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Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 3, P. 51-58
YAK 517.518.23+517.548.2 DOI 10.46698/z8419-0555-2432-n
ON THE GEHRING TYPE CONDITION AND PROPERTIES OF MAPPINGS*
S. K. Vodopyanov1
1 Sobolev Institute of Mathematics of the Siberian Branch of the RAS, 4 Ac. Koptyuga Ave., Novosibirsk 630090, Russia E-mail: [email protected]
Abstract. The goal of this work is to obtain an analytical description of mappings satisfying some capacity inequality (so called Gp-condition): we study mappings for which the Gp-condition holds for a cubical ring. In other words, we replace rings with concentric spheres in the Gp-condition by rings with concentric cubes. We obtain new analytic properties of homeomophisms in Rn meeting Gehring type capacity inequality. In this paper the capacity inequality means that the capacity of the image of a cubical ring is controlled by the capacity of the given ring. From the analytic properties we conclude some geometric properties of mappings under consideration. The method is new and is based on an equivalent analytical description of such mappings previously established by the author. Our arguments are based on assertions and methods discovered in author's recent papers [1] and [2] (see also some references inside). Then we obtain geometric properties of these mappings.
Keywords: quasiconformal analysis, Sobolev space, capacity inequality, pointwise condition. AMS Subject Classification: 30C65, 31B15, 46E35.
For citation: Vodopyanov, S. K. On the Gehring Type Condition and Properties of Mappings, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp. 51-58. DOI: 10.46698/z8419-0555-2432-n.
1. Introduction
In paper [3] F. W. Gehring studied some geometric properties of mappings in Rn, n ^ 2, meeting so called Gp-condition. More precisely, suppose that D and D' are domains in Rn and f : D ^ D' is a homeomorphism. Then f maps each ring U C D onto a ring f (U) C D'. Gehring says that f € Gp(K), 0 < K < to, if
capp(f (U)) < K capp(U) (1)
for all spherical rings U C Rn. When p = n, a homeomorphism is in Gn(K) for some K if and only if it is a quasiconformal mapping.
Recall that a bounded domain U C D is said to be a ring if Rn \ U has exactly two components: bounded component F1 and unbounded F0. Then for 1 ^ p < to we define the p-capacity of U as
capp(C0 = inf/|V,l№ £)•
u
# The study was carried out within the framework of the State contract of the Sobolev Institute of Mathematics, project № FWNF-2022-0006. © 2023 Vodopyanov, S. K.
where the infimum is taken over all functions u € Lp(Rn) n C(Rn) with u = 0 on F0 and u = 1 on Fi (called admissible). A function u : D — R belongs to the Sobolev class L^(D), if u € Lit\oc(D) and its weak derivative J^ € LP(D) for any i = 1,... ,n. The seminorm of u equals ||u | Lp(D)|| = ||Vu | Lp(D)||, 1 << to.
A ring U is said to be a spherical ring if it is bounded by two concentric spheres, that is, if U = {x : a < |x — P| < b}, where 0 < a < b < to and P € Rn is a center of spheres. Here and further |P| is the Euclidean norm of P € Rn.
The purpose of paper [3] is to establish some relations between the classes Gp(K) and Lip(K)1. They are given in the following statements of paper [3].
Theorem 1 [3, Theorem 2]. If f,f-1 € Gp(K), where p = n, then f,f-1 € Lip(K0), where K0 depends only on K, n and p.
Theorem 2 [3, Theorem 3]. If f € Gp(K), where n — 1 < p < n, then f € Lip(K0). If f € Gp(K), where n < p < to, then f-1 € Lip(K0). In both cases K0 depends only on K, n and p.
The goal of this work is to obtain an analytical description of mappings satisfying some capacity inequality similar to (1): we study mappings for which (1) holds whenever U is a cubical ring. In another words we replace rings with concentric spheres in the right hand side of (1) by rings with concentric cubes. Our arguments are based on assertions and methods discovered in recent papers [1] and [2] (see also some references inside). Then we obtain geometric properties of these mappings.
There is also another approach to this subject. For instance, authors of paper [4] study properties of homeomorptisms under stronger capacity inequality:
capp(^(F0 ),p(Fi); D') < Kp capp(F0,Fi; D), 1 <p< to,
for an arbitrary condenser (F0,F1) c D. However, method of paper (4) is not applicable to a minimal collection of rings (spherical or cubical). See [4] for more details.
1st STEP. The crucial result for our study is the following theorem proved in [2]. Before formulating this theorem we give some necessary definitions.
Definition 1. A ring U in Rra is called, cubical whenever U = Q(x,R) \ Q(x,r), where Q(x, R) = {z € Rn : | z < R} and 0 < r < R < to. Recall that |x|^ — maxk=1|Xfc |.
Definition 2. Suppose that D is an open set in Rn. Denote by Oc(D) some system of open sets in D with the following properties:
(a) if the closure Q of an open cube Q lies in D, then Q € &C{D)\
(b) if U1,...,Uk € Oc(D) is a disjoint system of open sets, then IJk=1 U € Oc(D) for arbitrary k € N.
(c) in the case n = 2, q = 1 we will consider an expanded family Oc(D) D Oc(D): we include additional rings of the following shape to this family:
U = ([a — r,a + r] x [b, c]) \ ({a} x [b + r,c — r]) c D, 2r < c — b,
and
U = ([s, t] x [d — t, d + t]) \ ([s + T,t — t] x {d}) c D, 2t < t — s.
Definition 3. A mapping ^ : Oc(D) — [0, to] is called a K-quasiadditive set function, whenever
1 Gehring says that / £ Lip(K), 0 < K < oc, if L(P, f) = lim < K whenever P £ D.
(a) for each point x € D there exists 5 with 0 < 5 < dist(x,dD), such that 0 < ^(Q(x,5)) < to, and if D = Rn, then the inequality 0 ^ ^(Q(x,5)) < to must hold for all 5 € (0,5(x)), where 5(x) > 0 may depend on x;
(b) for every finite disjoint collection of open sets U € Oc(D), where i = 1,..., l, with
i i
(J Ui C U, where U € Oc(D), we have ^ tf(Ui) < k^(U). (2)
i=1 i=1
If (2) holds with k = 1, then we refer to ^ as a quasiadditive set function instead of 1-quasi-additive. If for every finite collection (Ui € Oc(D)} of disjoint open sets we have
E ) = *( U ,
i=1 \i=1 )
then ^ is called finitely additive.
A function ^ is monotone whenever ^(U1) ^ ^(U2) provided that U1 C U2 C D and U1,U2 € Oc(D). It is obvious that every quasiadditive set function is monotone. A K-quasiadditive set function ^ : Oc(D) ^ [0, to] is called bounded, whenever supueOc(D) *(U) < to.
The Sobolev space Wp1(D) in a domain D C Rn consists of functions u € Lp(D) with the finite norm ||u | W1(D)|| = ||u | Lp(D)|| + ||Vu | Lp(D)||, 1 < p < to.
Let D and D' be domains in the Euclidean space Rn. Then a homeomorphism ^ : D ^ D' belongs to the Sobolev space W1loc(D) (Lp(D)), if its coordinate functions belong to W1loc(D) (Lp(D)). Then Jacobi matrix Dip(x) = (|fi)ij=1 n and its Jacobian detDip(x) are well defined at almost all points x € D.
Notice that Gehring's condition ^ € Lip(K) in D, 0 < K < to, is equivalent to ^ € L^D) and the norm ||^> | ¿^(D)! = ||D^> | L^(D)|| can be taken as K.
Theorem 3 [2, Theorems 18 and 23]. Given a homeomorphism ^ : D' ^ D of domains D', D c Rra, where n ^ 2, the following statements are equivalent:
(1) Every cubical ring U = Q(y,R) \ Q(y,r) C D with the preimage i!)~l(U) = ip~l(Q(y, R)) \ ip~l(Q(y,r)) in D' satisfies
,,-1(m) KpcapP(U), 1 < q = p< to,
^Ip(U) capp ([/), 1 < q / p < to,
capi(rW)<ri i (3)
where Kp € (0, to) is some constant and ^q,p is some bounded quasiadditive set function on the system ffc(D), and a is determined from \ = \ — ifl<q<p<oo and a = to, if 1 < q = p < to.
(2) The homeomorphism ^ : D' ^ D belongs to Wq1loc(D'), has finite distortion: D^(y) = 0 holds almost everywhere on Z = (y € D' | det D^(y) = 0}, and the operator distortion function
{ imy)l -L , det Dtp(y) / 0, D' 3 y Kq p(y, 1p) = < I det p (4)
[0, det D^(y) = 0,
belongs to La(D').
(3) The composition operator : ¿P(D) n Lipi(D) ^ L^(D'), ^*(f) = f o ^ if f € L1(D) n Lipi(D), where 1 < q ^ p < to, is bounded.
Moreover,
{R
"m,* \V7T <5»
7q q,p{D) ° , l<q<p<oo.
(4) Every ring U in D with the preimage ^-1(U) in D' satisfies
capM-HU)) < <1 7lnKf^iU) 1 1 < ' = P <
7in^qiP(U) capp (U), l<q^p<oo,
where Kp e (0, oo) and are from (3) and a is determined from ^ = ^ — if l<q<p<oo and a = œ, if 1 < q = p < œ.
(5) The claims of Theorem 3 remain valid in the case 1 = q ^ p < œ and n = 2, if (3)
~ n
holds for U € ÛC(D) (see Definition 2) with probably different constant instead ofl^n. Put Kq,p(t, D') = \\Kqp(-^) | La(D')\\.
Remark 1. In the case q = 1 analytic properties of ^ are proved in [2, Theorem 23]. Unfortunately, in Statement 5 of Theorem 18 of [2] the condition U € Oc(D) (see Definition 2) is missing.
We will apply Theorem 3 to mappings meeting capacity inequality (3) instead of (1). In another words we study mappings in Rn which control changing of capacity of cubical rings instead of spherical ones. The next statement is evident.
Proposition 1. Given a homeomorphism p : D — D' of domains D,D' C Rn, where n ^ 2, the inequality
(ttw fKpcap£(U), l^q = p<oo, cap{¡{p{U))^< " i (6)
holds for every cubical ring U = Q(y,R) \ Q(y,r) c D, iff inequality (3) holds for the homeomorphism ^ = p-1 : D' — D. Here Kp € (0, to) is some constant and ^q>p is some bounded quasiadditive set function on the system Oc(D).
Definition 4. Suppose that D' and D are domains in Rn and that ^ : D' — D is a homeomorphism.
1) We say that ^ € Qq,p, if
a) in the case q > 1 inequality (3) holds for each cubical ring U c Oc(D); _
b) in the case n = 2, q = 1 we ask for (3) to be true for an expanded family Oc(D) (see Definition 2).
2) We say that p = ^-1 € Gp,q, if ^ € Qq,p.
Theorem 3 implies Theorem 1:
Corollary 1. Given homeomorphism p : D — D' of domains D,D' c Rn, where n ^ 2, meeting conditions p € Gp,p and p-1 € Gp,p with 1 < p < to the following properties hold:
1) p, p-1 € Lip(K0), where K0 depends only on Kp, n, and p, p = n;
2) p is quasiconformal mapping, if p = n.
< If condition 1) holds, then the relation (4) holds for both p and p-1. The desired results are proved in [1, Subsections 1.2, 1.3]. >
Proposition 2. Given a homeomorphism ^ : D — D' of domains D,D' C Rn, where n ^ 2, of class Gp,q with n — 1 ^ q ^ p < to the following properties hold:
1) ip € Wj>loc(£>), where p' = if p > n - 1, and € L^D), if p = n - 1;
2) has the finite distortion;
3) the codistortion function
ImW*)^, det Dtf(x) / 0
let D</p(a;)| <?
0, det Dp(x) = 0
L> 9 a: ->■ ¿XrqtP{x,<p) = { IdetD^)!1 « ' ..........(7)
belongs to Lu{D), where a is determined from ^ = \ — ifn — 1 ^ q < p < oo, and a = 00, if n — 1 ^ q = p < to;
4) the distortion function
( lDip{x)l ± , det Dtf(x) / 0,
D3X Kp/yq/(x,ip) = I I det Dtp(x)\i' (8)
[0, det D^(x)=0,
belongs to Le(D), where q' = at q > n — 1, q' = oo at q = n — 1 and p is determined
from ^ = — ^r,ifn — l^q<p<oo, and q = 00, if n — 1 ^ q = p < 00.
< If € Gp,q then, according Proposition 1, ^ = € Qq,p. By Theorem 3 the homeomorphism ^ : D' ^ D
1) belongs to Wq1loc(D');
2) has the finite distortion: D^(y) = 0 holds almost everywhere on Z = {y € D' : det D^(y) = 0},
3) the distortion function
i i , det Dtp(y) / 0,
D' 3 ¡J Kqp(y,1p) = < i det p (9)
[0, det D^(y) = 0,
belongs to La(D'), where a is determined from ^ = ^ — ^,iil^q<p<oo, and a = 00, if 1 ^ q = p < to.
By [1, Theorem 4] we conclude that (p € W^1, loc(Z)), where p' = if p > n — 1,
^ € ¿^(D), if p = n — 1, and ^ has the finite distortion.
Take into account that Dtp(y) = ^and det Dtp(y) = (det Dlp{x))~1 at points y = ^>(x) = Z' n E', where Z' = {y € D' : det D^(y) = 0} and E' C D' is a maximal Borel null-set such that measure of Z = ^(E') is positive. Notice that up to a set of measure zero Z = {x € D : det D^>(x) = 0}, and E = ^(Z') C D is a null-set. The mapping ^ has Luzin property N outside of E.
By change of variable formula in the case q < p we get
\\JTU-M \La(DW = J I ' J ! ) dx
D\(ZnS)
D\(ZnS)
\adjD<p(x)\ det Dlp{x)\1~
I adj D<p(x)\ I det D<p(x)|1_p
| det D^>(x)| dx
„ y I det Dih(y)\p
JMïlM (10)
a
From the left hand side of this equality it follows (7): Kq,p(-,p) € La(D). In the case if n — 1 ^ q = p < to we have
\\^p(-,p)\L00(D)\\ = esssup
1 — i.
x&D\(zns)\detDp(x)\ i
= esssup m{y)][ 1 =\\KM{-^)\L00{DI)\\. y&D>\{z> ns')|det Dip(y)\p
Integrability Kp/,q/(-,p) € Lg(D) is proved in [1, Theorem 4]. > From Proposition 2 it follows a part of Theorem 2.
Corollary 2. Given a homeomorphism p : D ^ D' of domains D, D' C Rn, where n ^ 2, of class Gp,q with n — 1 ^ q ^ p < to the following properties hold:
1) p-1 = tp € LUD') and \\<p~l | LUD^W < \\Kp>p(;iP) | ¿oopOH^7 in the case n < q = p < to;
2) p € LUD) and
M 1 M M I, p'
\\p\Ll(D)\\^\\Kp/tq/(;p)\Le(D)\\P'-r> (11)
in the case n — 1 ^ q = p < n.
< Really, taking (9) into account at q = p > n and the inequality 1 ^ | d^t^D4>(y)| P°ints) where det D^(y) = 0 we have
V | det Dtp(y)\ J \detDip(y)\p It follows ^ = p-1 € Ll00(D') in the case n < q = p < to and
\\iP \ Ll(D')\\ < \\Kp>p(;iP)\L00(D')\\^.
In the case n — 1 < q = p < n we have integrability Kp(-,p) = L^(D) with p' > n. Therefore with above-mentioned arguments applied to (8) we obtain p € Ll00(D) and (11) holds.
Property p € Ll00(D) in the case q = p = n — 1 is just statement 1) of Proposition 2. >
2nd STEP. Proposition 3. Let p € Wl-1loc(D), p has the finite codistortion (adj Dp(x) = 0 almost everywhere on the set Z) and the codistortion function
{ det Dp(x) / 0,
D3X^ X~q}P(x,p) = I | det D(p(x)\ 1 (12)
[0, det Dp(x) = 0,
belongs to La(D'), where a is determined from ^ = ^ — if n — 1 ^ q < p < 00, and a = 00, if n — 1 ^ q = p < to. Then p € Gp,q.
< As soon as p € Wn1-1 loc(D) and has the finite codistortion then ^ = p-1 € W11loc(D') and has the finite distortion (see [1, Corollary 2]). Because of this we can apply (10) again for obtaining statement 4) of Proposition 2. It left to verify that ^ € Qq,p. For doing this we take an arbitrary cubical ring in U C D and an admissible function u for this ring, and evaluate
the norm of the composition u o As soon as u o ^ is an admissible function for the ring ^-1(U) we have:
capq(^-1(U)) < ||u o ^ | L1(^-1(U))\\q < [ \Vu(tp(y))\q\det Dtp(y)\p ■ q dy
h-i(L„ | det Dip(y)\p
4>-1(u )\z '
- —
p [ p / \ T~) I i \\ \ ® \ a
I \Vu(tp(y))\p\detDip(y)\dy I •( / ( , ) dy
'^-1(U)\(Z'US') 7 \-1(u)\(z' US')
I det Dtp(y)\p
|| Kqtp(;i>) \Ltt(i>-\U))\\q( J\Vu(x)\p dxY, q<p, ^ < ^ u /
||KP>P(-,^) | L^-1(U))||p( / |Vu(x)|p dxq = p. It follows (3) with bounded quasiadditive set function ^q>p equal to
D D U ^ *q,p(U) = ||Kq>p(.,p) | La(U)f = \\Kqp(-,^) | La(iP-1(U))\\c and Kp = \\Kpp(-, 4) I L^r1 (U))||.
^P - IP P,P V ! r/ | Ml
ce we proved ^ = € Gp,q.
3rd
3rd STEP. From Proposition 2 and 3 it follows the following criterium.
Theorem 4. A homeomorphism p : D ^ D' of domains D, D' C Rn, where n ^ 2,
(n— 1)2
belongs to class GPiQ with n — 1 ^ q ^ p < n_2 , iff the following properties hold:
1) p £ W!-i,ioc(D);
2) p has the finite codistortion;
3) the codistortion function
r^djM|r] det Dp(x) / 0, D3X^ J(rq>p(x,<p) = { I det D<p(x)\ 1 (13)
[0, det Dp(x) = 0,
belongs to Lfj(Dr), where a is determined from ^ = j — if n — 1 ^ q < p < , and
■f 1 ^ ^ (n-12)
a = oo, it n — 1 ^ q = p < y n_2 '.
< If a homeomorphism p : D ^ D' of domains D, D' C Rn, where n ^ 2, belongs to class
(n- 1)2
Gp>q with n — 1 ^ q ^ p < n_2 i then we apply Proposition 2 for obtaining that
1) p € Wp, [oc(D), where p' = p_(Pn_1), if P ^ n — 1. Since p' > n — 1 it follows that
p £ Wn-1;loc(D).
2) p has the finite distortion.
3) the codistortion function (13) is in La(D').
Thus the necessity is proved. The sufficiency is proved in Proposition 3. >
References
1. Vodopyanov, S. K. Regularity of Mappings Inverse to Sobolev Mappings, Sbornik: Mathematics, 2012, vol. 203, no. 10, pp. 1383-1410. DOI: 10.1070/SM2012v203n10ABEH004269.
2. Vodopyanov, S. K. The Regularity of Inverses to Sobolev Mappings and the Theory of Qq,p-Homeomor-phisms, Siberian Mathematical Journal, 2020, vol. 61, no. 6, pp. 1002-1038. DOI: 10.1134/S00374466 20060051.
3. Gehring, F. W. Lipschitz Mappings and the p-Capacity of Rings in n-Space, Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), 175-193, Annals of Mathematics Studies, vol. 66, Princeton, N. J., Princeton Univ. Press, 1971. DOI: 10.1515/9781400822492-013.
4. Salimov, R., Sevost'yanov, E. and Ukhlov, A. Capacity Inequalities and Lipschitz Continuity of Mappings, arXiv:2302.13302v1, 26 Feb 2023.
Received June 24, 2023 Sergey K. Vodopyanov
Sobolev Institute of Mathematics of the Siberian Branch of the RAS,
4 Ac. Koptyuga Ave., Novosibirsk 630090, Russia,
Principal Researcher
E-mail: [email protected]
https://orcid.org/0000-0003-1238-4956
Владикавказский математический журнал 2023, Том 25, Выпуск 3, С. 51-58
ОБ УСЛОВИИ ТИПА ГЕРИНГА И СВОЙСТВАХ ОТОБРАЖЕНИЙ
Водопьянов С. К.1
1 Институт математики им. С. Л. Соболева СО РАН, Россия, 630090, Новосибирск, пр. Ак. Коптюга, 4 E-mail: [email protected]
Аннотация. Целью данной работы является получение аналитического описания отображений, удовлетворяющих некоторому емкостному неравенству (так называемому Gp-условию); точнее, мы изучаем отображения, для которых выполнено Gp-условие для кубического кольца. Другими словами, мы заменяем кольца с концентрическими сферами в условии Gp кольцами с концентрическими кубами. Изучаются новые аналитические свойства гомеомофизмов в Rn, удовлетворяющих емкостному неравенству типа Геринга. В этой статье емкостное неравенство означает, что емкость образа кубического конденсатора контролируется емкостью исходного конденсатора. Из полученных аналитических свойств в качестве следствия получаем некоторые геометрические свойства рассматриваемых отображений. Метод является новым и основан на эквивалентном аналитическом описании таких отображений.
Ключевые слова: квазиконформный анализ, пространство Соболева, емкостное неравенство, поточечное условие.
AMS Subject Classification: 30C65, 31B15, 46E35.
Образец цитирования: Vodopyanov S. K. On the Gehring Type Condition and Properties of Mappings // Владикавк. мат. журн.—2023.—Т. 25, № 3.—C. 51-58 (in English). DOI: 10.46698/z8419-0555-2432-n.
