About one modulus inequality of the order p ≥ 1
The present paper is devoted to the study of space mappings which are more general than quasiregular. The so-called modulus inequalities of the order p, p ≥ 1, and it’s connections with space mappings are investigated. The analogue of the well-known Poletskii inequality has been proved for the mappi...
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irk-123456789-1240872017-09-20T03:03:04Z About one modulus inequality of the order p ≥ 1 Salimov, R.R. Sevost'yanov, E.A. The present paper is devoted to the study of space mappings which are more general than quasiregular. The so-called modulus inequalities of the order p, p ≥ 1, and it’s connections with space mappings are investigated. The analogue of the well-known Poletskii inequality has been proved for the mappings having N, N⁻¹ and L⁽²⁾p–property. Работа посвящена изучению пространственных отображений более общих, чем квазирегулярные. Предметом изучения работы являются так называемые модульные неравенства порядка p, p ≥ 1, и их взаимосвязь с пространственными отображениями. Для отображений, имеющих N; N⁻¹ и L⁽²⁾p-свойства доказано хорошо известное неравенство Полецкого. Роботу присвячено вивченню просторових вiдображень, бiльш загальних, нiж квазiрегулярнi. Предметом дослiдження статтi є так званi модульнi нерiвностi порядку p, p ≥ 1, та їх взаємозв’язок з просторовими вiдображеннями. Для вiдображень, що мають N, N⁻¹ i L⁽²⁾p-властивостi, доведено аналог добре вiдомої нерiвностi типу Полецького. 2012 Article About one modulus inequality of the order p ≥ 1 / R.R. Salimov, E.A. Sevost'yanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2012. — Т. 24. — С. 183-189. — Бібліогр.: 8 назв. — англ. 1683-4720 http://dspace.nbuv.gov.ua/handle/123456789/124087 531.38 en Труды Института прикладной математики и механики Інститут прикладної математики і механіки НАН України |
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| description |
The present paper is devoted to the study of space mappings which are more general than quasiregular. The so-called modulus inequalities of the order p, p ≥ 1, and it’s connections with space mappings are investigated. The analogue of the well-known Poletskii inequality has been proved for the mappings having N, N⁻¹ and L⁽²⁾p–property. |
| format |
Article |
| author |
Salimov, R.R. Sevost'yanov, E.A. |
| spellingShingle |
Salimov, R.R. Sevost'yanov, E.A. About one modulus inequality of the order p ≥ 1 Труды Института прикладной математики и механики |
| author_facet |
Salimov, R.R. Sevost'yanov, E.A. |
| author_sort |
Salimov, R.R. |
| title |
About one modulus inequality of the order p ≥ 1 |
| title_short |
About one modulus inequality of the order p ≥ 1 |
| title_full |
About one modulus inequality of the order p ≥ 1 |
| title_fullStr |
About one modulus inequality of the order p ≥ 1 |
| title_full_unstemmed |
About one modulus inequality of the order p ≥ 1 |
| title_sort |
about one modulus inequality of the order p ≥ 1 |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/124087 |
| citation_txt |
About one modulus inequality of the order p ≥ 1 / R.R. Salimov, E.A. Sevost'yanov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2012. — Т. 24. — С. 183-189. — Бібліогр.: 8 назв. — англ. |
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Труды Института прикладной математики и механики |
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AT salimovrr aboutonemodulusinequalityoftheorderp1 AT sevostyanovea aboutonemodulusinequalityoftheorderp1 |
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2025-07-09T00:49:48Z |
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2025-07-09T00:49:48Z |
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1837128421498421248 |
| fulltext |
ISSN 1683-4720 Труды ИПММ НАН Украины. 2012. Том 24
UDK 531.38
c©2012. R.R. Salimov, E.A. Sevost’yanov
ABOUT ONE MODULUS INEQUALITY OF THE ORDER p ≥ 1
The present paper is devoted to the study of space mappings which are more general than quasiregular.
The so-called modulus inequalities of the order p, p ≥ 1, and it’s connections with space mappings are
investigated. The analogue of the well-known Poletskii inequality has been proved for the mappings
having N, N −1 and L
(2)
p –property
Keywords: mappings with finite and bounded distortion, modulus of curves families, Poletskii inequality.
1. Introduction. Here we give some definitions. Everywhere below, D is a domain
in Rn, n ≥ 2, m is the Lebesgue measure in Rn, m1 is the linear Lebesgue measure in R.
The notation f : D → Rn assumes that f is continuous.
Recall that a mapping f : D → Rn is said to have the N -property (of Luzin) iff
m (f (S)) = 0 whenever m(S) = 0 for all measurable sets S ⊂ Rn. Similarly, f has the
N−1-property iff m
(
f −1(S)
)
= 0 whenever m(S) = 0.
A curve γ in Rn is a continuous mapping γ : ∆ → Rn where ∆ is an interval in
R. Its locus γ(∆) is denoted by |γ|. Given a family of curves Γ in Rn, a Borel function
ρ : Rn → [0,∞] is called admissible for Γ, abbr. ρ ∈ admΓ, if
∫
γ
ρ(x)|dx| ≥ 1
for each (locally rectifiable) γ ∈ Γ. Let p ≥ 1. The p-modulus Mp(Γ) of Γ is defined as
Mp(Γ) = inf
ρ∈adm Γ
∫
Rn
ρp(x)dm(x)
interpreted as +∞ if admΓ = ∅. Note that Mp(∅) = 0; Mp(Γ1) ≤ Mp(Γ2) whenever
Γ1 ⊂ Γ2, and Mp
( ∞⋃
i=1
Γi
)
≤
∞∑
i=1
Mp(Γi), see Theorem 6.2 in [8].
We say that a property P holds for p-almost every (p-a.e.) curves γ in a family Γ if
the subfamily of all curves in Γ for which P fails has p-modulus zero.
If γ : ∆ → Rn is a locally rectifiable curve, then there is the unique nondecreasing
length function lγ of ∆ onto a length interval ∆γ ⊂ R with a prescribed normalization
lγ(t0) = 0 ∈ ∆γ , t0 ∈ ∆, such that lγ(t) is equal to the length of the subcurve γ|[t0,t] of γ if
t > t0, t ∈ ∆, and lγ(t) is equal to −length (γ|[t,t0]) if t < t0, t ∈ ∆. Let g : |γ| → Rn be a
continuous mapping, and suppose that the curve γ̃ = g◦γ is also locally rectifiable. Then
there is a unique non-decreasing function Lγ,g : ∆γ → ∆γ̃ such that Lγ,g (lγ(t)) = lγ̃(t)
for all t ∈ ∆. A curve γ in D is called here a (whole) lifting of a curve γ̃ in Rn under
f : D → Rn if γ̃ = f ◦ γ.
183
R.R. Salimov, E.A. Sevost’yanov
We say that a mapping f : D → Rn satisfies the L
(2)
p -property for p-a.e. curve γ̃ in
f(D), if each lifting γ of γ̃ is locally rectifiable and the function Lγ,f has the N−1-pro-
perty.
Set
l
(
f ′(x)
)
= min
h∈Rn\{0}
|f ′(x)h|
|h| ,
KI,p(x, f) =
|J(x,f)|
l(f ′(x))p , J(x, f) 6= 0,
1, f ′(x) = 0,
∞, otherwise
.
On of the main results proved in the paper is the following.
Statement 1. Let a mapping f : D → Rn be differentiable a.e. and satisfies N, N−1
and L
(2)
p -properties. Then
Mp(f(Γ)) ≤
∫
D
KI,p(x, f) · ρp(x) dm(x) (1)
for every family of curves Γ in D and ρ ∈ admΓ.
Remark that an analog of the Statement 1 for p = n was proved in [4], see Theorem
8.6 (see also [1] and [3]).
2. Proof of the main result. Let I = [a, b]. Given a rectifiable path γ : I → Rn
we define a length function lγ(t) by the rule lγ(t) = S (γ, [a, t]) , where S(γ, [a, t]) is the
length of the path γ|[a,t]. Let α : [a, b] → Rn be a rectifiable curve in Rn, n ≥ 2, and l(α)
be its length. A normal representation α0 of α is defined as a curve α0 : [0, l(α)] → Rn
which can be got from α by change of parameter such that α(t) = α0 (S (α, [a, t])) for
every t ∈ [0, l(α)].
Suppose that α and β are curves in Rn. Then a notation α ⊂ β denotes that α is a
subpath of β. In what follows, I denotes an open, a closed or a semi-open interval on the
real axes. The following definition can be found in the section 5 of Ch. II in [6].
Let f : D → Rn be a mapping such that f−1(y) does not contain a non-degenerate
curve, β : I0 → Rn be a closed rectifiable curve and α : I → D such that f ◦ α ⊂ β. If
the length function lβ : I0 → [0, l(β)] is a constant on J ⊂ I, then β is a constant on J
and consequently a curve α to be a constant on J. Thus, there exists a unique function
α ∗ : lβ(I) → D such that α = α ∗ ◦ (lβ|I). We say that α ∗ to be a f -representation of α
by the respect to β if β = f ◦ α.
Remark 1. Given a closed rectifiable curve γ : [a, b] → Rn and t0 ∈ (a, b), let lγ(t)
denotes the length of the subcurve γ|[t0,t] of γ if t > t0, t ∈ (a, b), and lγ(t) is equal to
−l(γ|[t,t0]) if t < t0, t ∈ (a, b). Then we observe that properties of the Lγ,f connected with
the length functions lγ(t) and lγ̃(t), γ̃ = f ◦ γ, do not essentially depend on the choice of
t0 ∈ (a, b). Moreover, we may consider that in this case t0 = a because given t0 ∈ (a, b),
S(γ, [a, t]) = S(γ, [a, t0]) + lγ(t). Further, we use the notion lγ(t) for lγ(t) = S (γ, [a, t]) ,
where S(γ, [a, t]) is the length of the path γ|[a,t], and consider that t0 = 0 whenever a
curve γ is closed.
184
About one modulus inequality of the order p ≥ 1
The following statement gives the connection between L
(2)
p -property and some
properties of curves meaning above.
Lemma 1.A mapping f : D → Rn has L
(2)
p -property if and only if f−1(y) does not
contain a nondegenerate curve for every y ∈ Rn, and the f -representation γ ∗ is rectifiable
and absolutely continuous for p-a.e. closed curves γ̃ = f ◦ γ.
Proof. Suppose that f has L
(2)
p -property. Then γ ∗ is rectifiable for p-a.e. closed curves
γ̃ whenever γ̃ = f ◦ γ because (γ ∗) 0 = γ 0, see Theorem 2.6 in [8]. Moreover, we observe
that f−1(y) does not contain a nondegenerate curve for every y ∈ Rn because Lγ,f is
well-defined and has N −1-property for p-a.e. closed curves γ̃ and all γ with γ̃ = f ◦ γ.
For such γ and γ̃, we have
γ(t) = γ ∗ ◦ lγ̃(t) = γ 0 ◦ lγ(t) = γ 0 ◦ L−1
γ,f
(
lγ̃(t)
)
and, denoting by s := lγ̃(t) we obtain
γ ∗(s) = γ 0 ◦ L−1
γ,f (s) .
So γ ∗ is absolutely continuous because L−1
γ,f (s) is absolutely continuous, see section
2.10.13 in [2], and
|γ 0(s1)− γ 0(s2)| ≤ |s1 − s2|
for all s1, s2 ∈ [0, l(γ)].
Inversely, let f−1(y) does not contain a nondegenerate curve for every y ∈ Rn. Then
L−1
γ,f is well-defined for p-a.e. closed curve γ̃ and all γ with γ̃ = f ◦ γ. By assumption
curve γ ∗ is rectifiable for p-a.e. closed curve γ̃ = f ◦γ; in particular, γ ∗ 0 = γ 0. Moreover,
for all such γ̃, γ and γ ∗, lγ ∗(s) = L−1
γ,f (s), and absolutely continuity of L−1
γ,f (s) follows
from Theorem 1.3 in [8]. Let Γ1 be a family of all closed curves α̃ = f ◦ α in f(D) such
that α ∗ either is not rectifiable or L−1
α,f (s) is not absolutely continuous. Let Γ be a family
of all curves γ̃ = f ◦ γ in f(D) such that γ either is not locally rectifiable or L−1
γ,f (s) is
not locally absolutely continuous. Then Γ > Γ1 and, thus, Mp(Γ) ≤ Mp(Γ1) = 0 that
implies desired equality Mp(Γ) = 0. ¤
A mapping ϕ : X → Y between metric spaces X and Y is said to be a Lipschitzian
provided
dist (ϕ(x1), ϕ(x2)) ≤ M · dist(x1, x2)
for some M < ∞ and for all x1 and x2 ∈ X. The mapping ϕ is called bi-lipschitz if, in
addition,
M∗dist (x1, x2) ≤ dist (ϕ (x1) , ϕ (x2))
for some M∗ > 0 and for all x1 and x2 ∈ X. Later on, X and Y are subsets of Rn with
the Euclidean distance.
The following proposition can be found in [3], see Lemma 3.20, see also Lemma 8.3
Ch. VIII in [4].
Lemma 2. Let f : D → Rn be a differentiable a.e. in D, and have N - and N−1-pro-
perties. Then there is a countable collection of compact sets C∗
k ⊂ D such that m(B0) = 0
185
R.R. Salimov, E.A. Sevost’yanov
where B0 = D \
∞⋃
k=1
C∗
k and f |C∗k is one-to-one and bi-lipschitz for every k = 1, 2, . . . .
Moreover, f is differentiable at C∗
k and J(x, f) 6= 0.
Given a set E in Rn and a curve γ : ∆ → Rn, we identify γ ∩E with γ (∆) ∩E. If γ
is locally rectifiable, then we set
l (γ ∩ E) = m1(Eγ),
where Eγ = lγ
(
γ−1 (E)
)
; here lγ : ∆ → ∆γ as in the previous section. Note that
Eγ = γ−1
0 (E) , where γ0 : ∆γ → Rn is the natural parametrization of γ and
l (γ ∩ E) =
∫
∆
χE (γ (t)) |dx| :=
∫
∆γ
χEγ (s)ds .
The bellow statement can be found in Chapter IX of [4], see Theorem 9.1.
Lemma 3. Let E be a set in a domain D ⊂ Rn, n ≥ 2, p ≥ 1. Then E is measurable
if and only if γ ∩ E is measurable for p-a.e. curve γ in D. Moreover, m(E) = 0 if and
only if
l(γ ∩ E) = 0
on p-a.e. curve γ in D.
The following result is a generalization of the known Poletskii inequality for
quasiregular mappings, see Theorem 1 in [5] and Theorem 8.1 Ch. II in [6]. It’s analog
was also proved in [3-4] for the case p = n, see also [1].
Theorem 1. Let a mapping f : D → Rn be a differentiable a.e. in D, have N - and
N−1-properties, and L
(2)
p -property, too. Then the relation (1) holds for every curve family
Γ in D and a function ρ ∈ admΓ.
Proof. Let B0 and C∗
k , k = 1, 2, . . . , be as in Lemma 2. Setting by induction B1 = C∗
1 ,
B2 = C∗
2 \B1, . . . , and
Bk = C∗
k \
k−1⋃
l=1
Bl (2)
we obtain the countable covering of D consisting of mutually disjoint Borel sets Bk, k =
0, 1, 2, . . . with m(B0) = 0, B0 = D \
∞⋃
k=1
Bk. By the assumption, f has N -property in D
and, consequently, m(f(B0)) = 0. Let ρ ∈ admΓ and
ρ̃(y) = χf(D\B0) · sup
x∈f−1(y)∩D\B0
ρ∗(x) ,
where
ρ ∗(x) =
{
ρ(x)/l (f ′(x)) , for x ∈ D \B0,
0, otherwise.
186
About one modulus inequality of the order p ≥ 1
Note that ρ̃(y) = sup
k∈N
ρk(y) where
ρk(y) =
{
ρ∗(f−1
k (y)), for y ∈ f(Bk),
0, otherwise,
and every fk = f |Bk
, k = 1, 2, . . . , is injective. Thus, the function ρ̃ is Borel, see section
2.3.2 in [2].
Let γ̃ be a closed rectifiable curve such that γ̃ = f ◦ γ, γ̃0 be a normal representation
of γ̃ and γ∗ be f -representation of γ by the respect to γ̃, see above. Since m(f(B0)) = 0,
γ̃0(s) 6∈ f(B0) for p-a.e. curve γ̃ and a.e. s ∈ [0, l(γ̃)], see Lemma 3. For p-a.e. paths γ̃
and all γ with γ̃ = f ◦ γ, we have that
∫
γ̃
ρ̃(y)|dy| =
l(γ̃)∫
0
ρ̃(γ̃0(s)) ds =
=
l(γ̃)∫
0
sup
x∈f −1(γ̃0(s))∩D\B0
ρ ∗(x) ds ≥
l(γ̃)∫
0
ρ(γ ∗(s))
l (f ′(γ ∗(s)))
ds . (3)
Since γ̃ 0 is rectifiable, γ̃ 0(s) is differentiable a.e. Besides that, a curve γ ∗ is absolutely
continuous for p-a.e. γ̃ by Lemma 1. Since γ̃0(s) 6∈ f(B0) for a.e. s ∈ [0, l(γ̃)] and p-a.e.
curves γ̃, we have γ ∗(s) 6∈ B0 at a.e. s ∈ [0, l(γ̃)]. Thus, the derivatives f ′ (γ ∗(s))
and γ ∗′(s) exist for a.e. s. Taking into account the formula of the derivative of the
superposition of functions, and that the modulus of the derivative of the curve by the
natural parameter equals to 1, we have
1 =
∣∣(f ◦ γ ∗) ′ (s)
∣∣ =
∣∣f ′ (γ ∗(s)) γ ∗′(s)
∣∣ =
=
∣∣∣∣f ′ (γ ∗(s)) ·
γ ∗′(s)
|γ ∗′(s)|
∣∣∣∣ · |γ ∗′(s)| ≥ l
(
f ′ (γ ∗(s))
) · |γ ∗′(s)| . (4)
It follows from (4) that a.e.
ρ(γ∗(s))
l (f ′ (γ ∗(s)))
≥ ρ(γ∗(s)) · |γ ∗′(s)| . (5)
By absolutely continuity of γ ∗, definition of ρ and Theorem 4.1 in [8] we obtain
1 ≤
∫
γ
ρ(x)|dx| =
l(γ̃)∫
0
ρ (γ ∗(s)) · |γ ∗′(s)| ds . (6)
It follows from (3), (5) and (6) that
∫
γ̃
ρ̃(y)|dy| ≥ 1 for p-a.e. closed curve γ̃ in f(Γ). The
case of the arbitrary path γ̃ can be got from the taking of sup in
∫
γ̃ ′
ρ̃(y)|dy| ≥ 1 over all
187
R.R. Salimov, E.A. Sevost’yanov
closed subpaths γ̃ ′ of γ̃. Thus, ρ̃(y) ∈ adm f(Γ) \ Γ0, where Mp(Γ0) = 0. Hence
Mp (f (Γ)) ≤
∫
f(D)
ρ̃ p(y)dm(y) . (7)
Further, by 3.2.5 for m = n in [2] we have that
∫
Bk
KI,p(x, f) · ρp(x)dm(x) =
∫
Bk
|J(x, f)|
(l (f ′(x)))p · ρp(x)dm(x) =
=
∫
f(Bk)
ρp
(
f−1
k (y)
)
(
l
(
f ′
(
f −1
k (y)
)))p dm(y) =
∫
f(D)
ρp
k(y)dm(y). (8)
Finally, by the Lebesgue theorem, see Theorem 12.3 § 12 of Ch. I in [7], we obtain from
(7) and (8) the desired inequality
∫
D
KI,p(x, f) · ρp(x)dm(x) =
∞∑
k=1
∫
Bk
KI,p(x, f) · ρp(x)dm(x) =
=
∫
f(D)
∞∑
k=1
ρp
k(y)dm(y) ≥
∫
f(D)
sup
k∈N
ρp
k(y)dm(y) =
=
∫
f(D)
ρ̃ p(y)dm(y) = Mp (f(Γ)) .
¤
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Berlin etc.: Springer-Verlag, 1971.
188
About one modulus inequality of the order p ≥ 1
Р.Р. Салимов, E.A. Севостьянов
Об одном модульном неравенстве порядка p ≥ 1.
Работа посвящена изучению пространственных отображений более общих, чем квазирегулярные.
Предметом изучения работы являются так называемые модульные неравенства порядка p ≥ 1 и
их взаимосвязь с пространственными отображениями. Для отображений, имеющих N, N −1 и L
(2)
p -
свойства доказано хорошо известное неравенство Полецкого.
Ключевые слова: отображения с конечным и ограниченным искажением, модуль семейств
кривых, неравенство Полецкого.
Р.Р. Салiмов, Є.О. Севостьянов
Про одну модульну нерiвнiсть порядку p ≥ 1.
Роботу присвячено вивченню просторових вiдображень, бiльш загальних, нiж квазiрегулярнi.
Предметом дослiдження статтi є так званi модульнi нерiвностi порядку p ≥ 1, та їх взаємозв’язок
з просторовими вiдображеннями. Для вiдображень, що мають N, N −1 i L
(2)
p -властивостi, доведено
аналог добре вiдомої нерiвностi типу Полецького.
Ключовi слова: вiдображення зi скiнченним i обмеженим спотворенням, модуль сiм’ї кривих,
нерiвнiсть Полецького.
Ин-т прикл. математики и механики НАН Украины, Донецк
ruslan623@yandex.ru
esevostyanov2009@mail.ru
Received 19.04.12
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