BMO-quasiconformal mappings

BMO-QUASICONFORMAL

MAPPINGS

By V. RYAZANOV,U. SREBROAND E. YAKUBOV

To Olli Martio on his 60th birthday Abstract. Plane BMO-quasiconforrnal and BMO-quasiregular mappings are introduced, and their basic properties are studied. This includes distortion, existence, uniqueness, representation, integrability, convergenceand removability theorems, the reflectionprinciple, boundarybehaviorand mapping properties.

1

Introduction

We study here plane BMO-quasiconformal and BMO-quasiregular mappings, proving results which were announced in [RSY2]. These maps are considered also in [RSY1] and in [Sa]. For related but more general classes of mappings in the plane and in higher dimensions see the recent important papers [AIKM], [IKM] and [IM]. Let D be a domain in C and f : D ~ C an ACL sense-preserving open discrete mapping. Here ACL stands for absolutely continuous on lines (see [A] or [LV]), and discrete means that the preimage o f every point is a discrete set in D. Then f has partial derivatives

Of ---- fz = 89

- ifu )

and-Of

=

fe

=

l(fx + ifv)

a.e. in D; and since f is open, it is differentiable a.e. in D, by a result of Gehring and Lehto, see [A, p. 24]. At points z in D where f is differentiable, the complex dilatation #(z) is defined by (1.1)

/~(z) = 3f(z)/Of(z)

if Of(z) # O, and by #(z) = 0 if Of(z) = O. Then # is measurable and I#l < 1 a.e.; moreover, the dilatation of f, which is defined by (1.2)

K(z) =

1+ 1 1

JOURNAL"D'ANALYSE MATHEMATIQUE. ~b[. 83 (2001)

I (z)l

2

V. RYAZANOV, U. SREBRO AND E. YAKUBOV

is finite a.e. A sense preserving ACL embedding with a given # a.e. is called a #-homeomorphism. In some places, as in [D] and [T], the term is more restricted. Note that a #-homeomorphism f is quasiconformal (qc) if and only if I1#11~ < 1 or, equivalently, K E L~176 Given a measurable function Q : D --+ ]I~ and an ACL mapping f : D ~ C, we say that f is Q(z)-quasiregular (Q(z)-qr) i f f is constant or a sense-preserving open discrete mapping and K(z) < Q(z) a.e. in D. If, in addition, f is injective, we say that f is Q(z)-quasiconformal (Q(z)-qe). This term was probably first introduced by M. Schiffer and G. Schober for functions Q E L ~ . Earlier, TeichmiJller and later Volkovyskii, Andreian--Cazacu, Krushkal, Kiihnau and others (see the references in [KK]) considered Q(z)-qc maps or classes of Q(z)-qc maps. If f : D ~ C is Q(z)-qc, and Q belongs to a given class Y" of functions, we say that f is ~'quasiconformal (jr-qc). Jr-qr maps are defined similarly. In this paper we focus on ~'-qc and gZ-qr maps f : D ~ C, where ~- is the class BMO(D) of all real valued functions of bounded mean oscillation in D. BMO functions were introduced by John and Nirenberg [JN] and are related in various ways to qc maps; see, e.g., [AG], [As], [J], [G], [R] and [RR]. BMO-qc and BMOtoc-qc mappings are closely related to mappings which were considered by David I-D] and Tukia [T]. In [D], David considered ACL embeddings f : D ~ C with a complex dilatation/~ which satisfies an exponential condition (1.3)

I{z e D : I~(z)l > 1 - e}l < Ce -a/~

for all e E (0, e0], for some constants so 9 (0, 1], C > 0 and d > 0. Here IAI denotes the Lebesgue measure of the set A. In Tukia [T], (1.3) is replaced by the spherical exponential condition (1.4)

a({z 9 D: K(z) > t}) < Ce -~t

for all t > T, for some constants T > 0, C > 0 and a > 0 and domains D in C. Here a(A) denotes the spherical area of A. Note that by the John-Nirenberg lemma [JN], an embedding f is BMO-qc if and only if f has an exponentially integrable distortion, i.e., e K 9 L v for some p < oo; of. [IM]. One of the main results in [D] (page 27) says that if/z is measurable in C and satisfies the exponential condition (1.3), then the Beltrami equation (1.5)

~w(z) = #(z)OwCz)

' ~ for all has an ACL homeomorphic solution f = f~, : (; ~ C which belongs to w ,, l~o~ s < 2, and fixes the points 0, 1 and oo. David's proof of the existence goes along

BMO-QUASICONFORMAL MAPPINGS

the lines ofAhlfors [A] and Bojarski [B], involving a very fine analysis of singular integrals. Recently, Brakalova and Jenkins [B J] proved that, given a measurable function # in a domain D with I1 ,11 --- 1, there exists a #-homeomorphism if the following two conditions hold: 1

(1.6)

If

exp 1 + log(x_~)

dA < eB

B

for every bounded measurable set B in D, where e s is a constant which depends on B, and (1.7)

f/ll_-- dA=

O(R2),

R ~ c~.

{Izl 0 is Borel and

f pds > 1 "t

V. RYAZANOV, U. SREBRO AND E. YAKUBOV

for every locally rectifiable path 7 in F. Then (see [LV, V(6.6)])

(3.1)

M(yF) < f /

O(z)p2(z)dxdy.

C

Here M(F) stands for the conformal modulus o f F , i.e., M(F) =

inf

ffp2dxdy.

pEadm F JJ

C

In the following lemma, we prove a general property of a real-valued BMO function in a disk which is needed in the proof of the first distortion lemma. The lemma and its proof extend to BMO functions in balls in higher dimensions, which we plan to use elsewhere in the study of BMO-qc mappings in higher dimensions.

Lemma 3.2. For 0 < t < e -2, let A(t) = {z : t < [z] < e-a}. Let Q be a non-negative BMO function in A. Then (3.3)

r/(t) :=

f f

O(z)dxdy < cloglogl/t,

izl2(log izl) 2 _

A(t)

where c is a constant which depends only on the average Qa of Q over Izl < e -x and on the BMO norm IIQII. of Q in Zx.

Proof. Fix t E (0, e-2). For n = 1, 2, ..., let t,~ = e - " , A,={zeC:t,,+l_ k(f(z~),f(O)) > ~,

k(E~) >_k((:\f(A)) _> 5

and

k(Ei, E~) )~(k(f(z2), f(0))),

where A(t) = A~(t) is a strictly decreasing positive function with A(t) -4 ~ as t -4 0. Hence, by (3.13) and (3.14), k(f(z~),f(O)) > r where

has the required properties. As an immediate consequence of Lemma 3.12 and a known convergence theorem [R1~2] (see also [GI]), we obtain the following lemma.

Lemma 3.15. Let f , : D -4 (~, n = 1,2, ..., be a sequence o f q c mappings which are Q( z )-qcfor some Q( z ) E L~oc(D ). I f f,, -4 f locally uniformly, then either f is constant or f is Q(z)-qc, and Ofn and Ofn converge to Of and Of, respectively, weakly in L~oc. 1fin addition #I. -4 # a.e., then Of = #Of a.e. Proof. By Lemma 3.12, f is discrete if it is nonconstant, and then by an elementary and standard argument (cf. [LV]), f is injective and hence homeomorphic. Indeed, suppose that f ( g l ) ---- f(z2) for some gx ~ 2:2 in D. For small t > 0, let D~ be a disk of spherical radius t centered at zl such that Dt C D and z2 ~ D'--~. Then for all n, f,,(ODt) separates f,,(zl) from f,,(z2); hence k(f,(zl),f,(ODt)) < k(f,(zl),f,,(z2)). Since f(zl) = f(z2), there is a point zt on ODe such that f(zl) = f(z~), contradicting the discreteness o f f . All other assertions follow from [R1-2].

12

V. RYAZANOV, U. SREBRO AND E. YAKUBOV

4

Existence and representation

T h e o r e m 4.1. Let D be a domain in C and let # be a complex valued measurable function in D with [#(z)[ < 1 a.e. such that K(z) O. Indeed, the assertion about (fl,)-i follows from the fact that (fl,)-i E Wz~; see Theorem 4.1 and page 118 in [LV, III]. For the regularity o f f l ' a.e., let E denote the set of points of D where fl' is differentiable and Jr, (z) = 0, and suppose that [El > O. Then Ifl'(E)l > O, since E = ( f u ) - t ( f f ' ( E ) ) and (fu)-I preserves null sets. Clearly, (f~,)-i is not differentiable at any point of fU(E), contradicting the fact that (.f~,)-t is differentiable a.e.

13

BMO-QUASICONFORMAL MAPPINGS

Iwaniec and SverLk [IS] proved that every Wtlo'~ solution g of (1.5) with K 9 E L~o. has a representation g = h o f, where h is holomorphic and f is a homeomorphism; in particular, f is open and discrete. An extension of their result (concerning openness and discreteness) to higher dimensions can be found in [MV]. The following representation theorem can be obtained f r o m [IS] if it is combined with David's uniqueness theorem, which we present in Section 5 below. Instead, we use Theorem 4.1 (ii) to obtain a rather short and direct proof. T h e o r e m 4.3. Let #, Q and f~" be as in Theorem 4.1 and let g be a IT~lo~ solution of(1.5). Then g has the representation (4.4)

g = h o f~',

where h is holomorphic in f"(D). Proof. Let q0 = (f~,)-i and h = 9 o ~o. Since g E W,lo'~ and ~o E W,~ (see Theorem 4.1), it follows by Lemma 6.4 in [LV, HI] that h E W~,~(f(D)). Thus, by Weyl's lemma, it suffices to show that Oh = 0 a.e. in f~'(D). Let E denote the set of points z in D where either f~' or g does not satisfy (1.5) or d/, = 0. A direct computation (cf. [A, p. 9]) shows that Oh = 0 in I~'(D) \ I~'(E). To show that 3h = 0 a.e. in I~'(E), consider (4.5)

ff KS.(z)

1 - I~(z)l 2 E

(1 + I ~ ( z ) l ) 2 -<

E

ff Q(z)dxdy

= O,

E

which holds since IE] = 0 and Q E L~o~. On the other hand, ~o E W~o~; this permits a change of variables in the integral

ff

f-(~)

[Ocpl2dudv=

ff

dudv

dxdy

J ~ l - ,#(qo(w)), 2 = g

/-(E)

which by (4.5) implies that [09] = 0 a.e. o n Now, a.e. 10~l < 10~l and

l-,/~(z)l 2'

E

f"(E).

-oh = -a~o . a g o ~p + c3cp . c3g o cp;

hence [0~ol = 0 a.e. on f~'(E), implying 10hi = 0 a.e. on f~'(E). Thus 0h = 0 a.e. in f~'(D). Consequently, h is holomorpkic in fU(D) and (4.4) holds. 4.6. In a recent paper, Iwaniee and Martin [IM] showed that there are ACL solutions outside W~o~ which are not open and discrete and thus are not generated by a homeomorphic solution in the sense of (4.4). Remark

14

V. RYAZANOV, U. SREBRO AND E. YAKUBOV

5

Uniqueness and approximation

It is not hard to see that, by Stoilow's factorization theorem for open discrete mappings, one obtains the following factorization property for BMO-qr mappings.

Proposition

5.1. I f g is BMO-qr in a domain D, then g = h o ~, where ~ is BMO-qc in D with #~ = #g a.e. and h is meromorphic in ~(D). The following uniqueness theorem of David is used in the proof of some of the properties of BMO-qc and BMO-qr mappings which are presented in the sequel. T h e o r e m 5.2 ( D a v i d [ D , p. 55]). Let fi, i = 1, 2, be #-homeomorphisms in a domain D, where # satisfies David's condition (1.3). Then f2 o f l 1 is conformal. Based on Proposition 5.1, Theorem 5.2 and the John-Nirenberg lemma, one obtains the following corollary. C o r o l l a r y 5.3. Let ~ and g be, respectively, BMO-qc and BMO-qr mappings in a domain D with #~ = #g a.e. Then g = h o 9 f o r some meromorphic function h in ~(D). 5.4. G o o d approximation. Let f : D -~ C be an open discrete Q(z)-qr mapping. W e say that a sequence (f,) of Q(z)-qr mappings in a domain D with complex dilatations #,, n = I, 2,..., is a good approximation of f, ifthe following three conditions hold (cf. [LV, IV, 5.4]):

(i) II/Z,,II~ < 1 for all n; (ii) f , --+ f locally uniformly in D; (iii) /Z, ~ / z a.e. in D. T h e o r e m 5.5. Let D be a domain in C and f : D ~ C a Q(z)-qr mapping with complex dilatation #, where Q G BMOto~(D). Then f has a good approximation. P r o o f . Fix two points zx and z2 in D. For n E N, let #,~(z) = /z(z) if I/z.(z)l < 1 - 1/n and 0 otherwise. Let g,, be a homeomorphic solution of (1.5) with #n instead of/z, which fixes zl and z2. Since all gn are qc and Q(z)-qc and fix zl and z2, it follows by Corollary 3.11 that the sequence {g,,} has a subsequence, again denoted by {gn}, which converges locally uniformly in D to some mapping g in D. Then, by Lemma 3.15, g is Q(z)-qc and satisfies (1.5) a.e. By Corollary 5.3, f = hog for some holomorphic mapping h : g ( D ) -~ f(D). Then the sequence fn = h o g, is a good approximation of f. Based on Corollary 5.3 and Montel's normality criterion for analytic functions, one obtains by the good approximation theorem, Theorem 5.5, the following extension of Corollary 3.11.

BMO-QUASICONFORMALMAPPINGS

15

C o r o l l a r y 5.6. Given Q in BMOzoc(D) and two distinct points a and b in Co the family o f all Q(z)-qr mappings from D into C \ {a, b} is normal. The following theorem can be considered as an extension of Lemma 3.15. T h e o r e m 5.7. Let D be a domain in Co Q(z) E BMOto~(D), a and b two distinct points in Co E a set in D which has an accumulation point in D and {f~} a sequence o f Q(z)-qr mappings from D into C, \ {a, b}. I f { A } has a limit at every point e E E, then {fn} converges locally uniformly in D to a BMO-qr mapping in D. Moreover, Of~ and OA converge to Of and Of, respectively, weakly in L~o~. If, in addition, #j~ -r # a.e., then Of = #Of a.e.

Proof. It suffices to prove the statement for D = A and non-constant f,~. By Proposition 5.1, f,, = hn o 9,,, where hn is analytic and 9,~ is Q(z)-qc. In view of Picard's theorem and the Riemann Mapping Theorem, we may assume that 9,~(A) = A, 9n(O) = 0 and 9n(1) = 1, for all n. Here we use the fact that 9,~ e x t e n d s continuously to 0A, which follows from the reflection principle proved in 6.1. Then {h,~} is normal by Montel's theorem, and {9,~} is normal by Corollary 5.6. The result follows now by Vitali's condensation principle for analytic functions; see, e.g., [C, Sec. 191], and Lemma 3.15; cf. also Corollary 5.3 and Theorem 5.5.

6

Other properties

Let D be a domain in C, E a free boundary arc in OD and f : D ~ -C a (continuous) mapping. Then the cluster set C(f, E) of f on E is a continuum; and i f f is an embedding, then C(f, E) c Of(D) (of. [CL]). T h e o r e m 6.1 ( T h e reflection principle f o r B M O - q c mappings). Let D be a domain in the upper half plane IHI+ = {z E C :Im z > 0}, E a free boundary arc in OD and f : D -r C a BMO-qc mapping. Suppose that E C R and let D'={zEC:~ED}and[2=DUEUD*. l f C(I, E) C R and Re I(z) > Ofor z E D, then f has a BMO-qc extension on [2.

Proof. Since f is BMO-qc in D, there is a function Q in BMO(D) such that K(z) < Q(z) a.e. in D. Let ~) be the symmetric extension of Q to [2. Then Q E BMO([2) and 11811~ = IIQII.; see [RR, p. 8]. Let/2 : fZ ---rC be the symmetric extension to [2 of the complex dilatation of f in D, and let/~ denote the corresponding dilatation. Then K(z) < Q(z) a.e. in [2. Therefore, by Theorem 4.1, there exists a fi-homeomorphism ] : [2 ~ C; and by Corollary 5.3, there is a conformal map h of ](D) such that f(z) = h(i(z)) for all z e D. Since i ( E ) is a Jordan arc in ](fl)

16

v. RYAZANOV,U. SREBROAND E. YAKUBOV

and C(h, f(E)) = C(f, E) C R, it follows that h has a homeomorphic extension to f(D) U ](E) = f ( D U E), denoted again by h, and f(z) = h(f(z)) for all z E D t3 E. This shows that f is homeomorphic in D U E and hence that F is homeomorphic in ft. Now F is ACL in D t9 D*; and, since it is continuous on E, it is ACL in f~. Evidently, the dilatation of F is symmetric with respect to E; hence it coincides a.e. with/~" and is thus dominated by Q. Consequently, F is BMO-qc. C o r o l l a r y 6.2. Let D1 and D2 be domains in C and f a BMO-qc map of D1 onto D2. (i) (Carath6odory's theorem) I f Dx and D2 are Jordan domains, then f extends to a homeomorphism of D1 onto D2. (ii) If, in addition, each ODi is a quasicircle, then f has a BMO-qc extension to C. Proof. Let ~oibe a conformal mapping of Di onto A, i = 1, 2. Then the map F = ~2 o f o r is BMO-qc and hence, by Theorem 6.1, has a homeomorphic extension to A. Now, by Carath6odory's Theorem, for i = 1,2, ~pi has a homeomorphic extension to ODi; and the assertion follows. T h e o r e m 6.3. (i) Every BMO-qc mapping f in the punctured unit disk A0 = {z : 0 < [z[ < 1} has a BMO-qc extension on A. (ii) Every BMO'-qc mapping in C has a BMO-qc extension on C. P r o o f . We prove (i). The proofof(ii) is similar. Let f be a BMO-qc map with complex dilatation # and dilatation K. Then K(z)