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Владикавказский математический журнал 2022, Том 24, Выпуск 4, С. 58-69
УДК 517.518.23+517.548.2 DOI 10.46698/w5793-5981-8894-o
ON POLETSKY-TYPE MODULUS INEQUALITIES FOR SOME CLASSES OF MAPPINGS*
S. K. Vodopyanov1
1 Sobolev Institute of Mathematics, 4 Akademika Koptyuga Ave., Novosibirsk 630090, Russia E-mail: [email protected]
Abstract. It is well-known that the theory of mappings with bounded distortion was laid by Yu. G. Reshetnyak in 60-th of the last century [1]. In papers [2, 3], there was introduced the two-index scale of mappings with weighted bounded (q,p)-distortion. This scale of mappings includes, in particular, mappings with bounded distortion mentioned above (under q = p = n and the trivial weight function). In paper [4], for the two-index scale of mappings with weighted bounded (q,p)-distortion, the Poletsky-type modulus inequality was proved under minimal regularity; many examples of mappings were given to which the results of [4] can be applied. In this paper we show how to apply results of [4] to one such class. Another goal of this paper is to exhibit a new class of mappings in which Poletsky-type modulus inequalities is valid. To this end, for n = 2, we extend the validity of the assertions in [4] to the limiting exponents of summability: 1 < q < p < to. This generalization contains, as a special case, the results of recently published papers. As a consequence of our results, we also obtain estimates for the change in capacity of condensers.
Key words: quasiconformal analysis, Sobolev space, modulus of a family of curves, modulus estimate. AMS Subject Classification: 30C65 (26B35, 31B15, 46E35).
For citation: Vodopyanov, S. K. On Poletsky-type Modulus Inequalities for Some Classes of Mappings, Vladikavkaz Math. J., 2022, vol. 24, no. 4, pp. 58-69. DOI: 10.46698/w5793-5981-8894-o.
1. Introduction
The goal of this work is to show the application of results of [4] for output of Poletsky-type modulus inequalities for some classes of mappings. For doing this we formulate first the main result of [4], and then we provide how it can be applied for some concrete classes of mappings. The main classes of mappings studied in [4] were defined in [2, 3].
Definition 1. Let w: Rn ^ [0, to] be a measurable function, called a weight, with 0 < w < to holding Hn-almost everywhere, and Q C Rn is a domain in Rn. A mapping f : Q ^ Rn with n ^ 2 is called a mapping with (inner) bounded w-weighted (q,p)-codistortion, or briefly, f € ID(Q; q,p; w, 1), where n — 1 ^ q ^ p < to, whenever
(1) f is continuous, open and discrete;
(2) f belongs to the Sobolev class Wn\-1 ioc(Q);
(3) the Jacobian determinant satisfies det Df (x) ^ 0 for almost all x € Q;
# The study was carried out within the framework of the State contract of the Sobolev Institute of Mathematics, project № FWNF-2022-0006. © 2022 Vodopyanov, S. K.
(4) the mapping f has bounded codistortion: adj Df (x) = 0 a. e. on the set Z = {x € Q : det Df (x) = 0};
(5) the local u-weighted (q,p)-codistortion function
n-l
O. ^^ n ) U 9 {x)ladi^ix)l if detU/C^^O,
U 3 X ^ Jtiq f(x, f) = \ det Df(x) P (1)
otherwise,
belongs to Lg(Q), where g satisfies ^ = ^^ — while g = oo for q = p.
Put K,p1(f;Q) = HK^O.f) I
Definition 2. Let u: Rn ^ [0, to] be a measurable function, called a weight, with 0 < u < to holding -almost everywhere, and Q C Rn is a domain in Rn. A mapping f : Q ^ Rn with n ^ 2 is called a mapping with (outer) bounded u-weighted (q,p)-distortion, or briefly f € OD (Q; q,p; u, 1), with n — 1 ^ q ^ p < to, whenever:
(1)f is continuous, open and discrete;
(2) f belongs to the Sobolev class W?1_1 ioc(Q);
(3) the Jacobian determinant satisfies det Df (x) ^ 0 for a. e. x € Q;
(4) the mapping f has bounded distortion: Df (x) = 0 a. e. on the set Z = {x € Q : det Df (x) = 0};
(5) the local u-weighted (q,p)-distortion function
J(x)\Df(x)\
if det Df (x) = 0,
n3x^K^(x,f) = <| det£>/(*)* ^^ , (2)
otherwise,
belongs to LJQ), where x satisfies - = \ — while x = oo for q = p.
X q p
Put Kqwp1(f; Q) = |KqWp1(-,f) I Lk (Q)||. Remark 1. It is established in [3] that
OD(Q; q,p; u, 1) C ID(Q; q,p; u, 1) (3)
in case of n — 1 <q ^ p < to.
For justifying (3) we refer to [3, Theorem 8] where it is proved that every mapping f : Q ^ Q' of OD(Q; q,p; u, 1), n — 1 < q ^ p < to, belongs also to the class ID(Q; q,p; u, 1), and the estimate
IlK^OJ) I Le(Q)|| < HK^OJ) I L^f-1 (4)
holds. (Here g and k are defined after formulas (1) and (2) respectively). In [4] it was proved the following result.
Theorem 1 [4, Theorem 4.1]. Let n — 1 <q ^ p< to. Suppose that f : Q ^ Rn is a mapping with with inner bounded u-weighted (q,p)-codistortion (f € ID(Q; q,p; u, 1)),
n-l
while the weight function 0(x) = u «-(«-1) (x) is locally summable. IfT is a family of curves in the domain Q then we have the inequality
(mods f (r))1/s < Kqp1 (f; Q) (mod? r)1/r, (5)
with s = „ P -in and r =
p—(to—1) g—(to— 1) '
q
Below we recall the concept of the modulus of a family of curves (see [4] for more details).
A curve in Rn is a continuous mapping a: I ^ Rn, where I is an interval in R, that is, a set of the form (a, b), where each angular parenthesis can be either round or square, a, b € R with a ^ b. We also allow infinite intervals. A curve a is called closed (open) if the interval I is compact (open). Put |a| = a(1). The expression y' c y will mean that the curve y' is a restriction of the curve y to a subinterval or a point.
If a: I = [a, b] ^ Rn is a closed curve then its length is
i
|a(
i= 1
1(a) = sup£ |a(ti) - a(ii+i)!
where the supremum is taken over all finite partitions a = t1 ^ t2 ^ ... ^ tl ^ tl+1 = b. If a curve a is not closed then put its length equal to 1(a) = sup l(a| j), where the supremum is taken over all closed subintervals J of I.
A curve a: I ^ Rn is called rectifiable whenever 1(a) < to. A curve is called locally rectifiable if each closed subcurve of it is rectifiable.
Consider a closed curve a: [a, b] ^ Rn and suppose that it is rectifiable. Define a function sa: [a, b] ^ R by the equality sa(t) = l(a|[a,t]). For the rectifiable curve a there exists a unique curve a0: [0,l(a)] ^ Rn obtained from a by a monotonely increasing change of parameter such that sao(t) = t and a = a0 o sa [5, Section 2.4]. The curve a0 is called the positive natural parametrization of a.
Take a Borel set A c Rn and a Borel function p: A ^ [0, to]. The integral of p along a rectifiable curve a: [a, b] ^ Rn is defined as
£(a)
„0/
j pds = J p(a°(r)) dH 1(t)
a
0
with an usual Lebesgue integral in the right-hand side. If a is absolutely continuous then so is the function sa(t) = [a, b] ^ [0,l(a)]. Putting t = sa(t) in the last integral, using the change-of-variables theorem for Lebesgue integrals, and accounting for a(t) = ^a0(sQ,(i))sQ,(i) and ■£pa°(T) = 1, we infer that
b
J pds = j p(a(t))|a(t)| dH 1(t). (6)
a a
Observe that by the change of variable formula we can express this as
b
J pds = j p(a(t))|a(t)| dH 1(t) = y p(y)N (y,a, [a,b]) dH1(y), (7)
a a |a|
where N(y, a, [a, b]) = #{[a, b] n a-1(y)} is the Banach indicatrix. For a locally rectifiable curve a: I ^ Rn, put
J pds = sup J pds, (8)
a ^ £
where the supremum is taken over all closed subcurves ^ of a.
Consider a family r of curves in Rn, where n ^ 2. A Borel function p : Rn ^ [0, to] is called admissible for r whenever
J pds ^ 1 (9)
Y
for each locally rectifiable curve y € r. Denote the collection of all admissible functions by adm r. Given a weight function d : Rn ^ (0, to) and a number p € [1, to), define the d-weighted p-modulus of r as
modP r= inf /pP0dHn.
p p€adm r J
Properties of the weight function will be prescribed separately; at least, we assume that it is locally summable and 0 < d < to holds Hn-almost everywhere. For d = 1 we obtain the usual definition of p-modulus, and instead of modi r we write modp r. If admT = 0 then we put modp r = to; this case is realized only if r contains at least one curve determining a constant mapping.
Remark 2. The definition of modulus implies that every family of curves which are not locally rectifiable has zero modulus. Moreover, if r is a family of curves and r = {7 € r : Y is locally rectifiable} then modp(r) = modP(ri).
Suppose that a is a rectifiable closed curve in Rn. A mapping g: |a| — Rn is called absolutely continuous on a if the composition g o a0 is absolutely continuous on [0,1(a)].
Theorem 2 [5, Fuglede's Theorem; 6]. Suppose that f: Q — Rn is a mapping of class Wi(Q) with 1 ^ p < to, and r is a family of locally rectifiable curves in Q such that each curve has a closed subcurve on which f is not absolutely continuous. Then modp r = 0.
2. Modification of Theorem 1 in the Case of n = 2 and p = to
In this case parameters q,p may be taken within (1, to]: 1 < q ^ p ^ to. The case 1 <q ^ p< to is taken into consideration in Theorem 1.
Theorem 3. Let 1 < q < p = to. Suppose that Q C R2 is a domain, and f: Q ^ R2
is a mapping with inner bounded w-weighted (q, to)-codistortion (f € ID(Q; q, to; w, 1)1),
__
while the weight function 0(x) = uj 1-1 (x) is locally summable. IfT is a family of curves in the domain Q then we have the inequality
(modi f (r)) < ;Q)(mod? r)1/r (10)
with r = ~r~r.
q—1
In this theorem Ks°o(f; Q) = ||Ks°o(-,f) I Lr(Q)||.
Theorem 4. Suppose that Q C R2 is a domain, and f: Q R2 is a mapping belonging to the Sobolev class W11loc(Q) with the nonnegative Jacobian determinant: det Df (x) ^ 0 for almost all x € Q. Assume that
1) f is continuous, open and discrete;
(2) the mapping f has bounded codistortion: adj Df (x) =0 a. e. on the set Z = {x € Q : det Df (x) = 0}.
In the case p = oc we have to replace det Df(x) ? in (1) by 1.
Let, for a weight u: Rn ^ [0, to], (to, TO)-codistortion function
Q 3 x ^ K-L(x,f) = (u(x)|adj Df (x)| lf det Df (x) > 0, (11)
0 otherwise,
belongs to ll(Q) (in another words f € ID(Q; to, to; u, 1)). If the weight function 0(x) = u-1(x) is locally summable then, for any family of curves r in the domain Q, we have the inequality
mod1 f (r) < (f ;Q)mod? r. (12)
In this theorem (f; Q) = (■, f) | L-(Q)||.
Theorems 3 and 4 will be proved in Section 6.
3. Application
In paper [7, Example 32] the following class of mappings is considered. Suppose that n — 1 < p < to, and consider a continuous, open and discrete mapping f : D' ^ Rn of an open connected domain D' C Rn, where n ^ 2, such that
(1) f € Wn_1;ioc (D');
(2) det Df (y) ^ 0 and f has finite codistortion; i. e., adj Df (y) = 0 Hn-almost everywhere on Z = {y € D' : det Df (y) = 0};
(3) the inner operator distortion function
D' 3 y ^ K„_\Mf ) =
|adjD/fa)|
n — 1
det Df(y) —
if det Df (y) = 0, otherwise,
(13)
belongs to LPt\oc(D'), where | = j- — ^^ holds with s = > n — 1;
(4) the weight function a defined as
f I adj Df{y)\v = J detD/(y)P-i
if y € D'\Z', otherwise,
(14)
is in € L1,ioc (D'), here Z' = {y € D' : Df (y) = 0}.
Taking into acount saying above we see that f : D' ^ D meets the assumptions of Theorem 1 with D' instead of Q: (2a) f € Wn_1;ioc (D');
(2b) det Df (y) ^ 0 and f has finite codistortion;
(2c) f: D' ^ D is a mapping of bounded u-weighted (s, s)-codistortion with u(y) = __
<t p~1(y), that is, the w-weighted (s, s)-codistortion function
D' 3 y ^ «'W) =
^(y)\ad}Df(y)\
, n —1
det Df(y)~
if J(y, f) = 0,
otherwise,
belongs to Ll(D') and
|KSTs,1(-,f) I l-(D')|| = 1
0
0
(the last equality is proved in [7, Theorem 3] under more general assumption).
Taking into account saying above, by Theorem 1, we come to the following statement.
Proposition 1. Suppose that a continuous, open and discrete mapping f : D' ^ Rn of an open connected domain D' c Rn, where n ^ 2, has the following properties:
(1) f e WUioc (D');
(2) det Df (y) ^ 0 and f has finite codistortion (adj Df (y) = 0 Hn-almost everywhere on Z = {y e D' : det Df (y) = 0});
(3) the inner operator distortion function
\ |adJD/(^i if det Df(y) / 0, D' 3 y X^L\s{y, /) = { detD/^-r- (16)
I 0 otherwise,
belongs to LPi\oc(Dr) with some p > n — 1, where | = — ^ir holds with s = > n — 1.
If r is a family of curves in the domain D' then we have the inequality
modp f (r) < modp r (17)
where the weight function a is defined in (7).
< When deriving inequality (17) the properties (2a)-(2c) formulated above, should be
taken into account. Really, we see that f e ID(Q; q,p; w, 1) with q = p = s and w(y) = __
(7 p-i(y). Therefore, by Theorem 1, we get the inequality
mods' f (r))1/s' < K-'1(f; D')( modf' r)1/s'
with s' = ^^y (here ^/{f-D') = ||I ¿coPOID- Because of (15), s' = p and
n-1
6>(y) = u) (y) = cr(y) inequality (17) holds. >
Taking into account [2, Theorem 34] or [4, Theorem 5.2] and its proof we come to
Proposition 2. Suppose that for a continuous, open and discrete mapping f : D' ^ Rn of an open connected domain D' c Rn, where n ^ 2, conditions of Proposition 1 hold. If E = (A, C) is a condenser in Q, then the estimate holds: capp f (E) ^ capp E.
4. The Special Case of the Mappings Under Consideration: n = 2
In the case n = 2 we have the following modification of the results of the previous section. We have 1 < p < to and a continuous, open and discrete mapping f : D' ^ R2 of on open connected domains D' c R2 such that
(1) f e Wlioc (D');
(2) det Df (y) ^ 0 and f has finite codistortion; i. e., adj Df (y) = 0 H2-almost everywhere on Z = {y e D' : det Df (y) = 0};
(3) the inner operator distortion function
, |adJD/(& if detD/(y)/0, D' 9 y J^"/) = < detD/(y) P
'p_1 In if detDf{y) = 0,
belongs to Lp,loc(D').
(4) the weight function a defined as
C | adj Df(y)\p -f DM 7,
y 1 otherwise,
is in € Li,ioc (D'), here Z' = {y € D' : Df (y) = 0}.
It is not hard to see that the continuous, open and discrete mapping f : D' — R2 meets the assumptions of Proposition 1 under n = 2: (3a) f € Wlioc (D'); (3b) f has finite distortion;
(3c) f:D'—ïD is a mapping with bounded w-weighted (j/,^-distortion where p' = __
and w(y) = a v^1{y)1 that is the w-weighted (p', ^-distortion function
, J if det Df(y) / 0,
D 3 y M- (y, /) = < det Df(y)V
0 otherwise,
belongs to LM(D'), and
||K&(f) I L-(D')|| = 1. (19)
Taking into account saying above, by Proposition 1, we come to the following statement.
Corollary 1. Suppose that a continuous, open and discrete mapping f : D' — R2 of an open connected domain D' C R2 has the following properties:
(1) f € W^ioc (D');
(2) f has finite codistortion (adj Df (y) = 0 H2-almost everywhere on Z = {y € D' : det Df (y) = 0});
(3) the inner operator distortion function
|adJD/fa)l if det Df(y) /0,
f\ — J .
ay
D' J^y, /) = <( detD/(y)7 (20)
0 otherwise,
belongs to LPy\oc(D') with some p > 1, where ^ + ^7 = 1-
If r is a family of curves in the domain D' then we have the inequality
modp f (r) < modp r (21)
holds where the weight function a is defined in (18).
5. One More Special Case of the Mappings Under Consideration: n = 2 and p = 1
In this section we prove that Corollary 1 is valid also in the case p = 1. To show this we have to modify some arguments of the previous section. A counterpart of Corollary 1 is formulated in the following statement.
Proposition 3. Suppose that a continuous, open and discrete mapping f : D' — R2 of an open connected domain D' C R2 has the following properties:
(1) f € W^ioc (D');
(2) det Df (y) ^ 0 and f has finite codistortion (adj Df (y) = 0 H2-almost everywhere on Z = {y e D' | det Df (y) = 0});
(3) the inner operator codistortion function
D ^ y ^ K 1,1 (y f) / |adj Df (y)| if det Df (y) = 0, D3 y — K1 (yf . (22)
0 otherwise,
belongs to L1loc(D').
If r is a family of curves in D' then we have
mod1 f (r) < mod^ r (23)
with a defined in (25).
< We show that the proof of Proposition 3 can be reduced to Theorem 3. For doing this formulate first additional properties of f and ^ = f-1.
Properties of ^ = f-1. If f : D' — D is a homeomorphism then the inverse homeomorphism ^ = f-1 : D — D' enjoys the following properties:
(4) by [9, Theorem 4] or [7, Theorem 27] we have ^ e W11loc(D) (see also [10, Theorem 3.2]);
(5) has finite distortion by [7, Theorem 27] (see also [10, Theorem 3.3]);
(6) is differentiable a. e. in D by [7, Theorem 27]; while f : D' — D
(6) belongs to Q1;1 (D,D'; a) (see [4]), that is the distortion function
Kl'Ux, = [ -(^»dit %x) if det D<p(x) / 0, 1,1 v [0 if det Dp(x)=0,
of the inverse mapping ^ = f-1 with the weight function a e L1loc (D') defined as
a(y) = 11 adj (y)l 'fr°'\Z'' where Z' = {y e D : Df (y) = 0}, (25)
1 otherwise,
is in Lo(D) and KD) = UK| Lo(D)|| = 1 (see [4, Theorem 25, formulas (30) and (37); 8]).
Properties of f. Taking into account saying above, we see thatf : D' — D meets some additional properties:
(7) f e W1 ioc (D') and f is differentiable a.e.in D' by [7, Theorem 27];
(8) det Df (y) ^ 0 and f has finite distortion by [7, Theorem 27] (see also [10, Theorem 3.3]);
(9) f: D' — D is a mapping with bounded w-weighted (to, TO)-codistortion with the weight function w = a-1, that is the w-weighted (to, TO)-codistortion function
D * y - KSi(»,/) = (w(y)|adj B/(y)| =
0 otherwise,
belongs to L oo (D'), and
IIK&(-,f) I Lo(D')|| = UK1 ; r(.,¥>) | Lo(D)U = 1. (26)
Now it is evident that f enjoys the conditions of Theorem 3, and therefore (23) holds for f. >
6. Proof of Theorems 3 and 4
< We verify that the proof of Theorem 1 given in [4, Theorem 4.1] for mappings with bounded ^-weighted (q,p)-codistortion, where n — 1 < q ^ p< to, works also in the case 1 < q ^ p = to at n = 2. To do this we need properties of Poletsky function and Poletsky's Lemma in this case. We formulate and prove them below. >
1. Properties of Poletsky function. Take a continuous mapping f : Q — R2 and a domain D compactly embedded into Q, meaning that D is bounded and D C Q, written briefly as D d Q, and take y / f(dD). Denote by ^(y,f,D) the degree of f at y with respect to D. Say that f is sense-preserving whenever ^(y, f, D) > 0 for all domains D d Q and all points y € f (D)\f (<9D). For A C Q refer as the multiplicity function to R2 9 y — N(y, f, A) = ## {f-1(y) n A}. Moreover, put N(f, A) = sup^ N(y, f, A).
Suppose that f : Q — R2 is a continuous, open, and discrete mapping. A domain D d Q is called normal whenever f (dD) = df (D). A normal neighborhood of x € Q is a normal domain [/Cii such that IJ n /_1(/(a:)) = {x}. The quantity i(x,f) = /x(/(x),/, [/) is independent of the choice of a normal neighborhood U of x (see [11, Chapter II, §2] for instance) and is called the local index of f at x. A point x € Q is called a branch point of f whenever f is not a homeomorphism of any neighborhood of x. Denote the collection of all branch points of f by Bf. If D is a normal domain for a mapping f then ^(y, f, D) is independent of y € f (D). We will call this constant by ^(f, D).
In the following two lemmas we state propositions of interest in their own right. Both of them are applied in the proof of the main result of this section.
Lemma 1 [3, Lemma 10]. Assume that f : Q — R2 is a continuous, open and discrete mapping in W11ioc(Q) with finite distortion. Then for every open connected set U C Q the set {x € U\Bf : J(x, f) = 0} has positive measure.
< If, on the contrary, J(x, f) =0 a. e. on a connected set U C Q\Bf on which f is a homeomorphism then Df (x) = 0 a. e. on U because f has finite distortion. Then f is constant on U, and consequently, f cannot be open. >
Proposition 4. If f : Q — R2 is a continuous, open and discrete mapping in W11ioc(Q) with finite distortion, then f is differentiable a. e. on Q\Bf and sense-preserving.
< For a connected open set U C Q\Bf on which f is a homeomorphism, it is enough to apply the statement [9, Theorem 4] or [7, Theorem 27] twice. For the restriction f iu : U — f (U) it provides that the inverse homeomorphism (f |u)-1 : f (U) — U is in W11(f(U)), is of finite distortion, and is differentiable a.e. on f(U). Then applying [7, Theorem 27] to (f|u)-1 : f(U) — U we get similar properties to the given mapping f |u : U — f (U). By Lemma 1, det Df (x) ^ 0 and properties of degree we conclude that f is sense-preserving. >
Definition 3. For a sense-preserving, continuous, open and discrete mapping f : Q — R2 and a normal domain D d Q, define the Poletsky function go : V — R2 on V = f (D) [12] by putting
V 9 y — go (y)=A £ i(x,f )x, (27)
xef-1(y)nD
where A = ^(f, D).
The function of the form (27) was introduced by Poletsky in [12] for mappings with bounded distortion (p = q = n, w = 1). The next statement presents the properties of the Poletsky function for the classes of mappings under consideration.
Proposition 5 [2, 3]. Suppose that f : Q — R2 belongs to OD(Q; to, to; w, 1) (properties (4a)-(4c) hold). Then
(1) the function gD defined in (27) is continuous and belongs to ACL(V);
(2) DgD (y) =0 a.e. on Z' U £';
(3) Poletsky function gD defined in (27) is in W-^V); furthermore,
||DgD | L1(V)U < AUK-'^(■; f) | L„(D)\\ J a(x) dX.
D
We emphasize that the formulated statement is proved in [2, Theorem 18] for mappings f e ID(Q;p,p; w, 1), p e (1, to). The same proof works also in the case p = to at n = 2.
2. Poletsky's Lemma. Consider a continuous, open and discrete mapping f : Q — R2. Take a closed rectifiable curve 5 : /0 — Rn and a curve a : / — Q with f oa C 5. In particular, we have / C /0. If the function s^ : /0 — [0,1(5)] is constant on some interval J C /, then the mapping 5 is constant on J. In turn, since f is discrete, a is also constant on J. Consequently, there exists a unique mapping a* : s^(/) — Q satisfying a = a* o s^ |7. We can prove that a* is continuous and f o a* C 50. The curve a* is called an f-representative of a (with respect to 5 ) whenever 5 = f ◦ a. Suppose now that 5 = f ◦ a. The above arguments show that
f o a* = (f o a)0.
Therefore, the curve f oa* admits a positive natural parametrization, and hence it is Lipschitz. Thus we can integrate along this curve using (6) where (/ o a*) (t) | = 1 for Jf 1-almost all t e /.
The mapping f is called absolutely precontinuous on a provided that a* is absolutely continuous.
Lemma 2. Suppose that f : Q — R2 is a mapping of class ID(Q; to, to; w, 1). Consider a family r of curves in Q such that for every 7 e r the following holds: the curve f o 7 is locally rectifiable and 7 has a closed subcurve a on which f is not absolutely precontinuous. Then mod 1 f (r) = 0.
The formulated Lemma is proved in [4, Lemma 3.3] for mappings f e ID(Q;p,p; w, 1), p e (1, to). The same proof works also in the case p = to at n = 2.
In the proof of Lemma 2 we also need the following statement.
Lemma 3. Consider a homeomorphism ^ : Q — Q' of class ID(Q; q, to; 0,1), where Q, Q' C R2 and 1 < q ^ to. Then
(1) the inverse homeomorphism is ^>-1 e W^ loc(Q');
(2) ^>-1 has finite distortion: D^>-1 (y) = 0 almost everywhere on Z';
L1 , r
(3) K1 (-,^-1) € where
r=i^+î ifq< 00, 1-1 if q < 00,
if q = 00, [61-1 if q = to;
(4) if the weight function u is locally summable then the inverse homeomorphism induces, by the change-of-variable rule, the bounded operator
: L1(q; u) n W^ >loc ^ Lftfi').
We have the relations
IlK1;(,^-1 ) | Le(fi')|| = | Le(fi)||
and
(-^-1) | Le(Q')|| < ||^-1*|| < ||Ki;rw) | Le(Q')||, where 0q,^ is some constant.
< Properties (1) and (2) of ^ = f-1 were proved just after Proposition 3. Taking into account (1) and (2) Properties (3) and (4) can be proved by analogy with Theorem 9 of [2]. > Remark 3. By means of Theorems 3 and 4 for homeomorphisms ^ : Q — Q' of class ID(Q; q, to; 0,1), where Q, Q' C R2 and 1 < q ^ to, we can prove some more inequalities such that Vaisala inequality and the capacity inequality (see proofs in [4, Theorem 22] and [4, Theorem 28] respectively).
Remark 4. It is not hard to see that assumptions of Theorem 4 are weaker comparing with those in paper [13]. For instance, Theorem 1.3 of [13] is formulated under addition condition that the given mapping is closed. Therefore Theorem 4 with weaker assumptions contains the main result of paper [13].
Acknowledgements. I greatly appreciate the anonymous reviewers for critically reading and comments, which helped improve the initial manuscript.
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Received September 2, 2022
Sergey K. Vodopyanov Sobolev Institute of Mathematics,
4 Akademika Koptyuga Ave., Novosibirsk 630090, Russia, Principal Re,searcher E-mail: [email protected] https://orcid.org/0000-0003-1238-4956
Владикавказский математический журнал 2022, Том 24, Выпуск 4, С. 58-69
О МОДУЛЬНЫХ НЕРАВЕНСТВАХ ТИПА ПОЛЕЦКОГО ДЛЯ НЕКОТОРЫХ КЛАССОВ ОТОБРАЖЕНИЙ
Водопьянов С. К.1
1 Институт математики им. С. Л. Соболева, Россия, 630090, Новосибирск, пр-т Академика Коптюга, 4 E-mail: [email protected]
Аннотация. Хорошо известно, что теория отображений с ограниченным искажение была заложена Ю. Г. Решетняком в 60-е годы прошлого века [1]. В работах [2, 3] была введена двухиндексная шкала отображений с весовым ограниченным (д,р)-искажением. Эта шкала отображений включает в себя, в частности, отображения с ограниченным искажением, упомянутые выше (при q = p = n и тривиальной весовой функции). В работе [4] для двухиндексной шкалы отображений с весовым ограниченным ^,р)-искажения доказано модульное неравенство типа Полецкого при минимальной регулярности; приведено много примеров отображений, к которым можно применить результаты [4]. В этой статье мы приведем одно такое применение. Другая цель этой статьи — показать новый класс отображений, в которых выполняются модульные неравенства типа Полецкого. Для этого мы расширяем при n = 2 справедливость утверждений работы [4] на предельные показатели: 1 < q < p < то. Это обобщение содержит в качестве частного случая результаты недавно опубликованных работ. Как следствие результатов этой статьи мы получаем также оценки изменения емкости конденсаторов.
Ключевые слова: квазиконформный анализ, пространство Соболева, модуль семейства кривых, оценка модуля.
AMS Subject Classification: 30C65 (26B35, 31B15, 46E35).
Образец цитирования: Vodopyanov S. K. On Poletsky-Type Modulus Inequalities for Some Classes of Mappings // Владикавк. мат. журн.—2022.—Т. 24, № 4.—C. 58-69 (in English). DOI: 10.46698/w5793-5981-8894-o.
