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On the Ahlfors Class N in an Annulus

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ANNALES

UNIVERSITATIS MARIAE CURIE-SKLODOWSKA

LUBLIN —POLONIA

VOL. XL, 2 SECTIO A 1986

Instytut Matematyki Uniwersytet Marii Curie-SkłodowskieJ

W. CIEŚLAK, J. ZAJĄC

On the Ahlfors Class N in an Annulus

O klasie N Ahlforsa dla pierścienia

06 Aai>4>opcoBOM KJiacce N min kojiehh

Introduction, To show that the tneory of quasiconfor-nal mappings is not an ad hoc generalization of the theory of confor­

mal mappings, Out is, on tne contrary intimately tied to the classical theory Ahlfors [l] nas investigated tne class N of complex-valued L*° functions \) in the unit dish for wnicn the antilinear part of the Fr6chet differential of normalized ' quasiconformal mappings vanishes, where the mappings are generated by complex dilatation of tne form t-0 , t being a real parameter

.11

He gave there a tneory of this class and i showed that the complex structure of Teichmtlller space of closed Riemann surfaces of genus g 1 carries a natural complex analytic structure which can De' derived from tne corresponding structure of L* by means of generalized Riemann mapping theorem.

The theory of this class K has been used by Reicn and Strebel [dj in connection with one of tne most important extremal problems in the unit dish concerning tne functions with given ooundaiy values.

Very deep investigation of the class N has oeen given by

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Belch in W« wnere he considered also the class H in an annulus with "inward extension".

This class H has also been investigated by Ławrynowicz C2]

and Zając [5] and used as a tool to obtain a parametric representa­

tion of TeichmUller quasiconformal mappings of an annulus ¡6^. The results presented here have an expository character, lie present also some new results due to the first author concerning equiv­

alence condition for functions of this class.

1. Tne class Np . Let yu. be a complex-valued measurable function in an annulus = |z : r^lzl^l^ , 0 <fr £ 1 , which satisfies the condition

|UlL= inf sup |itiz)| <1

where the infimum is taken over all sets of the plane me.asure zero.

It is well-known tnat there exists exactly one number R , O^R^1»

and one Q-quasiconformal mapping f of the annulus Ar onto wnich satisfies tne Beltrami equation

U) f5 = /xfz with fO) = 1 ,

where Q= (1+lyU.II«, )/U-IlyUll«, ) .

Suppose now that yu. = t\> , wnere ll-Oll» 4.°° » and

0 t 4. 1/B-08m • Denote explicitly the dependence of f on 0 ! fQz,t) = f [<] (.z,t) , r lz| 41 • Let

(.2) ffr] (.z) = ^lim iz.t) - zj ,

wnich is a i'recnet differential of i'[-j] . Tnis expression is well defined and depends linearly on -¡) (.cf. [l]). irom f[V? r =

= t-Of£-o] it turns out that x‘C\?3 , regarded as 0 function of z , has partial derivatives almost everywhere, and in particular

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On the Ahlfors Class N in an Annulus 15

it satisfies the differential equation

fM 2 •

It is well-known that (.3) is satisfied only if

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1 p = - ÿ jj ^4^ dxdy + R<J )

witn nolomorpnic Ï . Tnus we nave (,cf.

15)

ür

- f» » _ 1+ r2kz >

z 1 - r2k j z 1 - r2lcz

1+r2kz 1—r2^z

dxdy ) -

He see that f is a linear continuous operator which maps every -0 ê L- (. Ar)' 0“ a function . As it is snown in [2]

the relations |f£\)} (.z,t)| = 1 for )z| = 1 , and |f[>)] (,z,t)J

= K[-0](.t) |z| = r yield

for |z| = 1 , r ç for |z| = r , (.6) Re zf[-û] (,z)j =

where = lim It) - rj .In analogy to tne aoove we

can verify that

for |z| = 1 , for |z| = r ,

wnere

t7) 8. U)} = P ,

5* = I R [ii>] (.t) - rj . Sor more details see [2].

tie recall [2J that

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by which

s'-# ft •

dp

following Ahlfors [l] let us decompose the Frechet differential iM defined by (.2) as follows

(10) f[<J = ^f[S7]+if[i<|j ♦ J pM.

where the first part is antilinear and tne second one is linear with respect to the complex multipliers. By the definition of to] we can see tuat pM ♦ = 0 i.e.

(11) + if[iS>]

is always a holomorphic function. The antilinearity is expressed by $ [i?] = - if [V] •

ii/e denote by Nr the subspace of 1 1 ¿r) which is formed by all with f W = 0 . It is a complex linear subspace of

. Now we can state

Theorem 1. An element •Q of L°° ( dr) belongs to Nr if and only if one of the following assumptions hold:

f _ for |j| = 1 ,

dxdy for lj| = r , (12) f i j) =

dxdy ,

(.14) -0(z)g(z)dxdy = dxdy J zg^z)dz

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On the Ahlfors Class N in an Annulus 17

for all g aolomorpalc la intAr wi th ll ,glz)| uxdy 4 00 .

>JAr

Jroof. The proof of 03) is presented in details iu [3] and (2]. Tae condition' 113) is an imaediate consequence of 13) and the definition of tne class Nf . To get the condition (.14) suppose tnat g is holomorphic in int Ar with finite 1? norm in Ar • Tnen by 112) and Green's formulae we have tne equality 114).

Conversely, if 114) is fulfilled, then we apply it to glz) =

= — 1J -z) , |^| = r , and next when = 1 . Because

zglz)dz , wnere 0 4 £ 4 1-r

l«l»r |zl=r+£

which follows by an approximation argument applied to classical Green's formulae. By this 114) is valid as soon as

zgQz)dz =

It shows that the right side of 114) has the same boundary values as it is given by 112). tasking use of the integral representation given by the formulae 14) we see that tne ooundary values of f[oj are those of an holomorphic function, wnich by tne normalization condition vanisnes at z=1 , so it mqst be identically zero and we conclude 112).

2, Other properties of tne class . suppose that

116) ¿)l<?elS)= £2 olnl 2 )einS . r4 <¿1

wnicn is the i'ourier series of -Q . Let now g oe as in Theorem, 1

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and let

(1?) 6la) = if = Jf a,,«’“'’1*6 . z-— ie= 5ei0 , r4,<1

be its Laurent series. Kow, uy the argument given in tne proof of Theorem 1, we may express (.14) in terms of the coefficients

<7 Q( ^) and aK , u,k=O, -1, -2,... . By this we have

(.16)

ci , r' f -+*» -1 n]

jj -Q(z)g(z)dxdy = 2 JT J ; jS.*»1 s >•-» ? J as

= 23Ł + »

SZ a-

a=-«o J

ior the right side of (.14) we have

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J zg(z) dz

|z| =r i

2V hxdy

_lLzl J

=" dr Jj dxdy

•'Ar |z| =r

z2«U) ff =

= - 2%a

f1 *2lg>

Jr S

Let

Au’k = £ d5 ’

then Dy (23) and (24) the equality (14) can be expressed in the form

(20) nfe, a’n Aq’“q = ' a"2 A2--

•2 *

Let h(b) denote tne Banach space of all holomorpnic func­

tions with finite L1 -norm in a domain D . If C ,

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On the Ahlfors Class N In an Annulus 19

then clearly H(D^) 3 H(D2) .

In the case of the unit disk it is easy to see Gnat the unit dick can be replaced by an aroitrary simply connected region D . If D,,CD2 and i)€N(D,,) , tnen "OtK(D2) , wnere

- v f"5(z) , zfcD.

‘'i(z) = <

[o ,

z

«D 2 \D1 .

Making use of (14) we see that previous implication is also true in the case of a doubly connected domain.

These results have a natural analog in the case r = 0 , i.e.

for mappings in tne unit disk with an additional invariant point zero.

i

B±jPiiiRiiłM C£S

[1] Ahlfors, L.V., Some remarks on TeicnmUller space of Kiemann surface , Ann, of Uath. 74(1961), 171-191 •

Krzyż, J., Ławrynowicz, J., The Parametrical Method for Quasiconformal Mappings in the Plane, Springer-Verlag, 1982.

J Reich, E., An extremum problem for analytic functions with area norms, Ann. Acad. Sci. Penn. Ser. A I, vol. 2(1976), 429-445.

[4j Reich, S., Strebel, K., On quasiconformal mappings with keep the boundary points fixed, Trans. Arner. Math. Soc. 158(1989)1 211-222;

[5J Zając, J., The Ahlfors class K and its connection with Teichmtlller quasiconformal mappings of an annulus, Ann. Univ.

Mariae Curie-Sklodowska, Sect. A, 12(1978), 155-182.

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[oj Zając, J., Ou tne paraniecrization of leicnnillller mappings in an annulus, Bull, de la 8oc. des dcien et des Lettves de Łódź Vol. XXVII (1977), 1-8.

‘ STRESZCZENIE

W pracy 1 Ahlfors wprowadził podklasę N klasy L°°

funkcji zespolonych V w kole jednostkowym, takich, że dla odwzo­

rowania ąuasikorforemnego generowanego przez dylatację t V , t fc IR, znika identycznie część antyllniowa jego różniczki Frócheta.

Autorzy badają własności funkcji należących do analogicznej klasy funkcji w pierścieniu.

PE3BME

5 paCoTe [lj AJib<{iopceM BBeaeH KJtacc NCLW KounJteKCHtiz 4>,yHKnnił V b enHHMHiioM Kpyre, t8khx no fljta KBa3HKOH<J>opuHoro OTOdpaneHns nopoacneitHoro KownJieKOHoii «M^baTamtefi tv, t ę.IR bhtm— JizHeinaH yacTb ero niicjiliepeHUMajia łpeme paBHa Hyjtio.

Abtoph saHMMaBTCH cBoflcTBauM ({lyHKUHii npMHafljiejtamiix k asaior«' ęecKOMy tcjiaccy b KOJtbpy.

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