On homeomorphisms with finite distortion in the plane
aa r X i v : . [ m a t h . C V ] N ov On homeomorphisms with finite distortionin the plane
Denis Kovtonyuk, Igor Petkov and Vladimir Ryazanov
August 13, 2018 (
KPR121110.tex ) Abstract
It is shown that every homeomorphism f of finite distortion in the plane is the so-calledlower Q -homeomorphism with Q ( z ) = K f ( z ) , and, on this base, it is developed the theory ofthe boundary behavior of such homeomorphisms. The concept of the generalized derivative was introduced by Sobolev in [31]. Given adomain D in the complex plane C , the Sobolev class W , ( D ) consists of all functions f : D → C in L ( D ) with first partial generalized derivatives which are integrable in D . A function f : D → C belongs to W , ( D ) if f ∈ W , ( D ∗ ) for every open set D ∗ with its compact closure D ∗ ⊂ D .Recall that a homeomorphism f between domains D and D ′ in C is called of finite distortion if f ∈ W , and || f ′ ( z ) || K ( z ) · J f ( z ) (1.1)with a.e. finite function K where || f ′ ( z ) || denotes the matrix norm of the Jacobianmatrix f ′ of f at z ∈ D and J f ( z ) = det f ′ ( z ) , see [10]. Later on, we use the notion K f ( z ) for the minimal function K ( z ) > in (1.1). Note that || f ′ ( z ) || = | f z | + | f ¯ z | and J f ( z ) = | f z | − | f ¯ z | at the points of total differentiability of f . Thus, K f ( z ) = || f ′ ( z ) || /J f ( z ) = ( | f z | + | f ¯ z | ) / ( | f z | − | f ¯ z | ) if J f ( z ) = 0 , K f ( z ) = 1 if f ′ ( z ) = 0 , i.e. | f z | = | f ¯ z | = 0 , and K f ( z ) = ∞ at the rest points.A continuous mapping γ of an open subset ∆ of the real axis R or a circle into D is called a dashed line , see e.g. Section 6.3 in [23]. Recall that every open set ∆ in R consists of a countable collection of mutually disjoint intervals. This is the motivationfor the term. iven a family Γ of dashed lines γ in complex plane C , a Borel function ̺ : C → [0 , ∞ ] is called admissible for Γ , write ̺ ∈ adm Γ , if Z γ ̺ ds > (1.2)for every γ ∈ Γ . The (conformal) modulus of Γ is the quantity M (Γ) = inf ̺ ∈ adm Γ Z C ̺ ( z ) dm ( z ) (1.3)where dm ( z ) corresponds to the Lebesgue measure in C . We say that a property P holds for a.e. (almost every) γ ∈ Γ if a subfamily of all lines in Γ for which P fails hasthe modulus zero, cf. [4]. Later on, we also say that a Lebesgue measurable function ̺ : C → [0 , ∞ ] is extensively admissible for Γ , write ̺ ∈ ext adm Γ , if (1.2) holdsfor a.e. γ ∈ Γ , see e.g. 9.2 in [23].The following concept was motivated by Gehring’s ring definition of quasiconfor-mality in [5]. Given domains D and D ′ in C = C ∪ {∞} , z ∈ D \ {∞} , and ameasurable function Q : D → (0 , ∞ ) , we say that a homeomorphism f : D → D ′ is a lower Q-homeomorphism at the point z if M ( f Σ ε ) > inf ̺ ∈ ext adm Σ ε Z D ∩ R ε ̺ ( x ) Q ( x ) dm ( x ) (1.4)for every ring R ε = { z ∈ C : ε < | z − z | < ε } , ε ∈ (0 , ε ) , ε ∈ (0 , d ) , where d = sup z ∈ D | z − z | , and Σ ε denotes the family of all intersections of the circles S ( r ) = S ( z , r ) = { z ∈ C : | z − z | = r } , r ∈ ( ε, ε ) , with the domain D .The notion can be extended to the case z = ∞ ∈ D in the standard way byapplying the inversion T with respect to the unit circle in C , T ( x ) = z/ | z | , T ( ∞ ) = 0 , T (0) = ∞ . Namely, a homeomorphism f : D → D ′ is a lower Q -homeomorphismat ∞ ∈ D if F = f ◦ T is a lower Q ∗ -homeomorphism with Q ∗ = Q ◦ T at . We alsosay that a homeomorphism f : D → C is a lower Q -homeomorphism in ∂D if f is a lower Q -homeomorphism at every point z ∈ ∂D .Further we show that every homeomorphism of finite distortion in the plane isa lower Q -homeomorphism with Q ( z ) = K f ( z ) and, thus, the whole theory of theboundary behavior in [12], see also Chapter 9 in [23], can be applied. Preliminaries
Recall first of all the following topological notion. A domain D ⊂ C is said to be locally connected at a point z ∈ ∂D if, for every neighborhood U of the point z ,there is a neighborhood V ⊆ U of z such that V ∩ D is connected. Note that everyJordan domain D in C is locally connected at each point of ∂D , see e.g. [35], p. 66. U VD ¶ D z We say that ∂D is weakly flat at a point z ∈ ∂D if, for every neighborhood U of the point z and every number P > , there is a neighborhood V ⊂ U of z suchthat M (∆( E, F ; D )) > P (2.1)for all continua E and F in D intersecting ∂U and ∂V . Here and later on, ∆( E, F ; D ) denotes the family of all paths γ : [ a, b ] → C connecting E and F in D , i.e. γ ( a ) ∈ E , γ ( b ) ∈ F and γ ( t ) ∈ D for all t ∈ ( a, b ) . We say that the boundary ∂D is weaklyflat if it is weakly flat at every point in ∂D . UV z E FD ¶ D We also say that a point z ∈ ∂D is strongly accessible if, for every neighbor-hood U of the point z , there exist a compactum E in D , a neighborhood V ⊂ U of z and a number δ > such that M (∆( E, F ; D )) > δ (2.2)for all continua F in D intersecting ∂U and ∂V . We say that the boundary ∂D is strongly accessible if every point z ∈ ∂D is strongly accessible. ere, in the definitions of strongly accessible and weakly flat boundaries, one cantake as neighborhoods U and V of a point z only balls (closed or open) centered at z or only neighborhoods of z in another fundamental system of neighborhoods of z .These conceptions can also be extended in a natural way to the case of C and z = ∞ .Then we must use the corresponding neighborhoods of ∞ .It is easy to see that if a domain D in C is weakly flat at a point z ∈ ∂D , thenthe point z is strongly accessible from D . Moreover, it was proved by us that if adomain D in C is weakly flat at a point z ∈ ∂D , then D is locally connected at z ,see e.g. Lemma 5.1 in [12] or Lemma 3.15 in [23].The notions of strong accessibility and weak flatness at boundary points of adomain in C defined in [11] are localizations and generalizations of the correspondingnotions introduced in [21]–[22], cf. with the properties P and P by V¨ais¨al¨a in [33] andalso with the quasiconformal accessibility and the quasiconformal flatness by N¨akki in[26]. Many theorems on a homeomorphic extension to the boundary of quasiconformalmappings and their generalizations are valid under the condition of weak flatness ofboundaries. The condition of strong accessibility plays a similar role for a continuousextension of the mappings to the boundary. In particular, recently we have provedthe following significant statements, see either Theorem 10.1 (Lemma 6.1) in [12] orTheorem 9.8 (Lemma 9.4) in [23]. Proposition 2.1.
Let D and D ′ be bounded domains in C , Q : D → (0 , ∞ ) a measurable function and f : D → D ′ a lower Q -homeomorphism in ∂D . Supposethat the domain D is locally connected on ∂D and that the domain D ′ has a (stronglyaccessible) weakly flat boundary. If δ ( z ) Z dr || Q || ( z , r ) = ∞ ∀ z ∈ ∂D (2.3) for some δ ( z ) ∈ (0 , d ( z )) where d ( z ) = sup z ∈ D | z − z | and || Q || ( z , r ) = Z D ∩ S ( z ,r ) Q ( z ) ds , then f has a (continuous) homeomorphic extension f to D that maps D (into) onto D ′ . Here as usual S ( z , r ) denotes the circle | z − z | = r .A domain D ⊂ C is called a quasiextremal distance domain , abbr. QED-domain , see [7], if M (∆( E, F ; C ) K · M (∆( E, F ; D )) (2.4) or some K > and all pairs of nonintersecting continua E and F in D .It is well known, see e.g. Theorem 10.12 in [33], that M (∆( E, F ; C )) > π log Rr (2.5)for any sets E and F in C intersecting all the circles S ( z , ρ ) , ρ ∈ ( r, R ) . Hence aQED-domain has a weakly flat boundary. One example in [23], Section 3.8, shows thatthe inverse conclusion is not true even among simply connected plane domains.A domain D ⊂ C is called a uniform domain if each pair of points z and z ∈ D can be joined with a rectifiable curve γ in D such that s ( γ ) a · | z − z | (2.6)and min i =1 , s ( γ ( z i , z )) b · d ( z, ∂D ) (2.7)for all z ∈ γ where γ ( z i , z ) is the portion of γ bounded by z i and z , see [24]. It isknown that every uniform domain is a QED-domain but there exist QED-domains thatare not uniform, see [7]. Bounded convex domains and bounded domains with smoothboundaries are simple examples of uniform domains and, consequently, QED-domainsas well as domains with weakly flat boundaries.A closed set X ⊂ C is called a null-set for extremal distances , abbr. NED-set , if M (∆( E, F ; C )) = M (∆( E, F ; C \ X )) (2.8)for any two nonintersecting continua E and F ⊂ C \ X . Remark 2.1.
It is known that if X ⊂ C is a NED-set, then | X | = 0 (2.9)and X does not locally separate C , see [34], i.e., dim X , (2.10)and hence they are totally disconnected, see e.g. p. 22 and 104 in [9]. Conversely, if aset X ⊂ C is closed and is of length zero, H ( X ) = 0 , (2.11)then X is a NED-set, see [34]. Note also that the complement of a NED-set in C is avery particular case of a QED-domain. ere H ( X ) denotes the 1-dimensional Hausdorff measure (length) of a set X in C . Also we denote by C ( X, f ) the cluster set of the mapping f : D → C for a set X ⊂ D , C ( X, f ) : = n w ∈ C : w = lim k →∞ f ( z k ) , z k → z ∈ X, z k ∈ D o . (2.12)Note that the inclusion C ( ∂D, f ) ⊆ ∂D ′ holds for every homeomorphism f : D → D ′ ,see e.g. Proposition 13.5 in [23]. Theorem 3.1.
Let f : D → C be a homeomorphism with finite distortion. Then f is a lower Q -homeomorphism at each point z ∈ D with Q ( z ) = K f ( z ) .Proof. Let B be a (Borel) set of all points z in D where f has a total differentialwith J f ( z ) = 0 a.e. It is known that B is the union of a countable collection of Borelsets B l , l = 1 , , . . . , such that f l = f | B l is a bi-Lipschitz homeomorphism, see e.g.Lemma 3.2.2 in [3]. With no loss of generality, we may assume that the B l are mutuallydisjoint. Denote also by B ∗ the set of all points z ∈ D where f has a total differentialwith f ′ ( z ) = 0 .Note that the set B = D \ ( B ∪ B ∗ ) has the Lebesgue measure zero in C byGehring–Lehto–Menchoff theorem, see [6] and [19]. Hence by Theorem 2.11 in [13],see also Lemma 9.1 in [23], length ( γ ∩ B ) = 0 for a.e. paths γ in D . Let us showthat length ( f ( γ ) ∩ f ( B )) = 0 for a.e. circle γ centered at z .The latter follows from absolute continuity of f on closed subarcs of γ ∩ D for a.e.such circle γ . Indeed, the class W , is invariant with respect to local quasi-isometries,see e.g. Theorem 1.1.7 in [25], and the functions in W , is absolutely continuous onlines, see e.g. Theorem 1.1.3 in [25]. Applying say the transformation of coordinates log( z − z ) , we come to the absolute continuity on a.e. such circle γ .Thus, length ( γ ∗ ∩ f ( B )) = 0 where γ ∗ = f ( γ ) for a.e. circle γ centered at z .Now, let ̺ ∗ ∈ a dm f (Γ) where Γ is the collection of all dashed lines γ ∩ D for suchcircles γ and ̺ ∗ ≡ outside f ( D ) . Set ̺ ≡ outside D and ̺ ( z ) : = ̺ ∗ ( f ( z )) ( | f z | + | f ¯ z | ) for a . e . z ∈ D Arguing piecewise on B l , we have by Theorem 3.2.5 under m = 1 in [3] that Z γ ̺ ds > Z γ ∗ ̺ ∗ ds ∗ > . e . γ ∈ Γ ecause length ( f ( γ ) ∩ f ( B )) = 0 and length ( f ( γ ) ∩ f ( B ∗ )) = 0 for a.e. γ ∈ Γ ,consequently, ̺ ∈ ext adm Γ .On the other hand, again arguing piecewise on B l , we have the inequality Z D ̺ ( x ) K f ( z ) dm ( z ) Z f ( D ) ̺ ∗ ( w ) dm ( w ) because J f ( z ) = | f z | − | f ¯ z | and K f ( z ) = ( | f z | + | f ¯ z | ) / ( | f z | − | f ¯ z | ) on B and K f ( z ) = 1 and ̺ ( z ) = 0 on B ∗ . Consequently, we obtain that M ( f Γ) > inf ̺ ∈ ext adm Γ Z D ̺ ( z ) K f ( z ) dm ( z ) , i.e. f is really a lower Q -homeomorphism with Q ( z ) = K f ( z ) . In view of Theorem 3.1 we obtain by Theorem 4.1 in [12] or Theorem 9.3 in [23] thefollowing statement.
Theorem 4.1.
Let D be a domain in C , z ∈ D , and f be a homeomorphismwith finite distortion of D \ { z } into C . Suppose that ε Z drr · k f ( r ) = ∞ (4.1) where ε < dist( z , ∂D ) and k f ( r ) = − Z | z − z | = r K f ( z ) | dz | . (4.2) Then f has a continuous extension to D in C . From here we have, in particular, the following consequences.
Corollary 4.1.
Let D be a domain in C and let f be a homeomorphism withfinite distortion of D \ { z } into C . If − Z | z − z | = r K f ( z ) | dz | = O (cid:18) log 1 r (cid:19) as r → , (4.3) hen f has a continuous extension to D in C . Corollary 4.2.
Let D be a domain in C , x ∈ D , and f be a homeomorphismwith finite distortion of D \ { z } into C . If − Z | z − z | = r K f ( z ) | dz | = O (cid:18) log 1 r · log log 1 r · . . . · log . . . log 1 r (cid:19) as r → , (4.4) then f has a continuous extension to D in C . In view of Theorem 3.1 we have by Theorem 6.1 in [12] or Lemma 9.4 in [23] the nextstatement.
Lemma 5.1.
Let D and D ′ be domains in C , z ∈ ∂D , and f : D → D ′ be a homeomorphism with finite distortion. Suppose that the domain D is locallyconnected at z ∈ ∂D and ∂D ′ is strongly accessible at least at one point of the clusterset C ( z , f ) . If ε Z dr || K f || ( r ) = ∞ (5.1) where < ε < d = sup z ∈ D | z − z | , and || K f || ( r ) = Z D ∩ S ( z ,r ) K f ds , (5.2) then f extends by continuity to z in C . In particular, we have the following consequence of Lemma 5.1.
Corollary 5.1.
Let D and D ′ be QED domains in C , z ∈ ∂D , and f : D → D ′ be a homeomorphism of finite distortion. If (5.1) holds, then f extends by continuityto z in C . Note that the complements of NED sets in C give very particular cases of QEDdomains. Thus, arguing locally, by Theorem 5.1, we obtain the following statement. heorem 5.1. Let D be a domain in C , X ⊂ D , and f be a homeomorphismwith finite distortion of D \ X into C . Suppose that X and C ( X, f ) are NED sets. If ε Z dr || K f || ( r ) = ∞ (5.3) where < ε < d = dist ( z , ∂D ) (5.4) and || K f || ( r ) = Z | z − z | = r K f ( z ) | dz | , (5.5) then f can be extended by continuity in C to z . The base of the proof for extending the inverse mappings for homeomorphisms of finitedistortion is the following lemma on the cluster sets.
Lemma 6.1.
Let D and D ′ be domains in C , z and z be distinct points in ∂D , z = ∞ , and let f be a homeomorphism with finite distortion of D onto D ′ . Supposethat the function K f is integrable on the dashed lines D ( r ) = { z ∈ D : | z − z | = r } = D ∩ S ( z , r ) (6.1) for some set E of numbers r < | z − z | of a positive linear measure. If D is locallyconnected at z and z and ∂D ′ is weakly flat, then C ( z , f ) ∩ C ( z , f ) = ∅ . (6.2)The of Lemma 6.1 follows by Theorem 3.1 from Lemma 9.1 in [12] or Lemma 9.5in [23].As an immediate consequence of Lemma 6.1, we have the following statement. Theorem 6.1.
Let D and D ′ be domains in C , D locally connected on ∂D and ∂D ′ weakly flat. If f is a homeomorphism with finite distortion of D onto D ′ with K f ∈ L ( D ) , then f − has an extension by continuity in C to D ′ .Proof. By the Fubini theorem, the set E = { r ∈ (0 , d ) : K f | D ( r ) ∈ L ( D ( r )) } (6.3) as a positive linear measure because K f ∈ L ( D ) . Remark 6.1.
It is clear from the proof that it is even sufficient to assume inTheorem 6.1 that K f is integrable only in a neighborhood of ∂D .Moreover, in view of Theorem 3.1 we obtain by Theorem 9.2 in [12] or Theorem9.7 in [23] the following conclusion. Theorem 6.2.
Let D and D ′ be domains in C , D locally connected on ∂D and ∂D ′ weakly flat, and let f : D → D ′ be a homeomorphism with finite distortion suchthat the condition δ ( z ) Z dr || K f || ( z , r ) = ∞ (6.4) holds for all z ∈ ∂D with some δ ( z ) ∈ (0 , d ( z )) where d ( z ) = sup z ∈ D | z − z | and || K f || ( z , r ) = Z D ( z ,r ) K f ds (6.5) is the L -norm of K f over D ( z , r ) = { z ∈ D : | z − z | = r } = D ∩ S ( z , r ) . Thenthere is an extension of f − by continuity in C to D ′ . Combining Lemma 5.1 and Theorem 6.2, we obtain the following statements.
Theorem 7.1.
Let D and D ′ be bounded domains in C and let f : D → D ′ bea homeomorphism with finite distortion in D . Suppose that the domain D is locallyconnected on ∂D and that the domain D ′ has a weakly flat boundary. If δ ( z ) Z dr || K f || ( z , r ) = ∞ ∀ z o ∈ ∂D (7.1) for some δ ( z ) ∈ (0 , d ( z )) where d ( z ) = sup z ∈ D | z − z | and || K f || ( z , r ) = Z D ∩ S ( z ,r ) K f ds , (7.2) then f has a homeomorphic extension to D . n particular, as a consequence of Theorem 7.1 we obtain the following general-ization of the well-known Gehring-Martio theorem on a homeomorphic extension tothe boundary of quasiconformal mappings between QED domains, see [7]. Corollary 7.1.
Let D and D ′ be bounded domains with weakly flat boundariesin C and let f : D → D ′ be a homeomorphism with finite distortion in D . If thecondition (7.1) holds at every point z ∈ ∂D , then f has a homeomorphic extensionto D . By Theorem 3.1 we have also the following, see Theorem 10.3 in [12] or Theorem9.10 in [23].
Theorem 7.2.
Let D be a bounded domain in C , X ⊂ D , and f : D \ { X } → C a homeomorphism with finite distortion. Suppose that X and C ( X, f ) are NED sets.If the condition (7.1) holds at every point z ∈ X for δ ( z ) < dist( z , ∂D ) where || K f || ( z , r ) = Z | z − z | = r K f ( z ) | dz | , (7.3) then f has a homeomorphic extension to D . Remark 7.1.
In particular, the conclusion of Theorem 7.2 is valid if X is a closedset with H ( X ) = 0 = H ( C ( X, f )) . (7.4) Recall theorems on interconnections between some integral conditions from [29] and[30]. For every non-decreasing function
Φ : [0 , ∞ ] → [0 , ∞ ] , the inverse function Φ − : [0 , ∞ ] → [0 , ∞ ] can be well defined by setting Φ − ( τ ) = inf Φ( t ) > τ t . (8.1)Here inf equal to ∞ if the set of t ∈ [0 , ∞ ] such that Φ( t ) > τ is empty. Note thatthe function Φ − is non-decreasing, too.Further, the integral in (8.4) is understood as the Lebesgue–Stieltjes integral andthe integrals in (8.3) and (8.5)–(8.8) as the ordinary Lebesgue integrals. In (8.3) and(8.4) we complete the definition of integrals by ∞ if Φ( t ) = ∞ , correspondingly, ( t ) = ∞ , for all t > T ∈ [0 , ∞ ) . Theorem 8.1.
Let
Φ : [0 , ∞ ] → [0 , ∞ ] be a non-decreasing function and set H ( t ) = log Φ( t ) . (8.2) Then the equality ∞ Z ∆ H ′ ( t ) dtt = ∞ (8.3) implies the equality ∞ Z ∆ dH ( t ) t = ∞ (8.4) and (8.4) is equivalent to ∞ Z ∆ H ( t ) dtt = ∞ (8.5) for some ∆ > , and (8.5) is equivalent to every of the equalities: δ Z H (cid:18) t (cid:19) dt = ∞ (8.6) for some δ > , ∞ Z ∆ ∗ dηH − ( η ) = ∞ (8.7) for some ∆ ∗ > H (+0) , ∞ Z δ ∗ dττ Φ − ( τ ) = ∞ (8.8) for some δ ∗ > Φ(+0) .Moreover, (8.3) is equivalent to (8.4) and hence (8.3)–(8.8) are equivalent eachto other if Φ is in addition absolutely continuous. In particular, all the conditions(8.3)–(8.8) are equivalent if Φ is convex and non-decreasing. Remark 8.1.
It is necessary to give one more explanation. From the right handsides in the conditions (8.3)–(8.8) we have in mind + ∞ . If Φ( t ) = 0 for t ∈ [0 , t ∗ ] , then H ( t ) = −∞ for t ∈ [0 , t ∗ ] and we complete the definition H ′ ( t ) = 0 for t ∈ [0 , t ∗ ] . Note,the conditions (8.4) and (8.5) exclude that t ∗ belongs to the interval of integrability ecause in the contrary case the left hand sides in (8.4) and (8.5) are either equalto −∞ or indeterminate. Hence we may assume in (8.3)–(8.6) that ∆ > t where t : = sup Φ( t )=0 t , t = 0 if Φ(0) > , and δ < /t , correspondingly. Theorem 8.2.
Let Q : D → [0 , ∞ ] be a measurable function such that Z D Φ( Q ( z )) dxdy < ∞ (8.9) where Φ : [0 , ∞ ] → [0 , ∞ ] is a non-decreasing convex function such that ∞ Z δ dττ Φ − ( τ ) = ∞ (8.10) for some δ > Φ(0) . Then Z drrq ( r ) = ∞ (8.11) where q ( r ) is the average of the function Q ( z ) over the circle | z | = r . Here D denotes the unit disk in C . Combining Theorems 8.1 and 8.2 we obtainalso the following. Corollary 8.1. If Φ : [0 , ∞ ] → [0 , ∞ ] is a non-decreasing convex function and Q : D → [0 , ∞ ] satisfies (8.9), then every of the conditions (8.3)–(8.8) implies (8.11). Integral conditions of the type Z D Φ( K ( x )) dm ( x ) < ∞ (9.1)are often applied in the mapping theory, see e.g. [1], [2], [8], [15]–[18], [27], [28] and[32]. Combining Theorem 8.2 with Lemma 5.1 and Theorem 7.1, we come to the fol-lowing statement. heorem 9.1. Let D and D ′ be bounded domains in C such that D is locallyconnected at ∂D and D ′ has a weakly flat (strongly accessible) boundary. Suppose that f : D → D ′ is a homeomorphism with finite distortion and Z D Φ( K f ( z )) dm ( z ) < ∞ (9.2) for a convex non-decreasing function Φ : [0 , ∞ ] → [0 , ∞ ] . If ∞ Z δ dττ Φ − ( τ ) = ∞ (9.3) for some δ > Φ(0) , then f has a homeomorphic (continuous) extension f to D thatmaps D onto (into) D ′ . Remark 9.1.
In particular, the conclusion on homeomorphic extension is validfor domains D and D ′ with smooth boundaries and for convex domains. Note alsothat by Theorem 8.1 the condition (9.3) can be replaced by each of the conditions(8.3) – (8.7). The example in [14] shows that each of the given conditions are not onlysufficient but also necessary for continuous extension of f to the boundary. References [1] Ahlfors L.: On quasiconformal mappings. J. Analyse Math. , 1–58 (1953/54).[2] Biluta P.A.: Extremal problems for mappings which are quasiconformal in themean. Sib. Mat. Zh. , 717–726 (1965).[3] Federer H.: Geometric Measure Theory. Springer-Verlag, Berlin (1969).[4] Fuglede B.: Extremal length and functional completion. Acta Math. , 171–219(1957).[5] Gehring F.W.: Rings and quasiconformal mappings in space. Trans. Amer. Math.Soc. , 353–393 (1962).[6] Gehring F.W., Lehto O.: On the total differentiability of functions of a complexvariable. Ann. Acad. Sci. Fenn. A1. Math. , 1–9 (1959).[7] Gehring F.W., Martio O.: Quasiextremal distance domains and extension of qua-siconformal mappings. J. Anal. Math. , 181–206 (1985).
8] Golberg A.: Homeomorphisms with finite mean dilatations. Contemporary Math. , 177–186 (2005).[9] Hurewicz W., Wallman H.: Dimension theory. Princeton Univ. Press, Princeton,NJ (1948).[10] Iwaniec T., Martin G.: Geometrical Function Theory and Non-linear Analysis.Clarendon Press, Oxford (2001).[11] Kovtonyuk D., Ryazanov V.: On boundaries of space domains. Proc. Inst. Appl.Math. & Mech. NAS of Ukraine , 110–120 (2006) [in Russian].[12] Kovtonyuk D., Ryazanov V.: On the theory of lower Q -homeomorphisms.Ukrainian Math. Bull. (2), 157–181 (2008).[13] Kovtonyuk D., Ryazanov V.: On the theory of mappings with finite area distor-tion. J. Anal. Math., , 291–306 (2008).[14] Kovtonyuk D., Ryazanov V.: On the boundary behavior of generalized quasi–isometries. ArXiv: 1005.0247, 20 p. (2010)[15] Kruglikov V.I.: Capacities of condensors and quasiconformal in the mean map-pings in space. Mat. Sb. (2) (1986), 185–206.[16] Krushkal’ S.L.: On mappings that are quasiconformal in the mean. Dokl. Akad.Nauk SSSR (3), 517–519 (1964).[17] Krushkal’ S.L., K¨uhnau R.: Quasiconformal mappings: new methods and appli-cations, Novosibirsk, Nauka (1984) (in Russian).[18] K¨uhnau R.: ¨Uber Extremalprobleme bei im Mittel quasiconformen Abbildungen.Lecture Notes in Math. , 113–124 (1983) (in German).[19] Menchoff D.: Sur les differentielles totales des fonctions univalentes. Math. Ann. , 75–85 (1931).[20] Martio O., Ryazanov V., Srebro U., Yakubov E.: Mappings with finite lengthdistortion. J. d’Anal. Math. , 215–236 (2004).[21] Martio O., Ryazanov V., Srebro U., Yakubov E.: Q -homeomorphisms. Contem-porary Math. , 193–203 (2004).[22] Martio O., Ryazanov V., Srebro U., Yakubov E.: On Q -homeomorphisms. Ann.Acad. Sci. Fenn. , 49–69 (2005).[23] Martio O., Ryazanov V., Srebro U., Yakubov E.: Moduli in Modern MappingTheory. Springer, New York (2009).
24] Martio O., Sarvas J.: Injectivity theorems in plane and space. Ann. Acad. Sci.Fenn. Ser. A1 Math. , 384–401 (1978/1979).[25] Maz’ya V.: Sobolev Classes. Springer-Verlag, Berlin (1985).[26] Nakki R.: Boundary behavior of quasiconformal mappings in n − space. Ann. Acad.Sci. Fenn. Ser. A1. Math. , 1–50 (1970).[27] Pesin I.N.: Mappings quasiconformal in the mean. Dokl. Akad. Nauk SSSR (4), 740–742 (1969).[28] Ryazanov V.I.: On mappings that are quasiconformal in the mean. Sibirsk. Mat.Zh. (2), 378–388 (1996).[29] Ryazanov V., Srebro U., Yakubov E.: Integral conditions in the theory of theBeltrami equations. ArXiv 1001.2821v11, 26 p. (2010)[30] Ryazanov V., Srebro U., Yakubov E.: Integral conditions in the mapping theory.Ukrainian Math. Bull. , 73–87 (2010).[31] Sobolev S.L.: Applications of functional analysis in mathematical physics. Izdat.Gos. Univ., Leningrad (1950); English transl, Amer. Math. Soc., Providence, R.I.(1963).[32] Ukhlov A., Vodopyanov S.K.: Mappings associated with weighted Sobolev spaces.Complex Anal. Dynam. Syst. III, Contemp. Math. , 369–382 (2008).[33] V¨ais¨al¨a J.: Lectures on n -Dimensional Quasiconformal Mappings. Lecture Notesin Math. . Springer–Verlag, Berlin etc. (1971).[34] V¨ais¨al¨a J.: On the null-sets for extremal distances. Ann. Acad. Sci. Fenn. Ser. A1.Math. , 1–12 (1962).[35] Wilder R.L.: Topology of Manifolds. AMS, New York (1949).Kovtonyk D., Petkov I. and Ryazanov V.,Institute of Applied Mathematics and Mechanics,National Academy of Sciences of Ukraine,74 Roze Luxemburg str., 83114 Donetsk, UKRAINEPhone: +38 – (062) – 3110145, Fax: +38 – (062) – 3110285denis [email protected], [email protected], [email protected], 1–12 (1962).[35] Wilder R.L.: Topology of Manifolds. AMS, New York (1949).Kovtonyk D., Petkov I. and Ryazanov V.,Institute of Applied Mathematics and Mechanics,National Academy of Sciences of Ukraine,74 Roze Luxemburg str., 83114 Donetsk, UKRAINEPhone: +38 – (062) – 3110145, Fax: +38 – (062) – 3110285denis [email protected], [email protected], [email protected]