(τ).
Similarly, if ϕ is non-decreasing, then
[ψ ◦ ϕ]−1 (τ) = inf
ψ(ϕ(t))≤τ
t = inf
ϕ(t)≥ψ−1(τ)
t = ϕ−1 ◦ ψ−1(τ).
Now, let us prove (2.5) and (2.1). If ϕ is non-increasing, then applying
the substitution η = ψ(t) we have
[ϕ ◦ ψ]−1 (τ) = inf
ϕ(ψ(t))≥τ
t
= inf
ϕ(η)≥τ
ψ−1(η) = ψ−1
(
sup
ϕ(η)≥τ
η
)
≤ ψ−1
(
inf
ϕ(η)≤τ
η
)
= ψ−1 ◦ ϕ−1 (τ) ,
i.e., (2.5) holds for all τ ∈ [0,∞]. It is evident that here the strict in-
equality is possible only for a countable collection of τ ∈ [0,∞] because
an interval of constancy of ϕ corresponds to every such τ. Hence (2.1)
holds for all τ ∈ [0,∞] if and only if ϕ is decreasing.
Similarly, if ϕ is non-decreasing, then
[ϕ ◦ ψ]−1 (τ) = inf
ϕ(ψ(t))≤τ
t
= inf
ϕ(η)≤τ
ψ−1(η) = ψ−1
(
sup
ϕ(η)≤τ
η
)
≤ ψ−1
(
inf
ϕ(η)≥τ
η
)
= ψ−1 ◦ ϕ−1 (τ) ,
i.e., (2.5) holds for all τ ∈ [0,∞] and again the strict inequality is possible
only for a countable collection of τ ∈ [0,∞]. In the case, (2.1) holds for
all τ ∈ [0,∞] if and only if ϕ is increasing.
Corollary 2.1. In particular, if ϕ : [0,∞] → [0,∞] is a monotone
function and ψ = j where j(t) = 1/t, then j−1 = j and
76 On integral conditions...
[j ◦ ϕ]−1(τ) = ϕ−1 ◦ j(τ) ∀ τ ∈ [0,∞] (2.6)
i.e.,
ϕ−1(τ) = Φ−1(1/τ) ∀ τ ∈ [0,∞] (2.7)
where Φ = 1/ϕ,
[ϕ ◦ j]−1(τ) ≤ j ◦ ϕ−1(τ) ∀ τ ∈ [0,∞] (2.8)
i.e., the inverse function of ϕ(1/t) is dominated by 1/ϕ−1, and except a
countable collection of τ ∈ [0,∞]
[ϕ ◦ j]−1(τ) = j ◦ ϕ−1(τ). (2.9)
1/ϕ−1 is the inverse function of ϕ(1/t) if and only if the function ϕ is
strictly monotone.
Further, in (2.11) and (2.12), we complete the definition of integrals
by ∞ if Φ(t) = ∞, correspondingly, H(t) = ∞, for all t ≥ T ∈ [0,∞).
The integral in (2.12) is understood as the Lebesgue–Stieltjes integral and
the integrals (2.11) and (2.13)–(2.16) as the ordinary Lebesgue integrals.
Theorem 2.1. Let Φ : [0,∞] → [0,∞] be a non-decreasing function and
let
H(t) = log Φ(t). (2.10)
Then the equality
∞
∫
∆
H ′(t)
dt
t
= ∞ (2.11)
implies the equality
∞
∫
∆
dH(t)
t
= ∞ (2.12)
and (2.12) is equivalent to
∞
∫
∆
H(t)
dt
t2
= ∞ (2.13)
for some ∆ > 0, and (2.13) is equivalent to every of the equalities:
V. Ryazanov, U. Srebro, E. Yakubov 77
δ
∫
0
H
(1
t
)
dt = ∞ (2.14)
for some δ > 0,
∞
∫
∆∗
dη
H−1(η)
= ∞ (2.15)
for some ∆∗ > H(+0),
∞
∫
δ∗
dτ
τΦ−1(τ)
= ∞ (2.16)
for some δ∗ > Φ(+0).
Moreover, (2.11) is equivalent to (2.12) and hence (2.11)–(2.16) are
equivalent each to other if Φ is in addition absolutely continuous. In
particular, all the conditions (2.11)–(2.16) are equivalent if Φ is convex
and non-decreasing.
Proof. The equality (2.11) implies (2.12) because except the mentioned
special case
T
∫
∆
dΨ(t) ≥
T
∫
∆
Ψ′(t) dt ∀T ∈ (∆, ∞)
where
Ψ(t) :=
t
∫
∆
dH(τ)
τ
, Ψ′(t) =
H ′(t)
t
,
see e.g. Theorem IV.7.4 in [25, p. 119], and hence
T
∫
∆
dH(t)
t
≥
T
∫
∆
H ′(t)
dt
t
∀T ∈ (∆, ∞)
The equality (2.12) is equivalent to (2.13) by integration by parts, see
e.g. Theorem III.14.1 in [25, p. 102]. Indeed, again except the mentioned
special case, through integration by parts we have
78 On integral conditions...
T
∫
∆
dH(t)
t
−
T
∫
∆
H(t)
dt
t2
=
H(T + 0)
T
−
H(∆ − 0)
∆
∀T ∈ (∆, ∞)
and, if
lim inf
t→∞
H(t)
t
<∞,
then the equivalence of (2.12) and (2.13) is obvious. If
lim
t→∞
H(t)
t
= ∞,
then (2.13) obviously holds, H(t)
t
≥ 1 for t > t0 and
T
∫
t0
dH(t)
t
=
T
∫
t0
H(t)
t
dH(t)
H(t)
≥ log
H(T )
H(t0)
= log
H(T )
T
+ log
T
H(t0)
→ ∞
as T → ∞, i.e. (2.12) holds, too.
Now, (2.13) is equivalent to (2.14) by the change of variables t→ 1/t.
Next, (2.14) is equivalent to (2.15) because by the geometric sense of
integrals as areas under graphs of the corresponding integrands
δ
∫
0
Ψ(t) dt =
∞
∫
Ψ(δ)
Ψ−1(η) dη + δ · Ψ(δ)
where Ψ(t) = H(1/t), and because by Corollary 2.1 the inverse function
for H (1/t) coincides with 1/H−1 at all points except a countable collec-
tion.
Further, set ψ(ξ) = log ξ. Then H = ψ ◦ Φ and by Lemma 2.1 and
Remark 2.2 H−1 = Φ−1 ◦ ψ−1, i.e., H−1(η) = Φ−1(eη), and by the
substitutions τ = eη, η = log τ we have the equivalence of (2.15) and
(2.16).
Finally, (2.11) and (2.12) are equivalent if Φ is absolutely continuous,
see e.g. Theorem IV.7.4 in [25, p. 119].
V. Ryazanov, U. Srebro, E. Yakubov 79
3. Connection with one more condition
In this section we establish useful connection of the conditions of
the Zorich–Lehto–Miklyukov–Suvorov type (3.23) further with one of the
integral conditions from the last section.
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 and t2 ∈ [0,∞] and λ ∈ [0, 1].
In what follows, B
n denotes the unit ball in the space R
n, n ≥ 2,
B
n = { x ∈ R
n : |x| < 1 }. (3.1)
Lemma 3.1. Let Q : B
n → [0,∞] be a measurable function and let
Φ : [0,∞] → [0,∞] be a non-decreasing convex function. Then
1
∫
0
dr
rq(r)
≥
1
n
∞
∫
λnM
dτ
τΦ−1(τ)
(3.2)
where q(r) is the average of the function Q(x) over the sphere |x| = r,
M =
∫
Bn
Φ(Q(x)) dx, (3.3)
λn = e/Ωn and Ωn is the volume of the unit ball in R
n.
Remark 3.1. In other words,
1
∫
0
dr
rq(r)
≥
1
n
∞
∫
eMn
dτ
τΦ−1(τ)
(3.4)
where
Mn : =
M
Ωn
= −
∫
Bn
Φ(Q(x)) dx (3.5)
is the mean value of the function Φ ◦ Q over the unit ball. Recall also
that by the Jacobi formula
Ωn =
ωn−1
n
=
2
n
·
π
n
2
Γ(n2 )
=
π
n
2
Γ(n2 + 1)
80 On integral conditions...
where ωn−1 is the area of the unit sphere in R
n, Γ is the well-known
gamma function of Euler, Γ(t + 1) = tΓ(t). For n = 2 we have that
Ωn = π, ωn−1 = 2π, and, thus, λ2 = e/Ω2 < 1. Consequently, we have
in the case n = 2 that
1
∫
0
dr
rq(r)
≥
1
2
∞
∫
M
dτ
τΦ−1(τ)
. (3.6)
In the general case we have that
Ω2m =
πm
m!
, Ω2m+1 =
2(2π)m
(2m+ 1)!!
,
i.e., Ωn → 0 and, correspondingly, λn → ∞ as n→ ∞.
Proof. Note that the result is obvious if M = ∞. Hence we assume
further that M <∞. Consequently, we may also assume that Φ(t) <∞
for all t ∈ [0,∞) because in the contrary case Q ∈ L∞(Bn) and then
the left hand side in (3.2) 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 (3.2) is equal to 0), Φ(t) is
(strictly) increasing, convex and continuous in a segment [t∗,∞] for some
t∗ ∈ [0,∞) and
Φ(t) ≡ τ0 = Φ(0) ∀ t ∈ [0, t∗]. (3.7)
Next, setting
H(t) := log Φ(t), (3.8)
we see by Proposition 2.1 and Remark 2.2 that
H−1(η) = Φ−1(eη), Φ−1(τ) = H−1(log τ). (3.9)
Thus, we obtain that
q(r) = H−1
(
log
h(r)
rn
)
= H−1
(
n log
1
r
+ log h(r)
)
∀ r ∈ R∗ (3.10)
where h(r) : = rnΦ(q(r)) and R∗ = {r ∈ (0, 1) : q(r) > t∗}. Then
also
q(e−s) = H−1
(
ns+ log h(e−s)
)
∀ s ∈ S∗ (3.11)
where S∗ = {s ∈ (0,∞) : q(e−s) > t∗}.
V. Ryazanov, U. Srebro, E. Yakubov 81
Now, by the Jensen inequality
∞
∫
0
h(e−s) ds =
1
∫
0
h(r)
dr
r
=
1
∫
0
Φ(q(r)) rn−1dr
≤
1
∫
0
(
−
∫
S(r)
Φ(Q(x)) dA
)
rn−1dr =
M
ωn−1
(3.12)
where we use the mean value of the function Φ ◦ Q over the sphere
S(r) = {x ∈ R
n : |x| = r} with respect to the area measure. Then
|T | =
∫
T
ds ≤
Ωn
ωn−1
=
1
n
(3.13)
where T = { s ∈ (0,∞) : h(e−s) > Mn}, Mn = M/Ωn. Let us show
that
q(e−s) ≤ H−1 (ns + log Mn) ∀ s ∈ (0,∞) \ T∗ (3.14)
where T∗ = T ∩ S∗. Note that (0,∞) \ T∗ = [(0,∞) \ S∗]∪ [(0,∞) \ T ] =
[(0,∞)\S∗]∪ [S∗ \T ]. The inequality (3.14) holds for s ∈ S∗ \T by (3.11)
because H−1 is a non-decreasing function. Note also that by (3.7)
ensMn = ens−
∫
Bn
Φ(Q(x)) dx > Φ(0) = τ0 ∀ s ∈ (0,∞). (3.15)
Hence, since the function Φ−1 is non-decreasing and Φ−1(τ0) = t∗, we
have by (3.9) that
t∗ < Φ−1 (Mn e
ns) = H−1 (ns + log Mn) ∀ s ∈ (0,∞). (3.16)
Consequently, (3.14) holds for s ∈ (0,∞) \ S∗, too. Thus, (3.14) is true.
Since H−1 is non-decreasing, we have by (3.13) and (3.14) that
1
∫
0
dr
rq(r)
=
∞
∫
0
ds
q(e−s)
≥
∫
(0,∞)\T∗
ds
H−1(ns+ ∆)
≥
∞
∫
|T∗|
ds
H−1(ns+ ∆)
≥
∞
∫
1
n
ds
H−1(ns+ ∆)
=
1
n
∞
∫
1+∆
dη
H−1(η)
(3.17)
82 On integral conditions...
where ∆ = logMn. Note that 1 + ∆ = log eMn = log λnM . Thus,
1
∫
0
dr
rq(r)
≥
1
n
∞
∫
log λnM
dη
H−1(η)
(3.18)
and, after the replacement η = log τ , we obtain (3.2).
Corollary 3.1. Let Q : B
n → [0,∞] be a measurable function and let
Φ : [0,∞] → [0,∞] be a non-decreasing convex function. Then
1
∫
0
dr
rqλ(r)
≥
1
n
∞
∫
λnM∗
dτ
τΦ−1(τ)
∀λ ∈ (0, 1) (3.19)
where q(r) is the average of the function Q(x) over the sphere |x| = r,
M∗ =
∫
Bn
Φ(Q∗(x)) dx, (3.20)
Q∗ is the lower cut-off function of Q, i.e., Q∗(x) = 1 if Q(x) < 1 and
Q∗(x) = Q(x) if Q(x) ≥ 1.
Indeed, let q∗(r) be the average of the function Q∗(x) over the sphere
|x| = r. Then q(r) ≤ q∗(r) and, moreover, q∗(r) ≥ 1 for all r ∈ (0, 1).
Thus, qλ(r) ≤ qλ∗ (r) ≤ q∗(r) for all λ ∈ (0, 1) and hence by Lemma 3.1
applied to the function Q∗(x) we obtain (3.19).
Theorem 3.1. Let Q : B
n → [0,∞] be a measurable function such that
∫
Bn
Φ(Q(x)) dx <∞ (3.21)
where Φ : [0,∞] → [0,∞] is a non-decreasing convex function such that
∞
∫
δ0
dτ
τΦ−1(τ)
= ∞ (3.22)
for some δ0 > τ0 = Φ(0). Then
1
∫
0
dr
rq(r)
= ∞ (3.23)
where q(r) is the average of the function Q(x) over the sphere |x| = r.
V. Ryazanov, U. Srebro, E. Yakubov 83
Remark 3.2. Note that (3.22) implies that
∞
∫
δ
dτ
τΦ−1(τ)
= ∞ (3.24)
for every δ ∈ [0,∞) but (3.24) for some δ ∈ [0,∞), generally speaking,
does not imply (3.22). Indeed, for δ ∈ [0, δ0), (3.22) evidently implies
(3.24) and, for δ ∈ (δ0,∞), we have that
0 ≤
δ
∫
δ0
dτ
τΦ−1(τ)
≤
1
Φ−1(δ0)
log
δ
δ0
< ∞ (3.25)
because Φ−1 is non-decreasing and Φ−1(δ0) > 0. Moreover, by the defini-
tion of the inverse function Φ−1(τ) ≡ 0 for all τ ∈ [0, τ0], τ0 = Φ(0), and
hence (3.24) for δ ∈ [0, τ0), generally speaking, does not imply (3.22). If
τ0 > 0, then
τ0
∫
δ
dτ
τΦ−1(τ)
= ∞ ∀ δ ∈ [0, τ0) (3.26)
However, (3.26) gives no information on the function Q(x) itself and,
consequently, (3.24) for δ < Φ(0) cannot imply (3.23) at all.
By (3.24) the proof of Theorem 3.1 is reduced to Lemma 3.1.
Corollary 3.2. If Φ : [0,∞] → [0,∞] is a non-decreasing convex func-
tion and Q satisfies the condition (3.21), then every of the conditions
(2.11)–(2.16) implies (3.23).
Moreover, if in addition Φ(1) < ∞ or q(r) ≥ 1 on a subset of (0, 1)
of a positive measure, then
1
∫
0
dr
rqλ(r)
= ∞ ∀λ ∈ (0, 1) (3.27)
and also
1
∫
0
dr
rαqβ(r)
= ∞ ∀α ≥ 1, β ∈ (0, α] (3.28)
84 On integral conditions...
Proof. First of all, by Theorems 2.1 and 3.1 every of the conditions
(2.11)–(2.16) implies (3.23). Now, if q(r) ≥ 1 on a subset of (0, 1) of
a positive measure, then also Q(x) ≥ 1 on a subset of B
n of a positive
measure and, consequently, in view of (3.21) we have that Φ(1) < ∞.
Then
∫
Bn
Φ(Q∗(x)) dx <∞ (3.29)
where Q∗ is the lower cut off function of Q from Corollary 3.1. Thus,
by Corollary 3.1 and Remark 3.2 we obtain that every of the conditions
(2.11)–(2.16) implies (3.27). Finally, (3.28) follows from (3.27) by the
Jensen inequality.
Remark 3.3. Note that if we have instead of (3.21) the condition
∫
D
Φ(Q(x)) dx <∞ (3.30)
for some measurable function Q : D → [0,∞] given in a domain D ⊂ R
n,
n ≥ 2, then also
∫
|x−x0|<r0
Φ(Q(x)) dx <∞ (3.31)
for every x0 ∈ D and r0 < dist (x0, ∂D) and by Theorem 3.1 and Corol-
lary 3.2, after the corresponding linear replacements of variables, we
obtain that
r0
∫
0
dr
rqλx0
(r)
= ∞ ∀λ ∈ (0, 1] (3.32)
where qx0
(r) is the average of the function Q(x) over the sphere |x−x0| =
r.
If D is a domain in the extended space Rn = R
n ∪ {∞} and ∞ ∈ D,
then in the neighborhood |x| > R0 of ∞ we may use the condition
∫
|x|>R0
Φ(Q(x))
dx
|x|2n
<∞ (3.33)
that is equivalent to the condition
∫
|x|<r0
Φ(Q′(x)) dx <∞ (3.34)
V. Ryazanov, U. Srebro, E. Yakubov 85
where r0 = 1/R0 and Q′(x) = Q(x/|x|2), i.e. Q′(x) is obtained from
Q(x) by the inversion of the independent variable x → x/|x|2, ∞ → 0,
with respect to the unit sphere |x| = 1.
Thus, by Theorem 3.1 and Corollary 3.2 the condition (3.33) imply
the equality
∞
∫
R0
dR
Rqλ∞(R)
= ∞ ∀λ ∈ (0, 1] (3.35)
where q∞(R) is the average of Q over the sphere |x| = R.
Finally, if D is an unbounded domain in R
n or a domain in Rn (3.30)
should be replaced by the following condition
∫
D
Φ(Q(x)) dS(x) <∞ (3.36)
where dS(x) = dx/(1 + |x|2)n is of a cell of the spherical volume. Here
the spherical distance
s(x, y) =
|x− y|
(1 + |x|2)
1
2 (1 + |y|2)
1
2
if x 6= ∞ 6= y, (3.37)
s(x,∞) =
1
(1 + |x|2)
1
2
if x 6= ∞.
It is easy to see that
dS(x) ≥ (1 + ρ2)−ndx
in every bounded part of D where |x| < ρ and
dS(x) ≥ 2−n
dx
|x|2n
in a neighborhood of ∞ where |x| ≥ 1. Hence the condition (3.36) implies
(3.31) as well as (3.33). Thus, under at least one of the conditions (2.12)–
(2.16) the condition (3.36) implies (3.32) and (3.35) if the function Φ is
convex and non-decreasing.
Recently it was established that the conditions (2.12)–(2.16) are not
only sufficient but also necessary for the degenerate Beltrami equations
with integral constraints of the type (3.21) on their characteristics to
have homeomorphic solutions of the class W 1,1
loc , see [5, 7, 9, 24].
Moreover, the above results will have significant corollaries to the
local and boundary behavior of space mappings in various modern classes
with integral constrains for dilatations, see e.g. [15].
86 On integral conditions...
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Contact information
Vladimir Ryazanov Institute of Applied Mathematics
and Mechanics, NAS of Ukraine,
ul. Roze Luxemburg 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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