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Di¤erentiability properties of some nonlinear operators associated to the conformal welding of Jordan curves in Schauder spaces

Massimo Lanza de Cristoforis and Luca Preciso

(Received November 16, 2001) (Revised June 25, 2002)

Abstract. As it is well-known, to a given plane simple closed curve z with non- vanishing tangent vector, one can associate a conformal welding homeomorphismw½z of the unit circle to itself, obtained by composing the restriction to the unit circle of a suitably normalized Riemann map of the domain exterior to zwith the inverse of the restriction to the unit circle of a suitably normalized Riemann map of the domain interior toz. Now we think the functionszandw½zas points in a Schauder function space on the unit circle, and we show that the correspondencewwhich takesztow½zis real di¤erentiable for suitable exponents of the Schauder spaces involved. Then we show that w has a right inverse which is the restriction of a holomorphic nonlinear operator.

1. Introduction

As it is well-known, given an element z of the set AqD of the complex- valued di¤erentiable injective functions, with nonvanishing first derivative, defined on the boundary qD of the open unit disk D of the complex plane C, the function z parametrizes a Jordan curve. To each zAAqD, one can associate a pair ðG;FÞ of Riemann maps, with G a suitably normalized holomorphic homeomorphism of the exterior CnclD of D onto the exterior E½z of z, and with F a suitably normalized holomorphic homeomorphism of D onto the interior I½z of z. It is also well-known that G and F can be extended with continuity to boundary homeomorphisms. Thus one can consider the so-called conformal welding homeomorphism Fð1ÞGjqD of qD, which we denote by w½z. Now let Cm;aðqD;CÞ be the Schauder space of m-times continuously di¤erentiable complex-valued functions on qD, whose m-th order derivative is a-Ho¨lder continuous, with aA0;1½,mb1. It is well- known that if zACm;aðqD;CÞVAqD, then w½zACm;aðqD;CÞVAqD. In this paper we first prove some di¤erentiability theorems for the nonlinear ‘con- formal welding operator’ w½. We note that such theorems can be shown to be optimal in the frame of Schauder spaces (cf. [19, Thm. 2.14].) Moreover,

2000 Mathematics Subject Classification. 30C99, 47H30.

Key words and phrases. Conformal welding, conformal sewing, di¤erentiability properties of nonlinear operators.

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we observe that by restricting w½to the set ofz’s which are boundary values of Riemann maps defined on CnD, the operator w½ becomes real analytic.

Next we turn to the problem of constructing a right inverse of w½. The problem of constructing a suitably normalized pair of functions ðG;FÞ as above, such that Fð1ÞGjqD¼f by a given regular orientation preserving homeomorphism f of qD to itself, a so-called ‘shift’ of qD, is known as the conformal sewing problem and is a particular type of boundary value problem with shift for sectionally holomorphic functions. By exploiting a classical method (cf. e.g., Lu [22]), one can show that to each shift fACm;aðqD;CÞV AqD, one can associate a unique suitably normalized pair of functions ðG;FÞas above. Then the nonlinear operator s, which takes f to s½f1GjqD is a right inverse of w, and will be called the ‘conformal sewing operator’. Next we prove that s½fACm;aðqD;CÞ if the shift fACm;aðqD;CÞVAqD. Then we analyze the di¤erentiability properties ofs. Since the domain ofs, namely the set of positively oriented fACm;aðqD;CÞVAqD such that fðqDÞ ¼qD is not open in the Banach space Cm;aðqD;CÞ, we construct an extension of s to the open set of orientation preserving elements of Cm;aðqD;CÞVAqD, and we show that such extension is complex-analytic. In other words, we show that the boundary values of the Riemann map G of the domain exterior to the curve s½f, depend complex-analytically on f. Then we consider the Riemann map F which is related to G by the equality FjqD¼Gfð1Þ¼s½f fð1Þ. We deduce the di¤erentiability properties of the dependence of the boundary values of Fupon f by ‘ad hoc’ variants of the di¤erentiability results on the inversion and on the composition operator of [15]. We note that the di¤erentiability results for the dependence of F on f can be shown to be sharp by means of inverse theorems. In particular, one can show thatFdoes not depend complex analytically on f (cf. [19, Thm. 2.17].)

The theory of boundary value problems with shift for sectionally holo- morphic functions, also called Haseman problems, is well-known and started with Haseman [9]. Kveselava [13] developped an existence and uniqueness theory in case f is of class C1;a. Later, other Haseman type problems have been studied, also for more general shifts (cf. Litvinchuk [21], Monakhov [23, pp. 357–367].) In the direction of the perturbation results however, the authors are only aware of the continuity result for the conformal welding operator of David [4], and of the continuity results for the conformal sewing operator of Monakhov [23, p. 363], and of Huber and Ku¨hnau [11], in di¤erent function space settings. We mention also the work of Nag [24], who has considered a one-parameter family fftg of shifts depending real analytically on a real parameter t, and who has provided an algorithm to compute the coe‰cients of the formal expansion of the corresponding families of curves s½ft and s½ft ftð1Þ, under the assumption that such expansions converge.

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We believe that our results could be employed in the perturbation analysis of other well-posed Haseman problems. Indeed, the operator which maps a shift to the corresponding solution of the Haseman problem can be expressed in terms of the conformal sewing operator and of operators of known regularity (cf. e.g., Gakhov [6, p. 129, § 14].)

This paper is organized as follows. Section 2 is a section of preliminaries and notation. Section 3 concerns the definition of the conformal welding map and contains di¤erentiability theorems for the conformal welding operator.

Section 4 is devoted to the definition of the conformal sewing operator and of its extension. Section 5 contains a complex di¤erentiability theorem for the conformal sewing operator.

2. Technical preliminaries and notation

Let X, Y be normed spaces over the field K, with K¼R or K¼C. We say that X is continuously imbedded in Y provided that XJY and that the inclusion map is continuous. We say that a map T of a subset of X to Y is compact, provided that it maps bounded sets to sets with compact closure.

For standard definitions of Calculus in normed spaces, we refer to Prodi and Ambrosetti [28] or to Berger [2]. Unless otherwise specified, we understand that a finite product of normed spaces is endowed with the supremum of the norms of the components. Let N be the set of nonnegative integers includ- ing zero. Throughout the paper, n denotes an element of Nnf0g. A com- plex normed space can be viewed naturally as a real normed space. Accord- ingly, we will say that a certain map between complex normed spaces is real linear, real di¤erentiable, or real analytic, to indicate that such map is linear, di¤erentiable or analytic between the corresponding underlying real spaces, respectively. To emphasize that we are retaining the complex structure, we will say that the map is complex linear, complex di¤erentiable, or complex analytic, respectively. The inverse function of a function f is denoted fð1Þ, as opposed to the reciprocal of a complex valued function g, which is denoted g1. For all subsetsBofRn, the closure ofBis denoted clB. We now define the Schauder spaces on the closure of an open subset of Rn. Let Wbe an open subset of Rn, mAN. We denote by CmðW;CÞ the space of m-times contin- uously real-di¤erentiable complex-valued functions on W, and by CmðclW;CÞ the subspace of those functions of CmðW;CÞsuch that for all h1ðh1;. . .;hnÞA Nn, with jhj1h1þ þhnam, the function Dhf1 qjhjf

qh1x1...qhnxn can be extended with continuity to clW. If Wis bounded, then CmðclW;CÞendowed with the norm defined by kfkCmðclW;1P

jhjamsupclWjDhfj is a Banach space. If W is bounded and if aA0;1, we denote by Cm;aðclW;CÞ the subspace of CmðclW;CÞ of those functions which have a-Ho¨lder contin-

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uous derivatives of order m. If f AC0;aðclW;CÞ, then we set jf :Wja1 supnjfðxÞfjxyjðayÞj:x;yAclW;x0yo

. The space Cm;aðclW;CÞ is endowed with its usual norm kfkCm;aðclW;CÞ1P

jhjamsupclWjDhfj þP

jhj¼mjDhf :Wja, and it is well-known to be a Banach space. If BJC, then Cm;aðclW;BÞ denotes the set ff ACm;aðclW;CÞ:fðclWÞJBg. By HðWÞ we understand the space of holomorphic functions of W to C. Finally, the space Cm;a;0ðclW;CÞ is defined as the closure of CyðclW;CÞ in Cm;aðclW;CÞ. Then we have the following.

Lemma 2.1. Let mAN, aA0;1. Let W be a bounded open con- nected subset of Rn of class Cmþ1. Then Cm;a;0ðclW;CÞ coincides with the closure in Cm;aðclW;CÞ of the set of restrictions to clW of the polynomials with complex coe‰cients in n real variables. Moreover, Cm;a;0ðclW;CÞ contains Cmþ1ðclW;CÞ and Cm;bðclW;CÞ, for all bAa;1.

Proof. Since W is of class Cmþ1, then all functions of Cmþ1ðclW;CÞ are restrictions of some element of Cmþ1ðRn;CÞ (cf. e.g., Troianiello [30, p. 13].) Then by Weierstrass Theorem (cf. e.g., Rohlin and Fuchs [29, p. 185]), all elements of Cmþ1ðRn;CÞ can be approximated in the Cmþ1ðclW;CÞ-norm by polynomials. Since clW is of class Cmþ1, then Cmþ1ðclW;CÞ is continu- ously imbedded in Cm;aðclW;CÞ (cf. e.g., [15, p. 460].) Then the first part of the statement and the inclusion Cmþ1ðclW;CÞJCm;a;0ðclW;CÞ follow. Now let f ACm;bðclW;CÞ. Since W is of class Cmþ1, then f admits an extension of class Cm;b and with compact support in a ball containing clW (cf. e.g., Troianiello [30, Thm. 1.3, p. 13].) By taking the convolution with a family of mollifiers, such extension can be approximated by a sequence of Cy functions bounded in Cm;bðclW;CÞ and convergent in Cm;aðclW;CÞ(cf. e.g., Troianiello [30, pp. 20, 21].) Then f ACm;a;0ðclW;CÞ. r We now define the Schauder spaces on plane Jordan curves, which are par- ticular compact subsets of C with no isolated points. With somewhat more generality, we define the Schauder spaces on a general compact subset K of C with no isolated points. We say that a function f of K to C is complex di¤erentiable atz0ACif limKCz!z0

fðzÞfðz0Þ

zz0 exists finite. We denote such limit by f0ðz0Þ. As usual the higher order derivatives, if they exist, are defined inductively. Let mAN. We denote by CmðK;CÞ the complex normed space of the m-times continuously complex di¤erentiable functions f of K to C endowed with the norm kfkCmðK;CÞ¼Pm

j¼0supKjfðj. If aA0;1, we denote by Cm;aðK;CÞ the subspace of CmðK;CÞ of those functions having a- Ho¨lder continuous m-th order derivative in K. If f AC0;aðK;CÞ, then we set jf :Kja1sup jfðzjz1Þfðz2Þj

1z2ja :z1;z2AK;z10z2

n o

. We endowCm;aðK;CÞwith the norm kfkCm;aðK;CÞ1kfkCmðK;þ jfðmÞ:Kja. If BJC, we set Cm;aðK;BÞ1

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ff ACm;aðK;CÞ:fðKÞJBg. We denote by Cm;a;0ðK;CÞ the closure of CyðK;CÞ in Cm;aðK;CÞ. Then the following variant of [14, Cor. 4.24, Prop. 4.29] holds (cf. [18, Lem. 2.5].)

Lemma 2.2. The following statements hold.

( i ) Let fAC1ðqD;CÞ. Then lqD½f1infnjfðxÞfðyÞjjxyj :x;yAqD;x0yo

>0 if and only if f is injective and f0ðxÞ00 for all x in qD.

( ii ) The function of C1ðqD;CÞ to R which maps f to lqD½f is continuous, and in particular, the set AqD1ffAC1ðqD;CÞ:lqD½f>0g is open in C1ðqD;CÞ.

(iii) minxAqDjf0ðxÞjblqD½f, for all fAC1ðqD;CÞ.

We are now ready to state the following, which collects a few facts which we need on the spaces Cm;aðK;CÞ. For a proof and for appropriate refer- ences, we refer to [18, Lems. 2.7, 2.8].

Lemma 2.3. Let mAN, a;bA0;1, fAAqD, L¼fðqDÞ. Then the fol- lowing statements hold.

( i ) Cmþ1ðL;CÞ is continuously imbedded in Cm;aðL;CÞ. If a<b, then Cm;bðL;CÞ is compactly imbedded in Cm;aðL;CÞ.

( ii ) The pointwise product is continuous in the Banach space Cm;aðL;CÞ.

( iii ) The reciprocal map in Cm;aðL;CÞ, which maps a nonvanishing function f to its reciprocal, is complex analytic from Cm;aðL;Cnf0gÞ to itself.

( iv ) Let f1AAqD, L1¼f1ðqDÞ. If f ACm;aðL1;CÞ and if gACm;bðL;L1Þ, then f gACm;gmða;bÞðL;CÞwith g0ða;bÞ ¼ab and gmða;bÞ ¼minfa;bg if m>0.

( v ) Let mb1. If gACm;aðL;CÞis injective and satisfies condition g0ðxÞ00, for all xAL, then gð1ÞACm;aðgðLÞ;LÞ.

( vi ) If I½f and E½f denote the bounded and the unbounded open connected component of CnfðqDÞ, respectively, then qI½f ¼qE½f ¼fðqDÞ.

(vii) If f AAqD, and if fðqDÞJqD, then fðqDÞ ¼qDand f is a homeomorphism of qD to itself.

We now introduce two di¤erentiability theorems, for the composition and for the inversion operator. To do so, we need the following, which we use to study the regularity of the operator w½, and the regularity of the dependence of F on the shift f.

Lemma 2.4. Let mAN, aA0;1, RA1;þy½. Let RD1fxAR2: jxj<Rg. Then there exists a linear and continuous extension operator E of Cm;aðqD;CÞ to Cm;aðclðRDÞ;CÞ such that the following statements hold.

( i ) ðE½fÞjqD¼ f , for all f ACm;aðqD;CÞ, and E½fACm;a;0ðclðRDÞ;CÞ for all f ACm;a;0ðqD;CÞ.

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(ii) Let 1ajam. For all f ACm;aðqD;CÞ, the real di¤erential of order j of the function E½f at tAqD satisfies the following equation

djE½fðtÞðs1;. . .;sjÞ ¼ fððtÞs1. . .sj; ð2:5Þ for all ðs1;. . .;sjÞACj. In particular, djE½fðtÞ is also a complex j- multilinear operator, whenever tAqD.

Proof. To prove statement (i), we first show that there exists a linear and continuous operator Z of Qm

l¼0Cml;aðqD;CÞ to Cm;aðclD;CÞ such that Z½fjqD;. . .;qmZ½f

qnm jqD

¼f for all fA Qm

l¼0Cml;aðqD;CÞ, where n is the outer unit normal to qD. If r;sAf0;. . .;mg, then we set drs¼1 if r¼s, drs¼0 if r0s. As a first step we fix an arbitrary lAf0;. . .;mg, and we show the existence of a linear and continuous operator Zl of Cml;aðqD;CÞ to Cm;aðclD;CÞ such that q

jZl½h

qnj jqD¼djll!h for 0ajal, and for all hA Cml;aðqD;CÞ. By a standard argument based on the partition of unity and on the use of local charts for qD, the existence of Zl follows from that of a linear and continuous operator ZZ~l of Cml;að½1;1;CÞ to Cm;aðclð1;

1½ 1;0½Þ;CÞsuch thatqjZZ~l½g

qx2j jx2¼0¼djll!g for 0ajal. LetKbe a linear and continuous operator of Cml;að½1;1;CÞtoCml;að½2;2;CÞwithK½g ¼gon

½1;1 and suppK½gJ2;2½, for all gACml;að½1;1;CÞ. Furthermore, one can choose K so that K maps Cmlþ1ð½1;1;CÞ to Cmlþ1ð½2;2;CÞ (cf. e.g., the construction of Troianiello [30, Thm. 1.3, p. 13] with k¼ mlþ1.) To construct ZZ~l, we take lþ1 distinct real numbers a0;. . .;al, and we determine b0;. . .;bl by solving the (Vandermonde) system Pl

s¼0asjbs¼ djll!, j¼0;. . .;l, and we set ZZ~l½gðx1;x2Þ1Pl

s¼0bsGl½gðx1þasx2Þ, where Gl½g is the m times di¤erentiable function of R to C determined by condi- tions dl

dtlGl½g ¼K½g, dtdjjjt¼0Gl½g ¼0 for 0aj<l. Then one can define Z by exploiting the operators Zl and formula (5.8) of Necˇas [25, p. 93]. It is also clear that Z maps Qm

l¼0Cmþ1lðqD;CÞ to Cmþ1ðclD;CÞ. Since clDis of class Cy, it is also known that there exists a linear and continuous exten- sion operator ER of Cm;aðclD;CÞ to Cm;aðclðRDÞ;CÞ such that ER½vjclD¼v, for all vACm;aðclD;CÞ. Furthermore, one can choose ER so that ER maps Cmþ1ðclD;CÞto Cmþ1ðclðRDÞ;CÞ(cf. e.g., the construction of Troianiello [30, Thm. 1.3, p. 13] with k¼mþ1.) Then we setE½f1ERZ½fðtÞ;tf0ðtÞ;. . .; tmfðmÞðtÞ. If f ACyðqD;CÞ, then E½fACmþ1ðclðRDÞ;CÞ and thus E½fA Cm;a;0ðclðRDÞ;CÞ by Lemma 2.1. We now prove (ii). By construction, the function E½f is m-times real di¤erentiable at each point tAqD, and the real di¤erential djE½fðtÞ is a real j-multilinear operator of R2j to R2. Also, the right hand side of equation (2.5) delivers a complex j-multilinear oper- ator, which we denote by MfððtÞ of Cj to C, and thus a real j-multilinear

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operator of R2j to R2. In order to prove equality (2.5), it su‰ces to show that djE½fðtÞ ¼MfððtÞ on the j-tuples of elements of a real basis of R2. To shorten our notation, we write v½l instead of v;zfflfflfflffl}|fflfflfflffl{l. . .terms;v in the argu- ment of a multilinear operator. Once t1ðt1;t2ÞAqD is fixed, we choose fðt1;t2Þ;ðt2;t1Þg as a real basis of R2. Note that t equals the exterior unit normal to qDat t, and that it¼ ðt2;t1Þlies in the tangent space to qDat t.

Since djE½fðtÞandMfððtÞ are multilinear and symmetric operators, it su‰ces to check that for 0alaj, we have

djE½fðtÞððt2;t1Þ½l;ðt1;t2Þ½jlÞ ¼ fððtÞðitÞltjl: ð2:6Þ We now prove (2.6) by induction on jAf1;. . .;mg. In case j¼1, it su‰ces to prove the following two equalities

dE½fðtÞððt1;t2ÞÞ ¼ f0ðtÞt; dE½fðtÞððt2;t1ÞÞ ¼ f0ðtÞit: ð2:7Þ The first equality of (2.7) follows by equality qnqE½fðtÞ ¼ f0ðtÞt, which holds by construction ofE. We now turn to prove the second equality of (2.7). We know that E½fðcosy;sinyÞ ¼ fðeiyÞ, for all yA½0;2p. Then by di¤er- entiating with respect to y, we obtain dE½fðcosy;sinyÞððsiny;cosyÞÞ ¼ f0ðeiyÞieiy, which implies the validity of the second equation of (2.7). If m¼1, the proof is complete, thus we can assume that m>1. We assume that equality (2.6) holds for jAf1;. . .;m1g, and for all 0alaj, and we prove (2.6) for jþ1, and for all 0alajþ1. If l¼0, then (2.6) follows by equality qnqjþ1jþ1E½fðtÞ ¼tjþ1fðjþ1ÞðtÞ, which holds by construction of E½f. Thus we can assume that lb1. By inductive assumption, we have

djE½fðtÞððt2;t1Þ½l1;ðt1;t2Þ½jlþ1Þ ¼ fððtÞðitÞl1tjlþ1: ð2:8Þ Now by setting t1ðt1;t2Þ ¼ ðcosy;sinyÞ in (2.8), and by di¤erentiating with respect to y, we obtain

djþ1E½fðtÞððt2;t1Þ½l;ðt1;t2Þ½jlþ1Þ

þ ðl1ÞdjE½fðtÞððt1;t2Þ;ðt2;t1Þ½l2;ðt1;t2Þ½jlþ1Þ þ ðjlþ1ÞdjE½fðtÞððt2;t1Þ½l1;ðt2;t1Þ;ðt1;t2Þ½jlÞ

¼ fðjþ1ÞðtÞðitÞltjlþ1þfððtÞðl1ÞðitÞl2ðtÞtjlþ1 þfððtÞðitÞl1ðjlþ1Þtjlit:

By exploiting the symmetry, the real j-multilinearity of djE½fðtÞ, and the inductive assumption, we obtain that (2.6) holds for jþ1, and for all

0alajþ1. r

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We now have the following variant of [15, Thm. 4.19]. See also Henry [10, p. 96]. For references to previous contributions on this issue by various authors, we refer to [15].

Theorem 2.9. Let m;rAN, a;bA0;1. Let gmða;bÞ be defined as in Lemma 2.3 (iv). Let RA1;þy½. Let E be the extension operator of Lemma 2.4. The operatorTT~ from Cmþr;a;0ðqD;CÞ Cm;bðqD;RDÞto Cm;gmða;bÞðqD;CÞ defined by setting TT~½f;g1ðE½fÞ g, for all ðf;gÞACmþr;a;0ðqD;CÞ Cm;bðqD;RDÞ is of class Cr in the real sense. The restriction of TT~ to Cmþr;a;0ðqD;CÞ Cm;bðqD;qDÞcoincides with the ordinary composition operator Tdefined byT½f;g1f g. The ordinary compositionTmaps Cm;a;0ðqD;CÞ Cm;b;0ðqD;qDÞ to Cm;gmða;bÞ;0ðqD;CÞ. If rb1, qAf1;. . .;rg, and if ðf0;g0ÞA Cmþr;a;0ðqD;CÞ Cm;bðqD;qDÞ, then the real di¤erential of order q of TT~ at ðf0;g0Þ is delivered by the formula

dqTT~½f0;g0ððv½1;w½1Þ;. . .;ðv½q;w½qÞÞ

¼ Xq

j¼1

ðvðq1Þ½j g0Þw½1. . .wwd½½jj. . .w½q

!

þ ðf0ðqÞg0Þw½1. . .w½q ð2:10Þ

for all ððv½1;w½1Þ;. . .;ðv½q;w½qÞÞAðCmþr;a;0ðqD;CÞ Cm;bðqD;CÞÞq, where the

‘b’ symbol on a factor denotes that such factor should not appear in the product.

Proof. We first prove that TT~ is of class Cr. It clearly su‰ces to show that given ðf#;g#ÞACmþr;a;0ðqD;CÞ Cm;bðqD;RDÞ, the map TT~ is of class Cr in an open neighborhood of ðf#;g#Þ. Now we set Ce1fzAC: j jzj 1j<eg for all e>0. By uniform continuity of E½g# on clðRDÞ and by the inclusion E½g#ðqDÞJRD, there exists e>0 such that E½g#ðclCeÞJ RD. Clearly, W#1fgACm;bðqD;RDÞ:E½gðclCeÞJRDg is an open neighborhood of g# in Cm;bðqD;CÞ. By [16, Thm. 5.3] and Lemma 2.1, T is of class Cr from Cmþr;a;0ðclðRDÞ;CÞ Cm;bðclCe;RDÞ to Cm;gmða;bÞðclCe;CÞ.

Furthermore, the restriction operator is easily seen to be linear and contin- uous from Cm;gmða;bÞðclCe;CÞ to Cm;gmða;ðqD;CÞ (for example, by arguing as in [18, Lem. 2.8 (ii)].) Thus, TT~ is of class Cr from Cmþr;a;0ðqD;CÞ W# to Cm;gmða;bÞðqD;CÞ. Formula (2.10) follows by formula (2.5) and by the formula for the derivatives of T of [16, Rmk. 5.4]. By definition of the space Cm;b;0ðclCe;CÞ and by continuity of T from Cm;a;0ðclðRDÞ;CÞ Cm;bðclCe;RDÞ to Cm;gmða;bÞðclCe;CÞ, and by Lemma 2.4 (i), T maps Cm;a;0ðqD;CÞ Cm;b;0ðqD;qDÞ to Cm;gmða;bÞ;0ðqD;CÞ. r We now turn to the study of the inversion operator by showing the validity of the following variant of [15, Thm. 5.9].

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Theorem 2.11. Let mANnf0g, rAN,aA0;1. Let J be the operator of Cmþr;a;0ðqD;qDÞVAqD to Cm;aðqD;qDÞdefined by equality J½f1fð1Þ, for all f ACmþr;a;0ðqD;qDÞVAqD. If r¼0, thenJis continuous and the image ofJis contained in Cm;a;0ðqD;qDÞ. If rb1, then for all f0ACmþr;a;0ðqD;qDÞVAqD, there exist an open neighborhood Wf0 of f0 in Cmþr;a;0ðqD;CÞVAqD, and an operator JJ~f0 of class Cr in the real sense from Wf0 to Cm;a;0ðqD;CÞ, such that

J~

Jf0½f ¼J½f; Ef AWf0VCmþr;a;0ðqD;qDÞ: ð2:12Þ Furthermore, the following formula holds

dJJ~f0½fðwÞ ¼ ðf0fð1ÞÞ1wfð1Þ; EwACmþr;a;0ðqD;CÞ; ð2:13Þ for all f AWf0VCmþr;a;0ðqD;qDÞ.

Proof. We first consider case r¼0. By Lemma 2.3 (v), (vii), the following inclusion holds JðCm;a;0ðqD;qDÞVAqDÞJCm;aðqD;CÞ. We now prove that J is continuous from Cm;a;0ðqD;qDÞVAqD to Cm;aðqD;CÞ.

To do so, we proceed by induction on m. Let m¼1, limj!y fj¼ f in C1;a;0ðqD;qDÞVAqD. Since the sequence ffjgjAN converges uniformly to f in qD, and qD is compact, a simple contradiction argument shows that limj!y fjð1Þ¼ fð1Þ pointwise on qD. By Lemma 2.2 (ii), (iii), there exists d>0 such that minqDjfj0jbd, minqDjf0jbd. Then a simple compu- tation shows that supjANkfjð1ÞkC1;a

ðqD;CÞ<y. Since C1;aðqD;CÞis compactly imbedded in C0;1ðqD;CÞ, and limj!y fjð1Þ¼ fð1Þ pointwise on qD, a simple contradiction argument shows that limj!y fjð1Þ¼ fð1Þ in C0;1ðqD;CÞ.

Then by Theorem 2.9, case r¼0, and by Lemma 2.3 (iii), we have limj!y½fj0ðfjð1ÞÞ1¼ ½f0ðfð1ÞÞ1 inC0;aðqD;CÞ. Since limj!y fjð1Þ¼ fð1Þ in C0;1ðqD;CÞ, we conclude that limj!y fjð1Þ¼ fð1Þ in C1;aðqD;CÞ. Now let the statement be true for mb1 and limj!y fj¼ f inCmþ1;a;0ðqD;CÞ. By case m we have limj!y fjð1Þ¼ fð1Þ in Cm;aðqD;CÞ. Then by Lemma 2.3 (iii), by Theorem 2.9, by the limiting relation limj!y fj0¼ f0 in Cm;a;0ðqD;CÞ, and by equality ½fjð1Þ0¼ ½fj0ðfjð1ÞÞ1, we conclude that the sequence f½fjð1Þ0gjAN converges to ½fð1Þ0 in Cm;aðqD;CÞ, and the proof of case r¼0 is complete. We now prove that JðCm;a;0ðqD;qDÞVAqDÞJCm;a;0ðqD;qDÞ.

Let f ACm;a;0ðqD;qDÞVAqD. Then there exists a sequence ffjgjAN in CyðqD;CÞVAqD such that limj!y fj¼ f in Cm;aðqD;CÞ. Since for j su‰ciently large we have jffj

jj ACyðqD;qDÞVAqD, then we have J jffj

jj

h i A CyðqD;qDÞVAqD for the same j’s. Now let wACyðR2;R2Þ be such that wðxÞ ¼xjxj1, for jxjb1=2. By [16, Thm. 5.3], and by Lemma 2.1, and by the continuity of the restriction to qD (cf. e.g., [18, Lem. 2.8]), we have

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limj!y fj

jfjj¼limj!y wðE½fjÞjqD¼wðfÞ ¼ f in Cm;aðqD;CÞ, with E as in Lemma 2.4. Then the continuity of J implies that J½fACm;a;0ðqD;qDÞ.

We now consider case rb1. We take R>1 and we define the operator Y of ðCmþr;a;0ðqD;CÞVAqDÞ Cm;a;0ðqD;RDÞ to Cm;a;0ðqD;CÞ by setting Y½f;g ¼ ðE½fÞ gidqD, where idqD denotes the identity map in qD. By Theorem 2.9, the operator Y is of class Cr. We now observe that for all f ACmþr;a;0ðqD;CÞVAqD such that fðqDÞ ¼qD, we have Y½f;g ¼0 if g¼ fð1Þ. We now apply the Implicit Function Theorem to equation Y½f;g ¼0 around the pair ðf0;f0ð1ÞÞ. By Theorem 2.9, the partial di¤er- ential of Y at ðf0;f0ð1ÞÞ with respect to the variable g is defined by dgY½f0;f0ð1ÞðhÞ ¼ f00ðf0ð1ÞÞh, for all hACm;a;0ðqD;CÞ. Since f00 belongs to Cmþr1;a;0ðqD;CÞ, which is contained in Cm;a;0ðqD;CÞ, and since f00ðtÞ00 for all tAqD, then Lemma 2.3 (ii), (iii) and Theorem 2.9 imply that dgY½f0;f0ð1Þ is a linear isomorphism of Cm;a;0ðqD;CÞ. Thus the Implicit Function Theorem implies the existence of an open neighborhood Wf0 of f0 in Cmþr;a;0ðqD;CÞVAqD, and of an open neighborhood Vfð1Þ

0

of f0ð1Þ in Cm;a;0ðqD;CÞ, and of a map JJ~f0 of class Cr from Wf0 to Vfð1Þ

0

such that the graph of JJ~f0 coincides with the set of zeros of Y in Wf0Vfð1Þ

0

. By the obvious inclusion of Cmþr;a;0ðqD;CÞ in Cm;a;0ðqD;CÞ, and by case r¼0,J is continuous onWf0VCmþr;a;0ðqD;qDÞ. Then by possibly shrinkingWf0, we can assume that Jmaps Wf0VCmþr;a;0ðqD;qDÞto Vfð1Þ

0

. Since Y½f;J½f ¼0, for all f AWf0VCmþr;a;0ðqD;qDÞ, we conclude that (2.12) holds. The validity of the formula (2.13) for the first di¤erential follows from formula (2.10) and by

the Implicit Function Theorem. r

If fAC1ðqD;CÞ, and if f is a function of L¼fðqDÞ to C, then we denote by Ð

f fðsÞds the line integral of the function f computed with respect to the parametrization y7!fðeiyÞ, with yA½0;2p, of fðqDÞ. Let fAAqD. We denote by ind½f the index of the curve y7!fðeiyÞ, yA½0;2p with respect to any of the points of I½f. Thus ind½f1 1

2pi

Ð

f dx

xz, for all zAI½f. The map ind½ is obviously constantly equal to 1 or to 1 on the open connected components of AqD in C1ðqD;CÞ. We set AqDþ1ffAAqD:ind½f>0g. The following Theorem collects known facts related to singular integrals with Cauchy kernels and to Cauchy type integrals.

Theorem 2.14. Let aA0;1½, mAN, fAC1;aðqD;CÞVAqDþ, L¼fðqDÞ.

Let Ibe the identity operator in C1;aðL;CÞ. Then the following statements hold.

( i ) For all f ACm;aðL;CÞ, the singular integral

Sf½fðtÞ1 1 pi

ð

f

fðsÞ

stds; EtAL; ð2:15Þ

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exists in the sense of the principal value, and Sf½fðÞACm;aðL;CÞ. The operator Sf defined by (2.15) is linear and continuous from Cm;aðL;CÞ to itself. If f coincides with the identity map idqD, then we set S1Sf. ( ii ) For all f ACm;aðL;CÞ, the function f½f of CnfLg to C defined by

f½fðzÞ1 1 2pi

ð

f

fðsÞ

szds; EzACnL;

is holomorphic. The function f½fjI½f admits a continuous extension to clI½f, which we denote by þf½f, and the function f½fjE½f admits a continuous extension toclE½f, which we denote byf½f. Then we have þf½fACm;aðclI½f;CÞVHðI½fÞ, f½fAC0ðclE½f;CÞVCm;aðL;CÞV HðE½fÞ, and the Plemelj formulas Gf½fðtÞ ¼G1

2fðtÞ þ12Sf½fðtÞ for all tAL hold. Furthermore, þf½defines a linear and continuous operator of Cm;aðL;CÞ to Cm;aðclI½f;CÞ. If f coincides with the identity map idqD, then we set 1f.

(iii) The function f ACm;aðL;CÞ satisfies equation ðISfÞ½f ¼0, if and only if there exists a function F ACm;aðclI½f;CÞVHðI½fÞ such that FðtÞ ¼ fðtÞ, for all tAL. The function F, if it exists, is unique.

(iv) The function f ACm;aðL;CÞsatisfies equation ðIþSfÞ½f ¼0, if and only if there exists a function F AC0ðclE½f;CÞVCm;aðL;CÞVHðE½fÞ such thatlimz!y FðzÞ ¼0, and FðtÞ ¼ fðtÞ, for alltAL. The function F, if it exists, is unique.

( v ) If f AC1;aðL;CÞ, then ðSf½fÞ0¼Sf½f0.

For case m¼0 of statements (i) and (ii), we refer to Hackbusch [8, Thm.

7.2.5]. Statement (v) follows by Gakhov [6, p. 30]. Casem>0 of statements (i) and (ii) follows by case m¼0 and by statement (v). Statements (iii) and (iv) follow by Gakhov [6, p. 27] together with statement (ii).

3. Di¤erentiability properties of the conformal welding operator

As we have said in the introduction, the conformal welding map is a composite function of Riemann maps. Thus we introduce the following Theorem, which summarizes some well-known properties of Riemann maps.

For a proof we refer to Ahlfors [1, Ch. 6, § 1] together with Pommerenke [26, Thms. 2.6, 3.5, 3.6].

Theorem 3.1. Let mANnf0g, aA0;1½. Let zACm;aðqD;CÞVAqD. Then the following statements hold.

( i ) There exists a unique homeomorphism g½zAC1ðCnD;CÞVHðCnclDÞ of CnD onto clE½z, such that g½zðyÞ1limz!yg½zðzÞ ¼y, g½z0ðyÞ1 limz!y g½z0ðzÞA0;þy½. Furthermore g½zjqDACm;aðqD;CÞVAqDþ.

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( ii ) Let a1;a2;a3 be three distinct points ofqD. Letzbe orientation preserving.

There exists a unique homeomorphism f½zACm;aðclD;CÞVHðDÞof clD onto clI½z such that f½zðajÞ ¼zðajÞ, for all j¼1;2;3. Furthermore,

f½zjqDAAqDþ.

(iii) For all bAI½z, there exists a unique homeomorphism ff~½z;b of clD onto clI½zsuch that ff~½z;bACm;aðclD;CÞVHðDÞand such that ff~½z;bð0Þ ¼b,

ff~½z;b0ð0ÞA0;þy½. For short, we set ff~½z ¼ff~½z;0 if b¼0.

We note that the map f½z of Theorem 3.1 (ii) depends also on a1;a2;a3. However, throughout the paper, we will assume the three points a1;a2;a3 to be fixed. Thus we have chosen not to display the dependence on a1;a2;a3 in the notation for f½z. By Lemma 2.3 (iv), (v), and by Theorem 3.1, the function f½g½zjqDð1Þg½zjqD belongs to Cm;aðqD;qDÞVAqDþ, for all zACm;aðqD;CÞV AqD, with mANnf0g, aA0;1½. Then we can introduce the following.

Definition 3.2. Let mANnf0g, aA0;1½. Let a1;a2;a3 be three distinct points of qD. If zACm;aðqD;CÞVAqD, then we define as conformal welding map associated to z (and to the triple ða1;a2;a3Þ), the map

w½z1f½g½zjqDð1Þg½zjqD: ð3:3Þ We define as conformal welding operator, the operator w½of Cm;aðqD;CÞVAqD

to Cm;aðqD;qDÞVAqDþ, which takes z to w½z.

Clearly, one can define the conformal welding map by normalizing f½z and g½z in a di¤erent way, and the corresponding w½z would di¤er from the w½z defined above by a suitable composition with Mo¨bius transformations.

Now we introduce the following Theorem, whose first statement has been proved in Lanza and Rogosin [20, Thm. 5.4]. Both statements can be con- sidered as a variant of [17, Thm. 3.10, Thm. 4.7].

Theorem 3.4. Let mANnf0g, aA0;1½. Then the following statements hold.

( i ) The map ðz;bÞ 7!ff~½z;bð1Þz is real analytic from

Em;a1fðz;bÞAðCm;aðqD;CÞVAqDÞ C:bAI½zg to Cm;aðqD;CÞ.

(ii) Let rAN. The map ðz;bÞ 7!ff~½z;b of Emþr;aVðCmþr;a;0ðqD;CÞ CÞ to Cm;a;0ðqD;CÞ is of class Cr.

Proof. Statement (i) is contained in Lanza and Rogosin [20, Thm. 5.4].

By Theorem 3.1, we deduce that ff~½z;bACy ðqD;CÞ if zACyðqD;CÞVAqD, bAI½z. Then by statement (i), we have ff~½z;bð1ÞzACm;a;0ðqD;CÞ if zA Cm;a;0ðqD;CÞVAqD,bAI½z. Then statement (ii) follows by statement (i), and

by Theorems 2.9, 2.11. r

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We now turn to consider the dependence of f½zð1Þzupon z. Since f½z has been normalized in a di¤erent way from ff~½z;b, we need the following Lemma in order to exploit the previous Theorem.

Lemma 3.5. Let a1;a2;a3 be three distinct points of qD. Let A1fðz;p1;p2;p3ÞAC4:p1;p2;p3 are distinct;

ðzp3Þðp1p2Þða1a3Þ ðzp2Þðp1p3Þða1a2Þ00g:

Let Q be the rational function of A to C defined by

Qðz;p1;p2;p3Þ1a2ðzp3Þðp1p2Þða1a3Þ a3ðzp2Þðp1p3Þða1a2Þ ðzp3Þðp1p2Þða1a3Þ ðzp2Þðp1p3Þða1a2Þ ; for all ðz;p1;p2;p3ÞAA. If p1;p2;p3 are three distinct points of qD, and if the triple ðp1;p2;p3Þ induces onqDthe same orientation of the triple ða1;a2;a3Þ, then ðz;p1;p2;p3ÞAA for all zAclD, and Qð;p1;p2;p3Þ is the unique homeo- morphism ofclDonto itself which is holomorphic in Dand which maps pj to aj, for j¼1;2;3.

Proof. By elementary Conformal Mapping Theory (cf. e.g., Ahlfors [1, p. 79]), the function Qð;p1;p2;p3Þ is the only linear fractional transfor- mation which maps pj to aj. Since p1;p2;p3 are three distinct points of qD, the function Qð;p1;p2;p3Þ is well-known to map qD onto itself. If the triple ðp1;p2;p3Þ induces on qD the same orientation of ða1;a2;a3Þ, then Qð;p1;p2;p3Þ is well-known to be a bijection of clD onto itself. The uniqueness follows by the Riemann Mapping Theorem. r We are now ready to prove the following.

Theorem 3.6. Let mANnf0g, aA0;1½. Let a1;a2;a3 be three distinct points of qD. Then the nonlinear operator z7! f½zð1Þzis real analytic from Cm;aðqD;CÞVAqDþ to Cm;aðqD;qDÞVAqDþ, and maps Cm;a;0ðqD;CÞVAqDþ to Cm;a;0ðqD;qDÞVAqDþ.

Proof. Let z0ACm;aðqD;CÞVAqDþ, z0AI½z0. Let W be an open neighborhood of z0 in Cm;aðqD;CÞVAqDþ such that z0AI½z, for all zAW.

By the uniqueness inferred by the Riemann Mapping Theorem, we have f½zz0 ¼ f½z z0. Thus there is no loss of generality in assuming that 0¼z0AI½z, for all zAW. By Lemma 3.5, and by the uniqueness inferred by the Riemann Mapping Theorem, we have

f½zð1ÞzðÞ ¼Qðff~½zð1ÞzðÞ;ff~½zð1Þzða1Þ;ff~½zð1Þzða2Þ;ff~½zð1Þzða3ÞÞ;

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where Q is as in Lemma 3.5. Thus the analyticity of f½zð1ÞzðÞ on z follows by Theorem 3.4 (i), and by Lemmas 2.3 (ii), (iii) and 3.5. By Theorem 3.1, we have f½zð1ÞzACyðqD;qDÞ if zACy ðqD;CÞVAqDþ. Then the last statement follows by the continuity ofz7! f½zð1Þz, and by the definition of

the spaces Cm;a;0. r

We now show that on a suitable subset Tm;a of Cm;aðqD;CÞVAqDþ we have g½zjqD¼z, for all zATm;a. Thus the conformal welding operator coincides with the nonlinear operator of the previous Theorem on Tm;a.

Proposition 3.7. Let mANnf0g, aA0;1½. Let Dm;a1

zACm;aðqD;CÞ:bZAHðCnclDÞVC0ðCnD;CÞ

such that z¼ZjqD;lim

z!yZ0ðzÞ1Z0ðyÞAR

: ð3:8Þ

Then the following statements hold.

( i ) If zADm;a, then the map Z of (3.8) is unique. Furthermore, Z0ðyÞ ¼

1 2pi

Ð

qD zðsÞ

s2 ds, and limz!y ZðzÞ 2piz Ð

qD zðsÞ

s2 ds2pi1 Ð

qD zðsÞ

s ds

n o

¼0.

( ii ) Dm;a is a real Banach subspace of Cm;aðqD;CÞ.

(iii) Let Z½z be the unique map of statement (i) corresponding to zADm;a. Then the set

Tm;a1fzADm;aVAqDþ :Z½z0ðyÞ>0g is open in Dm;a.

(iv) If zATm;a, then limz!y Z½zðzÞ ¼y, and Z½z ¼g½z.

Proof. We first prove statement (i). If ZAHðCnclDÞV C0ðCnD;CÞ, then standard properties of holomorphic functions imply that condition limz!yZ0ðzÞ1Z0ðyÞAC is equivalent to the existence of a;bAC and KAHðCnclDÞVC0ðCnD;CÞ such that limz!yKðzÞ ¼0 and ZðzÞ ¼azþbþKðzÞ. Moreover, it is easily checked that if such condi- tion holds, then Z0ðyÞ ¼a¼2pi1 Ð

qD ZðsÞ

s2 ds, b¼2pi1 Ð

qD ZðsÞ

s ds. Then by

Theorem 2.14 (iv), the membership of z in Dm;a is equivalent to condi- tion ðIþSÞ zðzÞ 2piz Ð

qD zðsÞ

s2 ds2pi1 Ð

qD zðsÞ

s ds

h i

¼0 together with condition

1 2pi

Ð

qD zðsÞ

s2 dsAR. Thus the uniqueness of Z follows. The completeness of Dm;a follows by the same argument, by Theorem 2.14 (i), and by the con- tinuous dependence of 2pi1 Ð

qD zðsÞ

s ds and of 2pi1 Ð

qD zðsÞ

s2 ds on zACm;aðqD;CÞ.

Statement (iii) follows by Lemma 2.2 (ii) and by the continuous dependence of Z0ðyÞ ¼2pi1 Ð

qD zðsÞ

s2 ds on zADm;a. We now prove statement (iv). By (i),

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we have limz!y Z½zðzÞ ¼y. Now let o0AI½z. Since Z½z0ðyÞ00, then the function Z½z is injective in a neighborhood of y. Since we also have zAAqDþ and o0AI½z, then a standard argument based on the Argument Principle shows that Z½zðÞ o0 does not vanish. Thus the function

½Z½zð1=ðÞÞ o01 extends to a holomorphic map in D. Since zAAqDþ, then the curve ½zð1=ðÞÞ o01 is one to one. Then again by the Argument Principle, ½Z½zð1=ðÞÞ o01 is one to one in clD, and a simple topological argument shows that ½Z½zð1=ðÞÞ o01 maps D onto I½½zð1=ðÞÞ o01. Accordingly, Z½z is one to one and Z½zðCnclDÞ ¼CnclI½z. Since Z½z0ðyÞ>0, Theorem 3.1 (i) implies that Z½z ¼g½z. r As an immediate Corollary of Theorem 3.6 and of Proposition 3.7, we obtain the following.

Theorem 3.9. Let mANnf0g, aA0;1½. Then the conformal welding operator is real analytic from the set Tm;a to Cm;aðqD;qDÞVAqDþ.

We now turn to the di¤erentiability properties of g½, by means of the following.

Theorem 3.10. Let mANnf0g, aA0;1½, rAN. Then the map which takes zto g½zjqD is of class Cr in the real sense from Cmþr;a;0ðqD;CÞVAqD to Cm;a;0ðqD;CÞVAqDþ.

Proof. Let z0ACmþr;a;0ðqD;CÞVAqD, z0AI½z0. Let W be an open neighborhood of z0 in Cmþr;a;0ðqD;CÞVAqD such that z0AI½z, for all zAW. By the uniqueness inferred by the Riemann Mapping Theorem, we have g½zz0 ¼g½z z0. Thus there is no loss of generality in assuming that z0¼0, for all zAW. Let ff~½ be as in Theorem 3.1. Clearly, g½zðzÞ ¼

½ff~½1=zð1=zÞ1, for all zACnclD, and for all zAW. In particular, Theorem 3.1 implies that g½zjqDACyðqD;CÞ if zACyðqD;CÞVAqD. Thus we can conclude the proof by Lemma 2.3 (iii), by Theorem 2.9, and by Theorem

3.4 (ii). r

As a consequence of Theorem 3.6, and of Theorem 3.10, we obtain the following.

Theorem 3.11. Let mANnf0g, aA0;1½, rAN. Then the conformal welding operator maps Cm;aðqD;CÞVAqD to Cm;aðqD;qDÞVAqDþ and is of class Cr in the real sense from Cmþr;a;0ðqD;CÞVAqD to Cm;a;0ðqD;qDÞVAqDþ. We note that it can be proved that Theorem 3.11 is optimal in the frame of Schauder spaces (cf. [19, Thm. 2.14].)

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