On global behavior of mappings with integral constraints
aa r X i v : . [ m a t h . C V ] J a n ON GLOBAL BEHAVIOR OF MAPPINGSWITH INTEGRAL CONSTRAINTS
E. SEVOST’YANOVJanuary 25, 2021
Abstract
This article is devoted to the study of mappings with branch points whose charac-teristics satisfy integral-type constraints. We have proved theorems concerning theirlocal and global behavior. In particular, we established the equicontinuity of families ofsuch mappings inside their definition domain, as well as, under additional conditions,equicontinuity of the families of these mappings in its closure.
This article is devoted to the study of mappings satisfying upper bounds for the distortionof the modulus of families of paths, see, for example, [Cr], [GRY], [MRV ]–[MRV ], [MRSY]and [RSY]. In particular, here we are dealing with mappings whose characteristics satisfythe so-called conditions of integral type, see, for example, [RSY], [RS] and [Sev ]. Themain purpose of the present manuscript is to study the local behavior of mappings havingbranch points whose characteristics are bounded only on the average. It is worth notingsome of the previous results in this direction. In particular, in [RS], homeomorphisms withsimilar conditions were investigated, and in [Sev ], mappings with branch points having ageneral characteristic Q. Unfortunately, the most general case when the mappings are nothomeomorphisms and also do not have a common majorant has been overlooked. Notethat the most interesting applications related to the study of the Beltrami equation and theDirichlet problem for it are associated with the absence of a general majorant for maximalcomplex characteristics, (see, e.g., [Dyb] and [L]).Let us move on to definitions and formulations of results. Given p > , M p denotes the p -modulus of a family of paths, and the element dm ( x ) corresponds to a Lebesgue measure1 N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... R n , n > , see [Va]. In what follows, we usually write M (Γ) instead of M n (Γ) . For thesets
A, B ⊂ R n we set, as usual, diam A = sup x,y ∈ A | x − y | , dist ( A, B ) = inf x ∈ A,y ∈ B | x − y | . Sometimes we also write d ( A ) instead of diam A and d ( A, B ) instead of dist ( A, B ) , if nomisunderstanding is possible. For given sets E and F and a given domain D in R n = R n ∪ {∞} , we denote by Γ( E, F, D ) the family of all paths γ : [0 , → R n joining E and F in D, that is, γ (0) ∈ E, γ (1) ∈ F and γ ( t ) ∈ D for all t ∈ (0 , . Everywhere below, unlessotherwise stated, the boundary and the closure of a set are understood in the sense of anextended Euclidean space R n . Let x ∈ D, x = ∞ ,S ( x , r ) = { x ∈ R n : | x − x | = r } , S i = S ( x , r i ) , i = 1 , ,A = A ( x , r , r ) = { x ∈ R n : r < | x − x | < r } . (1.1)Everywhere below, unless otherwise stated, the closure A and the boundary ∂A of the set A are understood in the topology of the space R n = R n ∪ {∞} . Let Q : R n → R n be aLebesgue measurable function satisfying the condition Q ( x ) ≡ for x ∈ R n \ D. Given p > , a mapping f : D → R n is called a ring Q -mapping at the point x ∈ D \ {∞} withrespect to p -modulus , if the condition M p ( f (Γ( S , S , D ))) Z A ∩ D Q ( x ) · η p ( | x − x | ) dm ( x ) (1.2)holds for all < r < r < d := sup x ∈ D | x − x | and all Lebesgue measurable functions η : ( r , r ) → [0 , ∞ ] such that r Z r η ( r ) dr > . (1.3)Inequalities of the form (1.2) were established for many well-known classes of mappings. So,for quasiconformal mappings and mappings with bounded distortion, they hold for p = n andsome Q ( x ) ≡ K = const (see, for example, [MRV , Theorem 7.1] and [Va, Definition 13.1]).Such inequalities also hold for many mappings with unbounded characteristic, in particular,for homeomorphisms belonging to the class W ,p loc , p > n − , the inner dilatation of the order α := pp − n +1 is locally integrable (see, for example, [MRSY, Theorems 8.1, 8.5] and [Sal ,Corollary 2], [Sal , Theorem 9, Lemma 5]).The concept of a set of capacity zero, used below, can be found in [MRV , Section 2.12]and is therefore omitted. A mapping f : D → R n is called discrete if the preimage { f − ( y ) } of each point y ∈ R n consist of isolated points, and open if the image of any open set U ⊂ D is an open set in R n . Let us formulate the main results of this manuscript. In what follows, h denotes the so-called chordal metric defined by the equalities h ( x, y ) = | x − y | q | x | q | y | , x = ∞ 6 = y , h ( x, ∞ ) = 1 q | x | . (1.4) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... E ⊂ R n , we set h ( E ) := sup x,y ∈ E h ( x, y ) , (1.5)The quantity h ( E ) in (1.5) is called the chordal diameter of the set E. For given sets
A, B ⊂ R n , we put h ( A, B ) = inf x ∈ A,y ∈ B h ( x, y ) , where h is a chordal metric defined in (1.4).Given a domain D ⊂ R n , a number M > , a set E ⊂ R n and a strictly increasingfunction Φ : R + → R + let us denote by F Φ M ,E ( D ) the family of all open discrete mappings f : D → R n \ E for which there exists a function Q = Q f ( x ) : D → [0 , ∞ ] such that (1.2)–(1.3) hold for any x ∈ D with p = n and, in addition, Z D Φ( Q ( x )) dm ( x )(1 + | x | ) n M < ∞ . (1.6)An analogue of the following statement was established for homeomorphisms in [RS, Theo-rem 4.1], and for mappings whose corresponding function Q is fixed, in [Sev , Theorem 1].However, let us note that it is in the form given below that the indicated statement seems tobe the most interesting from the point of view of applications to the problem of compactnessof solutions of the Beltrami equations and the Dirichlet problem (see, for example, [Dyb,Theorem 2]) and [L, Theorem 1]). Theorem 1.1.
Let D be a domain in R n , n > , and let cap E > . If ∞ Z δ dττ [Φ − ( τ )] n − = ∞ (1.7) holds for some δ > τ := Φ(0) , then F Φ M ,E ( D ) is equicontinuous in D. Note that the statement of Theorem 1.1 is much simpler and more elegant for the case p ∈ ( n − , n ) . Given p > , a domain D ⊂ R n , a number M > and a strictly increasingfunction Φ : R + → R + let us denote by F Φ M ,p ( D ) the family of all open discrete mappings f : D → R n for which there exists a function Q = Q f ( x ) : D → [0 , ∞ ] such that (1.2)–(1.3)hold for any x ∈ D and, in addition, (1.6) holds. The following statement is true. Theorem 1.2.
Let D be a domain in R n , n > , and let p ∈ ( n − , n ) . If (1.7) holdsfor some δ > τ := Φ(0) , then F Φ M ,p ( D ) is equicontinuous in D. Remark 1.1.
Let ( X, d ) and ( X ′ , d ′ ) be metric spaces with distances d and d ′ , respec-tively. A family F of mappings f : X → X ′ is said to be equicontinuous at a point x ∈ X, if for every ε > there is δ = δ ( ε, x ) > such that d ′ ( f ( x ) , f ( x )) < ε for all f ∈ F and x ∈ X with d ( x, x ) < δ . The family F is equicontinuous if F is equicontinuous at everypoint x ∈ X. In Theorem 1.1, the equicontinuity of the corresponding family of mappings should beunderstood in the sense of mappings of the metric spaces ( X, d ) and ( X ′ , d ′ ) , where X is a N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... D, and d is a Euclidean metric in D, besides, X ′ = R n , and h is a chordal metricdefined in (1.4). At the same time, in Theorem 1.2 the space X remains the same, and thespace X ′ is an usual Euclidean n -dimensional space with the Euclidean metric d ′ . A separate research topic is the equicontinuity of families of mappings in the closure of adomain. Results of this kind for fixed characteristics were obtained in some of our papers.In particular, in [Sev ] we considered the case of fixed domains between which the mappingsact, and in the papers [SevSkv ]–[SevSkv ] we considered the case when the mapped domaincan change. It should be noted that theorems on normal families of mappings are especiallyimportant in the study of the properties of solutions to the Dirichlet problem for the Beltramiequation (see, for example, [Dyb]). Note that the classical results on the equicontinuity ofquasiconformal mappings in the closure of a domain were obtained by N¨akki and Palka, seee.g. [NP, Theorem 3.3]. Let us formulate the main results related to this case.Let I be a fixed set of indices and let D i , i ∈ I, be some sequence of domains. Follow-ing [NP, Sect. 2.4], we say that a family of domains { D i } i ∈ I is equi-uniform with respect to p -modulus if for any r > there exists a number δ > such that the inequality M p (Γ( F ∗ , F, D i )) > δ (1.8)holds for any i ∈ I and any continua F, F ∗ ⊂ D such that h ( F ) > r and h ( F ∗ ) > r. It shouldbe noted that the condition of equi-uniformity of the sequence of domains implies strongaccessibility of the boundary of each of them with respect to p -modulus (see, for example,[SevSkv , Remark 1]). Given p > , u number δ > , a domain D ⊂ R n , n > , a continuum A ⊂ D and a strictly increasing function Φ : R + → R + denote F Φ ,A,p,δ ( D ) the family of allhomeomorphisms f : D → R n for which there exists a function Q = Q f ( x ) : D → [0 , ∞ ] such that: 1) relations (1.2)–(1.3) hold for any x ∈ D,
2) the relation (1.6) holds and 3) therelations h ( f ( A )) > δ and h ( R n \ f ( D )) > δ hold. The following statement is true. Theorem 1.3.
Let p ∈ ( n − , n ] , a domain D is locally connected at any x ∈ ∂D anddomains D ′ f = f ( D ) are equi-uniform with respect to p -modulus over all f ∈ F Φ ,A,p,δ ( D ) . If (1.7) holds for some δ > τ := Φ(0) , then any f ∈ F Q,A,p,δ ( D ) has a continuous extensionin D and, besides that, the family F Φ ,A,p,δ ( D ) of all extended mappings f : D → R n , isequicontinuous in D. As usual, we use the notation C ( f, x ) := { y ∈ R n : ∃ x k ∈ D : x k → x, f ( x k ) → y, k → ∞} . A mapping f between domains D and D ′ is called closed if f ( E ) is closed in D ′ for anyclosed set E ⊂ D (see, e.g., [Vu, Section 3]). Any open discrete closed mapping is boundarypreserving, i.e. C ( f, ∂D ) ⊂ ∂D ′ , where C ( f, ∂D ) = S x ∈ ∂D C ( f, x ) (see e.g. [Vu, Theorem 3.3]).Given p > , a domain D ⊂ R n , a set E ⊂ R n , a strictly increasing function Φ : R + → R + and a number δ > denote by R Φ ,δ,p,E ( D ) the family of all open discrete and closed N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... f : D → R n \ E such that: 1) relations (1.2)–(1.3) hold for any x ∈ D,
2) therelation (1.6) holds and 3) there exists a continuum K f ⊂ D ′ f such that h ( K f ) > δ and h ( f − ( K f ) , ∂D ) > δ > . The following statement is true.
Theorem 1.4.
Let p ∈ ( n − , n ] , a domain D is locally connected at any point x ∈ ∂D and, besides that, domains D ′ f = f ( D ) are equi-uniform with respect to p -modulus over all f ∈ R Φ ,δ,p,E ( D ) . Let cap
E > for p = n, and let E is any closed set for n − < p < n. If (1.7) holds for some δ > τ := Φ(0) , then any f ∈ R Φ ,δ,p,E ( D ) has a continuous extensionin D and, besides that, the family R Φ ,δ,p,E ( D ) of all extended mappings f : D → R n , isequicontinuous in D. Remark 1.2.
In Theorems 1.3 and 1.4, the equicontinuity should be understood interms of families of mappings between metric spaces ( X, d ) and ( X ′ , d ′ ) , where X = D, d is a chordal metric h, X ′ = R n and d ′ is a chordal (spherical) metric h, as well.Theorems 1.3 and 1.4 admit a natural generalization to the case of complex boundaries,when the maps do not have a continuous extension to points of the boundary of the domainin the usual sense, however, this extension holds in the sense of the so-called prime ends.Let us recall several important definitions associated with this concept. In the following, thenext notation is used: the set of prime ends corresponding to the domain D, is denoted by E D , and the completion of the domain D by its prime ends is denoted D P . The definition ofprime ends used below corresponds to the definition given in [IS ], and therefore is omitted.Consider the following definition, which goes back to N¨akki [Na ], see also [KR]. We say thatthe boundary of the domain D in R n is locally quasiconformal , if each point x ∈ ∂D has aneighborhood U in R n , which can be mapped by a quasiconformal mapping ϕ onto the unitball B n ⊂ R n so that ϕ ( ∂D ∩ U ) is the intersection of B n with the coordinate hyperplane.For a given set E ⊂ R n , we set d ( E ) := sup x,y ∈ E | x − y | . The sequence of cuts σ m , m = 1 , , . . . , is called regular, if σ m ∩ σ m +1 = ∅ for m ∈ N and, in addition, d ( σ m ) → as m → ∞ . Ifthe end K contains at least one regular chain, then K will be called regular . We say thata bounded domain D in R n is regular , if D can be quasiconformally mapped to a domainwith a locally quasiconformal boundary whose closure is a compact in R n , and, besides that,every prime end in D is regular. Note that space D P = D ∪ E D is metric, which can bedemonstrated as follows. If g : D → D is a quasiconformal mapping of a domain D witha locally quasiconformal boundary onto some domain D, then for x, y ∈ D P we put: ρ ( x, y ) := | g − ( x ) − g − ( y ) | , (1.9)where the element g − ( x ) , x ∈ E D , is to be understood as some (single) boundary point of thedomain D . The specified boundary point is unique and well-defined by [IS , Theorem 2.1,Remark 2.1], cf. [Na , Theorem 4.1]. It is easy to verify that ρ in (1.9) is a metric on D P , and that the topology on D P , defined by such a method, does not depend on the choice ofthe map g with the indicated property. The analogs of Theorems 1.3 and 1.4 for the case ofprime ends are as follows. N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... Theorem 1.5.
Let p ∈ ( n − , n ] and let D be a regular domain. Assume that D ′ f = f ( D ) are bounded equi-uniform domains with respect to p -modulus over all f ∈ F Φ ,A,p,δ ( D ) , which are domains with a locally quasiconformal boundary, as well. If (1.7) holds for some δ > τ := Φ(0) , then any f ∈ F Q,A,p,δ ( D ) has a continuous extension in D P and, besidesthat, the family F Φ ,A,p,δ ( D ) of all extended mappings f : D P → R n , is equicontinuous in D P . Theorem 1.6.
Let p ∈ ( n − , n ] and let D be a regular domain. Assume thatdomains D ′ f = f ( D ) are bounded equi-uniform domains with respect to p -modulus over all f ∈ R Φ ,δ,p,E ( D ) , which are domains with a locally quasiconformal boundary, as well. Let cap E > for p = n, and let E is any closed domain whenever n − < p < n. If (1.7)holds for some δ > τ := Φ(0) , then any f ∈ R Φ ,δ,p,E ( D ) has a continuous extension in D P and, besides that, the family f ∈ R Φ ,δ,p,E ( D ) of all extended mappings f : D P → R n , isequicontinuous in D P . Remark 1.3.
In Theorems 1.5 and 1.6, the equicontinuity should be understood in termsof families of mappings between metric spaces ( X, d ) and ( X ′ , d ′ ) , where X = D P , d is oneof the possible metrics, corresponding to the topological space D P , X ′ = R n and d ′ is achordal (spherical) metric. As in the article [RS], the key point related to the proof of the main statements of the articleis related to the connection between conditions (1.6)-(1.7) and the divergence of an integralof a special form (see, for example, [RS, Lemma 3.1]). Given a Lebesgue measurable function Q : R n → [0 , ∞ ] and a point x ∈ R n we set q x ( t ) = 1 ω n − r n − Z S ( x ,t ) Q ( x ) d H n − , (2.1)where H n − denotes ( n − -dimensional Hausdorff measure. The following lemma is ofparticular importance. Lemma 2.1.
Let p n, and let Φ : [0 , ∞ ] → [0 , ∞ ] be a strictly increasing convexfunction such that the relation ∞ Z δ dττ [Φ − ( τ )] p − = ∞ (2.2) holds for some δ > τ := Φ(0) . Let Q be a family of functions Q : R n → [0 , ∞ ] such that Z D Φ( Q ( x )) dm ( x )(1 + | x | ) n M < ∞ (2.3) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... for some < M < ∞ . Now, for any < r < and for every σ > there exists < r ∗ = r ∗ ( σ, r , Φ) < r such that r Z ε dtt n − p − q p − x ( t ) > σ , ε ∈ (0 , r ∗ ) , (2.4) for any Q ∈ Q . Proof.
Using the substitution of variables t = r/r , for any ε ∈ (0 , r ) we obtain that r Z ε drr n − p − q p − x ( r ) > r Z ε drrq p − x ( r ) = Z ε/r dttq p − x ( tr ) = Z ε/r dtt e q p − ( t ) , (2.5)where e q ( t ) is the average integral value of the function e Q ( x ) := Q ( r x + x ) over the sphere | x | = t, see the ratio (2.1). Then, according to [RS, Lemma 3.1], Z ε/r dtt e q p − ( t ) > n M ∗ ( ε/r rn εn Z eM ∗ ( ε/r ) dττ [Φ − ( τ )] p − , (2.6)where M ∗ ( ε/r ) = 1Ω n (1 − ( ε/r ) n ) Z A (0 ,ε/r , Φ ( Q ( r x + x )) dm ( x ) == 1Ω n ( r n − ε n ) Z A ( x ,ε,r ) Φ ( Q ( x )) dm ( x ) and A ( x , ε, r ) is defined in (1.1) for r := ε and r := r . Observe that | x | | x − x | + | x | r + | x | for any x ∈ A ( x , ε, r ) . Thus M ∗ ( ε/r ) β ( x )Ω n ( r n − ε n ) Z A ( x ,ε,r ) Φ( Q ( x )) dm ( x )(1 + | x | ) n , where β ( x ) = (1 + ( r + | x | ) ) n . Therefore, M ∗ ( ε/r ) β ( x )Ω n r n M for ε r / n √ , where M is a constant in (2.3). Observe that M ∗ ( ε/r ) > Φ(0) > , because Φ is increasing. Now, by (2.5) and (2.6) we obtain that r Z ε drr n − p − q p − x ( r ) > n Φ(0) rn εn Z β ( x M e Ω nrn dττ [Φ − ( τ )] p − . (2.7) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... ✷ Recall that a pair E = ( A, C ) , where A is an open set in R n , and C is a compact subsetof A, is called condenser in R n . Given p > , a quantity cap p E = cap ( A, C ) = inf u ∈ W ( E ) Z A |∇ u | p dm ( x ) , where W ( E ) = W ( A, C ) is a family of all nonnegative absolutely continuous on lines(ACL) functions u : A → R with compact support in A and such that u ( x ) > on C, iscalled p -capacity of the condenser E . We write cap E for cap n E. We also need the followingstatement given in [Ri, Proposition II.10.2].
Proposition 2.1.
Let E = ( A, C ) be a condenser in R n and let Γ E be the family of allpaths of the form γ : [ a, b ) → A with γ ( a ) ∈ C and | γ | ∩ ( A \ F ) = ∅ for every compact set F ⊂ A. Then cap E = M (Γ E ) . In what follows, we set a/ ∞ = 0 for a = ∞ , a/ ∞ for a > and · ∞ = 0 . One of themost important statements allowing us to connect the study of mappings in (1.2) with theconditions (1.6)–(1.7) is the following proposition. The principal points related to its proofwere indicated during the establishment of Lemma 1 in [SalSev ]; however, for the sake ofcompleteness of presentation, we will establish it in full in the text. Proposition 2.2.
Let D be a domain in R n , n > , let x ∈ D \ {∞} , let Q : D → [0 , ∞ ] be a Lebesgue measurable function and let f : D → R n be an open discrete mappingsatisfying relations (1.6)–(1.7) at a point x . If < r < r < sup x ∈ D | x − x | , then M p ( f (Γ( S ( x , r ) , S ( x , r ) , D ))) ω n − I p − , (2.8) where I = I ( x , r , r ) = r Z r drr n − p − q p − x ( r ) . (2.9) If, in addition, x ∈ D, < r < r < r = dist ( x , ∂D ) and E = (cid:16) B ( x , r ) , B ( x , r ) (cid:17) , then cap p f ( E ) ω n − I p − , (2.10) where f ( E ) = (cid:16) f ( B ( x , r )) , f (cid:16) B ( x , r ) (cid:17)(cid:17) . Proof.
We may consider that I = 0 , since (2.8) and (2.10) are obvious, in this case. Wealso may consider that I = ∞ . Otherwise, we may consider Q ( x ) + δ instead of Q ( x ) in (2.8)and (2.10), and then pass to the limit as δ → . Let I = ∞ . Let us first prove relation (2.8) for the case x ∈ D \ {∞} . Now q x ( r ) = 0 for a.e. r ∈ ( r , r ) . Set ψ ( t ) = ( / [ t n − p − q p − x ( t )] , t ∈ ( r , r ) , , t / ∈ ( r , r ) . N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... Z A Q ( x ) · ψ p ( | x − x | ) dm ( x ) = ω n − I , (2.11)where A = A ( r , r , x ) is defined in (1.1). Observe that a function η ( t ) = ψ ( t ) /I, t ∈ ( r , r ) , satisfies (1.3) because r R r η ( t ) dt = 1 . Now, by the definition of f in (1.2) M p ( f (Γ( S ( x , r ) , S ( x , r ) , D ))) Z A Q ( x ) · η p ( | x − x | ) dm ( x ) = ω n − I p − . (2.12)The first part of Proposition 2.2 is established. Let us prove the second part, namely,relation (2.10). Let Γ E and Γ f ( E ) be families of paths in the sense of the notation of Propo-sition 2.1. By this proposition cap p f ( E ) = cap p ( f ( B ( x , r )) , f ( B ( x , r ))) = M p (Γ f ( E ) ) . (2.13)Let Γ ∗ be a family of all maximal f -liftings of Γ f ( E ) starting in B ( x , r ) . Arguing similarlyto the proof of Lemma 3.1 in [Sev ], one can show that Γ ∗ ⊂ Γ E . Observe that Γ f ( E ) > f (Γ ∗ ) , and Γ E > Γ( S ( x , r − δ ) , S ( x , r ) , D ) for sufficiently small δ > . By (2.12), we obtain that M p (Γ f ( E ) ) M p ( f (Γ ∗ )) M p ( f (Γ E )) M p ( f (Γ( S ( x , r ) , S ( x , r − δ ) , A ( r , r − δ, x )))) ω n − r − δ R r dtt n − p − q p − x ( t ) ! p − . (2.14)Observe that a function g ψ ( t ) := ψ | ( r ,r ) = t n − p − q p − x ( t ) is integrable on ( r , r ) , because I = ∞ . Hence, by the absolute continuity of the integral, we obtain that r − δ Z r dtt n − p − q p − x ( t ) → r Z r dtt n − p − q p − x ( t ) (2.15)as δ → . By (2.14) and (2.15), we obtain that M p (Γ f ( E ) ) ω n − r R r dtt n − p − q p − x ( t ) ! p − . (2.16)Combining (2.13) and (2.16), we obtain (2.10). ✷ The next lemma contains an application of the previous Lemma 2.1 to mapping theory.
Lemma 2.2.
Let D be a domain in R n , let p n, let Φ : [0 , ∞ ] → [0 , ∞ ] be a strictlyincreasing convex function such that the relation and let x ∈ D. Denote by R Φ ,p ( D ) theN LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... family of all discrete open mappings for which there exists a Lebesgue measurable function Q = Q f ( x ) : R n → [0 , ∞ ] , Q ( x ) ≡ for x ∈ R n \ D, satisfying (1.2)–(1.3) for any x ∈ D, and, in addition, (2.3) holds for some < M < ∞ . Let < r < r < d = dist ( x , ∂D ) , and let E = ( B ( x , r ) , B ( x , r )) be a condenser. If the relation (2.2) holds for some δ > τ := Φ(0) , then cap p f ( E ) → as r → uniformly over f ∈ R Φ ,p ( D ) . Proof.
By Proposition 2.2 cap p f ( E ) ω n − I p − , (2.17)where ω n − denotes an area of the unit sphere S n − := S (0 , in R n , I := r R r drr n − p − q p − x ( r ) and q x is defined in (2.1). The rest of the statement follows from Lemma 2.1. ✷ The following statement was proved for p = n in [MRV , Lemma 3.11] (see also [Ri,Lemma 2.6, Ch. III]). Proposition 3.1.
Let F be a compact proper subset of R n with cap F > . Then forevery a > there exists δ > such that cap ( R n \ F, C ) > δ for every continuum C ⊂ R n \ F with h ( C ) > a. Proof of Theorem 1.1 largely uses the classical scheme used in the quasiregular case, as wellas applied by the author earlier, see, for example, [MRV , Theorem 4.1], [Ri, Theorem 2.9.III],[Cr, Theorem 8], [Sev , Lemma 3.1] and [SalSev , Lemma 4.2].Let x ∈ D, ε < d ( x , ∂D ) , and let E = ( A, C ) be a condenser, where A = B ( x , ε ) and C = B ( x , ε ) . As usual, ε := ∞ for D = R n . Let a > . Since cap
E > , by Proposition 3.1there exists δ = δ ( a ) > such that cap ( R n \ F, E ) > δ (3.1)for any continuum C ⊂ R n \ E such that h ( C ) > a. On the other hand, by Lemma 2.2 thereexists such that cap f ( E ) α ( ε ) , ε ∈ (0 , ε ) , for any f ∈ F Φ M ,E ( D ) , where α is some function such that α ( ε ) → as ε → . Now, for anumber δ = δ ( a ) there exists ε ∗ = ε ∗ ( a ) such that cap f ( E ) δ , ε ∈ (0 , ε ∗ ( a )) . (3.2) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... cap (cid:16) R n \ E, f ( B ( x , ε )) (cid:17) cap (cid:16) f ( B ( x , ε )) , f ( B ( x , ε )) (cid:17) δ for ε (0 , ε ∗ ( a )) . Now, by (3.1), h ( f ( B ( x , ε ))) < a. Finally, for any a > there is ε ∗ = ε ∗ ( a ) such that h ( f ( B ( x , ε ))) < a for ε ∈ (0 , ε ∗ ( a )) . Theorem is proved. ✷ To prove Theorem 1.2, we need the following most important statement (see [Ma, (8.9)]).
Proposition 3.2.
Given a condenser E = ( A, C ) and < p < n, cap p E > n Ω pn n (cid:18) n − pp − (cid:19) p − [ m ( C )] n − pn , where Ω n denotes the volume of the unit ball in R n , and m ( C ) is the n -dimensional Lebesguemeasure of C. The basic lower estimate of capacity of a condenser E = ( A, C ) in R n is given by cap p E = cap p ( A, C ) > (cid:18) b n ( d ( C )) p ( m ( A )) − n + p (cid:19) n − , p > n − , (3.3)where b n depends only on n and p and d ( C ) denotes the diameter of C (see [Kr, Proposition 6],cf. [MRV , Lemma 5.9]). Proof of Theorem 1.2 is based on the approach used in the proof of Lemma 2.4 in [GSS].Let < r < dist ( x , ∂D ) . Consider a condenser E = ( A, C ) with A = B ( x , r ) , C = B ( x , ε ) . By Lemma 2.2, there is a function α = α ( ε ) and < ε ′ < r such that α ( ε ) → as ε → and, in addition, cap p f ( E ) α ( ε ) for any ε ∈ (0 , ε ′ ) and f ∈ F Φ M ,p ( D ) . Applying Proposition 3.2, one obtains α ( ε ) > cap p f ( E ) > n Ω pn n (cid:18) n − pp − (cid:19) p − [ m ( f ( C ))] n − pn , where Ω n denotes the volume of the unit ball in R n , and m ( C ) stands for the n -dimensionalLebesgue measure of C. In other words, m ( f ( C )) α ( ε ) , where α ( ε ) → as ε → . The last relation implies the existence of a number ε ∈ (0 , , such that m ( f ( C )) , (3.4)where C = B ( x , ε ) . N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... E = ( A , C ε ) , A = B ( x , ε ) , and C ε = B ( x , ε ) , ε ∈ (0 , ε ) . By Lemma 2.2 there is a function α ( ε ) and a number < ε ′ < ε such that cap p f ( E ) α ( ε ) for any ε ∈ (0 , ε ′ ) , where α ( ε ) → as ε → . On the other hand, according to (3.3), c (cid:16) d ( f ( B ( x , ε ))) (cid:17) p ( m ( f ( B ( x , ε )))) − n + p n − cap p f ( E ) α ( ε ) . (3.5)By (3.4) and (3.5), one gets d ( f ( B ( x , ε ))) α ( ε ) , (3.6)where α ( ε ) → as ε → . The proof of Theorem 1.2 is completed, since the mapping f ∈ F Φ M ,p ( D ) participating in (3.6) is arbitrary. ✷ The proofs of these theorems are conceptually close to the proofs of Theorems 1–4 in [SevSkv ]and use the same approach. Let’s start with the following very useful remark (see, for ex-ample, [SevSkv , Remark 1]). Remark 4.1.
Let us show that, for a given domain D i , the relation (1.8) implies theso-called strong accessibility of its boundary with respect to p -modulus (see also [Na , The-orem 6.2]). Let i ∈ I, let x ∈ ∂D i and let U be some neighborhood of x . We mayassume that x = ∞ . Let ε > be such that V := B ( x , ε ) and V ⊂ U. If ∂U = ∅ and ∂V = ∅ , put ε := dist ( ∂U, ∂V ) > . Let F and G be continua in D i such that F ∩ ∂U = ∅ = F ∩ ∂V and G ∩ ∂U = ∅ = G ∩ ∂V. From the last relations it followsthat h ( F ) > ε and h ( G ) > ε . By the equi-uniformity of D i with respect to p -modulus, wemay find δ = δ ( ε ) > such that M p (Γ( F, G, D i )) > δ > . In particular, for any neigh-borhood U of x , there is a neighborhood V of the same point, a compact set F in D i anda number δ > such that M p (Γ( F, G, D i )) > δ > for any continuum G ⊂ D i such that G ∩ ∂U = ∅ = G ∩ ∂V. This property is called a strong accessibility of ∂D i at the point x with respect to p -modulus. Thus, this property is established for any domain D i which isan element of some equi-uniform family { D i } i ∈ I . Proof of Theorem 1.3 . The equicontinuity of the family F Φ ,A,p,δ ( D ) inside the domain D follows from [RS, Theorem 4.1] for p = n and Theorem 1.2 for p = n . Put f ∈ F Φ ,A,p,δ ( D ) and Q = Q f ( x ) . Set Q ′ ( x ) = Q ( x ) , Q ( x ) > , Q ( x ) < . N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... Q ′ ( x ) satisfies (1.6) up to a constant. Indeed, Z D Φ( Q ′ ( x )) dm ( x )(1 + | x | ) n = Z { x ∈ D : Q ( x ) < } Φ( Q ′ ( x )) dm ( x )(1 + | x | ) n ++ Z { x ∈ D : Q ( x ) > } Φ( Q ′ ( x )) dm ( x )(1 + | x | ) n M + Φ(1) Z R n dm ( x )(1 + | x | ) n = M ′ < ∞ . No, by [Sev , Theorem 2] and Remark 4.1, a mapping f ∈ F Φ ,A,p,δ ( D ) has a continu-ous extension to D for p = n. In addition, by Lemma 2.1, r R dtt n − p − q ′ p − x ( t ) = ∞ , where q ′ x ( t ) = ω n − r n − R S ( x ,t ) Q ′ ( x ) d H n − . In this case, a continuous extension of the mappingfrom f to ∂D can be established similarly to Theorem 1 in [Sev ]. Note that a rigorous proofof this fact was given in [IS , Theorem 1.2] for the case when the domains D and f ( D ) havecompact closures, and its proof in an arbitrary case can be presented completely by analogy.It remains to show that the family F Φ ,A,p,δ ( D ) is equicontinuous at ∂D. Suppose theopposite. Then there is x ∈ ∂D for which F Φ ,A,p,δ ( D ) is not equicontinuous at x . Due to the additional application of the inversion ϕ ( x ) = x | x | , we may assume that x = ∞ . Then there is a number a > with the following property: for any m = 1 , , . . . there is x m ∈ D and f m ∈ F Φ ,A,p,δ ( D ) such that | x − x m | < /m and, in addition, h ( f m ( x m ) , f m ( x )) > a. Since f m has a continuous extension at x , we may find a sequence x ′ m ∈ D, x ′ m → x as m → ∞ such that h ( f m ( x ′ m ) , f m ( x )) /m. Thus, h ( f m ( x m ) , f m ( x ′ m )) > a/ ∀ m ∈ N . (4.1)Since f m has a continuous extension to ∂D, we may assume that x m ∈ D. Since the domain D is locally connected, at the point x , there is a sequence of neighborhoods V m of the point x with h ( V m ) → for m → ∞ such that the sets D ∩ V m are domains and D ∩ V m ⊂ B ( x , − m ) . Without loss of the generality of reasoning, going to subsequences, if necessary, we mayassume that x m , x ′ m ∈ D ∩ V m . Join the points x m and x ′ m by the path γ m : [0 , → R n suchthat γ m (0) = x m , γ m (1) = x ′ m and γ m ( t ) ∈ V m for t ∈ (0 , , see Figure 1. We denote by C m the image of the path γ m ( t ) under the mapping f m . From the relation (4.1) it follows that h ( C m ) > a/ ∀ m ∈ N , (4.2)where h denotes the chordal diameter of the set.Let ε := dist ( x , A ) . Without loss of the generality of reasoning, one may assume thatthe continuum A participating in the definition of the class F Φ ,A,p,δ ( D ) , lies outside the balls B ( x , − m ) , m = 1 , , . . . , and B ( x , ε ) ∩ A = ∅ . In this case, the theorem on the propertyof connected sets that lie neither inside nor outside the given set implies the relation Γ m > Γ( S ( x , − m ) , S ( x , ε ) , D ) , (4.3) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... A m x x m D f m D C m f m ( A ) f m ( ) mm m x m n -m f m ( x m ) f m ( ) x m Figure 1: To the proof of Theorem 1.3see e.g. [Ku, Theorem 1.I.5.46]. Using Proposition 2.2 and by (2.4), (4.3), we obtain that M p ( f m (Γ m )) r Z − m drr n − p − q p − mx ( r ) → , m → ∞ , (4.4)where q mx ( t ) = ω n − r n − R S ( x ,t ) Q m ( x ) d H n − and Q m corresponds to the function Q of f m in (1.2). On the other hand, observe that f m (Γ m ) = Γ( C m , f m ( A ) , D ′ m ) . By the condition ofthe lemma, h ( f m ( A )) > δ for any m ∈ N . Therefore, by (4.2) h ( f m ( A )) > δ and h ( C m ) > δ , where δ := min { δ, a/ } . Taking into account that the domains D ′ m := f m ( D ) are equ-uniform with respect to p -modulus, we conclude that there exists σ > such that M p ( f m (Γ m )) = M p (Γ( C m , f m ( A ) , D ′ m )) > σ ∀ m ∈ N , which contradicts the condition (4.1). The resulting contradiction indicates that the assump-tion about the absence of equicontinuity of F Φ ,A,p,δ ( D ) was wrong. The resulting contradic-tion completes the proof. ✷ Proof of Theorem 1.4.
The equicontinuity of the family R Φ ,δ,p,E ( D ) inside the domain D follows from Theorem 1.1 for p = n and Theorem 1.2 for p = n . The possibility of continuousextension of any mapping f ∈ R Φ ,δ,p,E ( D ) to ∂D is established in the same way as at thebeginning of the proof of Theorem 1.3, and therefore the proof of this fact is omitted.It remains to show that the family R Φ ,δ,p,E ( D ) is equicontinuous at ∂D. Suppose theopposite. Then there is x ∈ ∂D for which R Φ ,δ,p,E ( D ) is not equicontinuous at x . Due tothe additional application of the inversion ϕ ( x ) = x | x | , we may assume that x = ∞ . Thenthere is a number a > with the following property: for any m = 1 , , . . . there is x m ∈ D and f m ∈ R Φ ,δ,p,E ( D ) such that | x − x m | < /m and, in addition, h ( f m ( x m ) , f m ( x )) > a. Since f m has a continuous extension at x , we may assume that x m ∈ D. Besides that, wemay find a sequence x ′ m ∈ D, x ′ m → x as m → ∞ such that h ( f m ( x ′ m ) , f m ( x )) /m. N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... D is locally connected, at the point x , thereis a sequence of neighborhoods V m of the point x with h ( V m ) → for m → ∞ such thatthe sets D ∩ V m are domains and D ∩ V m ⊂ B ( x , − m ) . Without loss of the generality ofreasoning, going to subsequences, if necessary, we may assume that x m , x ′ m ∈ D ∩ V m . Jointhe points x m and x ′ m by the path γ m : [0 , → R n such that γ m (0) = x m , γ m (1) = x ′ m and γ m ( t ) ∈ V m for t ∈ (0 , , see Figure 2. We denote by C m the image of the path γ m under m x x m D f m D C m f m ( ) f m ( ) mm m x m n -m f m ( ) x m f m ( ) x m K m -1 K m * * m = Figure 2: To the proof of Theorem 1.4the mapping f m . It follows from the relation (4.1) that a condition (4.2) is satisfied, where h denotes a chordal diameter of the set.By the definition of the family of mappings R Φ ,δ,p,E ( D ) , for any m = 1 , , . . . , any f m ∈ R Φ ,δ,p,E ( D ) and any domain D ′ m := f m ( D ) there is a continuum K m ⊂ D ′ m such that h ( K m ) > δ and h ( f − m ( K m ) , ∂D ) > δ > . Since, by the hypothesis of the lemma, thedomains D ′ m are equi-uniform with respect to p -modulus, by (4.2) we obtain that M p (Γ( K m , C m , D ′ m )) > b . (4.5)for any m = 1 , , . . . and some b > . Let Γ m be a family of all paths β : [0 , → D ′ m such that β (0) ∈ C m and β ( t ) → p ∈ K m as t → . Recall that a path α : [ a, b ) → R n iscalled a (total) f -lifting of a path β : [ a, b ) → R n starting at x , if ( f ◦ α )( t ) = β ( t ) for any t ∈ [ a, b ) . Let Γ ∗ m be a family of all total f m -liftings α : [0 , → D of Γ m starting at γ m . Sucha family is well-defined by [Vu, Theorem 3.7]. Since the mapping f m is closed, we obtainthat α ( t ) → f − m ( K m ) as t → b − , where f − m ( K m ) denotes the pre-image of K m under f m . Since R n is a compact metric space, the set C δ := { x ∈ D : h ( x, ∂D ) > δ } is compact in D for any δ > and, besides that, f − m ( K m ) ⊂ C δ . By [Sm, Lemma 1] the set C δ can beembedded in the continuum E δ lying in the domain D. In this case, we may assume that dist ( x , E δ ) > ε by decreasing ε . By the property of connected sets that lie neither insidenor outside the given set, we obtain that Γ ∗ m > Γ( S ( x , − m ) , S ( x , ε ) , D ) , (4.6) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... M p ( f m (Γ ∗ m )) M p ( f m (Γ( S ( x , − m ) , S ( x , ε ) , D ))) r Z − m drr n − p − q p − mx ( r ) → , m → ∞ , (4.7)where q mx ( t ) = ω n − r n − R S ( x ,t ) Q m ( x ) d H n − and Q m corresponds to f m in (1.2). Observethat f m (Γ ∗ m ) = Γ m and M p (Γ m ) = M p (Γ( K m , C m , D ′ m )) , so that M p ( f m (Γ ∗ m )) = M p (Γ( K m , C m , D ′ m )) . (4.8)However, the relations (4.7) and (4.8) together contradict (4.5). The resulting contradic-tion indicates that the original assumption (4.1) was incorrect, and therefore the family ofmappings R Φ ,δ,p,E ( D ) is equicontinuous at every point x ∈ ∂D. ✷ Proof of Theorem 1.5 . The equicontinuity of the family F Φ ,A,p,δ ( D ) inside the domain D follows from [RS, Theorem 4.1] for p = n and Theorem 1.2 for p = n . The existenceof a continuous extension of each f ∈ F Φ ,A,p,δ ( D ) to a continuous mapping in D followsfrom [Sev , Lemma 3]. In particular, the strong accessibility of D ′ f = f ( D ) with respect to p -modulus follows by Remark 4.1.Let us show the equicontinuity of the family F Φ ,A,p,δ ( D ) at E D , where E D denotes thespace of prime ends in D. Suppose the contrary, namely, that the family F Φ ,A,p,δ ( D ) is notequicontinuous at some point P ∈ E D . Then there is a number a > , a sequence P k ∈ D P ,k = 1 , , . . . , and elements f k ∈ F Q,A,p,δ ( D ) such that d ( P k , P ) < /k and h ( f k ( P k ) , f k ( P )) > a ∀ k = 1 , , . . . , . (4.9)Since f k has a continuous extension to D P , for any k ∈ N there is x k ∈ D such that d ( x k , P k ) < /k and h ( f k ( x k ) , f k ( P k )) < /k. Now, by (4.9) we obtain that h ( f k ( x k ) , f k ( P )) > a/ ∀ k = 1 , , . . . , . (4.10)Similarly, since f k has a continuous extension to D P , there is a sequence x ′ k ∈ D, x ′ k → P as k → ∞ for which h ( f k ( x ′ k ) , f k ( P )) < /k for k = 1 , , . . . . Now, it follows from (4.10)that h ( f k ( x k ) , f k ( x ′ k )) > a/ ∀ k = 1 , , . . . , (4.11)where x k and x ′ k belong to D and converge to P as k → ∞ , see Figure 3. By [IS , Lemma 3.1],cf. [KR, Lemma 2], a prime end P of a regular domain D contains a chain of cuts σ k lyingon spheres S k centered at some point x ∈ ∂D and with Euclidean radii r k → as k → ∞ .Let D k be domains associated with the cuts σ k , k = 1 , , . . . . Since the sequences x k and x ′ k converge to the prime end P as k → ∞ , we may assume that x k and x ′ k ∈ D k for any N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... D A f k f A k ( ) C k f k ( ) D n x k x k D k k k k k x Figure 3: To the proof of Theorem 1.5 k = 1 , , . . . , . Let us join the points x k and x ′ k by the path γ k , completely lying in D k . Onecan also assume that the continuum A from the definition of the class F Φ ,A,p,δ ( D ) does notintersect with any of the domains D k , and that dist ( ∂D, A ) > ε . We denote by C k the image of the path γ k under the mapping f k . It follows from therelation (4.11) that h ( C k ) > a/ ∀ k ∈ N , (4.12)where h is a chordal diameter of the set.Let Γ k be a family of all paths joining | γ k | and A in D. By [Ku, Theorem 1.I.5.46], Γ k > Γ( S ( x , r k ) , S ( x , ε ) , D ) . (4.13)Using Proposition 2.2 and by (2.4), (4.13) we obtain that M p ( f k (Γ k )) M p ( f k (Γ( S ( x , r k ) , S ( x , ε ) , D ))) r Z r k drr n − p − q p − kx ( r ) → , k → ∞ , (4.14)where q kx ( t ) = ω n − r n − R S ( x ,t ) Q k ( x ) d H n − and Q k corresponds to the function Q of f k in (1.2).On the other hand, note that f k (Γ k ) = Γ( C k , f k ( A ) , D ′ k ) , where D ′ k = f k ( D ) . Since bythe hypothesis of the lemma h ( f k ( A )) > δ for any k ∈ N , by (4.12), h ( f k ( A )) > δ and h ( C k ) > δ , where δ := min { δ, a/ } . Using the fact that the domains D ′ k are equ-uniformwith respect to p -modulus, we conclude that there is σ > such that M p ( f k (Γ k )) = M p (Γ( C k , f k ( A ) , D ′ k )) > σ ∀ k ∈ N , N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... F Φ ,A,p,δ ( D ) was wrong. The resultingcontradiction completes the proof of the theorem. ✷ Proof of Theorem 1.6.
The equicontinuity of the family R Φ ,δ,p,E ( D ) inside the domain D follows from [RS, Theorem 4.1] for p = n and Theorem 1.2 for p = n . The existenceof a continuous extension of each f ∈ R Φ ,δ,p,E ( D ) to a continuous mapping in D followsfrom [Sev , Lemma 3]. In particular, the strong accessibility of D ′ f = f ( D ) with respect to p -modulus follows by Remark 4.1.It remains to show that the family R Φ ,δ,p,E ( D ) is equicontinuous at ∂ P D := D P \ D. Suppose the opposite. Arguing as in the proof of Theorem 1.5, we construct two sequences x k and x ′ k ∈ D, converging to the prime end P as k → ∞ , for which a relation (4.11) holds.Let us join the points x k and x ′ k of the path γ k : [0 , → R n such that x ′ k ∈ D, γ k (0) = x k ,γ k (1) = x ′ k and γ k ( t ) ∈ D for t ∈ (0 , . Denote by C k the image of γ k under the mapping f k . It follows from the relation (4.11) that h ( C k ) > a/ ∀ k = 1 , , . . . . (4.15)By [IS , Lemma 3.1], cf. [KR, Lemma 2], a prime end P of a regular domain D contains achain of cuts σ k lying on spheres S k centered at some point x ∈ ∂D and with Euclideanradii r k → as k → ∞ . Let D k be domains associated with the cuts σ k , k = 1 , , . . . . Sincethe sequences x k and x ′ k converge to the prime end P as k → ∞ , we may assume that x k and x ′ k ∈ D k for any k = 1 , , . . . , . By the definition of the family R Φ ,δ,p,E ( D ) , for every f k ∈ R Q,δ,p,E ( D ) and any domain D ′ k := f k ( D ) there is a continuum K k ⊂ D ′ k such that h ( K k ) > δ and h ( f − ( K k ) , ∂D ) > δ > . Since, by the condition of the lemma, the domains D ′ k are equi-uniform with respectto p -modulus, by (4.15) we obtain that M p (Γ( K k , C k , D ′ k )) > b . (4.16)for any k = 1 , , . . . and some b > . Let Γ k be a family of all paths β : [0 , → D ′ k , where β (0) ∈ C k and β ( t ) → p ∈ K k as t → . Let Γ ∗ k be a family of all total liftings α : [0 , → D of Γ k under the mapping f k starting at γ k . Such a family i well-defined by [Vu, теорема 3.7].Since f k is closed, α ( t ) → f − k ( K k ) as t → , where f − k ( K k ) denotes the pre-image of K k under f k . Since R n is a compact metric space, the set C δ := { x ∈ D : h ( x, ∂D ) > δ } iscompact in D for any δ > and, besides that, f − k ( K k ) ⊂ C δ . By [Sm, Lemma 1] the set C δ can be embedded in the continuum E δ lying in the domain D. In this case, we may assumethat dist ( x , E δ ) > ε by decreasing ε . Let Γ k be a family of all paths joining | γ k | and A in D. By [Ku, Theorem 1.I.5.46], Γ ∗ k > Γ( S ( x , r k ) , S ( x , ε ) , D ) . (4.17) N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... M p ( f k (Γ ∗ k )) M p ( f k (Γ( S ( x , r k ) , S ( x , ε ) , D ))) r Z r k drr n − p − q p − kx ( r ) → , k → ∞ , (4.18)where q kx ( t ) = ω n − r n − R S ( x ,t ) Q k ( x ) d H n − and Q k corresponds to the function Q of f k in (1.2). Observe that f k (Γ ∗ k ) = Γ k and, simultaneously, M p (Γ k ) = M p (Γ( K k , C k , D ′ k )) . Now M p ( f k (Γ k )) = M p (Γ( K k , C k , D ′ k )) . (4.19)Combining (4.18) and (4.19), we obtain a contradiction with (4.16). The resulting contra-diction indicates that the initial assumption (4.1) was incorrect, and, therefore, the familyof mappings R Φ ,δ,p,E ( D ) is equicontinuous at any point x ∈ E D . ✷ References [Cr]
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Analogs of the Ikoma-Schwartz lemma andLiouville theorem for mappings with unbounded characteristic. - Ukrainian Math. J. 63:10,2012, 1551–1565.[Sev ] Sevost’yanov, E.A.:
Towards a theory of removable singularities for maps with unboundedcharacteristic of quasiconformity. - Izvestiya: Mathematics 74:1, 2010, 151–165.[Sev ] Sevost’yanov, E.A.:
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On spatial mappings with integral restrictions on the characteristic.- St. Petersburg Math. J. 24:1, 2013, 99–115.
N LOCAL BEHAVIOR OF MAPPINGS WITH INTEGRAL CONSTRAINTS... [Sev ] Sevost’yanov, E.A.:
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Boundary behavior and equicontinuity for families of mappings interms of prime ends. - St. Petersburg Math. J. 30:6, 2019, 973–1005.[SevSkv ] Sevost’yanov, E.A. and S.A. Skvortsov:
On the equicontinuity of families of map-pings in the case of variable domains. - Ukrainian Mathematical Journal 71:7, 2019, 1071–1086.[SevSkv ] Sevost’yanov, E.A. and S.A. Skvortsov:
Equicontinuity of families of mappingswith one normalization condition. - Math. Notes 109:4, 2021, 1–12 (accepted for print).[Sm]
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Zhytomyr Ivan Franko State University,40 Bol’shaya Berdichevskaya Str., 10 008 Zhytomyr, UKRAINE2.