On the Gehring type condition and properties of mappings

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 G Gp(K), 0 < K < to, if capp(f (U)) ^ K capp(U)

for all spherical rings U C R n . When p = n , a homeomorphism is in G n (K ) for some K if and only if it is a quasiconformal mapping.

Recall that a bounded domain U ⊂ D is said to be a ring if Rn \ U has exactly two components: bounded component F and unbounded F. Then for 1 ^ p < to we define the p-capacity of U as capp(U) = inf

|∇ u | ^p dx,

U

∂u     ∂u

* ^u l Д , . . . , я ) , ∂x 1      ∂x n

• ^# The study was carried out within the framework of the State contract of the Sobolev Institute of Mathematics, project № FWNF-2022-0006.

where the infimum is taken over all functions u E L p ( R ⁿ ) П C ( R ⁿ ) with u = 0 on F and u = 1 on F i (called admissible ). A function u : D ^ R belongs to the Sobolev class L p (D) , if u E L i,i_oc (D) and its weak derivative ddu E L _p (D) for any i = 1,... ,n . The seminorm of u equals ||u | L p (D) | = ||Vu | L _p (D) | , 1 C p C 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 E R ⁿ is a center of spheres. Here and further | P | is the Euclidean norm of P E R ⁿ .

The purpose of paper [3] is to establish some relations between the classes G _p (K ) and Lip(K) 1 . They are given in the following statements of paper [3].

Theorem 1 [3, Theorem 2] . If f,f ^-1 E G _p (K ) , where p = n, then f,f ^-1 E Lip(K o ) , where K 0 depends only on K , n and p.

Theorem 2 [3, Theorem 3] . If f E G _p (K ) , where n — 1 < p < n, then f E Lip(K o ) . If f E G _p (K ) , where n < p <  to , then f ^-1 E Lip(K o ) . In both cases K o 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(^(Fo ),^(Fi); D‘) C Kp capp(Fo,Fi; D),  1

1 ^st 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 R ⁿ is called, cubical whenever U = Q(x,R) \ Q(x,r) , where Q(x, R) = { z E R ⁿ : | z — x | ^ <  R } and 0 < r < R <  to . Recall that | x | ^ = max k=_1v..,_n | x k | .

Definition 2. Suppose that D is an open set in R ⁿ . Denote by O _c (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 E O _c ( D);

• (b)    if U i ,... ,U k E O c (D') is a disjoint system of open sets, then U k=i U i E O c (D) for arbitrary k E N .

• (c)    in the case n = 2 , q = 1 we will consider an expanded family O c (D) D O c (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 + т ]) \ ([s + t, t — т ] x { d } ) C D, 2т < t — s.

Definition 3. A mapping Ф : O _c (D) ^ [0, to ] is called a к -quasiadditive set function, whenever

• ¹    Gehring says that f E Lip( K ) , 0 < K <  ro, if L ( P, f ) = lim ^{| f} ( xX - P ^ P ) | C K whenever P E D .

• (a)    for each point x G D there exists 6 with 0 < 5 < dist(x, dD) , such that 0 < Ф(Q(x,д)) <  to , and if D = R n , then the inequality 0 С Ф(Q(x,5)) <  to must hold for all 5 G (0, 6(x)) , where 6(x) > 0 may depend on x ;

• (b)    for every finite disjoint collection of open sets U i G O _c (D) , where i = 1,..., l , with

ll

У U i C U, where U G O c (D), we have ^^ Ф(U i ) С кФ(U).            (2)

i=1                                             i=1

If (2) holds with к = 1 , then we refer to Ф as a quasiadditive set function instead of 1 -quasi-additive. If for every finite collection { U i G O _c (D) } of disjoint open sets we have

l

(У Ui)

^ Ф(Ui) = Ф i=1

then Ф is called finitely additive.

A function Ф is monotone whenever Ф(U 1 ) С Ф(^) provided that U i C U 2 C D and U 1 , U 2 G O _c (D) . It is obvious that every quasiadditive set function is monotone. A к -quasiadditive set function Ф : O _c (D) ^ [0, to ] is called bounded, whenever ^suP u e O c (D) ^Ф(U) <  ^to.

The Sobolev space W p (D) in a domain D C R n consists of functions u G L p (D) with the finite norm || u | W p (D) | = ||u | L p (D) | + ||Vu | L p (D) | , 1 С p С to .

Let D and D ‘ be domains in the Euclidean space R n . Then a homeomorphism ^ : D ^ D ‘ belongs to the Sobolev space Wp i_oc (D) ( L ^ (D) ), if its coordinate functions belong to W_p i_oc (D) ( L p (D) ). Then Jacobi matrix D^(x) = ( q j ) i j=i _n and its Jacobian det D^(x) are well defined at almost all points x G D .

Notice that Gehring’s condition ^ G Lip(K) in D , 0 < K <  to , is equivalent to ^ G L^D) and the norm | ^ | L {_O (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 R n , where n ^ 2 , the following statements are equivalent:

• (1)    Every cubical ring U = Q(y, R) \ Q(y,r) C D with the preimage ^ ^-1 (U) = ^ ^-1 (Q(y,R)) \ ^ ^-1 (Q(y,r)) in D ‘ satisfies

cap q (^ ^-1 (U)) С <

K p cap p p (U), Ф q - _p (U ) cap p p (U),

1 < q = p <  to ,

1 < q = p <  to ,

where K _p G (0, to ) is some constant and Ф _q ,_p is some bounded quasiadditive set function on the system O _c (D), and a is determined from p = q — p , if 1 < q < p <  to and a = to , if 1 < q = p <  to .

(2) The homeomorphism ^ : D ‘ ^ D belongs to W qi_oc (D ‘ ) , has finite distortion: D^(y) = 0 holds almost everywhere on Z = { y G D ‘ | det D^(y) = 0 } , and the operator distortion function

D ‘ 9 y ^ K q ^У,^) = <

—^|D^ ^{( y ) |} 1 , det D^(y) = 0, | det Dψ(y) | ^p

0,

det D^(y) = 0,

belongs to L _a (D ‘ ) .

• (3)    The composition operator c * : L p (D) П Lip i (D) ^ L q (D ' ) , ^ * (f) = f о ^ if f g L p (D) П Lip i (D) , where 1 < q С p <  to , is bounded.

  Moreover,

  
  

  Н Ф * \1 C ^ K q,p Ы) I L(D ) \ C

  
  

  ⁿ

  7 p nK p ,

  7 q n^ q,p (D) ^Ь ,

  
  

  1 < q = p <  to ,

  1 < q < p <  to .

  
  
  
  

• (4)    Every ring U in D with the preimage ф 1 (U ) in D satisfies

  cap q (ф ^-1 (U )) C <

  
  

  7 ^p nK _p cap p (U ),

  n 1                ¹

  7 ^q n^ p (U ) cap p p (U ),

  
  

  1 < q = p <  to ,

  1 < q = p <  to ,

  
  

where K _p E (0, to ) and ^ _q,_p are from (3) and a is determined from T = q — p , if 1 < q < p <  to and a = to , if 1 < q = p <  to .

• (5)    The claims of Theorem 3 remain valid in the case 1 = q C p <  to and n = 2 , if (3) holds for U E O _c (D) ( see Definition 2) with probably different constant instead of 7 p n.

Put K q,p (^; D) = \\ K q,pM ) I L t (D ‘ ) \ .

Remark 1. In the case q = 1 analytic properties of ф are proved in [2, Theorem 23]. Unfortunately, in Statement 5 of Theorem 18 of [2] the condition U E O _c (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 R ⁿ which control changing of capacity of cubical rings instead of spherical ones.

The next statement is evident.

Proposition 1. Given a homeomorphism у : D ^ D of domains D,D C R n , where n ^ 2, the inequality

K p cap p p (U ),

Ф q,p (U) cap p p (U),

1 C q = p <  to,

1 C q < p <  to,

holds for every cubical ring U = Q(y, R") \ Q(y, r) C D, iff inequality (3) holds for the homeomorphism ф = у ^-1 : D ^ D. Here K _p E (0, to ) is some constant and ^ _q,_p is some bounded quasiadditive set function on the system O _c (D).

Definition 4. Suppose that D and D are domains in R n and that ф : D ^ D is a homeomorphism.

• 1)    We say that ф E Q _q,p , if

• a)    in the case q >  1 inequality (3) holds for each cubical ring U C O _c (D) ;

• b)    in the case n = 2 , q = 1 we ask for (3) to be true for an expanded family O _c (D) (see Definition 2).

• 2)    We say that у = ф -1 E G _p,_q , if ф E Q _q,_p .

• Theorem 3 implies Theorem 1:

Corollary 1. Given homeomorphism у : D ^ D of domains D,D C R n , where n ^ 2 , meeting conditions у E G _p,_p and у ^-1 E G _p,_p with 1 < p <  to the following properties hold:

• 1)    у, у ^-1 E Lip(K o ) , where K depends only on K _p , n, and p, p = n;

• 2)    у is quasiconformal mapping, if p = n .

• <1 If condition 1) holds, then the relation (4) holds for both у and у ^-1 . The desired results are proved in [1, Subsections 1.2, 1.3]. >

Proposition 2. Given a homeomorphism у : D ^ D of domains D,D C R ⁿ , where n ^ 2 , of class G _p,q with n — 1 C q C p <  ro the following properties hold:

• 1)    у E W^Jd ) , where p = _{р -}(П_{- 1)} , if P>n - 1 , and у E L^D), if p = n — 1;

• 2)    у has the finite distortion;

• 3)    the codistortion function

  D 9 x ^ K q,p (x, у) = <

  ' 1 adj D^x) 1 ! , det Dу(x) = 0, | det Dϕ(x) | 1- q                                                 (7) 0,              det Dу(x) = 0,

belongs to L _a (D), where a is determined from P = q — p , if n — 1 C q < p <  ro , and ст = ro , if n — 1 C q = p <  ro ;

• 4)    the distortion function

  D 9 x ^ K p ' ,q ' (x, у) = *

  
  

  | Dϕ(x) |

  | det Dϕ(x) | ^{q ′}

  0,

  
  

  det Dy(x) = 0,

  det Dу(x) = 0,

  
  
  
  

belongs to L _e (D), where q ' = q — n — ij at q > n — 1 , q' = ro at q = n — 1 and p is determined from P = p y — I , if n — 1 C q < p <  ro , and g = ro , if n — 1 C q = p <  ro .

<1 If у E G _p,_q then, according Proposition 1, ф = у ^-1 E Q _q,_p . By Theorem 3 the homeomorphism ф : D' ^ D

• 1)    belongs to Wqj oc CD ' ) ;

• 2)    has the finite distortion: Dф(y) = 0 holds almost everywhere on Z = { y E D ' : det Dф(y) = 0 } ,

• 3)    the distortion function

  D ' 9 y ^ K q,p ⁽ y,^ ⁾ = <

  
  

  | Dψ(y) |

  1 | det Dψ(y) | ^p

  0 ,

  
  

  det Dф(y) = 0,

  det Dф(y) = 0,

  
  
  
  

belongs to L _CT (D ' ) , where a is determined from 1 = q — p , if 1 C q < p <  ro , and a = ro , if 1 C q = p <  ro .

By [1, Theorem 4] we conclude that у E W p ’ ioc (D) , where p ' = p - n - ij if p > n — 1 , у E L ^ (D) , if p = n — 1 , and у has the finite distortion.

Take into account that Dф(y) = ddD^(X) and det Dф(y) = (det Dу(x))-1 at points y = у(х) = Z' П S', where Z' = {y E D : det Dф(y) = 0} and X' C D' is a maximal Borel null-set such that measure of Z = ф(Хф is positive. Notice that up to a set of measure zero Z = {x E D : det Dу(x) = 0}, and S = ф(Z') C D is a null-set. The mapping у has Luzin property N outside of S.

By change of variable formula in the case q < p we get hKq,p(•,у) | LCT(D)T = у f

D \ (Z ∩ Σ)

-( i D \ (Z ∩ Σ)

| adj Dу(x) |

| det Dу(x) | ¹

1 dx q

| adj Dу(x) |

| det Dу(x) | ¹

—   | det Dу(x) | dx

p

I (

D ^′ \ (Z ^′ ∩ Σ ^′ )

| Dф(y) |

| det Dф(y) | P

dy = h K q,p ( • ,Ф) | D' T . (10)

From the left hand side of this equality it follows (7): K q_JP ( • ,у) G L _CT (D) . In the case if n — 1 C q = p <  to we have

KpM | L ~ (D )|| =

= esssup D'y = _Wp,_q (.^ | L ^ (D ’ ) ^ . yeD ‘ \⁽Z ‘ ^ns ’ ) | det D^(y) | ^p

Integrability K _p ‘ ,_q ‘ (,у) G L _e (D) is proved in [1, Theorem 4]. >

From Proposition 2 it follows a part of Theorem 2.

Corollary 2. Given a homeomorphism у : D ^ D ' of domains D, D' C R ⁿ , where n ^ 2 , of class G _p,_q with n — 1 C q C p <  to the following properties hold:

• 1)    у ^-1 = ^ G L ^ (D ' ) and ||у^-1 | L ^ D ) | C ||K p,p (n ^) | L_x (D ‘ ) H p-n in the case n < q = p <  to ;

• ²)    у ^G L L ^{( D ) and}

p ^′

| у | LUD) ^ C ^ K p ‘ ,q ‘ (,у) | L _e (D) \\ p’-n                        (11)

in the case n — 1 C q = p < n.

<1 Really, taking (9) into account at q = p > n and the inequality 1 C JdDDrDy^yj y in points, where det D^(y) = 0 we have

I DW)^ C (1«^Ю ^p = JD*^ C HK p, (,^) I L „ (D ' )H.

• V     | ^det DW)    7      | det D^(y) | p

It follows ^ = у^-1 G L ^ (D ' ) in the case n < q = p <  to and

|| ^ | L ^ (D ' ) || C \\ K ppM) | L ^ (D ' ) | p - pn .

In the case n — 1 < q = p < n we have integrability K _p ‘ (^ у) = L ^ (D) with p > n . Therefore with above-mentioned arguments applied to (8) we obtain у G L ^ (D) and (11) holds.

Property у G L ^ (D) in the case q = p = n — 1 is just statement 1) of Proposition 2. >

2 ^nd STEP. Proposition 3. Let у G W, — i_oc (D) , V has the finite codistortion (adj Dy(x) = 0 almost everywhere on the set Z ) and the codistortion function

| adj Dϕ(x) |

D 9 x ^ K q,_p (x, v) =

| det Dϕ(x) | ^q

0 ,

det Dy(x) = 0, det Dy(x) = 0,

belongs to L _a (D ^r ), where a is determined from ^d = 1 — p , if n — 1 C q < p <  to , and а = to , if n — 1 C q = p <  to . Then у G G _p,_q .

cap q (ф - 1 (и)) *| u о ф | L> -1 (U )) || ^q

*  /

ψ ^-1 (U) \ Z ^′

q

|V u(ф(y)) | ^q | det Dф(y) | p

| Dф(y) | ^q

| det Dф(y) | p

dy

*(        [      |V u(ф(y))|^p | det Dф(y) | dy) p J [      (    ^{| DФ ( У )|} i V

^ ^-1 (U )\( z ‘ u s ’ )                                            V^{- 1} (U )\( z ‘ u s ’ ) ^{1 det Dф(y) |} p

σ ^q dy

*

IlKqpM) | L.(ф-1(и))||q ( / |Vu(x)|p dx) p, lKp,p(; ф) | L. < и)) ||p ( / | Vu(x)|p dx) , q < p, q = p-

It follows (3) with bounded quasiadditive set function Ф _q,_p equal to

D D U — Фq,p(U) = W^v) | L.(U ' = МК^ф) | L.(ф-1(и ', and Kp = ||Кр,р(^,ф) | Lx(r 1 (U))H.

Hence we proved v = ф -1 E G _p,_q . >

3 ^rd STEP. From Proposition 2 and 3 it follows the following criterium.

Theorem 4. A homeomorphism v : D — D ' of domains D, D C R n , where n ^ 2 , belongs to class G _p,_q with n — 1 * q * p <  (П - 12 , iff the following properties hold:

• i)    v e WUloM

• 2)    v has the finite codistortion;

• 3)    the codistortion function

  D 5 X —— ^Kq,p (x, v) — *

  
  

  | adj Dϕ ( x ) |

  | det Dϕ ( x ) | ^q

  0,

  
  

  det Dv(x) = 0,

  det Dv ( x ) = 0 ,

  
  
  
  

belongs to L . (D ‘ ) , where ст is determined from .1 = q — p , if n — 1 * q < p <  (nn - 1^) , and ( n - 1 ² )

ст = го , it n — 1 * q = p <  ^v_n-2 ' .

<1 If a homeomorphism v : D — D ‘ of domains D, D' C R n , where n ^ 2 , belongs to class G p,q with n — 1 * q * p <  (nn - 12 , then we apply Proposition 2 for obtaining that

• 1)    v E W p ’ ioc (D) , where p ' = _{р -}(П-1) , if p Ф n — 1 . Since p ' > n — 1 it follows that

v E W, 1 _₀ oc (D) .

• 2)    v has the finite distortion.

• 3)    the codistortion function (13) is in L . (D ' ) .

Thus the necessity is proved. The sufficiency is proved in Proposition 3. >