On compact classes of solutions of the Dirichlet problem with integral restrictions
aa r X i v : . [ m a t h . C V ] N ov ON COMPACT CLASSES OF SOLUTIONS OFTHE DIRICHLET PROBLEM WITHINTEGRAL RESTRICTIONS
OLEKSANDR DOVHOPIATYI, EVGENY SEVOST’YANOVNovember 11, 2020
Abstract
This article is devoted to the study of the problem of compactness of solutions of thedifferential Beltrami equation with degeneration. We study the case when the complexcharacteristic of the equations satisfies the constraints of integral type. In this case,we have proved a theorem on the compactness of the class of homeomorphic solutionsof the Beltrami equation, which satisfy the hydrodynamic normalization condition atinfinity. Another important result is the compactness theorem for the class of opendiscrete solutions of the Dirichlet problem for the Beltrami equations in the Jordandomain, the imaginary part of which vanishes at some predetermined inner point.
As is known, the problems of compactness of classes of mappings are among the most im-portant in modern analysis. In particular, these theorems have applications to the existenceof solutions of some differential equations, and also have an important role in the considera-tion of many extremal problems (see, for example, [Dyb ], [GR ]–[GR ], [KPRS], [LGR] and[RSY]). We will point out the relatively little-known publication of Dybov [Dyb ], publishedin an inaccessible journal. In this publication, the compactness of the class of homeomorphicsolutions of the Dirichlet problem for the Beltrami equation in the unit disc is proved. Notethat the proof of Dybov’s result essentially uses the geometry of the unit disk, and thereforecannot be transferred to the case of more general domains by analogy.The main purpose of this manuscript is to obtain new results related to the compactnessof the classes of solutions of the Beltrami equations. Separately, we give a result on the1 N COMPACT CLASSES... f (0) = 0 , f (1) = 1 and f ( ∞ ) = ∞ were obtained in [L, Theorem 3].With regard to the present manuscript, we consider a different normalization of solutions,namely, when the behavior of the mappings at infinity is close to the identity mapping (theso-called hydrodynamic normalization). From the point of view of further applications toextremal problems, such a normalization is the most important. Regarding the compactnessof solutions to the Dirichlet problem, as we noted above, there is a corresponding result byDybov for the unit disk (see e.g. [Dyb , Theorem 2]). We also note that all the main resultsof this manuscript relate to the situation when complex dilations of solutions satisfy integral-type constraints. In other words, these complex dilations are bounded only on the average,and this ”averaging” is associated with the integral of some given convex increasing function.The latter essentially distinguishes the result of [Dyb ] from the present manuscript, wherewere the restrictions on dilations of a slightly different type.In what follows, a mapping f : D → C is assumed to be sense-preserving, moreover,we assume that f has partial derivatives almost everywhere. Put f z = ( f x + if y ) / and f z = ( f x − if y ) / . The complex dilatation of f at z ∈ D is defined as follows: µ ( z ) = µ f ( z ) = f z /f z for f z = 0 and µ ( z ) = 0 otherwise. The maximal dilatation of f at z is thefollowing function: K µ ( z ) = K µ f ( z ) = 1 + | µ ( z ) | − | µ ( z ) | . (1.1)Given a Lebesgue measurable function µ : D → D , D = { z ∈ C : | z | < } , we definethe maximal dilatation of f at z the function K µ ( z ) in (1.1). максимальною дилатацiєювiдповiдної функцiї µ. Note that the Jacobian of f at z ∈ D is calculated by the formula J ( z, f ) = | f z | − | f z | . It is easy to see that K µ f ( z ) = | f z | + | f z || f z |−| f z | whenever partial derivatives of f exist at z ∈ D and,in addition, J ( z, f ) = 0 . We will call the
Beltrami equation the differential equation of the form f z = µ ( z ) · f z , (1.2)where µ = µ ( z ) is a given function with | µ ( z ) | < a.a. The regular solution of (1.2) in thedomain D ⊂ C is a homeomorphism f : D → C of the class W , ( D ) such that J ( z, f ) = 0 for almost all z ∈ D. In the extended Euclidean space R n = R n ∪ {∞} , we use the so-called chordal metric h defined by the equalities h ( x, y ) = | x − y | q | x | q | y | , x = ∞ 6 = y , h ( x, ∞ ) = 1 q | x | , (1.3) N COMPACT CLASSES... E ⊂ R n , we set h ( E ) := sup x,y ∈ E h ( x, y ) . (1.4)The quantity h ( E ) in (1.4) is called the chordal diameter of the set E. As usual, the family F of mappings f : D → C is called normal, if from each sequence f n ∈ F , n = 1 , , . . . , onecan choose subsequence f n k , k = 1 , , . . . , converging to some mapping f : D → C locallyuniformly with respect to the metric h. If, in addition, f ∈ F , the family F is called compact .Let K be a compact set in C , M (Ω) be a function of the open set Ω , and Φ : R + → R + be a non-decreasing function. Denote by F M Φ ( K ) the class of all regular solutions f : C → C of the equation (1.2) with complex coefficients µ equal to zero outside K such that f ( z ) = z + o (1) при z → ∞ , (1.5)wherein Z Ω Φ( K µ ( z )) · dm ( z )(1 + | z | ) M (Ω) (1.6)for each open set Ω ⊂ C . The following statement is true.
Theorem 1.1.
Let
Φ : R + → R + be an increasing convex function that satisfies thecondition ∞ Z δ dττ Φ − ( τ ) = ∞ (1.7) for some δ > Φ(0) . Suppose, in addition, that the function M is bounded. Then the family F M Φ ( K ) is compact in C . Let us now turn to the problem of the compactness of the classes of solutions of theDirichlet problem for Beltrami equation. Consider the following Dirichlet problem: f z = µ ( z ) · f z , (1.8) lim ζ → z Re f ( ζ ) = ϕ ( z ) ∀ z ∈ ∂D , (1.9)where ϕ : ∂D → R is a predefined continuous function. In what follows, we assume that D is some simply connected Jordan domain in C . 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 theimage of any open set U ⊂ D is an open set in R n . The solution of the problem (1.8)–(1.9)will be called regular if one of two is fulfilled: either f ( z ) = const in D, or f is an opendiscrete mapping of class W , ( D ) , such that J ( z, f ) = 0 for almost all z ∈ D. Let us fix the point z ∈ D and the function ϕ. Let F M ϕ, Φ ,z ( D ) denotes the class of allregular solutions f : D → C of the Dirichlet problem (1.8)–(1.9) that satisfies the condition Im f ( z ) = 0 and, in addition, Z Ω Φ( K µ ( z )) · dm ( z )(1 + | z | ) M (Ω) (1.10) N COMPACT CLASSES... Ω ⊂ D. The following statement generalizes [Dyb , Theorem 2] to the caseof simply connected Jordan domains. Theorem 1.2.
Let D be some simply connected Jordan domain in C , Φ : R + → R + isan increasing convex function such that the condition (1.7) holds for some δ > Φ(0) . Assumethat the function M is bounded, and the function ϕ in (1.9) is continuous. Then the family F M ϕ, Φ ,z ( D ) is compact in D. Further, the manuscript is structured as follows. In the following sections, auxiliary resultsare presented that concern convergence of mappings, closure of classes, local and boundarybehavior. The proofs of the main results, Theorems 1.1 and 1.2, are located in the last twosections.
In what follows, M denotes the n -modulus of a family of paths, and the element dm ( x ) corresponds to a Lebesgue measure in R n , n > , see [Va]. Set S ( x , r ) = { x ∈ R n : | x − x | = r } , B ( x , r ) = { x ∈ R n : | x − x | < r } , B n := B (0 , , Ω n = m ( B n ) . In what follows, we set h ( A, B ) = inf x ∈ A,y ∈ B h ( x, y ) , where h is a chordal distance defined in (1.3). In addition, given domains A, B ⊂ R n we put dist ( A, B ) = inf x ∈ A,y ∈ B | x − y | . Given sets E and F and a domain D in R n = R n ∪ {∞} , we denote by Γ( E, F, D ) the familyof 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, unless otherwise stated, the boundary and the closure ofa set are understood in the sense of an extended Euclidean space R n . Let x ∈ D, x = ∞ ,A = A ( x , r , r ) = { x ∈ R n : r < | x − x | < r } . Let Q : R n → R n be a Lebesgue measurable function satisfying the condition Q ( x ) ≡ for x ∈ R n \ D. The mapping f : D → R n is called a ring Q -mapping at the point x ∈ D \ {∞} ,if the condition M ( f (Γ( S ( x , r ) , S ( x , r ) , A ∩ D ))) Z A ∩ D Q ( x ) · η n ( | x − x | ) dm ( x ) (2.1) N COMPACT CLASSES... < 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 > . (2.2)We prove the following important lemma, which in the case of the corresponding fixedfunctions Q = Q m ( x ) , m = 1 , , . . . , is proved earlier in [RSS, Theorem 4.4]. Lemma 2.1.
Let D be a domain in R n , n > , and let f k , k = 1 , , . . . be a sequence ofhomeomorphisms of the domain D in R n , which converges locally uniformly in D to somemapping f : D → R n by chordal metric h. Suppose, moreover, that
Φ : [0 , ∞ ] → [0 , ∞ ] isa strictly increasing convex function. Suppose that each map f k , k = 1 , , . . . satisfies therelation (2.1) at each point x ∈ D with some function Q = Q k ( x ) such that Z D Φ ( Q k ( x )) dm ( x )(1 + | x | ) n M < ∞ , k = 1 , , . . . . (2.3) If ∞ Z δ dττ [Φ − ( τ )] n − = ∞ , (2.4) for some δ > τ := Φ(0) , then f is either a homeomorphism f : D → R n , of a constant c ∈ R n . Proof.
We use Lemma 4.1 in [RSS] (see also the estimates of the integrals used in the proofof Theorem 1 in [Sev ]). As above, we put A = A ( x , r, R ) = { x ∈ R n : r < | x − x | < R } . To use [RSS, Lemma 4.1] it is necessary to establish the existence of sequences < r m
Rnmεn Z βm ( x M e Ω nRnm dττ [Φ − ( τ )] n − . (2.10)From the conditions (1.7) and (2.10) it follows that there exists a number < r m < R m suchthat R m Z r m drrq n − k x ( r ) > m . (2.11)Finally, from (2.6) and (2.11) it follows that there are infinitesimal positive sequences r m and R m satisfying the condition (2.5). Then by [RSS, Lema 4.1] the map f is either ahomeomorphism in R n , or a constant in R n , which was required to prove. ✷ According to [GM], a domain D ⊂ R n is called the quasiextremal distance domain , abbr. QED -domain , if there exists a number A > such that the inequality M (Γ( E, F, R n )) A · M (Γ( E, F, D )) (2.12) N COMPACT CLASSES... E and F in D. The next assertion was established by the secondco-author in [Sev , Lemmas 3.1, 3.2] for the case of a fixed function Q. However, it isfundamentally important for us to prove a similar assertion when the functions Q can vary,but are subject to condition (2.3). Lemma 2.2.
Let D and D ′ be domains in R n , n > , b ∈ D, b ′ ∈ D ′ , and let f k : D → D ′ , k = 1 , , . . . , be a family of homeomorphisms of the domain D onto D ′ with f k ( b ) = b ′ , k = 1 , , . . . . Suppose that each mapping f k , k = 1 , , . . . satisfies therelation (2.1) at any point x ∈ D and some function Q = Q k ( x ) > such that Z D Φ ( Q k ( x )) dm ( x )(1 + | x | ) n M < ∞ , k = 1 , , . . . . (2.13) Let D be locally connected on the boundary, and let D ′ be a QED -domain containing atleast one finite boundary point. If ∞ Z δ dττ [Φ − ( τ )] n − = ∞ (2.14) for some δ > τ := Φ(0) , then each f k , k = 1 , , . . . , has a continuous extension f k : D → D ′ and, besides that, the family of all extended mappings f k , k = 1 , , . . . , is equicontinuous in D. Proof.
The equicontinuity of { f k } ∞ k =1 at inner points of D follows by [RS, Theorem 4.1].It is only necessary to prove the possibility of continuous boundary extension of each f k ,k = 1 , , . . . , , as well as the equicontinuity of the family of extended mappings f k at ∂D. We may assume that all functions Q k ( x ) , k = 1 , , . . . , are extended by an identical unitoutside the domain D. Let x ∈ ∂D, and let < ε < δ ( x ) := sup x ∈ D | x − x | , < ε < ε . Consider the function I k ( ε, ε ) = ε R ε ψ k ( t ) dt, де ψ k ( t ) = ( / [ tq n − k x ( t )] , t ∈ ( ε, ε ) , , t / ∈ ( ε, ε ) , (2.15)where q k x is defined by (2.7). Not that I k ( ε, ε ) < ∞ for any ε ∈ (0 , ε ) . Arguing similarlyto the proof of the relation (2.10), we may show that b Z a drrq n − k x ( r ) > n Φ(0) bnan Z β ( x M e Ω nbn dττ [Φ − ( τ )] n − , (2.16)for every < b < ε and sufficiently small < a < b, where β ( x ) = (1 + ( b + | x | ) ) n . By (2.16) and (2.14), I k ( ε, ε ) > for ε ∈ (0 , ε ′ ) and some < ε ′ < ε . Using direct
N COMPACT CLASSES... Z ε< | x − x | <ε Q k ( x ) · ψ n ( | x − x | ) dm ( x ) = ω n − · I k ( ε, ε ) . Now, by [Sev , Lemma 3.1] there exists e ε = e ε ( x ) ∈ (0 , ε ) such that σ ∈ (0 , e ε ) h ( f k ( E )) α n δ exp (cid:8) − βI k ( σ, ε ) · ( α ( σ )) − / ( n − (cid:9) , k = 1 , , . . . , (2.17)for any continuum E ⊂ B ( x , σ ) ∩ D, where h ( f k ( E ) in the left part of (2.17) is definedin (1.4). Here α ( σ ) = ε R e ε ψ k ( t ) dt e ε R σ ψ k ( t ) dt n , (2.18) δ = · h ( b ′ , ∂D ′ ) , α n is some constant depending only on n, besides that, A is a constantfrom the definition of QED -domain D ′ (see (2.12)) and β = (cid:0) A (cid:1) n − . Now, by (2.16) and(2.17) we obtain that h ( f k ( E )) α n δ exp − β n Φ(0) εn σn Z β ε M e Ω nεn dττ [Φ − ( τ )] n − · ( α ( σ )) − / ( n − , k = 1 , , . . . , (2.19) α ( σ ) = n Φ(0) εn f ε n Z β f ε ,ε M e Ω nεn dττ [Φ − ( τ )] n − n Φ(0) f ε nσn Z β f ε M e Ω n f ε n dττ [Φ − ( τ )] n − − n . (2.20)One can show that the inequalities β ( ε ) M e Ω n ε n > , β ( e ε ,ε ) M e Ω n ε n > Φ(0) and β ( e ε ) M e Ω n e ε n > Φ(0) hold in (2.19) and (2.20). Then, due to the condition (2.14), the relation (2.19) implies theexistence of a non-negative function ∆ = ∆( σ ) such that h ( f k ( E )) ∆( σ ) → , σ → , k = 1 , , . . . . (2.21)Note that QED -domains have the so-called strongly accessible boundaries (see, e.g., [MRSY,Remark 13.10]). Further considerations are made similarly to the proof of Lemma 3.2in [Sev ]. Namely, the possibility of extension of f k to a continuous mapping f k : D → D ′ follows from [Sev , Theorem 2]. Let us to prove the equicontinuity of { f k } ∞ k =1 at ∂D. Supposethe contrary, namely, that the family of mappings { f k } ∞ k =1 is not equicontinuous at somepoint x ∈ ∂D. We can consider that x = ∞ . Then there is a number a > such that foreach m = 1 , , . . . there is x m ∈ D and f k m such that | x − x m | < /m and, simultaneously, h ( f k m ( x m ) , f k m ( x )) > a . (2.22) N COMPACT CLASSES... f k have a continuous extension on ∂D, we may consider that x m ∈ D, m = 1 , , . . . . Besides that, by continuity of f k m at x there exists a sequence x ′ m ∈ D such that h ( f k m ( x ′ m ) , f k m ( x )) a/ . Now, by (2.22) and by the triangle inequality we obtain that h ( f k m ( x m ) , f k m ( x ′ m )) > a/ . (2.23)Since D is locally connected at x , there is a sequence of open neighborhoods V m of x such that the sets W m := D ∩ V m are connected and W m ⊂ B ( x , − m ) . Turning to thesubsequence, if necessary, we may assume that x m , x ′ m ∈ W m . Put < ε < sup x ∈ D | x − x | . We may consider that B ( x , − m ) ⊂ B ( x , ε ) for any m = 1 , , . . . . Since W m is openand connected, it is also path connected (see, e.g., [MRSY, Proposition 13.1]). Thus, thepoints x m and x ′ m may be joined by a path γ m in W m . Set E m := | γ m | , where, as usual, | γ | = { x ∈ R n : ∃ t ∈ [ a, b ] : γ ( t ) = x } denotes the locus of γ. Now, by (2.21) h ( f k m ( E m )) ∆(2 − m ) → , m → ∞ . The last relation contradicts (2.23)), which completes the proof. ✷ I. First of all, we prove that the family F M Φ ( K ) is equicontinuous. Fix f ∈ F M Φ ( K ) , arbitrarycompact set C ⊂ C and put e f = f (1 /z ) . Since f ( z ) = z + o (1) as z → ∞ , we have that lim z →∞ f ( z ) = ∞ . Now e f (0) = 0 . Since f ( z ) = z + o (1) as z → ∞ , there is a neighborhood U of the origin and a function ε : U → C such that f (1 /z ) = 1 /z + ε (1 /z ) , where z ∈ U and ε (1 /z ) → as z → . Thus, e f (∆ z ) − e f (0)∆ z = 1∆ z · / (∆ z ) + ε (1 / ∆ z ) = 11 + (∆ z ) · ε (1 / ∆ z ) → as ∆ z → . This proves that there exists e f ′ (0) , while, e f ′ (0) = 1 . Since µ ( z ) vanishes outside K, the mapping f is conformal in some neighborhood V := C \ B (0 , /r ) of infinity point,and the number /r depends only on K, and K ⊂ B (0 , /r ) . Without loss of generality, wemay assume that the compact set C also satisfies the condition C ⊂ B (0 , /r ) . In this case,the mapping e f = f (1 /z ) is conformal in B (0 , r ) . In addition, the mapping F ( z ) := r · e f ( r z ) is a homeomorphism of the unit disk in C such that F (0) = 0 and F ′ (1) = 1 . By Koebe’stheorem about 1/4 (see, e.g., [CG, Theorem 1.3], cf. [GR , Theorem, Section 1.3, Ch. 1]) F ( D ) ⊃ B (0 , / . Then e f ( B (0 , r )) ⊃ B (0 , r / . (3.1)By (3.1) (1 /f )( C \ B (0 , /r )) ⊃ B (0 , r / . (3.2) N COMPACT CLASSES... f ( C \ B (0 , /r )) ⊃ C \ B (0 , /r ) . (3.3)Indeed, let y ∈ C \ B (0 , /r ) . Now, y ∈ B (0 , r / . By (3.2), y = (1 /f )( x ) , x ∈ C \ B (0 , /r ) . Thus, y = f ( x ) , x ∈ C \ B (0 , /r ) , which proves (3.3).Since f is a homeomorphism in C \ B (0 , /r ) , by (3.3) we obtain that f ( B (0 , /r )) ⊂ C \ B (0 , /r ) . Set ∆ := h ( C \ B (0 , /r )) , where h ( C \ B (0 , /r )) is a chordal diameterof C \ B (0 , /r ) . By [LSS, Theorem 3.1] f is a ring Q -homeomorphism in C for Q = K µ ( z ) , where µ is defined in (1.2), and K µ is defined in (1.1). In this case, F M Φ ( K ) is equicontinuousin B (0 , /r ) by [RS, Theorem 4.1]. Let f n ∈ F M Φ ( K ) , n = 1 , , . . . . By the Arzela-Ascolitheorem (see, e.g., [Va, Theorem 20.4]) there is a subsequence f n k ( z ) of f n , k = 1 , , . . . , anda continuous mapping f : B (0 , /r ) → C such that f n k converges to f in B (0 , /r ) locallyuniformly as k → ∞ . In particular, since C belongs to B (0 , /r ) , a sequence f n k convergesto f uniformly in C. Since the compact set C was chosen arbitrarily, we proved that thefamily of mappings f n k converges to the mapping f locally uniformly. II.
To complete the proof of Theorem 1.1, it remains to establish that f ∈ F M Φ ( K ) . First ofall, we prove that the limit mapping f satisfies the condition f ( z ) = z + o (1) as z → ∞ . Notethat the family mappings F n k ( z ) := r · f nk ( r z ) is compact in the unit disk (see, e.g., [CG,Theorem 1.10], cf. [GR , Theorem 1.2 Ch. I]). Without loss of generality, we may considerthat F n k converges locally uniformly in D . Now F ( z ) = r · f ( r z ) belongs to the class S, consisting of conformal mappings F of the unit disk that satisfy the conditions F (0) = 0 ,F ′ (0) = 1 . Then the expansion in the Taylor series of the function F near zero has the form F ( z ) = z + o ( z ) , z → . It follows from F ( z ) = r · f ( r z ) that f ( t ) = 1 r r t + o ( r t ) ! , t ∈ W , (3.4)at some neighborhood W of the infinity. From the relation F ( z ) = r · f ( r z ) it follows that f is also a homeomorphism in some neighborhood of infinity. In addition, from (3.4) it followsthat f ( t ) − t = 1 r r t + o ( r t ) ! − t = ε ( r t )1 + ε ( r t ) → as t → ∞ , where ε is some function such that ε ( r t ) → as t → ∞ . Thus, f ( z ) = z + o (1) as z → ∞ . Now we show that f is a homeomorphism of the complex plane. We put µ k := µ g k . By [LSS, Theorem 3.1] g k is a ring Q -homeomorphism at each point z ∈ C , where Q = K µ k ( z ) . By the definition of the class of mappings F M Φ ( K ) , there exists M > such that M ( C ) M , (3.5) N COMPACT CLASSES... M is a function from (1.6). By Lemma 2.1 the following alternative holds: either f is a homeomorphism with values in C , or f is a constant with values in C . As shown abovein step I , the mapping f is homeomorphism in some neighborhood of infinity. Therefore, f is a homeomorphism of the whole complex plane, which takes only finite complex values.By [L, Lemma 1 and Theorem 1] f ∈ W , loc ( C ) , beside that, f is a regular solutions ofthe equation (1.2) for some function µ : C → D , and for the corresponding function K µ therelation (1.6) is fulfilled. By the Gehring-Lehto theorem, the map f is almost everywheredifferentiable (see [LV, Theorem III.3.1]). Therefore, µ ( z ) = 0 for almost all z ∈ C \ K (seeTheorem 16.1 in [RSS]). Thus, f ∈ F M Φ ( K ) . ✷ Compared to Theorem 1.1, Theorem 1.2 is a much more complicated result, and for itsproof we need a number of auxiliary statements. The most important of them relate tomappings whose inverse satisfy an inequality of the form (2.1). However, for greater gen-erality, we will establish the corresponding assertions even in the case when the mappingsunder consideration have no inverse at all, although, at the same time, the correspondinginequality, containing the distortion of the modulus of families of paths under the mappings,is quite meaningful. Note that results similar to those presented below were obtained insome individual situations by us in [SevSkv ] and [SSD].Let y ∈ R n , < r < r < ∞ , and let A = A ( y , r , r ) = { y ∈ R n : r < | y − y | < r } . (4.1)Let, as above, M (Γ) denotes a conformal modulus of families of paths Γ in R n (see, e.g.,[Va, Ch. 6]). Let f : D → R n , n > , be some mapping, and let Q : R n → [0 , ∞ ] be aLebesgue measurable function satisfying the condition Q ( x ) ≡ for x ∈ R n \ f ( D ) . Let A = A ( y , r , r ) , and let Γ f ( y , r , r ) denotes the family of all paths γ : [ a, b ] → D suchthat f ( γ ) ∈ Γ( S ( y , r ) , S ( y , r ) , A ( y , r , r )) , i.e., f ( γ ( a )) ∈ S ( y , r ) , f ( γ ( b )) ∈ S ( y , r ) , and f ( γ ( t )) ∈ A ( y , r , r ) for any a < t < b. We say that f satisfies the inverse Poletskyinequality at y ∈ f ( D ) if the relation M (Γ f ( y , r , r )) Z A Q ( y ) · η n ( | y − y | ) dm ( y ) (4.2)holds for an arbitrary Lebesgue measurable function η : ( r , r ) → [0 , ∞ ] such that r Z r η ( r ) dr > . (4.3) N COMPACT CLASSES... g = f − , provided that it exists. Indeed, remark that g (Γ( S ( y , r ) , S ( y , r ) , f ( D ))) = Γ f ( y , r , r ) . (4.4)In fact, if γ ∈ g (Γ( S ( y , r ) , S ( y , r ) , f ( D ))) , then γ : [ a, b ] → R n , where γ = g ◦ α, α :[ a, b ] → R n and α ( a ) ∈ S ( y , r ) , α ( b ) ∈ S ( y , r ) , α ( t ) ∈ f ( D ) for a t b. Now, γ ( t ) ∈ D for a t b and f ( γ ) = α ∈ Γ( S ( y , r ) , S ( y , r ) , f ( D )) , i.e., γ ∈ Γ f ( y , r , r ) . Thus, g (Γ( S ( y , r ) , S ( y , r ) , f ( D ))) ⊂ Γ f ( y , r , r ) . The inverse inclusion is proved similarly.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]). The boundary of the domain D is called weaklyflat at the point x , if for every number P > and for every neighborhood U of this pointthere is a neighborhood V of point x such that M (Γ( E, F, D )) > P for arbitrary continua E and F, satisfying conditions F ∩ ∂U = ∅ = F ∩ ∂V. The boundary of domain D is called weakly flat if it is such at each point of its boundary.Given δ > , M > , domains D, D ′ ⊂ R n , n > , and a continuum A ⊂ D ′ denoteby S δ,A,M ( D, D ′ ) the family of all open discrete and closed mappings f of the domain D onto D ′ , for which the condition (4.2) is fulfilled for each y ∈ D ′ with some function Q = Q f ∈ L ( D ′ ) such that R D ′ Q ( y ) dm ( y ) M and, in addition, h ( f − ( A ) , ∂D ) > δ. An analogue of the following statement is proved earlier for the case of a fixed function Q (see [SSD, Theorem 1.2]). Note that the proof given in [SSD] practically does not differ fromthe one given below, however, for the sake of completeness of presentation, we give it in fullin this text. Theorem 4.1.
Let D and D ′ be domains in R n , n > . Assume that D has aweakly flat boundary, and D ′ is locally connected at the boundary. If Q ∈ L ( D ′ ) , theneach f ∈ S δ,A,M ( D, D ′ ) has a continuous boundary extension f : D → D ′ , such that f ( D ) = D ′ , in addition, the family S δ,A,M ( D, D ′ ) of all extended mappings f : D → D ′ isequicontinuous in D. Remark 4.1.
In Theorem 4.1, the equicontinuity must be understood with respect to thechordal metric, that is, for every ε > there is δ = δ ( ε, x ) > such that h ( f ( x ) , f ( x )) < ε for all x ∈ D, h ( x, x ) < δ, and all f ∈ S δ,A,M ( D, D ′ ) . Proof of Theorem 4.1.
Let f ∈ S δ,A,M ( D, D ′ ) . By [SSD, Theorem 3.1] a mapping f has a continuous extension f : D → D ′ , bedside that, f ( D ) = D ′ . The equicontinuity of S δ,A,M ( D, D ′ ) in D follows from [SSD, Theorem 1.1]. It remains to establish the equicon-tinuity of the family S δ,A,M ( D, D ′ ) on ∂D. We carry out the proof by contradiction. Suppose that the above conclusion does not hold.Then there is a point x ∈ ∂D, a positive number ε > , a sequence x m ∈ D, converging toa point x and a map f m ∈ S δ,A,M ( D, D ) such that h ( f m ( x m ) , f m ( x )) > ε , m = 1 , , . . . . (4.5) N COMPACT CLASSES... f m := f m | D . Since the map f m has a continuous extension to ∂D ′ , we may assumethat x m ∈ D. and, therefore, f m ( x m ) = f m ( x m ) . In addition, there is a sequence x ′ m ∈ D with z ′ m → z as m → ∞ , such that x ′ m → x as m → ∞ . Since R n is a compact metricspace, we may consider that f m ( x m ) and f m ( x ) converge as m → ∞ . Let f m ( x m ) → x and f m ( x ) → x as m → ∞ . By the continuity of the metric in (4.5), x = x . Withoutloss of generality, we may consider that x = ∞ . Since f m are closed, they are boundarypreserving (see [Vu, Theorem 3.3]). Thus, x ∈ ∂D. Let e x and e x be diferent points of A, none of which is the same as x . By [SevSkv , Lemma 2.1], cf. [SevSkv , Lemma 2.1], wecan join pairs of points e x , x and e x , x using the paths γ : [0 , → D and γ : [0 , → D so that | γ | ∩ | γ | = ∅ , γ ( t ) , γ ( t ) ∈ D for t ∈ (0 , , γ (0) = e x , γ (1) = x , γ (0) = e x and γ (1) = x . Since D ′ is locally connected on ∂D ′ , there are neighborhoods U and U of points x and x , respectively, whose closures do not intersect, and, moreover, sets W i := D ′ ∩ U i are path-connected. Without loss of generality, we may assume that U ⊂ B ( x , δ ) i B ( x , δ ) ∩ | γ | = ∅ = U ∩ | γ | , B ( x , δ ) ∩ U = ∅ . (4.6)We may also consider that f m ( x m ) ∈ W and f m ( x ′ m ) ∈ W for all m ∈ N . Let a and a are arbitrary points belonging to | γ | ∩ W and | γ | ∩ W . Let < t , t < be suchthat γ ( t ) = a and γ ( t ) = a . Join the points a and f m ( x m ) by means of the path α m : [ t , → W such that α m ( t ) = a and α m (1) = f m ( x m ) . Similarly, let us join a and f m ( x ′ m ) by means of the path β m : [ t , → W , β m ( t ) = a and β m (1) = f m ( x ′ m ) (seeFigure 1). Put D D Ax x a f x m m ( ) a C m C m x x f m m x x m x m D m f x m m ( ) f A m ( ) -1 b m b m D m * Figure 1: To the proof of Theorem 4.1 C m ( t ) = ( γ ( t ) , t ∈ [0 , t ] ,α m ( t ) , t ∈ [ t , , C m ( t ) = ( γ ( t ) , t ∈ [0 , t ] ,β m ( t ) , t ∈ [ t , . Recall that a path α : [ a, b ] → D is called a whole f -lifting of a path β : [ a, b ] → R n startingat x ∈ D, if f ( α ( t )) = β ( t ) for any t ∈ [ a, b ] and, in addition, α ( a ) = x . Let D m and D m be whole f m -liftings of paths C m and C m starting at points x m and x ′ m , respectively (such N COMPACT CLASSES... h ( f − m ( A ) , ∂D ) > δ > participating in the definition of class S δ,A,M ( D, D ′ ) , the end points of paths D m and D m , which we denote by b m and b m , distant from the boundary D by at least δ. As usual, wedenote by | C m | and | C m | the loci of the paths C m and C m , respectively. Set l = min { dist ( | γ | , | γ | ) , dist ( | γ | , U \ {∞} ) } . Consider the coverage A := S x ∈| γ | B ( x, l / by balls of | γ | . Since | γ | is a compact set,you may choose a finite number of indexes N < ∞ and the corresponding points z , . . . , z N ∈ | γ | such that | γ | ⊂ B := N S i =1 B ( z i , l / . In this case, | C m | ⊂ U ∪ | γ | ⊂ B ( x , δ ) ∪ N [ i =1 B ( z i , l / . Let Γ m be a family of paths joining the sets | C m | and | C m | in D ′ . Now we obtain that Γ m = N [ i =0 Γ mi , (4.7)where Γ mi is a family of all paths γ : [0 , → D ′ such that γ (0) ∈ B ( z i , l / ∩ | C m | and γ (1) ∈ | C m | for i N . Similarly, let Γ m be a family of all paths γ : [0 , → D ′ suchthat γ (0) ∈ B ( x , δ ) ∩ | C m | and γ (1) ∈ | C m | . By (4.6) there is some σ > δ > such that B ( x , σ ) ∩ | γ | = ∅ = U ∩ | γ | , B ( x , σ ) ∩ U = ∅ . By [Ku, Theorem 1.I.5.46], we may show that Γ m > Γ( S ( x , δ ) , S ( x , σ ) , A ( x , δ , σ )) , Γ mi > Γ( S ( z i , l / , S ( z i , l / , A ( z i , l / , l / . (4.8)We may chose x ∗ ∈ D ′ , δ ∗ > and σ ∗ > such that A ( x ∗ , δ ∗ , σ ∗ ) ⊂ A ( x , δ , σ ) . Thus, Γ( S ( x , δ ) , S ( x , σ ) , A ( x , δ , σ )) >> Γ( S ( x ∗ , δ ∗ ) , S ( x ∗ , σ ∗ ) , A ( x ∗ , δ ∗ , σ ∗ )) . (4.9)Set η ( t ) = ( /l , t ∈ [ l / , l / , , t [ l / , l / ,η ( t ) = ( / ( σ ∗ − δ ∗ ) , t ∈ [ δ ∗ , σ ∗ ] , , t [ δ ∗ , σ ∗ ] . Let Γ ∗ m := Γ( | D m | , | D m | , D ) . Observe that f m (Γ ∗ m ) ⊂ Γ m . Now, by (4.7), (4.8) and (4.9) weobtain that Γ ∗ m > N [ i =1 Γ f m ( z i , l / , l / ! ∪ Γ f m ( x ∗ , δ ∗ , σ ∗ ) . (4.10) N COMPACT CLASSES... f m satisfy the relation (4.2) for Q = Q m in D ′ , we obtain by (4.10) that M (Γ ∗ m ) (4 n N /l n + (1 / ( σ ∗ − δ ∗ )) n ) k Q m k c < ∞ . (4.11)We show that the relation (4.11) contradicts the condition of the weakly flatness of themapped domain. In fact, by construction h ( | D m | ) > h ( x m , b m ) > (1 / · h ( f − m ( A ) , ∂D ) > δ/ ,h ( | D m | ) > h ( x ′ m , b m ) > (1 / · h ( f − m ( A ) , ∂D ) > δ/ (4.12)for all m > f M and some f M ∈ N . We put U := B h ( x , r ) = { y ∈ R n : h ( y, x ) < r } , where < r < δ/ and the number δ refers to the ratio (4.12). Note that | D m | ∩ U = ∅ = | D m | ∩ ( D \ U ) for every m ∈ N , because h ( | D m | ) > δ/ and x m ∈ | D m | , x m → x as m → ∞ . Similarly, | D m | ∩ U = ∅ = | D m | ∩ ( D \ U ) . Since | D m | and | D m | are continua, | D m | ∩ ∂U = ∅ , | D m | ∩ ∂U = ∅ , (4.13)see, e.g., [Ku, Theorem 1.I.5.46]. Let c be the number from (4.11). Since ∂D is weakly flat,for P := c > there is a neighborhood V ⊂ U of x such that M (Γ( E, F, D )) > c (4.14)for any continua E, F ⊂ D such that E ∩ ∂U = ∅ = E ∩ ∂V and F ∩ ∂U = ∅ = F ∩ ∂V. Let us show that | D m | ∩ ∂V = ∅ , | D m | ∩ ∂V = ∅ . (4.15)for sufficiently large m ∈ N . Indeed, x m ∈ | D m | and x ′ m ∈ | D m | , where x m , x ′ m → x ∈ V as m → ∞ . In this case, | D m | ∩ V = ∅ = | D m | ∩ V for sufficiently large m ∈ N . Observe that h ( V ) h ( U ) r < δ/ . By (4.12) h ( | D m | ) > δ/ . Thus, | D m | ∩ ( D \ V ) = ∅ and, there-fore, | D m | ∩ ∂V = ∅ (see, e.g., [Ku, Theorem 1.I.5.46]). Similarly, h ( V ) h ( U ) r < δ/ . By (4.12) we obtain that h ( | D m | ) > δ/ . Thus, | D m |∩ ( D \ V ) = ∅ . By [Ku, Theorem 1.I.5.46]we obtain that | D m | ∩ ∂V = ∅ . Thus, the relation (4.15) is proved. Combining the rela-tions (4.13), (4.14) i (4.15), we obtain that M (Γ ∗ m ) = M (Γ( | D m | , | D m | , D )) > c. The lastrelation contradicts the inequality (4.11), which proves the theorem. ✷ The following lemma was also proved earlier in somewhat other situations, in particular,in the case of a fixed function Q (see, e.g., [SevSkv , Lemma 4.1], [SevSkv , Lemma 4.1]and [SSD, Lemma 6.1]). Lemma 4.1.
Let n > , and let D and D ′ be domains in R n such that D ′ is locallyconnected on ∂D ′ , D has a weakly flat boundary, and, moreover, no connected componentof ∂D does not degenerate into a point. Let A be a nondegenerate continuum in D ′ , andlet δ, M > . Assume that f m , m = 1 , , . . . , be a sequence of discrete, open and closedmappings of D onto D ′ , satisfying the relation (4.2) in D for some function Q = Q m , suchN COMPACT CLASSES... that f m ( A m ) = A for some continuum A m ⊂ D with h ( A m ) > δ > , m = 1 , , . . . . If R D ′ Q ( y ) dm ( y ) M for m = 1 , , . . . , then there is δ > such that h ( A m , ∂D ) > δ > ∀ m ∈ N . Proof.
Due to the compactness of the space R n the boundary of the domain D is notempty and is compact, so that the distance h ( A m , ∂D ) is well-defined.We carry out the proof by contradiction. Suppose that the conclusion of the lemma is nottrue. Then for each k ∈ N there is some number m = m k ∈ N such that h ( A m k , ∂D ) < /k. We may assume that m k increases on k = 1 , , . . . . Since A m k is a compact set, thereare x k ∈ A m k and y k ∈ ∂D, k = 1 , , . . . , such that h ( A m k , ∂D ) = h ( x k , y k ) < /k (seeFigure 2). Since ∂D is a compact set, we may consider that y k → y ∈ ∂D as k → ∞ . A k G DU k U k DD D z k k g w k x k y y k f m k (| k |) f m k f m k ( k )y k A m k Figure 2: To the proof of Lemma 4.1Then x k → y ∈ ∂D при k → ∞ . Let K be a connected component of ∂D, containing y . Obviously, K is a continuum in R n . Since ∂D is weakly flat, by [SSD, Theorem 3.1] f m k has a continuous extension f m k : D → D ′ . Moreover, f m k is uniformly continuous in D forany fixed k, because f m k is continuous on a compact set D. Now, for every ε > there is δ k = δ k ( ε ) < /k such that h ( f m k ( x ) , f m k ( x )) < ε (4.16) ∀ x, x ∈ D, h ( x, x ) < δ k , δ k < /k . Pick ε > such that ε < (1 / · h ( ∂D ′ , A ) . (4.17)Denote B h ( x , r ) = { x ∈ R n : h ( x, x ) < r } . Given k ∈ N , put B k := [ x ∈ K B h ( x , δ k ) , k ∈ N . Since B k is a neighborhood of the continuum K , by [HK, Lemma 2.2] there is a neighborhood U k of K such that U k ⊂ B k and U k ∩ D is connected. We may consider that U k is open, so N COMPACT CLASSES... U k ∩ D is path connected (see [MRSY, Proposition 13.1]). Let h ( K ) = m . Now, we canfind z , w ∈ K such that h ( K ) = h ( z , w ) = m . Thus, there are sequences y k ∈ U k ∩ D,z k ∈ U k ∩ D and w k ∈ U k ∩ D such that z k → z , y k → y and w k → w as k → ∞ . We mayconsider that h ( z k , w k ) > m / ∀ k ∈ N . (4.18)Since U k ∩ D is path connected, we can join the points z k , y k and w k using some path γ k ∈ U k ∩ D. As usual, we denote by | γ k | a locus of the path γ k in D. In this case, f m k ( | γ k | ) is a compact set in D ′ . If x ∈ | γ k | , there exists x ∈ K such that x ∈ B ( x , δ k ) . Put ω ∈ A ⊂ D. Since x ∈ | γ k | and, in addition, x is an inner point of D, we may use thenotation f m k ( x ) instead of f m k ( x ) . By (4.16) and (4.17), and by the triangle inequality, weobtain that h ( f m k ( x ) , ω ) > h ( ω, f m k ( x )) − h ( f m k ( x ) , f m k ( x )) >> h ( ∂D ′ , A ) − (1 / · h ( ∂D ′ , A ) = (1 / · h ( ∂D ′ , A ) > ε (4.19)for sufficiently large k ∈ N . Taking inf in the relation (4.19) over all x ∈ | γ k | and ω ∈ A, weobtain that h ( f m k ( | γ k | ) , A ) > ε, k = 1 , , . . . . Since h ( x, y ) | x − y | for any x, y ∈ R n , weobtain that dist ( f m k ( | γ k | ) , A ) > ε, ∀ k = 1 , , . . . . (4.20)We cover the continuum A with balls B ( x, ε/ , x ∈ A. Since A is a compact set, we mayconsider that A ⊂ N S i =1 B ( x i , ε/ , x i ∈ A, i = 1 , , . . . , N , N < ∞ . By the definition, N depends only on A, in particular, N does not depend on k. Set Γ k := Γ( A m k , | γ k | , D ) . (4.21)Let Γ ki := Γ f mk ( x i , ε/ , ε/ , in other words, Γ ki consists from all paths γ : [0 , → D suchthat f m k ( γ (0)) ∈ S ( x i , ε/ , f m k ( γ (1)) ∈ S ( x i , ε/ and γ ( t ) ∈ A ( x i , ε/ , ε/ for < t < . Let us show that Γ k > N [ i =1 Γ ki . (4.22)Indeed, let e γ ∈ Γ k , in other words, e γ : [0 , → D, e γ (0) ∈ A m k , e γ (1) ∈ | γ k | and e γ ( t ) ∈ D for t . Then γ ∗ := f m k ( e γ ) ∈ Γ( A, f m k ( | γ k | ) , D ′ ) . Since the balls B ( x i , ε/ , i N , form a covering of the compactum A, there is i ∈ N such that γ ∗ (0) ∈ B ( x i , ε/ and γ ∗ (1) ∈ f m k ( | γ k | ) . By (4.20), | γ ∗ | ∩ B ( x i , ε/ = ∅ = | γ ∗ | ∩ ( D ′ \ B ( x i , ε/ . Thus,by [Ku, Theorem 1.I.5.46] there is < t < such that γ ∗ ( t ) ∈ S ( x i , ε/ . We mayconsider that γ ∗ ( t ) B ( x i , ε/ for t > t . Set γ := γ ∗ | [ t , . By (4.20) it follows that | γ | ∩ B ( x i , ε/ = ∅ = | γ | ∩ ( D \ B ( x i , ε/ . Thus, by [Ku, Theorem 1.I.5.46] there is t < t < such that γ ∗ ( t ) ∈ S ( x i , ε/ . We may consider that γ ∗ ( t ) ∈ B ( x i , ε/ forany t < t . Putting γ := γ ∗ | [ t ,t ] , we observe that γ is a subpath of γ ∗ , which belongs to Γ( S ( x i , ε/ , S ( x i , ε/ , A ( x i , ε/ , ε/ . N COMPACT CLASSES... e γ has a subpath e γ := e γ | [ t ,t ] such that f m k ◦ e γ = γ , wherein γ ∈ Γ( S ( x i , ε/ , S ( x i , ε/ , A ( x i , ε/ , ε/ . Thus, the relation (4.22) is fulfilled. Set η ( t ) = ( /ε, t ∈ [ ε/ , ε/ , , t [ ε/ , ε/ . Observe that η satisfies (4.3) for r = ε/ and r = ε/ . Since a mapping f m k satisfies therelation (4.2) for Q = Q m k , k = 1 , , . . . , putting here y = x i , we obtain that M (Γ f mk ( x i , ε/ , ε/ (4 /ε ) n · k Q m k k (4 /ε ) n M < ∞ , (4.23)where c is some positive constant and k Q m k k is L -norm of the function Q m k in D ′ . By (4.22)and (4.23), taking into account the subadditivity of the modulus of families of paths, weobtain that M (Γ k ) n N ε n Z D ′ Q m k ( y ) dm ( y ) c < ∞ . (4.24)Arguing similarly to the proof of relations (4.12), and using the condition (4.18), we obtainthat M (Γ k ) → ∞ as k → ∞ , that contradicts to (4.24). The contradiction obtained aboveprove the lemma. ✷ Given domains
D, D ′ ⊂ R n , points a ∈ D, b ∈ D ′ and a number M > denote by S a,b,M ( D, D ′ ) the family of all open, discrete, and closed mappings f of the domain D onto D ′ , satisfying the condition (4.2) with some Q = Q f , k Q k L ( D ′ ) M , for every y ∈ f ( D ) , such that f ( a ) = b. The following statement is proved in [SSD, Theorem 7.1] in the case ofa fixed function Q. Theorem 4.2.
Suppose that a domain D has a weakly flat boundary, none of theconnected components of which is degenerate. If D ′ is locally connected at its boundary,then each map f ∈ S a,b,M ( D, D ′ ) has a continuous boundary extension f : D → D ′ such that f ( D ) = D ′ and, in addition, the family S a,b,M ( D, D ′ ) of all extended mappings f : D → D ′ is equicontinuous in D. Proof.
The equicontinuity of the family S a,b,M ( D, D ′ ) , the possibility of continuousboundary extension of f ∈ S a,b,M ( D, D ′ ) and the equality f ( D ) = D ′ follow from [SSD,Theorems 1.1 and 3.1]. It remains to establish the equicontinuity of the family of extendedmappings f : D → D ′ at the boundary points of the domain D. We will prove this statement from the contrary. Assume that the family S a,b,M ( D, D ′ ) is not equicontinuous at some point x ∈ ∂D. Then there are points x m ∈ D, m = 1 , , . . . , and mappings f m ∈ S a,b,M ( D, D ′ ) , such that x m → x as m → ∞ and h ( f m ( x m ) , f m ( x )) > ε , m = 1 , , . . . , (4.25) N COMPACT CLASSES... ε > . Choose y ∈ D ′ , y = b, and join it with the point b by some path α in D ′ . Put A := | α | . Let A m be a whole f m -lifting of α starting at a (it exists by [Vu, Lemma 3.7]).Note that h ( A m , ∂D ) > by the closeness of the mapping f m . The following cases are nowpossible: either h ( A m ) → as m → ∞ , or h ( A m k ) > δ > as k → ∞ for some increasingsequence of numbers m k , k = 1 , , . . . , and some δ > . In the first case, obviously, h ( A m , ∂D ) > δ > for some δ > . Then the family { f m } ∞ m =1 is equicontinuous at x by Theorem 4.1, which contradicts the condition (4.25).In the second case, if h ( A m k ) > δ > as k → ∞ , by Lemma 4.1 we also have that h ( A m k , ∂D ) > δ > for some δ > . Again, by Theorem 4.1, the family { f m k } ∞ k =1 isequicontinuous at x , and this contradicts the condition (4.25).The contradiction obtained above indicates that the assumption concerning the absenceof equicontinuity of the family S a,b,M ( D, D ′ ) on D is incorrect. Theorem proved. ✷ I . Let f m ∈ F M ϕ, Φ ,z ( D ) , m = 1 , , . . . . By Stoilow’s factorization theorem (see, e.g., [St,5(III).V]) a mapping f m ∈ F M ϕ, Φ ,z ( D ) has a representation f m = ϕ m ◦ g m , (5.1)where g m is some homeomorphism, and ϕ m is some analytic function. By Lemma 1 in [Sev ],the mapping g m belongs to the Sobolev class W , ( D ) and has a finite distortion. Moreover,by [A, (1).C, Ch. I] f mz = ϕ mz ( g m ( z )) g mz , f mz = ϕ mz ( g m ( z )) g mz (5.2)for almost all z ∈ D. Therefore, by the relation (5.2), J ( z, g m ) = 0 for almost all z ∈ D, inaddition, K µ fm ( z ) = K µ gm ( z ) . II.
We prove that ∂g m ( D ) contains at least two points. Suppose the contrary. Then either g m ( D ) = C , or g m ( D ) = C \{ a } , where a ∈ C . Consider first the case g m ( D ) = C . By Picard’stheorem ϕ m ( g m ( D )) is the whole plane, except perhaps one point ω ∈ C . On the other hand,for every m = 1 , , . . . the function u m ( z ) := Re f m ( z ) = Re ( ϕ m ( g m ( z ))) is continuous onthe compact set D under the condition (1.9) by the continuity of ϕ. Therefore, there exists C m > such that | Re f m ( z ) | C m , but this contradicts the fact that ϕ m ( g m ( D )) contains allpoints of the complex plane except, perhaps, one. Now g m ( D ) = C \ { a } , a ∈ C . The point a is either a removable or an essential singularity for ϕ m . If a is a removable singularity for ϕ m , then ϕ m extends to holomorphic mapping ϕ m : C → C . The set ϕ m ( C ) coincides withthe whole complex plane except, perhaps, one point, so that the mapping ϕ m takes all valuesin C except maybe two. The latter contradicts the condition | Re f m ( z ) | C m . Finally, if a is an essential singularity of ϕ m , then by Picard’s theorem ϕ m takes all possible values from N COMPACT CLASSES... C in any neighborhood of a, except perhaps one point. The latter contradicts the condition | Re f m ( z ) | C m . Therefore, the boundary of the domain g m ( D ) contains at least two points. Then, ac-cording to Riemann’s mapping theorem, we may transform the domain f g m ( D ) onto the unitdisk D using the conformal mapping ψ m . Let z ∈ D be a point from the condition of thetheorem. By using an auxiliary conformal mapping f ψ m ( z ) = z − ( ψ m ◦ g m )( z )1 − z ( ψ m ◦ g m )( z ) of the unit disk onto itself we may consider that ( ψ m ◦ g m )( z ) = 0 . Now, by (5.1) we obtainthat f m = ϕ m ◦ g m = ϕ m ◦ ψ − m ◦ ψ m ◦ g m = F m ◦ G m , m = 1 , , . . . , (5.3)where F m := ϕ m ◦ ψ − m , F m : D → C , and G m = ψ m ◦ g m . Obviously, a function F m isanalytic, and G m is a regular Sobolev homeomorphism in D. In particular, Im F m (0) = 0 forany m ∈ N . III.
We prove that the L -norms of the functions K µ Gm ( z ) are bounded from above bysome universal positive constant C > over all m = 1 , , . . . . Indeed, by the convexity ofthe function Φ in (1.10) and by [Bou, Proposition 5, I.4.3], the slope [Φ( t ) − Φ(0)] /t is anon-decreasing function. Hence there exist constants t > and C > such that Φ( t ) > C · t ∀ t ∈ [ t , ∞ ) . (5.4)Fix m ∈ N . By (1.10) and (5.4), we obtain that Z D K µ Gm ( z ) dm ( z ) = Z { z ∈ D : K µGm ( z )
We prove that each map G m , m = 1 , , . . . , has a continuous extension to ∂D, inaddition, the family of extended maps G m , m = 1 , , . . . , is equicontinuous in D. Indeed, asproved in item
III , K µ Gm ∈ L ( D ) . By [KPRS, Theorem 3] (see also [LSS, Theorem 3.1])each G m , m = 1 , , . . . , is a ring Q -homeomorphism in D for Q = K µ Gm ( z ) , where µ is N COMPACT CLASSES... K µ my be calculated by the formula (1.1). Thus, the desired conclusionis the statement of Lemma 2.2. V. Let us prove that the inverse homeomorphisms G − m , m = 1 , , . . . , have a continuousextension to ∂ D , and { G − m } ∞ m =1 is equicontinuous in D . Since by the item IV mappings G m ,m = 1 , , . . . , a ring K µ Gm ( z ) -homeomorphisms in D, the corresponding inverse mappings G − m satisfy (4.2) ( D corresponds to the unit disk D in (4.3), in addition, f G m , Q K µ Gm ( z ) , and f ( D ) corresponds D in (4.2)). Since G − m (0) = z for any m = 1 , , . . . , thecontinuous extension of G − m to ∂ D , and the equicontinuity of { G − m } ∞ m =1 on D follows fromTheorem 4.2. VI.
Since, as proved above the family { G m } ∞ m =1 is equicontinuous in D, according toArzela-Ascoli criterion there exists an increasing subsequence of numbers m k , k = 1 , , . . . , such that G m k converges locally uniformly in D to some continuous mapping G : D → C as k → ∞ (see, e.g., [Va, Theorem 20.4]). By Lemma 2.1, either G is a homeomorphism withvalues in R n , or a constant in R n . Let us prove that the second case is impossible. Let’suse the approach used in proof of the second part of Theorem 21.9 in [Va]. Suppose thecontrary: let G m k ( x ) → c = const as k → ∞ . Since G m k ( z ) = 0 for all k = 1 , , . . . , wehave that c = 0 . By item
III , the family of mappings G − m , m = 1 , , . . . , is equicontinuousin D . Then h ( z, G − m k (0)) = h ( G − m k ( G m k ( z )) , G − m k (0)) → as k → ∞ , which is impossible because z is an arbitrary point of the domain D. The obtainedcontradiction refutes the assumption made above.
VII.
According to V , the family of mappings { G − m } ∞ m =1 is equicontinuous in D. Bythe Arzela-Ascoli criterion (see, e.g., [Va, Theorem 20.4]) we may consider that G − m k ( y ) ,k = 1 , , . . . , converges to some mapping e F : D → D as k → ∞ locally uniformly in D. Let us to prove that e F = G − . as k → ∞ . For this purpose, we show that G ( D ) = D . Fix y ∈ D . Since G m k ( D ) = D for every k = 1 , , . . . , we obtain that G m k ( x k ) = y for some x k ∈ D. Since D is bounded, we may consider that x k → x ∈ D as k → ∞ . By the triangleinequality and the equicontinuity of { G m } ∞ m =1 in D, see IV , we obtain that | G ( x ) − y | = | G ( x ) − G m k ( x k ) | | G ( x ) − G m k ( x ) | + | G m k ( x ) − G m k ( x k ) | → as k → ∞ . Thus, G ( x ) = y. Observe that x ∈ D, because G is a homeomorphism. Since y ∈ D is arbitrary, the equality G ( D ) = D is proved. In this case, G − m k → G − locallyuniformly in D as k → ∞ (see, e.g., [RSS, Lemma 3.1]). Thus, e F ( y ) = G − ( y ) for every y ∈ D . Finally, since e F ( y ) = G − ( y ) for any y ∈ D and, in addition, e F has a continuous extensionon ∂ D , due to the uniqueness of the limit at the boundary points we obtain that e F ( y ) = G − ( y ) for y ∈ D . Therefore, we have proved that G − m k → G − uniformly in D as k → ∞ . N COMPACT CLASSES... VIII. By VII, for y = e iθ ∈ ∂D Re F m k ( e iθ ) = ϕ ( G − m k ( e iθ )) → ϕ ( G − ( e iθ )) (5.6)as k → ∞ uniformly on θ ∈ [0 , π ) . Since by the construction Im F m k (0) = 0 for any k = 1 , , . . . , by the Schwartz formula (see, e.g., [GK, §
8, Ch. III, part 3]) the analyticfunction F m k is uniquely restored by its real part, namely, F m k ( y ) = 12 πi Z S (0 , ϕ ( G − m k ( t )) t + yt − y · dtt . (5.7)Set F ( y ) := 12 πi Z S (0 , ϕ ( G − ( t )) t + yt − y · dtt . (5.8)Let K ⊂ D be an arbitrary compact set. By (5.7) and (5.8) we obtain that | F m k ( y ) − F ( y ) | π Z S (0 , | ϕ ( G − m k ( t )) − ϕ ( G − ( t )) | (cid:12)(cid:12)(cid:12)(cid:12) t + yt − y (cid:12)(cid:12)(cid:12)(cid:12) | dt | . (5.9)Since K is compact, there is < R = R ( K ) < such that K ⊂ B (0 , R ) . By the triangleinequality | t + y | R and | t − y | > | y | − | t | > − R for y ∈ K and any t ∈ S . Then (cid:12)(cid:12)(cid:12)(cid:12) t + yt − y (cid:12)(cid:12)(cid:12)(cid:12) R − R := M = M ( K ) . (5.10)Put ε > . By (5.6), for a number ε ′ := εM there is N = N ( ε, K ) ∈ N such that | ϕ ( G − m k ( t )) − ϕ ( G − ( t )) | < ε ′ for all k > N ( ε ) . Now, by (5.9) | F m k ( y ) − F ( y ) | < ε ∀ k > N . (5.11)It follows from (5.11) that the sequence F m k converhes to F as k → ∞ locally uniformly inthe unit disk. In particular, we obtain that Im F (0) = 0 . Note that F is analytic function in D (see remarks made at the end of item 8.III in [GK]), and Re F ( re iψ ) = 12 π π Z ϕ ( G − ( θ )) 1 − r − r cos( θ − ψ ) + r dθ for z = re iψ . By [GK, Theorem 2.10.III.3] lim ζ → z Re F ( ζ ) = ϕ ( G − ( z )) ∀ z ∈ ∂ D . (5.12)Observe that F either is a constant or open and discrete (see, e.g., [St, Ch. V, I.6 and II.5]).Thus, f m k = F m k ◦ G m k converges f = F ◦ G locally uniformly as k → ∞ , where f = F ◦ G either is a constant or open and discrete. Moreover, by (5.12) Re f ( z ) = Re F ( G ( z )) = ϕ ( G − ( G ( z ))) = ϕ ( z ) . N COMPACT CLASSES... IX.
Since by VI G is a homeomorphism, by [L, Lemma 1 and Theorem 1] G is a regularsolution of the equation (1.8) for some function µ : C → D . Since the set of points of thefunction F, where its Jacobian is zero, can consist only of isolated points (see [St, Ch. V,5.II and 6.II]), f is regular whenever F const. Note that the relation (1.6) holds for thecorresponding function K µ = K µ f . Therefore, f ∈ F M ϕ, Φ ,z ( D ) . ✷ References [A]
Ahlfors, L.V.:
Lectures on Quasiconformal Mappings. - Van Nostrand, Toronto,1966.[Bou]
Bourbaki, N.:
Functions of a real variable. - Springer, Berlin, 2004.[CG]
Carleson, L., T.W. Gamelin:
Complex dynamics. - Universitext: Tracts in Math-ematics, Springer-Verlag, New York etc., 1993.[Dyb ] Dybov, Yu.P.:
Compactness of classes of solutions of the Dirichlet problem for theBeltrami equations // Proc. Inst. Appl. Math. and Mech. of NAS of Ukraine 19, 2009,81-89 (in Russian).[Dyb ] Dybov, Yu.P.:
On regular solutions of the Dirichlet problem for the Beltramiequations // Complex Variables and Elliptic Equations 55:12, 2010, 1099–1116.[GM]
Gehring, F.W. and O. Martio:
Quasiextremal distance domains and extensionof quasiconformal mappings. - J. d’Anal. Math. 24, 1985, 181-206.[GR ] Gutlyanskii, V.Ya. and V.I. Ryazanov:
Quasiconformal mappings with integralconstraints on M. A. Lavrent’ev’s characteristic. - Siberian Mathematical Journal 31,1990, 202–215.[GR ] Gutlyanskii, V.Ya. and V.I. Ryazanov:
Geometric theory of functions and map-pings. - Naukova Dumka, Kiev, 2011 (in Russian).[GK]
Hurwitz, A. and R. Courant:
The Function Theory. - Nauka, Moscow, 1968 (inRussian).[HK]
Herron, J. and P. Koskela.
Quasiextremal distance domains and conformal map-pings onto circle domains. - Compl. Var. Theor. Appl. 15, 1990, 167-179.[Ku]
Kuratowski, K.:
Topology, v. 2. – Academic Press, New York–London, 1968.[LV]
Lehto, O., and K. Virtanen:
Quasiconformal Mappings in the Plane. - Springer,New York etc., 1973.
N COMPACT CLASSES...
Kovtonyuk, D.A., I.V. Petkov, V.I. Ryazanov, R.R. Salimov:
The bound-ary behavior and the Dirichlet problem for the Beltrami equations. - St. PetersburgMath. J. 25:4, 2014, 587-603.[LGR]
Lomako, T., V. Gutlyanskii and V. Ryazanov.:
To the theory of variationalmethod for Beltrami equations. - Journal of Mathematical Sciences 182:1, 2012, 37-54.[L]
Lomako, T.:
On the theory of convergence and compactness for Beltrami equations.- Ukrainian Mathematical Journal 63:3, 2011, 393–402.[LSS]
Lomako, T., R. Salimov R. and E. Sevost’yanov:
On equicontinuity of solu-tions to the Beltrami equations. - Ann. Univ. Bucharest (math. series) V. LIX:2, 2010,261-271.[MRSY]
Martio, O., V. Ryazanov, U. Srebro, and E. Yakubov:
Moduli in modernmapping theory. - Springer Science + Business Media, LLC, New York, 2009.[RS]
Ryazanov, V., E. Sevost’yanov:
Equicontinuity of mappings quasiconformal inthe mean. - Ann. Acad. Sci. Fenn. 36, 2011, 231-244.[RSS]
Ryazanov, V., R. Salimov and E. Sevost’yanov:
On Convergence Analysis ofSpace Homeomorphisms. - Siberian Advances in Mathematics 23:4, 2013, 263-293.[RSY]
Ryazanov, V., U. Srebro, and E. Yakubov:
Integral conditions in the theoryof the Beltrami equations. - Complex Variables and Elliptic Equations 57:12, 2012,1247-1270.[Sev ] Sevost’yanov, E.A.:
On spatial mappings with integral restrictions on the charac-teristic. - St. Petersburg Math. J. 24:1, 2013, 99-115.[Sev ] Sevost’yanov, E.A.:
Analog of the Montel Theorem for Mappings of the SobolevClass with Finite Distortion. - Ukrainian Math. J. 67:6, 2015, 938–947.[Sev ] Sevost’yanov, E.A.:
Equicontinuity of homeomorphisms with unbounded charac-teristic. - Siberian Advances in Mathematics 23:2, 2013, 106-122.[Sev ] Sevost’yanov, E.A.:
On the boundary behavior of open discrete mappings withunbounded characteristic. - Ukrainian Math. J. 64:6, 2012, 979-984.[SevSkv ] Sevost’yanov, E.A., S.A. Skvortsov:
On the local behavior of a class ofinverse mappings. - J. Math. Sci. 241:1, 2019, 77-89.[SevSkv ] Sevost’yanov, E.A., S.A. Skvortsov:
On mappings whose inverse satisfy thePoletsky inequality. - Ann. Acad. Scie. Fenn. Math. 45, 2020, 259-277.
N COMPACT CLASSES...
Sevost’yanov, E.A., S.O. Skvortsov and O.P. Dovhopiatyi:
On non-homeomorphic mappings with inverse Poletskii inequality. - Ukr. Mat. Visnyk, 2020(accepted for print).[St]
Stoilow, S.:
Principes Topologiques de la Th´eorie des Fonctions Analytiques. -Gauthier-Villars, Paris, 1956.[Va]
V ¨ais ¨al ¨a, J.:
Lectures on n -dimensional quasiconformal mappings. - Lecture Notes inMath. 229, Springer-Verlag, Berlin etc., 1971.[Vu] Vuorinen, M.:
Exceptional sets and boundary behavior of quasiregular mappings in n -space. - Ann. Acad. Sci. Fenn. Ser. A 1. Math. Dissertationes 11, 1976, 1-44. Evgeny Sevost’yanov1.
Zhytomyr Ivan Franko State University,40 Bol’shaya Berdichevskaya Str., 10 008 Zhytomyr, UKRAINE Institute of Applied Mathematics and Mechanicsof NAS of Ukraine,1 Dobrovol’skogo Str., 84 100 Slavyansk, [email protected]