NORMAL FAMILIES OF RING FMO

NORMAL FAMILIES OF RING FMO

-QUASIREGULAR MAPPINGS

VICTORIA STANCIU

We generalize the normality criterium obtained in [6] for BMO_loc-quasiregular mappings between Riemann surfaces to the class of ring FMO_loc-quasiregular mappings.

AMS 2000 Subject Classification: 30C62.

Key words: FMO_loc, qc, qr, ring FMO_loc-qr mapping, Riemann surface.

1. INTRODUCTION

LetG be a domain in Cand f :G→Can ACL (absolutely continuous on lines in G, cf. [2], p. 127) sense-preserving open mapping. Then f has partial derivatives

∂f =fz = 1

2(fx−ify) and ∂f =fz= 1

2(fx+ ify)

a.e. inG. Sincef is open, by a result of Gehring and Lehto [2], p. 128, Theo- rem 3.1, it is a.e. differentiable inG. At pointsz∈Gwheref is differentiable, thecomplex dilatation µ(z) is defined by

(1.1) µ(z) =∂f(z)/∂f(z)

if∂f(z)= 0, and byµ(z) = 0 if∂f(z) = 0 (cf. [4], p. 1). Thenµis measurable and|µ|<1 a.e. Moreover, thedilatation of f, defined by

(1.2) K_f(z) = 1 +|µ(z)| 1− |µ(z)|, is a.e. finite.

A function Q : G → R is called of finite mean oscillation at a point z₀ ∈G ifQis integrable in a neighbourhood of z₀ and

dQ(z0) = lim

ε→0

1

|D(z0, ε)|

D(z0,ε)

|Q(z)−Q_ε(z0)|dxdy <∞,

MATH. REPORTS9(59),4 (2007), 369–376

where

Q_ε(z0) = 1

|D(z₀, ε)|

D(z0,ε)

Q(z)dxdy

is the mean value of the functionQ(z) over D(z₀, ε) ={z∈C:|z−z₀|< ε}, see [5], and|A|denotes the Lebesgue measure of A⊂C.

We call dQ(z0) the dispersion of the function Q at z0. We say that an L¹_loc(G) function Q : G→ R is of finite mean oscillation in a domain G, for shortQ∈FMO(G) or, simply,Q∈ FMO, if Q has finite dispersion at every pointz∈G. A functionQ∈FMO_loc(G) ifQ|_U ∈FMO(U) for every relatively compact subdomainU ofG.

Given a domainGand two sets E andF in ˆC, denote by Γ(E, F, G) the family of all paths γ : [a, b] → Cˆ which join E and F in G, i.e., γ(a) ∈ E, γ(b) ∈F, and γ(t) ∈ Gfor a < t < b. Let R =R(C1, C2) be a ring domain, i.e., a doubly connected domain R in ˆC with connected components C1 and C2 of ˆC\R. The capacityof R is defined by

capR(C1, C2) = Mod(Γ(C1, C2, R)), see [5], p. 82.

Given z0 ∈ G\ {∞}, ε0 > 0 with D(z0, ε0) ⊂ G, and Q : D(z0, ε0) → [1,∞], we say that a homeomorphism f : G → f(G) ⊂ Cˆ is a ring Q- homeomorphism at a pointz₀∈Gif

(1.3) Mod Γ(f C₁, f C₂, f R)≤

R

Q(z)η²(|z−z₀|)dxdy

for all circlesC_j ={z∈C:|z−z₀|=ε_j},j= 1,2, inGwith 0< ε₁ < ε₂< ε₀ and for every measurable functionη : (ε₁, ε₂)→[0,∞] such that

(1.4)

ε2

ε1

η(r)dr = 1.

Embedding means any homeomorphism which is into while homeomor- phism designates a homeomorphism onto.

Given ∆>0, and a measurable function Q:G→[1,∞], denote byR^∆_Q the class of all ringQ-homeomorphismf :G→f(G)⊂Cˆ such that

(1.5) δ( ˆC−f(G))≥∆,

where δ(A) is the spherical diameter of A, and let F_Q^∆ be the class of all qc mappingsf satisfying (1.5) and K_f(z)≤Q(z) a.e. in G.

Remark. F_Q^∆⊂R^∆_Q.

We shall consider these classes of mappings on Riemann surfaces and give a definition of the FMO-functions on a Riemann surface R by means of the universal covering ( ˆR,Π, R), where ˆR=D– the unit disk (but sometimes also a half-plane H = {zˆ∈ Cˆ : ˆx < a, a ∈ R} or a vertical strip S = {zˆ∈ C:s1 <x < sˆ 2, s1 ands2inR}) and ˆR=C, according as R is of hyperbolic or parabolic type. Thus, FMO(R) ={Q :R → R: ˆQ=Q◦Π∈ FMO( ˆR)}.

Similarly,

FMO_loc(R) ={Q:R →R: ˆQ=Q◦Π∈FMO_loc( ˆR)}.

Let R and R be homeomorphic Riemann surfaces, ( ˆR,Π, R) and ( ˆR,Π, R) their universal coverings, f : R → R a homeomorphism (em- bedding) and ˆf : ˆR → Rˆ a lifting of f to the universal coverings. Let Q∈FMO_loc(R) and ˆQ=Q◦Π∈FMO_loc( ˆR).

By definition, f is a ring Q(z)-homeomorphism (embedding) iff ˆf is a ring ˆQ(ˆz)-homeomorphism (embedding),z= Π(ˆz). The definition in the case Q∈FMO(R) is similar.

LetQ:G→Rbe a FMO_locfunction inG, andf :G→Can ACL sense- preserving mapping. By definition, a ringQ(z)-homeomorphism f solution of (1.1) is aringQ(z)-quasiconformal(Q(z)-qc)homeomorphismifK_f(z)≤Q(z) a.e. in G. We say that f is a ring Q(z)-quasiregular (Q(z)-qr) mapping if it is constant or allows (see [8]) a Stoilow representation f = ϕ◦ w, where w:G→G =w(G)⊂C is a ring Q(z)-quasiconformal homeomorphism and ϕis a non-constant analytic function inG.

Recently, Ryazanov, Srebro and Yakubov [5] have extended the existence and uniqueness theorem for the solution of Beltrami equation using normality and compacity criteria, to the case of ring FMO_loc-homeomorphisms in the plane belonging to the classR^∆_Q. In what follows, we give a normality criterion for families of ring FMO_loc-qr mappings on Riemann surfaces.

2. NORMALITY CRITERIA FOR RING FMOloc-qr MAPPINGS

For families of ringQ(z)-qc homeomorphisms between Riemann surfaces, the normality criterion below holds.

Theorem 2.1 ([7], p. 802, Theorem 2.1). Let R and R be two homeo- morphic Riemann surfaces, where R is non-conformally equivalent to either Cˆ or C, z₀ ∈ R, z₀ ∈ R and Q ∈ FMO_loc(R). Then any family F of ring Q(z)-qc homeomorphisms f :R→R with

(2.1) f(z0) =z₀

is normal. (In paper[7] radial Q(z)-homeomorphism means ring Q(z)-homeo- morphism.)

This criterion does not hold for the class of ring Q(z)-qr mappings, be- cause it does not hold for BMO-qr mappings, as follows from an example in [6],§2, p.682.

Theorem2.2. Let Q∈FMO_loc(G) be given.

(i)The familyF of all ringQ-qrmappings fromGintoC\{a_f, b_f} such that the spherical distanceskbetween af, bf and∞ are minorated by a strictly positive constant, is normal; or

(ii) There exist three distinct points z1, z2, z3 ∈ G such that the preim- age of f(z_i) reduces to {z_i} for every f ∈ F, while the spherical distances k(f(zi), f(zj)),k(f(zi),∞)are minorated by a strictly positive constant. Then the familyF is normal.

Proof. (i) As normality is a local property, it is sufficient to consider that G = D–the unit disk. Every ring Q(z)-qr mapping f has the Stoilow factorization

(1.6) f =ϕ◦w,

whereϕis analytic andwis a ringQ(z)-qc homeomorphism. By Picard’s the- orem,w(D)⊂

=C. Now, by Riemann’s representation theorem we can suppose w(D) =D.

Everyw omits C\D and, according to Theorem 1.1, p. 801 in [7], the familyW of these ringQ(z)-qc homeomorphismsw:D→Dis normal, while the family Φ of the analytical functions ϕ : D → C\ {a_f, b_f} is normal by TheoremM, p. 683 in [6].

(ii) A threefold application of (i) yields a locally uniformly subsequence convergent onG= (G\ {z₁, z₂})∪(G\ {z₁, z₃})∪(G\ {z₂, z₃}).

Corollary2.1. LetGbe a domain inC,Qa function inFMO_loc(G),E a set inGwhich has at least one accumulation point inGand{f_n}a sequence of ring Q(z)-qrmappings such that every fn omits two valuesan, bn∈C, and the spherical distances between the points a_n, b_n and ∞ are minorated by a strictly positive constant. If{f_n}is pointwise convergent at every point e∈E, then{fn}converges locally uniformly inGto a ringFMO_loc-qrmapping inG. Proof. This is a straightforward consequence of Theorem 2.2 (cf. the proof of Theorem 5.7, p. 15 in [4]).

3. A NORMALITY CRITERION FOR RING FMOloc-qr MAPPINGS BETWEEN RIEMANN SURFACES

We have

Theorem 3.1. Let R and R be two homeomorphic Riemann surfaces, withR not conformally equivalent to eitherCˆ orC, zj ∈R,ζj ∈R,j= 0,1, z₀ = z₁, ζ₀ = ζ₁ and Q ∈ FMO_loc(R). If F is a family of ring Q(z)-qr mappingsf :R→R such that f⁻¹(ζj) =zj, j= 0,1, thenF is normal.

The proof of this theorem uses two auxiliary results, Lemmas 3.1 and 3.2 below.

Lemma 3.1. Let F be a family of ring Q(z)-qr mappings of an open hyperbolic Riemann surface R into the unit disk D, z_j ∈R, ζ_j ∈D, j = 0,1, z₀ = z₁, ζ₀ = ζ₁ and Q ∈ FMO_loc(R) and f⁻¹(ζ_j) = z_j, j = 0,1. If F is normal with respect to the spherical distance k, then it is normal with respect to the hyperbolic distanced on D.

Proof. Let {f_n} be a sequence in F and {f_ni} be a subsequence locally uniformly convergent to a mapping f : R → D. By Stoilow’s factorization theorem, we havef_ni =ϕ_i◦w_i,where ϕ_i is analytic and w_i is a ringQ(z)-qc embedding. Thenw_i(R) is a hyperbolic Riemann surface. Denoting by ˆR the universal covering of R, we may suppose without any loss of generality that ˆz₀ = ˆζ₀ = 0.

Let ˆw_i : ˆR → D and ˆϕ_i : D → D be the lifting of w_i normalized by wˆi(0) = 0, respectively the lifting of ϕi normalized by ˆϕi(0) = 0. Then each wˆ_iis a ringQ(z)-qc embedding omitting every point ofC\D. By Theorem 1.1, p. 801 in [7], the sequence {wˆi} admits a subsequence locally uniformly con- vergent to a mapping ˆw, which by Lemma 3.15 in [4] and Theorem 3.2 in [5]

is either a constant or a ring ˆQ(z)-qc embedding such that ˆw( ˆR) is an open subset of D. But, ˆw cannot be a constant, since this would contradict the assumption ζ1 =ζ0. The sequence {ϕˆi} consists of analytic functions and is uniformly bounded on D. Hence, by Montel’s classical result in [3], p. 21, Ch. I, 10, there is a locally uniformly convergent subsequence {ϕˆ_kn} having a limit ˆϕ which is either holomorphic or a constant. ˆϕ cannot be a constant as before. Hence ˆf( ˆR) is an open subset ofD, implying the same property of f(R). This shows that the sequence {f_ni} is locally uniformly convergent to f with respect tod.

Lemma 3.2. Under the hypotheses of Theorem 3.1, let Rˆ and Rˆ be the universal coverings of R and R. Choose a pair of points zˆ_j ∈ Π⁻¹(z_j) and ζˆ_j ∈ Π⁻¹(ζ_j), j = 0,1, with Π and Π being the canonical projections and

denote by Fˆ the family of the ring Qˆ(z)-qr mappings fˆ: ˆR → Rˆ, where fˆ is the lifting of a ring Q(z)-qr mapping f ∈ F normalized by the condition fˆ(ˆz₀) = ˆζ₀. ThenFˆ is normal forR of hyperbolic type while forR of parabolic type Fˆ is normal at each point except for the points belonging to Π⁻¹(z₁).

Proof. Case1. R of hyperbolic type: ˆR =D. By Theorem 1.1, p. 801 in [7], the lifted family ˆF is normal with respect to k. By Lemma 3.1, ˆF is normal with respect tod.

Case2. R of parabolic type: ˆR =C.

Subcase2.1. Rand R-tori, hence ˆR=C, too. AssumeR=C/_Zω1+Zω2 andR =C/_Zω2+Zω1, withωj, ω_j ∈C^∗,j = 1,2, and Im(ω2/ω1),Im(ω₁/ω₂)>

0. The covering groups G and G are generated by the translations T_j : ˆz → zˆ+ωj and T_j : ˆz → zˆ +ω_j, j = 1,2, respectively. Without any loss of generality we can take ˆz0 = ˆζ0 = 0. If ˆf ∈ Fˆ, the homeomorphism Θ_fˆ:G→G defined by Θ_fˆ(T) =T yields

T : ˆz →zˆ+mω₁ +nω₂ forT : ˆz→zˆ+mω₁+nω₂,m, n, m, n ∈Z. The equation

(3.1) f Tˆ =Tfˆ

implies

(3.2) fˆ(ˆz+mω₁+nω₂) = ˆf(ˆz) +mω₁ +nω₂.

Since z0 =z1 we get ˆz1 = ˆz0 = 0. Denote by F^∗ the family of the restricted mappingsf^∗ = ˆf|_C^∗_\{Π⁻¹_(z1)}. Everyf^∗omits two values belonging to different fibers 0 and ˆf(ˆz1), since ˆf(ˆz1)∈/ Π⁻¹(ζ0). Define

G=C^∗\ {Π⁻¹(z1)},

pick a fixed point ˆz ∈ G, and let f_n^∗ : G → C^∗ \ {fˆn(ˆz1)} be a sequence of mappings fromF^∗.

By Theorem 2.2, in order to show the normality of ˆF at ˆz, it will suffice to fixδ >0 such that

(3.3) k( ˆf_n(ˆz₁),∞)> δ.

Since Π⁻¹(ζ₁) is a discrete set, we deduce the existence of aδ₁ >0 such that if fˆn(ˆz1)= ˆfm(ˆz1) thenk( ˆfn(ˆz1),fˆm(ˆz1))> δ1.By considering a subsequence of the sequence{fˆn}we reduce the problem to the case wherek( ˆfn(ˆz1),fˆm(ˆz1))>

δ₁form=n.

This clearly implies inequality (3.3). Now, the family ˆF is easily seen to be normal in C\

Π⁻¹(z1)

. Indeed, every sequence {fn} ⊂Fˆ determines a sequence{f_n^∗} ⊂ F^∗. The family F^∗ being normal, there exists a subsequence f_n^∗_k

, locally uniformly convergent inGto a limit functionf^∗ :G→C^∗which

is a ringQ(z)-qr mapping by Corollary 2.1. Since fn(0) = 0, the subsequence {f_nk} converges inG∪ {0} to a mappingf :G∪ {0} →Cwithf(0) = 0 and f|G = f^∗. By Corollary 2.1 again, f is a ring Q(z)-qr mapping. It remains to show the normality at ˆz₀ = 0. By the local character of normality, it will suffice to check it in a diskD of radiusεand center 0. This case follows from Lemma 3, p. 686 in [6].

Subcase 2.2. R =C^∗,R is either C^∗ or a punctured disk D^∗ of radius 1 < r < ∞ and center 0, or an annulus A = {z∈C:r1 <|z|< r2} with 0 < r₁ < 1 < r₂ < ∞, in which cases ˆR = C, ˆR = {zˆ ∈ C : Re ˆz < logr}

or ˆR ={zˆ∈ C: logr₁ <Re ˆz <logr₂}, respectively. Consider the universal coverings ( ˆR,exp, R) and (C,exp,C^∗) and choose ˆz₀ = ˆζ₀ = 0. By (3.1), for any ˆf ∈Fˆ we have

(3.4) fˆ(ˆz+ 2mπi) = ˆf(ˆz)±2mπi, m∈Z.

Similarly to Subcase 2.1, we can deduce that the family ˆF is normal with respect tok.

In both Subcases 2.1 and 2.2, the normality ofFwith respect todfollows by similar arguments to those in Lemma 3.1. Normalization implies that the limit mapping ˆf is not the constant ∞. As ˆf is a ring ˆQ(z)-qr mapping (cf.

Corollary 2.1), it follows from (3.2) and (3.4), respectively, that ∞ ∈/ fˆ( ˆR).

Hence ˆf( ˆR) ⊂Rˆ. Finally, one verifies that ˆf_n converges uniformly to ˆf on every compactK ⊂Cwith respect to both distances kand d.

Proofof Theorem 3.1. IfRis conformally equivalent toD– the unit disk, one takes R = D. Then F, considered as family of ring Q(z)-qr mappings f : R → C, is normal by Theorem 1.1, p. 801 in [7], since every mapping f ∈ F omits C\D. Hence one can apply Lemma 3.1 in order to deduce the convergence with respect to the hyperbolic metric.

IfRis of parabolic type, thenF is normal inR\{z₁}by Lemma 3.2 and Proposition 1.2, pp. 410–411, in [1]. By Lemma 3.2 again, reverting the roles of ˆz₀ and ˆz₁, the conclusion follows from Theorem 2.2 (ii) and Proposition 1.2, pp. 410–411 in [1]. In the other case, Theorem 3.1 follows from Lemma 3.2, Theorem 2.2 and Proposition 1.2 in [1].

Remark 3.1. The following examples show that Theorem 3.1 does not hold if either (i)R= ˆCor (ii)R =C.

(i) If fn : ˆC→ Cˆ, fn(z) =nz, then fn are Q(z)-qr mappings such that f_n⁻¹(0) = 0, f_n⁻¹(∞) =∞, but

f(z) =

0, z= 0

∞, z = 0.

Hence,f(z) being discontinuous, the family of BMO-qr mappings from ˆC to Cˆ having 0 and∞ as fixed points, is not normal.

(ii) The sequence {f_n}, f_n : C → C, f_n(z) = ze^n(1−z), f_n⁻¹(0) = 0, f_n⁻¹(1) = 1, has the limit

f(z) =

1, z= 1

∞, z = 1.

REFERENCES

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Received 8 January 2007 University “Politechnica” of Bucharest Faculty of Applied Sciences

Splaiul Independentei 313 060032 Bucharest, Romania

victoriastanciu@yahoo.com