### 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 F^{ð1Þ}G_{jqD} of qD,
which we denote by w½z. Now let C_{}^{m;}^{a}ðqD;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 zAC_{}^{m;a}ðqD;CÞVAqD, then w½zAC_{}^{m;a}ðqD;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.

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 F^{ð1Þ}GjqD¼f 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 fAC_{}^{m;a}ðqD;CÞV
AqD, one can associate a unique suitably normalized pair of functions ðG;FÞas
above. Then the nonlinear operator s, which takes f to s½f1G_{jqD} is a right
inverse of w, and will be called the ‘conformal sewing operator’. Next we
prove that s½fAC_{}^{m;}^{a}ðqD;CÞ if the shift fAC_{}^{m;a}ðqD;CÞVAqD. Then we
analyze the di¤erentiability properties ofs. Since the domain ofs, namely the
set of positively oriented fAC_{}^{m;a}ðqD;CÞVAqD such that fðqDÞ ¼qD is not
open in the Banach space C_{}^{m;a}ðqD;CÞ, we construct an extension of s to the
open set of orientation preserving elements of C_{}^{m;a}ðqD;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 F_{jqD}¼Gf^{ð1Þ}¼s½f f^{ð1Þ}. 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 C_{}^{1;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 ff_{t}g 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
s½f_{t} and s½f_{t} f_{t}^{ð1Þ}, under the assumption that such expansions converge.

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 f^{ð1Þ},
as opposed to the reciprocal of a complex valued function g, which is denoted
g^{1}. For all subsetsBofR^{n}, the closure ofBis denoted clB. We now deﬁne
the Schauder spaces on the closure of an open subset of R^{n}. Let Wbe an open
subset of R^{n}, mAN. We denote by C^{m}ðW;CÞ the space of m-times contin-
uously real-di¤erentiable complex-valued functions on W, and by C^{m}ðclW;CÞ
the subspace of those functions of C^{m}ðW;CÞsuch that for all h1ðh_{1};. . .;h_{n}ÞA
N^{n}, with jhj1h_{1}þ þh_{n}am, the function D^{h}f1 ^{q}^{jhj}^{f}

q^{h}^{1}x1...q^{hn}xn can be
extended with continuity to clW. If Wis bounded, then C^{m}ðclW;CÞendowed
with the norm deﬁned by kfkC^{m}ðclW;CÞ1P

jhjamsup_{cl}_{W}jD^{h}fj is a Banach
space. If W is bounded and if aA0;1, we denote by C^{m;}^{a}ðclW;CÞ the
subspace of C^{m}ðclW;CÞ of those functions which have a-Ho¨lder contin-

uous derivatives of order m. If f AC^{0;}^{a}ðclW;CÞ, then we set jf :Wja1
supn^{j}^{f}^{ðxÞf}_{jxyj}^{ð}a^{yÞj}:x;yAclW;x0yo

. The space C^{m;}^{a}ðclW;CÞ is endowed with
its usual norm kfk_{C}^{m;}^{a}ðclW;CÞ1P

jhjamsup_{cl}_{W}jD^{h}fj þP

jhj¼mjD^{h}f :Wja, and it
is well-known to be a Banach space. If BJC, then C^{m;a}ðclW;BÞ denotes
the set ff AC^{m;a}ðclW;CÞ:fðclWÞJBg. By HðWÞ we understand the space
of holomorphic functions of W to C. Finally, the space C^{m;}^{a;0}ðclW;CÞ is
deﬁned as the closure of C^{y}ðclW;CÞ in C^{m;a}ðclW;CÞ. Then we have the
following.

Lemma 2.1. Let mAN, aA0;1. Let W be a bounded open con-
nected subset of R^{n} of class C^{mþ1}. Then C^{m;}^{a;0}ðclW;CÞ coincides with the
closure in C^{m;a}ðclW;CÞ of the set of restrictions to clW of the polynomials with
complex coe‰cients in n real variables. Moreover, C^{m;a;0}ðclW;CÞ contains
C^{mþ1}ðclW;CÞ and C^{m;b}ðclW;CÞ, for all bAa;1.

Proof. Since W is of class C^{mþ1}, then all functions of C^{mþ1}ðclW;CÞ are
restrictions of some element of C^{mþ1}ðR^{n};CÞ (cf. e.g., Troianiello [30, p. 13].)
Then by Weierstrass Theorem (cf. e.g., Rohlin and Fuchs [29, p. 185]), all
elements of C^{mþ1}ðR^{n};CÞ can be approximated in the C^{mþ1}ðclW;CÞ-norm by
polynomials. Since clW is of class C^{mþ1}, then C^{mþ1}ðclW;CÞ is continu-
ously imbedded in C^{m;}^{a}ðclW;CÞ (cf. e.g., [15, p. 460].) Then the ﬁrst part of
the statement and the inclusion C^{mþ1}ðclW;CÞJC^{m;a;0}ðclW;CÞ follow. Now
let f AC^{m;}^{b}ðclW;CÞ. Since W is of class C^{mþ1}, then f admits an extension
of class C^{m;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 C^{y} functions
bounded in C^{m;}^{b}ðclW;CÞ and convergent in C^{m;a}ðclW;CÞ(cf. e.g., Troianiello
[30, pp. 20, 21].) Then f AC^{m;a;}^{0}ðclW;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 f^{0}ðz0Þ. As usual the higher order derivatives, if they exist, are deﬁned
inductively. Let mAN. We denote by C_{}^{m}ðK;CÞ the complex normed space
of the m-times continuously complex di¤erentiable functions f of K to C
endowed with the norm kfkC_{}^{m}ðK;CÞ¼Pm

j¼0sup_{K}jf^{ð}^{jÞ}j. If aA0;1, we denote
by C_{}^{m;}^{a}ðK;CÞ the subspace of C_{}^{m}ðK;CÞ of those functions having a-
Ho¨lder continuous m-th order derivative in K. If f AC_{}^{0;a}ðK;CÞ, then we set
jf :Kja1sup ^{j}^{f}^{ðz}_{jz}^{1}^{Þf}^{ðz}^{2}^{Þj}

1z2j^{a} :z1;z2AK;z10z2

n o

. We endowC_{}^{m;a}ðK;CÞwith the
norm kfkC_{}^{m;a}ðK;CÞ1kfkC_{}^{m}ðK;CÞþ jf^{ðmÞ}:Kja. If BJC, we set C_{}^{m;}^{a}ðK;BÞ1

ff AC_{}^{m;a}ðK;CÞ:fðKÞJBg. We denote by C_{}^{m;}^{a;0}ðK;CÞ the closure of
C_{}^{y}ðK;CÞ in C_{}^{m;a}ðK;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 fAC_{}^{1}ðqD;CÞ. Then l_{qD}½f1infn^{jfðxÞfðyÞj}_{jxyj} :x;yAqD;x0yo

>0 if
and only if f is injective and f^{0}ðxÞ00 for all x in qD.

( ii ) The function of C_{}^{1}ðqD;CÞ to R which maps f to lqD½f is continuous,
and in particular, the set AqD1ffAC_{}^{1}ðqD;CÞ:l_{qD}½f>0g is open in
C_{}^{1}ðqD;CÞ.

(iii) min_{x}AqDjf^{0}ðxÞjbl_{qD}½f, for all fAC_{}^{1}ðqD;CÞ.

We are now ready to state the following, which collects a few facts which
we need on the spaces C_{}^{m;}^{a}ðK;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 ) C_{}^{mþ1}ðL;CÞ is continuously imbedded in C_{}^{m;a}ðL;CÞ. If a<b, then
C_{}^{m;b}ðL;CÞ is compactly imbedded in C_{}^{m;a}ðL;CÞ.

( ii ) The pointwise product is continuous in the Banach space C_{}^{m;a}ðL;CÞ.

( iii ) The reciprocal map in C_{}^{m;a}ðL;CÞ, which maps a nonvanishing function f to
its reciprocal, is complex analytic from C_{}^{m;a}ðL;Cnf0gÞ to itself.

( iv ) Let f_{1}AAqD, L1¼f_{1}ðqDÞ. If f AC_{}^{m;a}ðL1;CÞ and if gAC_{}^{m;}^{b}ðL;L1Þ,
then f gAC^{m;}^{g}^{m}^{ða;bÞ}ðL;CÞwith g_{0}ða;bÞ ¼ab and g_{m}ða;bÞ ¼minfa;bg if
m>0.

( v ) Let mb1. If gAC_{}^{m;a}ðL;CÞis injective and satisﬁes condition g^{0}ðxÞ00,
for all xAL, then g^{ð1Þ}AC_{}^{m;a}ðgðLÞ;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 RD1fxAR^{2}:
jxj<Rg. Then there exists a linear and continuous extension operator E of
C_{}^{m;a}ðqD;CÞ to C^{m;}^{a}ðclðRDÞ;CÞ such that the following statements hold.

( i ) ðE½fÞ_{jqD}¼ f , for all f AC_{}^{m;a}ðqD;CÞ, and E½fAC^{m;a;0}ðclðRDÞ;CÞ for
all f AC_{}^{m;}^{a;0}ðqD;CÞ.

(ii) Let 1ajam. For all f AC_{}^{m;a}ðqD;CÞ, the real di¤erential of order j of
the function E½f at tAqD satisﬁes the following equation

d^{j}E½fðtÞðs1;. . .;s_{j}Þ ¼ f^{ð}^{jÞ}ðtÞs1. . .s_{j}; ð2:5Þ
for all ðs1;. . .;sjÞAC^{j}. In particular, d^{j}E½fðtÞ 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

l¼0C_{}^{ml;}^{a}ðqD;CÞ to C^{m;}^{a}ðclD;CÞ such
that Z½f_{jqD};. . .;^{q}^{m}^{Z½f}

qn^{m} jqD

¼f for all fA Qm

l¼0C_{}^{ml;a}ðqD;CÞ, where n is the
outer unit normal to qD. If r;sAf0;. . .;mg, then we set d_{rs}¼1 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 Z_{l} of C_{}^{ml;a}ðqD;CÞ
to C^{m;a}ðclD;CÞ such that ^{q}

jZl½h

qn^{j} jqD¼djll!h for 0ajal, and for all hA
C_{}^{ml;}^{a}ðqD;CÞ. By a standard argument based on the partition of unity and
on the use of local charts for qD, the existence of Z_{l} follows from that of
a linear and continuous operator ZZ~l of C^{ml;a}ð½1;1;CÞ to C^{m;a}ðclð1;

1½ 1;0½Þ;CÞsuch that^{q}^{j}^{Z}^{Z}^{~}^{l}^{½g}

qx_{2}^{j} jx2¼0¼d_{jl}l!g for 0ajal. LetKbe a linear and
continuous operator of C^{ml;a}ð½1;1;CÞtoC^{ml;a}ð½2;2;CÞwithK½g ¼gon

½1;1 and suppK½gJ2;2½, for all gAC^{ml;a}ð½1;1;CÞ. Furthermore,
one can choose K so that K maps C^{mlþ1}ð½1;1;CÞ to C^{mlþ1}ð½2;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 b_{0};. . .;b_{l} by solving the (Vandermonde) system Pl

s¼0a_{s}^{j}b_{s}¼
djll!, j¼0;. . .;l, and we set ZZ~l½gðx1;x2Þ1Pl

s¼0b_{s}Gl½gðx1þasx2Þ, where
G_{l}½g is the m times di¤erentiable function of R to C determined by condi-
tions ^{d}^{l}

dt^{l}G_{l}½g ¼K½g, _{dt}^{d}^{j}jjt¼0G_{l}½g 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

l¼0C_{}^{mþ1l}ðqD;CÞ to C^{mþ1}ðclD;CÞ. Since clDis of
class C^{y}, it is also known that there exists a linear and continuous exten-
sion operator E_{R} of C^{m;}^{a}ðclD;CÞ to C^{m;}^{a}ðclðRDÞ;CÞ such that E_{R}½v_{jcl}_{D}¼v,
for all vAC^{m;a}ðclD;CÞ. Furthermore, one can choose E_{R} so that E_{R} maps
C^{mþ1}ðclD;CÞto C^{mþ1}ðclðRDÞ;CÞ(cf. e.g., the construction of Troianiello [30,
Thm. 1.3, p. 13] with k¼mþ1.) Then we setE½f1E_{R}Z½fðtÞ;tf^{0}ðtÞ;. . .;
t^{m}f^{ðmÞ}ðtÞ. If f AC_{}^{y}ðqD;CÞ, then E½fAC^{mþ1}ðclðRDÞ;CÞ and thus E½fA
C^{m;}^{a;0}ðclðRDÞ;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 d^{j}E½fðtÞ is a real j-multilinear operator of R^{2j} to R^{2}. Also,
the right hand side of equation (2.5) delivers a complex j-multilinear oper-
ator, which we denote by M_{f}ðjÞðtÞ of C^{j} to C, and thus a real j-multilinear

operator of R^{2j} to R^{2}. In order to prove equality (2.5), it su‰ces to show
that d^{j}E½fðtÞ ¼M_{f}ðjÞðtÞ on the j-tuples of elements of a real basis of R^{2}.
To shorten our notation, we write v^{½l} instead of v;zﬄﬄﬄﬄ}|ﬄﬄﬄﬄ{^{l}. . .^{terms};v in the argu-
ment of a multilinear operator. Once t1ðt1;t2ÞAqD is ﬁxed, we choose
fðt1;t_{2}Þ;ðt2;t_{1}Þg as a real basis of R^{2}. Note that t equals the exterior unit
normal to qDat t, and that it¼ ðt2;t1Þlies in the tangent space to qDat t.

Since d^{j}E½fðtÞandM_{f}ðjÞðtÞ are multilinear and symmetric operators, it su‰ces
to check that for 0alaj, we have

d^{j}E½fðtÞððt2;t_{1}Þ^{½l};ðt1;t_{2}Þ^{½}^{jl}Þ ¼ f^{ð}^{jÞ}ðtÞðitÞ^{l}t^{jl}: 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;t_{2}ÞÞ ¼ f^{0}ðtÞt; dE½fðtÞððt2;t_{1}ÞÞ ¼ f^{0}ðtÞit: ð2:7Þ
The ﬁrst equality of (2.7) follows by equality _{qn}^{q}E½fðtÞ ¼ f^{0}ðtÞt, which holds
by construction ofE. We now turn to prove the second equality of (2.7). We
know that E½fðcosy;sinyÞ ¼ fðe^{iy}Þ, for all yA½0;2p. Then by di¤er-
entiating with respect to y, we obtain dE½fðcosy;sinyÞððsiny;cosyÞÞ ¼
f^{0}ðe^{iy}Þie^{iy}, 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 _{qn}^{q}^{jþ1}jþ1E½fðtÞ ¼t^{jþ1}f^{ð}^{jþ1Þ}ðtÞ, which holds by construction of E½f.
Thus we can assume that lb1. By inductive assumption, we have

d^{j}E½fðtÞððt2;t_{1}Þ^{½l1};ðt1;t_{2}Þ^{½}^{jlþ1}Þ ¼ f^{ð}^{jÞ}ðtÞðitÞ^{l1}t^{jlþ1}: ð2:8Þ
Now by setting t1ðt1;t2Þ ¼ ðcosy;sinyÞ in (2.8), and by di¤erentiating with
respect to y, we obtain

d^{jþ1}E½fðtÞððt2;t_{1}Þ^{½l};ðt1;t_{2}Þ^{½}^{jlþ1}Þ

þ ðl1Þd^{j}E½fðtÞððt1;t2Þ;ðt2;t1Þ^{½l2};ðt1;t2Þ^{½}^{jlþ1}Þ
þ ðjlþ1Þd^{j}E½fðtÞððt2;t_{1}Þ^{½l1};ðt2;t_{1}Þ;ðt1;t_{2}Þ^{½}^{jl}Þ

¼ f^{ð}^{jþ1Þ}ðtÞðitÞ^{l}t^{jlþ1}þf^{ð}^{jÞ}ðtÞðl1ÞðitÞ^{l2}ðtÞt^{jlþ1}
þf^{ð}^{jÞ}ðtÞðitÞ^{l1}ðjlþ1Þt^{jl}it:

By exploiting the symmetry, the real j-multilinearity of d^{j}E½fðtÞ, and the
inductive assumption, we obtain that (2.6) holds for jþ1, and for all

0alajþ1. r

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 g_{m}ða;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 C_{}^{mþr;a;}^{0}ðqD;CÞ C_{}^{m;}^{b}ðqD;RDÞto C^{m;}^{g}^{m}^{ða;bÞ}ðqD;CÞ
deﬁned by setting TT~½f;g1ðE½fÞ g, for all ðf;gÞAC_{}^{mþr;a;0}ðqD;CÞ
C_{}^{m;b}ðqD;RDÞ is of class C^{r} in the real sense. The restriction of TT~ to
C_{}^{mþr;}^{a;0}ðqD;CÞ C_{}^{m;b}ðqD;qDÞcoincides with the ordinary composition operator
Tdeﬁned byT½f;g1f g. The ordinary compositionTmaps C_{}^{m;a;}^{0}ðqD;CÞ
C_{}^{m;}^{b;0}ðqD;qDÞ to C^{m;}^{g}^{m}^{ða;bÞ;0}ðqD;CÞ. If rb1, qAf1;. . .;rg, and if ðf0;g0ÞA
C_{}^{mþr;}^{a;0}ðqD;CÞ C_{}^{m;b}ðqD;qDÞ, then the real di¤erential of order q of TT~ at
ðf0;g0Þ is delivered by the formula

d^{q}TT~½f0;g0ððv½1;w_{½1}Þ;. . .;ðv½q;w_{½q}ÞÞ

¼ X^{q}

j¼1

ðv^{ðq1Þ}_{½}_{j} g0Þw½1. . .wwd½½jj. . .w½q

!

þ ðf_{0}^{ðqÞ}g0Þw½1. . .w½q ð2:10Þ

for all ððv½1;w_{½1}Þ;. . .;ðv½q;w_{½q}ÞÞAðC_{}^{mþr;a;}^{0}ðqD;CÞ C_{}^{m;}^{b}ðqD;CÞÞ^{q}, 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 C^{r}. It clearly su‰ces to
show that given ðf^{#};g^{#}ÞAC_{}^{mþr;}^{a;0}ðqD;CÞ C_{}^{m;b}ðqD;RDÞ, the map TT~ is of
class C^{r} 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^{#}1fgAC_{}^{m;b}ðqD;RDÞ:E½gðclCeÞJRDg is an open
neighborhood of g^{#} in C_{}^{m;b}ðqD;CÞ. By [16, Thm. 5.3] and Lemma 2.1, T is
of class C^{r} from C^{mþr;a;0}ðclðRDÞ;CÞ C^{m;b}ðclCe;RDÞ to C^{m;}^{g}^{m}^{ða;bÞ}ðclCe;CÞ.

Furthermore, the restriction operator is easily seen to be linear and contin-
uous from C^{m;g}^{m}^{ða;bÞ}ðclCe;CÞ to C^{m;g}^{m}^{ða;}^{bÞ}ðqD;CÞ (for example, by arguing as
in [18, Lem. 2.8 (ii)].) Thus, TT~ is of class C^{r} from C_{}^{mþr;a;}^{0}ðqD;CÞ W^{#}
to C^{m;}^{g}^{m}^{ða;bÞ}ðqD;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 C^{m;}^{b;}^{0}ðclC_{e};CÞ and by continuity of T from C^{m;a;}^{0}ðclðRDÞ;CÞ
C^{m;}^{b}ðclCe;RDÞ to C^{m;}^{g}^{m}^{ða;bÞ}ðclCe;CÞ, and by Lemma 2.4 (i), T maps
C_{}^{m;}^{a;0}ðqD;CÞ C_{}^{m;}^{b;0}ðqD;qDÞ to C^{m;g}^{m}^{ða;}^{bÞ;}^{0}ðqD;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].

Theorem 2.11. Let mANnf0g, rAN,aA0;1. Let J be the operator of
C_{}^{mþr;a;}^{0}ðqD;qDÞVAqD to C_{}^{m;a}ðqD;qDÞdeﬁned by equality J½f1f^{ð1Þ}, for all
f AC_{}^{mþr;a;0}ðqD;qDÞVAqD. If r¼0, thenJis continuous and the image ofJis
contained in C_{}^{m;a;0}ðqD;qDÞ. If rb1, then for all f_{0}AC_{}^{mþr;}^{a;0}ðqD;qDÞVAqD,
there exist an open neighborhood Wf0 of f0 in C_{}^{mþr;a;0}ðqD;CÞVAqD, and
an operator JJ~_{f}_{0} of class C^{r} in the real sense from Wf0 to C_{}^{m;a;}^{0}ðqD;CÞ, such
that

J~

J_{f}_{0}½f ¼J½f; Ef AWf0VC_{}^{mþr;a;0}ðqD;qDÞ: ð2:12Þ
Furthermore, the following formula holds

dJJ~_{f}_{0}½fðwÞ ¼ ðf^{0}f^{ð1Þ}Þ^{1}wf^{ð1Þ}; EwAC_{}^{mþr;}^{a;0}ðqD;CÞ; ð2:13Þ
for all f AWf0VC_{}^{mþr;a;0}ðqD;qDÞ.

Proof. We ﬁrst consider case r¼0. By Lemma 2.3 (v), (vii),
the following inclusion holds JðC_{}^{m;a;}^{0}ðqD;qDÞVAqDÞJC_{}^{m;a}ðqD;CÞ. We
now prove that J is continuous from C_{}^{m;}^{a;0}ðqD;qDÞVAqD to C_{}^{m;}^{a}ðqD;CÞ.

To do so, we proceed by induction on m. Let m¼1, limj!y fj¼ f in
C_{}^{1;a;}^{0}ðqD;qDÞVAqD. Since the sequence ff_{j}g_{j}AN converges uniformly to
f in qD, and qD is compact, a simple contradiction argument shows
that limj!y f_{j}^{ð1Þ}¼ f^{ð1Þ} pointwise on qD. By Lemma 2.2 (ii), (iii), there
exists d>0 such that min_{qD}jf_{j}^{0}jbd, min_{qD}jf^{0}jbd. Then a simple compu-
tation shows that sup_{j}ANkf_{j}^{ð1Þ}k_{C}^{1;}^{a}

ðqD;CÞ<y. Since C_{}^{1;a}ðqD;CÞis compactly
imbedded in C_{}^{0;1}ðqD;CÞ, and limj!y f_{j}^{ð1Þ}¼ f^{ð1Þ} pointwise on qD, a simple
contradiction argument shows that limj!y f_{j}^{ð1Þ}¼ f^{ð1Þ} in C_{}^{0;}^{1}ðqD;CÞ.

Then by Theorem 2.9, case r¼0, and by Lemma 2.3 (iii), we have
lim_{j!}y½f_{j}^{0}ðf_{j}^{ð1Þ}Þ^{1}¼ ½f^{0}ðf^{ð1Þ}Þ^{1} inC_{}^{0;a}ðqD;CÞ. Since lim_{j!}y f_{j}^{ð1Þ}¼ f^{ð1Þ}
in C_{}^{0;1}ðqD;CÞ, we conclude that limj!y f_{j}^{ð1Þ}¼ f^{ð1Þ} in C_{}^{1;}^{a}ðqD;CÞ. Now
let the statement be true for mb1 and limj!y f_{j}¼ f inC_{}^{mþ1;}^{a;0}ðqD;CÞ. By
case m we have limj!y f_{j}^{ð1Þ}¼ f^{ð1Þ} in C_{}^{m;a}ðqD;CÞ. Then by Lemma 2.3
(iii), by Theorem 2.9, by the limiting relation limj!y f_{j}^{0}¼ f^{0} in C_{}^{m;a;}^{0}ðqD;CÞ,
and by equality ½f_{j}^{ð1Þ}^{0}¼ ½f_{j}^{0}ðf_{j}^{ð1Þ}Þ^{1}, we conclude that the sequence
f½f_{j}^{ð1Þ}^{0}g_{j}AN converges to ½f^{ð1Þ}^{0} in C_{}^{m;a}ðqD;CÞ, and the proof of case r¼0
is complete. We now prove that JðC_{}^{m;}^{a;0}ðqD;qDÞVAqDÞJC_{}^{m;a;}^{0}ðqD;qDÞ.

Let f AC_{}^{m;a;}^{0}ðqD;qDÞVAqD. Then there exists a sequence ff_{j}g_{j}AN in
C_{}^{y}ðqD;CÞVAqD such that limj!y fj¼ f in C_{}^{m;a}ðqD;CÞ. Since for j
su‰ciently large we have _{j}^{f}_{f}^{j}

jj AC_{}^{y}ðqD;qDÞVAqD, then we have J _{j}^{f}_{f}^{j}

jj

h i
A
C_{}^{y}ðqD;qDÞVAqD for the same j’s. Now let wAC^{y}ðR^{2};R^{2}Þ be such that
wðxÞ ¼xjxj^{1}, 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

limj!y fj

jfjj¼limj!y wðE½fjÞ_{jqD}¼wðfÞ ¼ f in C_{}^{m;a}ðqD;CÞ, with E as in
Lemma 2.4. Then the continuity of J implies that J½fAC_{}^{m;a;}^{0}ðqD;qDÞ.

We now consider case rb1. We take R>1 and we deﬁne the operator
Y of ðC_{}^{mþr;a;0}ðqD;CÞVAqDÞ C_{}^{m;a;}^{0}ðqD;RDÞ to C_{}^{m;a;0}ðqD;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 C^{r}. We now observe that for all
f AC_{}^{mþr;a;0}ðqD;CÞVAqD such that fðqDÞ ¼qD, we have Y½f;g ¼0 if
g¼ f^{ð1Þ}. We now apply the Implicit Function Theorem to equation
Y½f;g ¼0 around the pair ðf0;f_{0}^{ð1Þ}Þ. By Theorem 2.9, the partial di¤er-
ential of Y at ðf0;f_{0}^{ð1Þ}Þ with respect to the variable g is deﬁned by
dgY½f0;f_{0}^{ð1Þ}ðhÞ ¼ f_{0}^{0}ðf_{0}^{ð1Þ}Þh, for all hAC_{}^{m;a;0}ðqD;CÞ. Since f_{0}^{0} belongs to
C_{}^{mþr1;}^{a;0}ðqD;CÞ, which is contained in C_{}^{m;}^{a;0}ðqD;CÞ, and since f_{0}^{0}ðtÞ00
for all tAqD, then Lemma 2.3 (ii), (iii) and Theorem 2.9 imply that
dgY½f0;f_{0}^{ð1Þ} is a linear isomorphism of C_{}^{m;a;}^{0}ðqD;CÞ. Thus the Implicit
Function Theorem implies the existence of an open neighborhood Wf0 of f_{0}
in C_{}^{mþr;a;}^{0}ðqD;CÞVAqD, and of an open neighborhood V_{f}^{ð1Þ}

0

of f_{0}^{ð1Þ} in
C_{}^{m;a;}^{0}ðqD;CÞ, and of a map JJ~_{f}_{0} of class C^{r} from Wf0 to V_{f}^{ð1Þ}

0

such that the
graph of JJ~_{f}_{0} coincides with the set of zeros of Y in Wf0V_{f}^{ð1Þ}

0

. By the
obvious inclusion of C_{}^{mþr;}^{a;0}ðqD;CÞ in C_{}^{m;a;0}ðqD;CÞ, and by case r¼0,J is
continuous onWf0VC_{}^{mþr;a;}^{0}ðqD;qDÞ. Then by possibly shrinkingWf0, we can
assume that Jmaps Wf0VC_{}^{mþr;}^{a;0}ðqD;qDÞto V_{f}^{ð1Þ}

0

. Since Y½f;J½f ¼0, for
all f AWf0VC_{}^{mþr;a;0}ðqD;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 fAC_{}^{1}ðqD;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!fðe^{iy}Þ, with yA½0;2p, of fðqDÞ. Let fAAqD. We
denote by ind½f the index of the curve y7!fðe^{iy}Þ, 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 C_{}^{1}ðqD;CÞ. We set A_{qD}^{þ}1ffAAqD: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, fAC_{}^{1;}^{a}ðqD;CÞVA_{qD}^{þ}, L¼fðqDÞ.

Let Ibe the identity operator in C_{}^{1;a}ðL;CÞ. Then the following statements hold.

( i ) For all f AC_{}^{m;}^{a}ðL;CÞ, the singular integral

S_{f}½fðtÞ1 1
pi

ð

f

fðsÞ

stds; EtAL; ð2:15Þ

exists in the sense of the principal value, and S_{f}½fðÞAC_{}^{m;a}ðL;CÞ. The
operator S_{f} deﬁned by (2.15) is linear and continuous from C_{}^{m;a}ðL;CÞ to
itself. If f coincides with the identity map idqD, then we set S1S_{f}.
( ii ) For all f AC_{}^{m;}^{a}ðL;CÞ, the function _{f}½f of CnfLg to C deﬁned by

_{f}½fðzÞ1 1
2pi

ð

f

fðsÞ

szds; EzACnL;

is holomorphic. The function _{f}½f_{jI½f} admits a continuous extension to
clI½f, which we denote by ^{þ}_{f}½f, and the function _{f}½f_{jE½f} admits a
continuous extension toclE½f, which we denote by^{}_{f}½f. Then we have
^{þ}_{f}½fAC^{m;}^{a}ðclI½f;CÞVHðI½fÞ, ^{}_{f}½fAC^{0}ðclE½f;CÞVC_{}^{m;a}ðL;CÞV
HðE½fÞ, and the Plemelj formulas ^{G}_{f}½fðtÞ ¼G^{1}

2fðtÞ þ^{1}_{2}S_{f}½fðtÞ for all
tAL hold. Furthermore, ^{þ}_{f}½deﬁnes a linear and continuous operator of
C_{}^{m;a}ðL;CÞ to C^{m;a}ðclI½f;CÞ. If f coincides with the identity map idqD,
then we set 1_{f}.

(iii) The function f AC_{}^{m;}^{a}ðL;CÞ satisﬁes equation ðIS_{f}Þ½f ¼0, if and
only if there exists a function F AC^{m;}^{a}ðclI½f;CÞVHðI½fÞ such that
FðtÞ ¼ fðtÞ, for all tAL. The function F, if it exists, is unique.

(iv) The function f AC_{}^{m;}^{a}ðL;CÞsatisﬁes equation ðIþS_{f}Þ½f ¼0, if and only
if there exists a function F AC^{0}ðclE½f;CÞVC_{}^{m;a}ðL;CÞVHðE½fÞ such
thatlimz!y FðzÞ ¼0, and FðtÞ ¼ fðtÞ, for alltAL. The function F, if it
exists, is unique.

( v ) If f AC_{}^{1;a}ðL;CÞ, then ðSf½fÞ^{0}¼S_{f}½f^{0}.

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 zAC_{}^{m;a}ðqD;CÞVAqD.
Then the following statements hold.

( i ) There exists a unique homeomorphism g½zAC^{1}ðCnD;CÞVHðCnclDÞ
of CnD onto clE½z, such that g½zðyÞ1limz!yg½zðzÞ ¼y, g½z^{0}ðyÞ1
limz!y g½z^{0}ðzÞA0;þy½. Furthermore g½z_{jqD}AC_{}^{m;a}ðqD;CÞVA_{qD}^{þ}.

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

There exists a unique homeomorphism f½zAC^{m;a}ðclD;CÞVHðDÞof clD
onto clI½z such that f½zðajÞ ¼zðajÞ, for all j¼1;2;3. Furthermore,

f½z_{jqD}AA_{qD}^{þ}.

(iii) For all bAI½z, there exists a unique homeomorphism ff~½z;b of clD onto
clI½zsuch that ff~½z;bAC^{m;}^{a}ðclD;CÞVHðDÞand such that ff~½z;bð0Þ ¼b,

ff~½z;b^{0}ð0ÞA0;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 a_{1};a_{2};a_{3} in the
notation for f½z. By Lemma 2.3 (iv), (v), and by Theorem 3.1, the function
f½g½z_{jqD}^{ð1Þ}g½z_{jqD} belongs to C_{}^{m;}^{a}ðqD;qDÞVA_{qD}^{þ}, for all zAC_{}^{m;a}ðqD;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 zAC_{}^{m;}^{a}ðqD;CÞVAqD, then we deﬁne as conformal welding
map associated to z (and to the triple ða1;a2;a3Þ), the map

w½z1f½g½z_{jqD}^{ð1Þ}g½z_{jqD}: ð3:3Þ
We deﬁne as conformal welding operator, the operator w½of C_{}^{m;a}ðqD;CÞVAqD

to C_{}^{m;a}ðqD;qDÞVA_{qD}^{þ}, 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;b^{ð1Þ}z is real analytic from

Em;a1fðz;bÞAðC_{}^{m;a}ðqD;CÞVAqDÞ C:bAI½zg
to C_{}^{m;a}ðqD;CÞ.

(ii) Let rAN. The map ðz;bÞ 7!ff~½z;b of Emþr;aVðC_{}^{mþr;a;0}ðqD;CÞ CÞ to
C_{}^{m;a;0}ðqD;CÞ is of class C^{r}.

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

By Theorem 3.1, we deduce that ff~½z;bAC^{y}_{} ðqD;CÞ if zAC_{}^{y}ðqD;CÞVAqD,
bAI½z. Then by statement (i), we have ff~½z;b^{ð1Þ}zAC_{}^{m;}^{a;0}ðqD;CÞ if zA
C_{}^{m;}^{a;0}ðqD;CÞVAqD,bAI½z. Then statement (ii) follows by statement (i), and

by Theorems 2.9, 2.11. r

We now turn to consider the dependence of f½z^{ð1Þ}zupon 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;p3ÞAC^{4}:p1;p2;p3 are distinct;

ðzp3Þðp1p2Þða1a3Þ ðzp2Þðp1p3Þða1a2Þ00g:

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

Qðz;p_{1};p_{2};p_{3}Þ1a_{2}ðzp_{3}Þðp1p_{2}Þða1a_{3}Þ a_{3}ðzp_{2}Þðp_{1}p_{3}Þða1a_{2}Þ
ðzp3Þðp1p2Þða1a3Þ ðzp2Þðp1p3Þða1a2Þ ;
for all ðz;p_{1};p_{2};p_{3}ÞAA. If p_{1};p_{2};p_{3} 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;p_{1};p_{2};p_{3}ÞAA for all zAclD, and Qð;p_{1};p_{2};p_{3}Þ 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;p_{1};p_{2};p_{3}Þ 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;p_{1};p_{2};p_{3}Þ 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 a_{1};a_{2};a_{3} be three distinct
points of qD. Then the nonlinear operator z7! f½z^{ð1Þ}zis real analytic from
C_{}^{m;a}ðqD;CÞVA_{qD}^{þ} to C_{}^{m;}^{a}ðqD;qDÞVA_{qD}^{þ}, and maps C_{}^{m;a;0}ðqD;CÞVA_{qD}^{þ} to
C_{}^{m;a;}^{0}ðqD;qDÞVA_{qD}^{þ}.

Proof. Let z_{0}AC_{}^{m;a}ðqD;CÞVA_{qD}^{þ}, z0AI½z_{0}. Let W be an open
neighborhood of z_{0} in C_{}^{m;a}ðqD;CÞVA_{qD}^{þ} such that z_{0}AI½z, for all zAW.

By the uniqueness inferred by the Riemann Mapping Theorem, we have
f½zz_{0} ¼ f½z z_{0}. Thus there is no loss of generality in assuming that
0¼z_{0}AI½z, for all zAW. By Lemma 3.5, and by the uniqueness inferred by
the Riemann Mapping Theorem, we have

f½z^{ð1Þ}zðÞ ¼Qðff~½z^{ð1Þ}zðÞ;ff~½z^{ð1Þ}zða1Þ;ff~½z^{ð1Þ}zða2Þ;ff~½z^{ð1Þ}zða3ÞÞ;

where Q is as in Lemma 3.5. Thus the analyticity of f½z^{ð1Þ}zðÞ on z
follows by Theorem 3.4 (i), and by Lemmas 2.3 (ii), (iii) and 3.5. By Theorem
3.1, we have f½z^{ð1Þ}zAC_{}^{y}ðqD;qDÞ if zAC^{y}_{} ðqD;CÞVA_{qD}^{þ}. Then the last
statement follows by the continuity ofz7! f½z^{ð1Þ}z, and by the deﬁnition of

the spaces C_{}^{m;}^{a;0}. r

We now show that on a suitable subset Tm;a of C_{}^{m;a}ðqD;CÞVA_{qD}^{þ} we have
g½z_{jqD}¼z, 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

zAC_{}^{m;a}ðqD;CÞ:bZAHðCnclDÞVC^{0}ðCnD;CÞ

such that z¼Z_{jqD};lim

z!yZ^{0}ðzÞ1Z^{0}ðyÞAR

: ð3:8Þ

Then the following statements hold.

( i ) If zADm;a, then the map Z of (3.8) is unique. Furthermore, Z^{0}ðyÞ ¼

1 2pi

Ð

qD zðsÞ

s^{2} ds, and limz!y ZðzÞ _{2pi}^{z} Ð

qD zðsÞ

s^{2} ds_{2pi}^{1} Ð

qD zðsÞ

s ds

n o

¼0.

( ii ) Dm;a is a real Banach subspace of C_{}^{m;a}ðqD;CÞ.

(iii) Let Z½z be the unique map of statement (i) corresponding to zAD_{m;}_{a}.
Then the set

Tm;a1fzADm;aVA_{qD}^{þ} :Z½z^{0}ðyÞ>0g
is open in D_{m;}_{a}.

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

Proof. We ﬁrst prove statement (i). If ZAHðCnclDÞV
C^{0}ðCnD;CÞ, then standard properties of holomorphic functions imply
that condition lim_{z!}yZ^{0}ðzÞ1Z^{0}ðyÞAC is equivalent to the existence of
a;bAC and KAHðCnclDÞVC^{0}ðCnD;CÞ such that limz!yKðzÞ ¼0 and
ZðzÞ ¼azþbþKðzÞ. Moreover, it is easily checked that if such condi-
tion holds, then Z^{0}ðyÞ ¼a¼_{2pi}^{1} Ð

qD ZðsÞ

s^{2} ds, b¼_{2pi}^{1} Ð

qD ZðsÞ

s ds. Then by

Theorem 2.14 (iv), the membership of z in D_{m;a} is equivalent to condi-
tion ðIþSÞ zðzÞ _{2pi}^{z} Ð

qD zðsÞ

s^{2} ds_{2pi}^{1} Ð

qD zðsÞ

s ds

h i

¼0 together with condition

1 2pi

Ð

qD zðsÞ

s^{2} 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 _{2pi}^{1} Ð

qD zðsÞ

s ds and of _{2pi}^{1} Ð

qD zðsÞ

s^{2} ds on zAC_{}^{m;}^{a}ðqD;CÞ.

Statement (iii) follows by Lemma 2.2 (ii) and by the continuous dependence
of Z^{0}ðyÞ ¼_{2pi}^{1} Ð

qD zðsÞ

s^{2} ds on zAD_{m;a}. We now prove statement (iv). By (i),

we have limz!y Z½zðzÞ ¼y. Now let o0AI½z. Since Z½z^{0}ðyÞ00, then
the function Z½z is injective in a neighborhood of y. Since we also have
zAA_{qD}^{þ} and o0AI½z, then a standard argument based on the Argument
Principle shows that Z½zðÞ o_{0} does not vanish. Thus the function

½Z½zð1=ðÞÞ o0^{1} extends to a holomorphic map in D. Since zAA_{qD}^{þ}, then
the curve ½zð1=ðÞÞ o0^{1} is one to one. Then again by the Argument
Principle, ½Z½zð1=ðÞÞ o0^{1} is one to one in clD, and a simple topological
argument shows that ½Z½zð1=ðÞÞ o0^{1} maps D onto I½½zð1=ðÞÞ o0^{1}.
Accordingly, Z½z is one to one and Z½zðCnclDÞ ¼CnclI½z. Since
Z½z^{0}ðyÞ>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 C_{}^{m;a}ðqD;qDÞVA_{qD}^{þ}.

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 g½z_{jqD} is of class C^{r} in the real sense from C_{}^{mþr;a;}^{0}ðqD;CÞVAqD to
C_{}^{m;a;}^{0}ðqD;CÞVA_{qD}^{þ}.

Proof. Let z_{0}AC_{}^{mþr;}^{a;0}ðqD;CÞVAqD, z0AI½z_{0}. Let W be an open
neighborhood of z_{0} in C_{}^{mþr;}^{a;0}ðqD;CÞVAqD such that z_{0}AI½z, for all
zAW. By the uniqueness inferred by the Riemann Mapping Theorem, we
have g½zz_{0} ¼g½z z_{0}. 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=zÞ^{1}, for all zACnclD, and for all zAW. In particular, Theorem
3.1 implies that g½z_{jqD}AC_{}^{y}ðqD;CÞ if zAC_{}^{y}ðqD;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 C_{}^{m;a}ðqD;CÞVAqD to C_{}^{m;a}ðqD;qDÞVA_{qD}^{þ} and is of
class C^{r} in the real sense from C_{}^{mþr;a;}^{0}ðqD;CÞVAqD to C_{}^{m;a;}^{0}ðqD;qDÞVA_{qD}^{þ}.
We note that it can be proved that Theorem 3.11 is optimal in the frame
of Schauder spaces (cf. [19, Thm. 2.14].)