SUFFICIENT AND NECESSARY CONDITIONS FOR CONFORMALITY. PART II. ANALYTIC VIEWPOINT

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1 Annales Academiæ Scientiaum Fennicæ Mathematica Volumen 35, 00, SFFICIENT AND NECESSARY CONDITIONS FOR CONFORMALITY. PART II. ANALYTIC VIEWPOINT Melkana A. Bakalova Fodham nivesity, Depatment of Mathematics Bonx, New Yok 0543,.S.A.; This pape is dedicated in loving memoy to V. V. Alexandov and A. A. Gol dbeg. Abstact. This is the second of two papes devoted to the topic of confomality at a point and elated notions in the plane. We deive epesentation fomulas and estimates fo the modules of families of cuves that ae images of cicles, adial segments and acs of spials unde a µ-homeomophism. We use them to convet the extemal-length type sufficient and necessay conditions fo confomality at a point fom Pat I to analytic sufficient conditions, that depend on diectional dilatations and bypass the assumption of K-quaisconfomality. Ou esults extend the Teichmülle Wittich Belinskii theoem, esults of Reich and Walczak, the autho and Jenkins, and Gutlyanskii and Matio.. Definitions and main esults Confomality at a point (see (.3)) and elated notions in the plane ae ich popeties that have had applications in the theoy of Riemann sufaces, Nevanlinna theoy, in the study of egulaity popeties of the bounday coespondence, local modulus of continuity popeties and othes (e.g. [0, 5, 6, 7,, 6]). Ou main esults, Theoems..4, povide sufficient conditions fo confomality at a point and elated notions fo ACL-homeomophisms, not necessaily K-quasiconfomal, that depend on the diectional dilatations in two o thee diections. Let Ω C be a domain and let f be an ACL (absolutely continuous on lines), sense-peseving homeomophism f : Ω f(ω). Let µ be a Lebesgue-measuable complex-valued function in Ω C with µ < a.e. and µ. f is called a µ-homeomophism if (.) f z = µf z fo a.e. z Ω. Then f has complex patial deivatives f z and f z a.e., and the Jacobian J f = f z f z 0 a.e. In this pape we make the additional egulaity assumptions that f is locally absolutely continuous and J f > 0 a.e., thus f is egula a.e. in Ω. Recall that µ is efeed to as the Beltami coefficient. If µ <, (.) is efeed to as the (complex) Beltami equation. By the Existence theoem (measuable Riemann mapping theoem [, 5, 5]) fo such µ we can solve (.) and its solutions doi:0.586/aasfm Mathematics Subject Classification: Pimay 30C6, 30C99. Key wods: Confomality at a point, Beltami equation, quasiconfomal mapping, degeneate Beltami equation, µ-homeomophism, diectional dilatation. The wok was patially suppoted by a Faculty Fellowship (Sping 008) and a Faculty Reseach Gant, Fodham nivesity.

2 36 Melkana A. Bakalova ae quasiconfomal mappings. If µ = 0 a.e. in Ω then f is confomal in Ω [, Coollay ]. If µ =, one efes to (.) as the degeneate Beltami equation. nde vaious conditions on µ, one can solve the Beltami equation in the degeneate case (see [4, 4, 8,, 9, 3] and othes.) Fo the most ecent eseach developments concening solutions of the degeneate Beltami equation in the plane, in highe dimensions and on metic spaces, see the ecent monogaphs [0, 7] and efeences theein. One says that z 0 Ω is a egula point if f is totally diffeentiable at z 0, i.e. f(z) = f(z 0 ) + f z (z 0 )(z z 0 ) + f z (z 0 )( z z 0 ) + o( z z 0 ) and J f (z 0 ) > 0. It is a well-known fact, in the theoy of K-quasiconfomal mappings, [5, ], that an ACLhomeomophism is totally diffeentiable a.e. At the egula points of f, the complex dilatation of f is defined as µ f = f z, and the eal dilatation as D f = + µ f f z µ f, thus D f at such a point. Without loss of geneality we may assume that D f = and µ f = 0 outside of the set of egula points. Let α be a eal numbe (the diection). At a egula point z Ω the diectional dilatation D f,α of f in diection α is defined as (.) D f,α = f α J f, whee f α = f z + e iα f z is the diectional deivative of f in diection α. Note that a.e. in Ω, D f,α D f and that D f,α may assume any value in the inteval D f (0, ). Check (3.) fo an equivalent expession to (.). Futhe we nomalize f so that it is defined in a neighbohood of the oigin, say the unit disk, = {z : z < }, f(0) = 0. We say that f is confomal at the point 0 (at the oigin) if (.3) lim z 0 f(z) z = A 0,. Ou main esults on confomality hold unde an additional assumption that: Condition.. Let t > be fixed. Thee exists a constant C 0 = C 0 (t) such that da z lim sup µ f z < C 0(t) <. 0 < z <t The main esults of this pape ae as follows. Theoem.. If (.4) lim 0 < z < (D f,θ+α ) da z z Instead of the integal condition involving µ f one could conside a weake one, namely D f,θ+ j da z = o(), 0, j = 0,. It is clea that if Condition. holds fo z < z <t some t >, then it holds fo any othe t >, and that it holds in case f is K-quasiconfomal.

3 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 37 is finite fo α = 0, /, then (.5) lim z 0 f(z) z = A 0 0,, f is asymptotically a otation on cicles at z = 0, namely (.8) holds, and f peseves asymptotically adial segments of any fixed aspect atio t > at the oigin, namely (.9) holds. Theoem.3. If lim 0 < z < (D f,θ+α ) da z z is finite fo α = 0, / and α = α 0, α 0 k, k any intege, then f is confomal at z = 0. Theoem.4. If (.4) is finite fo α = 0,, and if fo some fixed θ 0, lim 0 ag f(e iθ 0 ) θ 0 = const, then f is confomal at z = 0. The above theoems extend seveal ealie esults on confomality and elated notions, within the class of K-quasiconfomal mappings and to the lage class of µ-homeomophisms. Section 5 povides a detailed account of the esults we have in mind and some examples. Note that Theoem.4 gives paticulaly weak confomality conditions, povided that f is known to peseve a diection in the limit. Late on, in Theoems and Theoem 4.9 we state the above esults in equivalent foms that depend explicitly on the complex dilatation. In the special case when f is a smooth adial stetching, i.e. f(e iθ ) = ρ()e iθ, ρ(0) = 0, ρ() inceasing and continuously diffeentiable, (.6) (D f ) da z z < is a sufficient and necessay condition fo confomality at a point (see [5, p. ]). In the case of K-quasiconfomal mappings, by the well-known Teichmülle Wittich Belinskii theoem (Theoem 5.), (.6) is a sufficient condition. Howeve it is not necessay. It is impossible to find, in the geneal case, sufficient and necessay conditions fo confomality that depend on the eal dilatation alone. To show this, we conside two K-quasiconfomal mappings in, f and g, f(0) = 0, g(0) = 0, such that D f = D g, o equivalently µ f = µ g in, and such that f is confomal at 0 and g is not. In Example., [] we constucted a adial stetching f(e iθ ) = ρ()e iθ defined in, f(0) = 0 such that f is confomal at the oigin and µ f (e iθ ) = 3 eiθ. Define g(e iθ ) = e i(θ+ i log ), 0 < <, and g(0) = 0. Since µ g (e iθ ) = 4 + i eiθ, it follows that µ f = µ g. Clealy g is not confomal at the oigin. Note that, since f fom Example., [] is confomal at the oigin and (.4) is not finite fo any α, the conditions in Theoem.3 and Theoem.4 ae not necessay fo confomality eithe. (.8) and (.9) ae intoduced futhe in this section.

4 38 Melkana A. Bakalova Both sufficient and necessay conditions fo µ-homeomohisms, in tems of modules of families of cuves, ae obtained in []. Some of these esults ae stated in section and ae used to pove ou main theoems. Notions elated to the study of confomality at a point ae intoduced below. We say that a µ-homeomophism f is cicle-like at the point 0 if it maps cicles centeed at the oigin onto almost such cicles, namely (.7) lim f(z) z = =, min f(z) 0 max z = in othe wods the cicula dilatation (see [5]), H f (0) =. A µ-homeomophism confomal at 0 is cicle-like at 0. The convese is not tue, as the example f = z z shows. A µ-homeomophism f is asymptotically a otation on cicles at the oigin if (.5) holds and if fo an appopiate choice of a banch of the agument (.8) ag f(e iθ ) ag f(e iθ ) (θ θ ) 0 as 0, unifomly in θ, θ. This notion was intoduced in [9]. A µ-homeomophism, confomal at 0, is asymptotically a otation on cicles at 0. Howeve, the convese does not hold as the example f(z) = ze i log z, 0 < z <, f(0) = 0, shows. A µ-homeomophism is weakly confomal if it is cicle-like, i.e. (.7) holds, and if fo an appopiate choice of a banch of the agument, f satisfies (.8). This notion was intoduced fo K-quasiconfomal maps in [8]. Weak confomality is a weake notion than asymptotical otation on cicles and stonge than cicle-like behavio. Let 0 < < <, and θ be a eal numbe. Denote by σ f,θ (, ) the image unde f of the adial segment σ θ (, ) = {z : < z <, ag z = θ}. We define the angula oscillation of σ f,θ (, ) at the oigin as ω σf,θ (, ) = sup ag z inf ag z. σ f,θ (, ) σ f,θ (, ) In the above definition we choose a single-valued, continuous detemination of the agument. ω σf,θ (, ) measues the deviation of σ f,θ (, ) fom a segment on a line though the oigin. ω σf,θ (, ) = 0 if and only if σ f,θ (, ) is such a segment. Let t >, and be such that 0 < t <. We say that a µ-homeomophism f peseves asymptotically adial segments of fixed aspect atio t at 0, if (.7) holds and unifomly in θ, (.9) ω σf,θ (, t) 0 as 0. A µ-homeomophism confomal at 0 peseves asymptotically adial segments of fixed aspect atio t at 0. The adial stetching f = z( log z ), 0 < z <, f(0) = 0, does that without being confomal. The est of the pape is stuctued as follows. In Section we state the sufficient and necessay conditions fo confomality at a point fom [] that ae applied in this pape. In Section 3 we use diectional dilatations to povide epesentation fomulas and estimates fo modules of families of cuves that ae images unde a µ-homeomophism of adial segments, cicula acs and acs of spials. In Section 4 we pove Theoems..4 and deive equivalent esults, Theoems , and Theoem 4.9 that depend explicitly on the complex dilatation. In Section 5 we give

5 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 39 an oveview of peviously known esults on confomality and elated notions at a point and discuss how we extend them.. Geometic esults on confomality In this section we state the sufficient and necessay conditions fo confomality at a point fom [] that ae applied in this pape. Fo a definition of the module of a family of cuves and elated teminology, see [, ], [, Definition.3]. Let 0 < < <. A f = A f (, ) denotes the image unde f of A(, ) = {z : < z < }. M(A(, )), M(A f (, )) ae thei modules, whee M(A(, )) = log. Let θ < θ θ +. Q(,, θ, θ ) = {z = e iθ : < <, θ < θ < θ } denotes a quadilateal with a-sides the cicula acs. Its image unde f is denoted by Q f (,, θ, θ ). The a-sides of Q f (,, θ, θ ) ae the images of the a-sides of Q(,, θ, θ ). Let M(Q(,, θ, θ )) M(Q f (,, θ, θ )) be the modules of the family of cuves connecting the a-sides of each quadilateal, espectively. Hee M(Q(,, θ, θ )) = θ θ log. Let β be a fixed eal numbe. Fo θ [θ, θ + ) we conside the acs of logaithmic spials of inclination β, s β θ (, ) = {z : z = e i( β log +θ), < <, ag z = θ}. Then the family S β (, ) = s β θ (, ) sweeps out A(, ). θ [θ,θ +) sually, θ = 0. S β f (, ) denotes its image, M(S β (, )) and M(S β f (, )) the modules of the coesponding families of cuves. As shown in Coollay 3., M(S β (, )) = ( + β ) log. Theoem.. [, Lemma 4.] Let t > be a fixed numbe. Assume that (.) M(A f (, )) M(A(, )) = o() as 0, and (.) lim 0 M(Q f (, t, θ, θ )) = θ θ log t f(z) unifomly in θ and θ. Then lim = A 0 0,, f is asymptotically a otation z 0 z on cicles at z = 0, namely (.8) holds, and f peseves asymptotically adial segments of fixed aspect atio t > at the oigin, namely (.9) holds. Conditions (.) and (.) ae necessay fo confomality at the oigin. Combining the esults of Lemma 3., Lemma 4., and Lemma 5. fom [] one obtains the following esult. Theoem.. Assume that (.) and (.) fom Theoem. hold, and that fo some β 0 and fo any ε > 0 thee exists R = R(ε) such that fo 0 < < < R (.3) M(S β f (, )) M(S β (, )) < ε.

6 40 Melkana A. Bakalova Then f is confomal at z = 0. Conditions (.) and (.) ae necessay fo confomality at the oigin. Theoem.3. [, Theoem.] Assume that (.) and (.) fom Theoem. hold. If fo some fixed θ 0, lim ag f(e iθ 0 ) θ 0 = const, then f is confomal at z = 0. 0 The above conditions ae necessay fo confomality at the oigin. 3. Repesentation fomulas and estimates fo modules of families of cuves and ing domains In this section we use diectional dilatations to povide epesentation fomulas and estimates fo modules of families of cuves that ae images unde a µ- homeomophism of adial segments, cicula acs and acs of spials. Let α be a eal numbe. Let z = e iθ be a egula point. Then the diectional dilatation (.) in diection α can be witten as (3.) D f,α = + e iα µ f µ f. Fo α = θ, θ + / we have (3.) D f,θ = + e iθ µ f µ f, D f,θ+/ = e iθ µ f µ f. Since f = e iθ ( f z + e iθ f z ) and fθ = ie iθ ( f z e iθ f z ) it follows that f = f z +e iθ µ f, f θ = f z e iθ µ f. Thus we obtain the following equivalent expessions at a egula point z = e iθ : (3.3) D f,θ = f J f, D f,θ+ = f θ J f. In the liteatue, D f,θ has been efeed to as the adial dilatation, and D f,θ+/ as the angula dilatation, and denoted by D µ and D µ, espectively (see e.g. [8] and [9]). In addition, we obseve that fom (3.) follows that at the egula point z = e iθ (3.4) D f,θ = µ f + R(e iθ µ f ) µ f, D f,θ+ = µ f R(e iθ µ f ) µ f. Let 0 < < < <, 0 < <, and θ < θ θ +. Denote by C (θ, θ ) = {z : z =, θ < ag z < θ } a cicula ac of the cicle C = {z : z = }. Let C (,, θ, θ ) = C (θ, θ ) be the family of cicula acs. (, ) Denote thei images by C f,, C f, (θ, θ ), C f (,, θ, θ ). M(C(,, θ, θ )) and M(C f (,, θ, θ )) ae the coesponding modules. Note that M(C (,, θ, θ )) = log θ θ. Fo θ [θ, θ ), we denote by σ θ (, ) = {z : < z <, ag z = θ} a adial segment in A(, ), and its image by σ f,θ (, ) (see Section.) Denote Σ(,, θ, θ ) = σ θ (, ) the family of adial segments and the family of θ (θ,θ )

7 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 4 thei images by Σ f (,, θ, θ ). M(Σ(,, θ, θ )) and M(Σ f (,, θ, θ )) ae the modules of the coesponding families of cuves. Note that M(Σ(,, θ, θ )) = θ θ log. Lemma 3.. Let z = e iθ Q(,, θ, θ ). If D f,θ+/ L (A(, )), then (3.5) M(C f (,, θ, θ )) = If D f,θ L (A(, )), then (3.6) M(Σ f (,, θ, θ )) = θ θ θ d θ D f,θ+ dθ. dθ. d D f,θ Poof. We only show the validity of (3.5). The poof of (3.6) is done in a simila manne. We use popeties of the Lebesgue integal (see [8]) that hold tue due to ou egulaity assumptions, including that f is locally absolutely continuous and egula a.e. When necessay one should conside that the equalities below hold only a.e. Conside ρ = f θ = f θ dθ f θ θ J f J f θ D f,θ+ θ θ D f,θ+ dθ defined a.e. in Q(,, θ, θ ), and ρ 0 = ρ f defined a.e. in Q f (,, θ, θ ). The ρ 0 -length of C f, (θ, θ ) defined fo almost evey (, ) is l ρ0 (C f, (θ, θ )) = C f, (θ,θ ) ρ 0 dw = θ θ ρ f θ dθ =. Thus ρ 0 is an admissible function fo the module poblem defined by the family of cuves C f (,, θ, θ ), whee C f (,, θ, θ ) Q f (,, θ, θ ). The ρ 0 -aea of Q f (,, θ, θ ) is A ρ0 (Q f (,, θ, θ )) = ρ 0 da w = ρ J f da z Q f (,,θ,θ ) Q(,,θ,θ ) = θ θ ρ J f d dθ = d = θ f θ dθ θ J f θ d θ D f,θ+ dθ. Let ρ 0 be any othe admissible function fo the module poblem defined by the family of cuves C f (,, θ, θ ). Then l ρ 0 (C f, (θ, θ )) = ρ 0 dw. C f, (θ,θ )

8 4 Melkana A. Bakalova Since C f, (θ,θ ) (ρ 0 ρ 0 ) dw 0, θ θ f θ J f dθ θ θ (ρ ρ) f θ dθ 0, whee ρ = ρ 0 f. Theefoe ( ρ ρ ρ ) J f da z 0 Q(,,θ,θ ) and thus ( ) ρ 0 ρ 0 ρ 0 daw 0. Q f (,,θ,θ ) The Cauchy Schwaz inequality implies (ρ 0) da w (ρ 0 ) da w and theefoe Q f (,,θ,θ ) Q f (,,θ,θ ) M(C f (,, θ, θ )) = A ρ0 (Q f (,, θ, θ )) = θ d θ D f,θ+ dθ. Remak 3.. Fom (3.3) follows that (3.5) and (3.6) can be ewitten in the following way. Fist, and second, M(C f (,, θ, θ )) = M(Σ f (,, θ, θ )) = θ θ d, θ f θ dθ θ J f dθ f J f The poof of Lemma 3. is simila to the one used by Rodin in [30] fo the case of quadilateals, whee f is assumed to be sufficiently smooth. Geneal esults in this diection wee obtained in the woks of Andeian Cazacu (see [3] and the efeences to he ealie esults theein.) Coollay 3.. Let z = e iθ Q(,, θ, θ ), D f,θ+α L (A(, )) fo α = 0, /. Then θ dθ da z (3.7) M(Q θ d f (,, θ, θ )) ( D f,θ log ) D f,θ+ z, Q(,,θ,θ ) d.

9 (3.8) and (3.9) and Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 43 θ d θ D f,θ+ (M(Q f (,, θ, θ ))) dθ 0 Poof. We have d D f,θ+ dθ M(A f (, )) () (θ θ ) A(, ) Q(,,θ,θ ) D f,θ da z z. M(Σ f (,, θ, θ )) M(Q f (,, θ, θ )) (M(C f (,, θ, θ ))) M(C f (,, 0, )) M(A f (, )) (M(Σ f (,, 0, ))) by the Compaison pinciple, see [, ]. inequality, Coollay 3. follows. D f,θ da z z, sing (3.5), (3.6), and Cauchy Schwaz (3.9) is well-known if f is a K-qusiconfomal mapping (see [6, 9]), though the estimates ae usually expessed in tems of the complex dilatation o patial deivatives. The uppe estimate in (3.9) was poven fo the geneal class of µ-homeomophisms by a diffeent method in [9]. Let β be a fixed eal numbe. In the next two coollaies we give a epesentation fomula and an estimate fo the module of a family of acs of logaithmic spials of inclination β and its image, defined in Section. Coollay 3.. We have (3.0) M(S β (, )) = ( + β ) log. Poof. Let z = e iθ, 0 θ <, h β (e iθ ) = e i( β log +θ) be an aea peseving map of A(, ) onto itself. By (3.3), D hβ,θ = (h β) = + β. Moeove, J hβ M(Σ hβ (,, 0, )) = 0 dθ d D hβ,θ by (3.6). Since M(Σ hβ (,, 0, )) = M(S β (, )), (3.0) follows. Coollay 3.3. Assume that D f,θ+α0 L (A(, )), whee α 0 = tan β. Then (3.) M(S β f (, )) ( + β ) da z () D f,θ+α0 z. A(, ) Poof. Let z = e iθ, 0 θ <, and h β (e iθ ) = e i( β log +θ) be the map defined in the Coollay 3. above. Conside the map g = f h β : A(, ) A f (, ).

10 44 Melkana A. Bakalova Since (h β ) z + e iθ (h β ) z = e iθ (h β ) and (3.) A(, ) D g,θ da z z = A(, ) A(, ) g z + e iθ g z J f J g (h β ) z + e iθ (h β ) z = e iθ (h β ), one has da z z = A(, ) f ζ (h β ) + f ζ (h β ) J g da z z. Afte change of vaiables ζ = h β (z), using the identity + iβ iβ = e iα 0, the fact that J hβ =, and (h β ) = (h β )( iβ), (h β ) = (h β )( + iβ), the last tem in (3.) is equal to ( + β ) f ζ + + iβ ζ iβ f ζ ζ da ζ ζ = da z D f,θ+α0 z. Thus A(, ) D g,θ da z z = A(, ) A(, ) D f,θ+α0 da z z and D g,θ L (A(, )). By (3.6) of Lemma 3. and the Cauchy Schwaz inequality (3.3) M(Σ g (,, 0, )) da z () D g,θ z. A(, ) Since M(Σ g (,, 0, )) = M(S β f (, )), (3.) follows fom (3.3). Coollay 3.4. Let z = e iθ, and let β be a fixed numbe. Assume that D f,θ+α L (A(, )) fo α =, α 0, whee α 0 = tan β. Then (3.4) 0 d D f,θ+ dθ M(A f (, )) ( + β ) () A(, ) D f,θ+α0 da z z. Poof. By the Compaison pinciple, M(C f (,, 0, )) M(A f (, )) (. M(S β f (, ))) sing (3.5) and (3.) we obtain (3.4). (3.9) is a special case of (3.4) and the latte may povide, depending on the choice of β, moe pecise uppe estimate fo M(A f (, )). 4. Poof of the main esults In this section we pove Theoems..4 and deive equivalent esults, Theoems Lemma 4.. Let f be a µ-homeomophism, D f,θ+α L (A(, )) fo α = 0, /. Denote by Q = Q(,, θ, θ ), Q f = Q f (,, θ, θ ), and by M(Q f ) the

11 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 45 module of the latte. Then log M(Q f ) θ θ and M(Q f ) θ θ log (θ θ ) max j=0, ( log ) max j=0, Q Q ( Df,θ+ ( Df,θ+ j ) da z z j ) da z z. and Poof. Let m = θ θ log. Then M(Q f ) m ( log ) Q M(Q f ) m (θ θ ) follow fom (3.7) and (3.8) of Coollay 3.. This yields m M(Q f ) M(Q f) ) m and m M(Q f ) ( log Q (D f,θ+ )da z z (D f,θ ) da z z Q m M(Q f ) (θ θ ) Q (D f,θ ) da z z (D f,θ+ )da z z. Consideing sepaately the case when M(Q f ) m > 0 (and theefoe and the case when M(Q f ) m < 0 (and theefoe M(Q f) m inequalities we obtain the desied estimates. m M(Q f ) < ) < ) fom the above Lemma 4.. Let D f,θ+α L (A(, )) fo α = 0, /. Then log M(A f (, )) () max ( Df,θ+ j=0, j ) da z. z A(, ) The poof of Lemma 4. is identical to the poof of Lemma 4.. Lemma 4.3. Let f satisfy Condition.. Let t > be a fixed numbe, 0 < < t <, and A(, t) = {z : < z < t}. Assume that µ f da z (4.) = o() as 0. µ f z A(,t)

12 46 Melkana A. Bakalova Then (4.) A(,t) R(e iθ µ f ) da z = o() as 0. µ f z Poof. sing Cauchy Schwaz inequality and Condition. we have R(e iθ µ f ) da z µ f da z da z µ f z µ f z µ f z A(,t) A(,t) C 0 (t) A(,t) fo small enough, and (4.) follows fom (4.). A(,t) µ f da z µ f z Poof of Theoem.. Let 0 < < <. The assumptions in Theoem. imply (4.3) (D f,θ+α ) da z z = o() as 0, A(, ) fo α = 0, /. By Lemma 4. M(A f(, )) log () and, due to (4.3), (.) holds: max j=0, A(, ) ( Df,θ+ j ) da z z M(A f (, )) log = o() as 0. Let t > be a fixed numbe. Denote fo 0 < < t <, A(, t) = {z : < z < t}. Next we show that fo j = 0,, (4.4) D f,θ+ j da z = o() as 0. z Fom (3.4) follows (4.5) Q(,t,θ,θ ) A(,t) Df,θ+ j da z z A(,t) µ f + R(e iθ µ f ) µ f da z z. Since (.4) is finite fo α = 0, /, we have (4.). Lemma 4.3 implies (4.). Fom this and (4.5) follows (4.4). By Lemma 4. M(Q f(, t, θ, θ )) θ θ log t (log t) max Df,θ+ j=0, j da z z. sing (4.4) we have (.) M(Q f (, t, θ, θ )) θ θ log t A(,t) = o() as 0

13 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 47 unifomly in θ, θ. Thus the assumptions of Theoem. imply (.) and (.) of Theoem.. Theoem. follows. Poof of Theoem.3. Fom the assumptions in Theoem.3 and the poof of Theoem. follow (.) and (.) of Theoem.. We only need to show (.3). Take β = tan α 0. Since (.4) is finite fo α = α 0, fo any sufficiently small ε > 0 we can choose R = R(ε) such that fo 0 < < < R, + β () (D f,θ+α0 ) da z < ε. z A(, ) Fom (3.) of Coollay 3.3 follows that M(S β f (, )) ( + β ) log + β () A(, ) (D f,θ+α0 ) da z z. Theefoe, by Coollay 3., (.3) holds. Theoem.3 follows fom Theoem. and thus f is confomal at z = 0. Poof of Theoem.4. Fom the assumptions in Theoem.4 and by Theoem., lim = A 0 0, and ag f(e iθ f(z) ) ag f(e iθ ) (θ θ ) 0 as z 0 z 0. Since fo some fixed θ 0, lim ag f(e iθ 0 ) θ 0 = const, f is confomal at the 0 oigin. (4.6) (4.7) The following Theoems ae equivalent to Theoems..4, espectively. Theoem 4.4. If µ f µ f da z z <, R(e iθ µ f ) da z µ f z f(z) exists in the sense of pincipal value, then lim = A 0 0,, f is asymptotically z 0 z a otation on cicles at z = 0, namely (.8) holds, and f peseves asymptotically adial segments of any fixed aspect atio t > at the oigin, namely (.9) holds. Theoem 4.5. If (4.6) holds and µ f da z (4.8) µ f z exists in the sense of pincipal value, then f is confomal at z = 0. Theoem 4.6. If (4.6) holds, (4.7) exists in the sense of pincipal value, and if fo some fixed θ 0, lim 0 ag f(e iθ 0 ) θ 0 = const, then f is confomal at z = 0. The next esults establish the equivalence between Theoems..4 and Theoems , espectively.

14 48 Melkana A. Bakalova Lemma 4.7. If (4.9) lim 0 < z < R ( e i(θ+α) µ f ) µ f da z z is finite 3 fo any two eal numbes α = α, α = α, α α k, k any intege, then (4.9) is finite fo any α. Poof. The poof follows fom the popeties of the Lebesgue integal and the elementay identity R(ze iα ) = ar(ze iα ) + br(ze iα ), valid fo any z. Hee a = sin ((α α )) sin ((α α )), b = sin ((α α)) sin ((α α )). Theoem 4.8. Let ag z = θ. Fist, the limit (4.0) lim (D f,θ+α ) da z 0 z < z < is finite fo α = 0, / if and only if (4.6) holds and R(e iθ µ f ) da z (4.) µ f z exists in the sense of pincipal value. Second, the limit (4.0) is finite fo α = 0, / and α 0, α 0 k, k an intege, if and only if (4.6) holds and µ f da z (4.) µ f z exists in the sense of pincipal value. Poof. The fist sufficient and necessay pat of Theoem 4.8 follows immediately fom (3.4). Now we pove the necessity in the second pat of the theoem. Since (4.0) is finite fo α = 0, / and α 0, α 0 k, k an intege, then (4.6) holds and (4.3) R ( e i(θ+α) µ f ) µ f da z z exists in the sense of pincipal value fo α = 0, α 0, α 0 k, k any intege. By Lemma 4.7, R ( ) e i(θ+α) µ f da z µ f z exists in the sense of pincipal value fo any α, in paticula α =, and this implies 4 that (4.) exists in the sense of pincipal value. The necessity follows. The sufficient pat can be poven in a simila manne. The above consideations yield that the following sufficient condition fo confomality is equivalent to Theoem 4.5. R ( ) e i(θ+α) µ f 3 i.e. µ f da z z exists in the sense of pincipal value.

15 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 49 Theoem 4.9. If (4.6) holds and (4.3) exists in the sense of pincipal value fo α = α, α, α α k, k an intege, then f is confomal at z = Some comments on the histoy of the study of confomality at a point In this section we give an account of peviously known esults on confomality at a point and how we extend these esults. Fist we state the Teichmülle Wittich Belinskii theoem. Theoem 5.. [3, 3, 33] Let f be a quasiconfomal mapping in such that f(0) = 0. If (5.) (D f ) da z z <, then f is confomal at 0. Remak 5.. (5.) can be ewitten in the equivalent fom (5.) µ f da z z <. The poof of Theoem 5. was accomplished in seveal stages. Fist, in 938, Teichmülle [3] showed that (5.3) lim z 0 f(z) z = A 0 0, unde the assumption that f is a diffeomophism and satisfies a condition slightly stonge than (5.). In [3] he also obtained sufficient and necessay esults on cicle-like behavio involving modules of ing domains. Late, Wittich [33] showed that (5.3) holds fo quasiconfomal maps satisfying (5.). This esult is sometimes efeed to as Teichmülle Wittich theoem. In 954 Belinskii [3] poved that fo a geneal quasiconfomal mapping f with eal dilatation D f, (5.) implies (5.3) and fo an appopiate choice of the agument (5.4) lim ag f(z) = a, z 0 z as well, thus poving confomality at a point when (5.) holds. Reich and Walczak [9] studied confomality at a point as well. Among othe inteesting esults they showed the following theoem. Theoem 5.3. [9] Let f be a quasiconfomal mapping and θ = ag z. If dθ d (5.5) D f,θ < and Df,θ+ dθ d <, then (5.3) holds. Theoem 5.3 extends Teichmülle Wittich theoem. The authos [9] show that (5.5) is not sufficient to claim convegence of the agument (5.4), as ag f(e iθ ) may go to as 0 at a ate as big as o( log(/)), whee the estimate is shap. It seems that one could easily extend this esult to the weake case of µ-homeomophisms when (.4) is finite fo α = 0, /.

16 50 Melkana A. Bakalova Regulaity conditions fo K-quasiconfomal mappings wee also studied by Lehto [3, 5]. Cicle-like behavio and egulaity popeties wee studied in 988 by the autho [6, 7]. An equivalent vesion of a genealization of the Teichmülle Wittich theoem, fom these woks, in the notations of this pape is stated below. Theoem 5.4. [7] Let f be a continuously diffeentiable µ-homeomophism. If µ f da z (5.6) µ f z <, then f is cicle-like, i.e. (.7) holds. In addition, thee exists a constant A > 0 such that f(e iθ da z ) A exp D f,θ z as 0, < z < whee D f,θ is the adial dilatation at z = e iθ. As shown in [6, 7], if (5.) holds then thee exist a constant A 0 such that da z exp D f,θ z A 0 as 0, < z < and the Teichmülle Wittich theoem follows in the case of continuously diffeentiable µ-homeomophisms. Conside the adial stetching f(e iθ ) = eiθ + log with µ f (e iθ ) = eiθ. Fo this map the integal in (5.) diveges, while (5.6) holds. + log Sufficient conditions fo asymptotic otation on cicles fo µ-homeomophisms, wee fist obtained by the autho and Jenkins in 994 [9]. Theoem 5.5. [9]. Assume that f is an a.e. egula µ-homeomophism in, f(0) = 0. Let θ = ag z. Assume that D f,θ+α L (A(, )) fo α = 0, /. If µ f + Re iθ µ f da z (5.7) µ f z <, then (5.3) holds and f is asymptotically a otation on cicles at z = 0. A stip vesion of Theoem 5. that applies to a geneal class of µ-homeomophisms is poven in [9] as Lemma 6.. Fom that esult follows the extension of the Teichmülle Wittich Belinskii Theoem to the class of µ-homeomophisms, asseting confomality unde the condition (5.8) µ da z µ z <. A ecent beakthough (003) in elaxing the analytic conditions fo confomality at a point, in the case of a K-quasiconfomal mapping, was obtained in a pape of Gutlyanskii and Matio [8].

17 Sufficient and necessay conditions fo confomality. Pat II. Analytic viewpoint 5 Theoem 5.6. [8] Let f be a K-quasiconfomal homeomophism in such that f(0) = 0. If µ f (5.9) z da z < and if the singula integal (5.0) µ f z da z exists in the sense of pincipal value, then f is confomal at z = 0. The poof of Theoem 5.6 follows a somewhat unusual path. Fist, the authos show that (5.9) implies weak confomality at the oigin (see Section ). If, in addition, (5.0) holds, then one has convegence of the agument (5.4). In the poof of Theoem 5.6 one uses deep esults fom the theoy of K-quasiconfomal mappings that do not necessaily extend to the class of µ-homeomophisms consideed hee. The following emak, based on [8, Theoem.7], shows to what extent Theoem 5.6 extends Theoem 5. within the class of K-quasiconfomal mappings. Remak 5.7. [8] Fo each K-quasiconfomal mapping f whose complex dilatation µ f satisfies (5.9) and does not satisfy (5.) one can constuct a K-quasiconfomal mapping g such that µ f = µ g a.e. in such that g satisfies (5.9) and (5.0) (and thus is confomal at the oigin). sing any of the existence theoems fo the degeneate complex Beltami equation in the plana case, e.g. [8,, 9, 0, 3], and othes, one can modify the poof of Remak 5.7 so that it holds fo the geneal class of µ-homeomophisms fo which the existence theoems apply. Note that Theoem 5.6 is the quasiconfomal vesion of Theoem 4.5 (Theoem.3, espectively), since if f is K-quasiconfomal, (4.6) and (4.8) ae equivalent to (5.9) and (5.0), espectively. The quasiconfomal countepat of Theoem 4.4, and thus Theoem., implying (5.3), was also fist poven in [8]. An unifom vesion of this esult is given in [7]. Popeties like asymptotic otation on cicles, which is stonge than weak confomality, (.9), and the quasiconfomal countepat of Theoem.4 and Theoem 4.6, espectively, seem to have not been diectly addessed in [8]. Below we give a couple of examples of µ-homeomophisms, confomal at the oigin, such that lim sup µ f (e iθ ) and thus µ f = in any neighbohood of 0 the oigin. Example 5.8. Let G = G n, whee G n = z < n n e/n3. We conside the n= adial stetching f(z) = ρ() e iθ, whee ρ() = if z \ G, and if z G n. {[ ρ() = e n 3 n n n + ( n (e /n3 ) + n ) ] } + n + n + n + n n

18 5 Melkana A. Bakalova Obseve that ρ() simplifies to (n ) + (e /n3 )( + ), µ f (e iθ ) = 0 in (e /n3 )(n + ) \ G and µ f (e iθ ) = ρ ρ ρ + ρ eiθ in G. In each G n, µ f attains its maximum value of µ f = n at z = /n. Thus lim sup µ f (e iθ ) and theefoe f is not 4 + n 0 K-quasiconfomal. Since (5.8) holds, µ f da z µ f z µ f n 3 n= n= + n/4 n 3 <, it follows that f is confomal at the oigin. The confomality of this adial stetching follows diectly fom the constuction as well. Example 5.9. Define a deceasing sequence { n } n= as follows: =, =, e n+ = (n ) n e, n+ = n+ e /n3 fo n =,,.... Define a Beltami coefficient µ in such that µ = 0 in (, ), e µ(eiθ ) = ( )n+ e iθ in (n+, n ], µ(e iθ ) = ln n n eiθ in (n+, n+ ], and µ(0) = 0. Since µ is subexponentially integable [0], thee exists a µ-homeomophism f, f(0) = 0, [4, 8, 0], such that µ f = µ a.e. Clealy, in the above example, lim sup µ f (e iθ ). Afte some computations 0 one can veify that µ f satisfies the conditions of Theoems , Theoems..3, espectively, and theefoe f is confomal at 0. Also f does not satisfy any of the conditions (5.), (5.5), (5.7) and since, in addition, f is not K-quasiconfomal, one can not apply Theoem 5., Theoem 5.3, Theoem 5.5 and Theoem 5.6. Obseve that (5.8) does not hold eithe, while (5.6) holds. Thus we have extended most of the esults on confomality and elated notions discussed in this section within the class of K-quasiconfomal maps and to a geneal class of µ-homeomophisms. Refeences [] Ahlfos, L. V.: Lectues on quasiconfomal mappings. - niv. Lectue Se. 38, nd ed. (with additional chaptes by C. J. Eale and I. Ka, M. Shishikua, J. Hubbad), Ame. Math. Soc., 006. [] Ahlfos, L. V.: Confomal invaiants: Topics in geometic function theoy. - McGaw-Hill, 973. [3] Belinskii, P. P.: Behavio of a quasiconfomal mapping at an isolated singula point. - Lvov Gos. niv. chen. Zap. Se. Meh.-Mat. 9, 954, (in Russian). [4] Belinskii, P.: Geneal popeties of quasiconfomal mappings. - Nauka, Novosibisk, 974 (in Russian). [5] Bojaski, B. V.: Genealized solutions of a system of diffeential equations of the fist ode and elliptic type with discontinuous coefficients. - Repot nivesity of Jyväskylä Dept. of Math. and Stat. 8, tansl. fom the 957 Russian oiginal, with a fowad by E. Saksman, 009. [6] Bacalova, M.: A genealization of the Teichmülle theoem. - In: Complex Analysis (Joensuu 987), Lectue Notes in Math. 35, Spinge, 988, [7] Bakalova, M.: On the asymptotic behavio of some confomal and quasiconfomal mappings. - Thesis, Sofia nivesity, Sofia, 988, 5.

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20 54 Melkana A. Bakalova [3] Teichmülle, O.: ntesuchungen übe konfome und qusiconfome Abbildung. - Deutsche Math. 3, 938, [33] Wittich, H.: Zum Beweis eines Satzes übe quasikonfome Abbildungen. - Math. Z. 5, 948, Received 3 Febuay 009