Analysis on quasidisks; a unified approach through transmission and jump problems
aa r X i v : . [ m a t h . C V ] S e p ANALYSIS ON QUASIDISKSA UNIFIED APPROACH THROUGH TRANSMISSION AND JUMPPROBLEMS
ERIC SCHIPPERS AND WOLFGANG STAUBACHDedicated to our friend Ian Graham
Abstract.
We give an exposition of results from a crossroad between geometric func-tion theory, harmonic analysis, boundary value problems and approximation theory, whichcharacterize quasicircles. We will specifically expose the interplay between the jump decom-position, singular integral operators and approximation by Faber series. Our unified pointof view is made possible by the the concept of transmission.
Contents
1. Introduction 22. Function spaces and boundary values 52.1. Dirichlet and Bergman spaces 52.2. Quasisymmetries and quasiconformal maps 82.3. Null sets 102.4. Dirichlet space of the disk and boundary values 132.5. Conformally non-tangential boundary values 143. Transmission of harmonic functions in quasicircles 163.1. Vodop’yanov-Nag-Sullivan theorem 163.2. Transmission (Overfare) 193.3. The bounce operator and density theorems 224. Schiffer and Cauchy operator 284.1. Schiffer operators 284.2. Cauchy operator 324.3. The Schiffer isomorphisms and the Plemelj-Sokhotski isomorphisms 355. Faber and Grunsky operator 415.1. The Faber operator and Faber series 415.2. Grunsky inequalities 446. Notes and literature 486.1. Notes on the Introduction 486.2. Notes on Section 2 496.3. Notes on Section 3 496.4. Notes on Section 4 506.5. Notes on Section 5 52References 53
Mathematics Subject Classification.
Primary: , Secondary:
Key words and phrases.
Conformally non-tangential boundary value, Bergman space, Dirichlet space,Faber operator, Grunsky operator, Jump problem, Schiffer operator, Transmission, Quasisymmetry,Quasicircles. . Introduction
A quasiconformal map in the plane is a homeomorphism between planar domains whichmaps small circles to small ellipses of bounded eccentricity. A quasicircle is by definitionthe image of the circle S under a quasiconformal map and a quasidisk is the interior ofa quasicircle. In geometric function theory quasicircles play a fundamental role in the de-scription of the universal Teichm¨uller space. They also play an important role in complexdynamical systems. The reader is referred to the book by F. Gehring and K. Hag [26] for anice introduction to various ramifications of this topic.It is a familiar fact in the field that quasicircles have an unusually large number of charac-terizations which are not obviously equivalent, and indeed are qualitatively quite different.See e.g. [26, Chapters 8,9] for some of the classical and also some less well-known ones. Itis somewhat astonishing that these continue to be found. In this paper, we will focus onthe relatively recent ones, due to A. C¸ avu¸s [16], Y. Y. Napalkov and R. S. Yulmukhametov[43], Y. Shen [63], and the authors [58, 59]. Indeed, our purpose here is to highlight a char-acterization based on an interplay between geometric function theory, harmonic analysis,boundary value problems and approximation theory. This point of view was investigated bythe authors in a series of papers, and in these works, it emerged that the key to a unifiedapproach is the method of transmission of harmonic functions (or forms).The goal of this paper is to give an essentially self-contained and unified exposition of thiscircle of ideas and the method of transmission, not least because of its potential applicationsoutside geometric function theory. In doing so we have also refined and improved many ofour theorems in previous papers.To define the notion of transmission, let Γ be a Jordan curve separating the Riemannsphere C into two components Ω and Ω . Given a harmonic function h on Ω which extendscontinuously to Γ, there is a harmonic function on Ω with the same continuous extension onΓ. We call the new function the transmission of h . We generalize the concept of transmissionto Dirichlet bounded harmonic functions. For such harmonic functions, the transmission ex-ists and is bounded with respect to the Dirichlet semi-norm if and only if the curve Γ is aquasicircle.Returning to the problem of characterization of quasicircles, it came to light that in thesetting of Dirichlet bounded harmonic functions, a number of perfect equivalences arise,which make a unified treatment of a number of topics possible. To begin with, given aJordan curve Γ as above, the following three statements are equivalent.(1) Γ is a quasicircle.(2) There is a bounded transmission from the Dirichlet space of harmonic functions onΩ to the Dirichlet space of harmonic functions on Ω , which agrees with transmissionof continuous functions.(3) The linear operator taking the boundary values of a Dirichlet bounded harmonicfunction to its Plemelj-Sokhotski jump decomposition is a bounded isomorphism. hese results are due to the authors [58, 59].The first three equivalent statements also are closely related to approximability by Faberseries, the Faber and Grunsky operators, and the Schiffer operator. We thus have the fol-lowing further equivalent statements. Attributions in brackets refers to the first proofs ofthe equivalence with (1), unless clarified below.(4) The Faber operator corresponding to Ω is an isomorphism (authors [59]).(5) The sequential Faber operator is an isomorphism (C¸ avu¸s [16], Shen [63]).(6) Every element of the holomorphic Dirichlet space of Ω is uniquely approximable bya Faber series (C¸ avu¸s [16], Shen [63]).(7) The Schiffer operator is an isomorphism (Napalkov and Yulmukhametov [43]).The implications (1) ⇒ (5) and (1) ⇒ (6) are due to C¸ avu¸s, and later independently byShen, while the reverse implications are due to Shen. For the special case of rectifiableJordan curves, the equivalence of (1) and (4) is due to H. Y. Wei, M. L. Wang, and Y. Hu[73]. In this paper, we give proofs of the equivalence of (4) - (7) with (1) which rely on thetransmission result (2). The proofs given here that (4)-(7) imply (1) are new.Finally, all of these results are closely connected to the classical result that Γ is a quasicircleif and only if(8) The norm of the Grunsky operator is strictly less than one.The implication (1) ⇒ (8) is due to R. K¨uhnau [35] and (8) ⇒ (1) is due to C. Pommerenke[45]. In the literature, all proofs of the implication (k) ⇒ (1) for k = 4 , . . . , ⇒ (1). However the proofs given in thispaper do not.An important issue in connection to transmission is that some notion of boundary values isnecessary in order to define the transmission in a sensible way. To this end we also include anexposition of a conformally invariant notion of non-tangential boundary value, which we callconformally non-tangential (CNT for short). This was developed by the authors for Jordancurves in Riemann surfaces [60]. The existence of such boundary values for the Dirichletspace of a simply connected domain is an automatic consequence of a well-known result ofA. Beurling. On the other hand, it is not true in general that the boundary values of aharmonic function in one connected component of the complement of Γ are boundary valuesof a harmonic function in the other component. Even potential-theoretically negligible setsare not obviously the same: for example, sets of harmonic measure zero with respect to oneside are not necessarily harmonic measure zero with respect to the other, even for quasicircles.At any rate, we give a general framework for the application of the CNT boundary values tosewing and transmission. Aside from the bounded transmission theorem mentioned above,the most important of these results are: i) for quasicircles, the potential-theoretically negligible sets on the boundary of Ω arealso negligible for Ω ;(ii) the operator (what we call the bounce operator) taking a Dirichlet-bounded harmonicfunction on a doubly-connected region in Ω , one of whose boundaries is Γ, to theharmonic function on Ω with the same boundary values, is bounded for any Jordancurve;(iii) limiting integrals taken over level curves of Green’s function are the same for any twoDirichlet bounded harmonic function in a collar near Γ which have the same CNTboundary values (the anchor lemma).The precise statements are given in Theorems 2.18, 3.17, and Theorem 3.21 respectively.To conclude, we strive in this paper to show the clarifying power of the transmission the-orem for understanding approximation by Faber series, the Grunsky operator, the Plemelj-Sokhotski jump theorem, and Schiffer operators. The results should have many applicationsin the investigation of the behaviour of function spaces, boundary value problems, and re-lated operators under sewing. The results here are also the basis for a scattering theory ofharmonic functions and one-forms for general Riemann surfaces [62].The paper is organized as follows. In Section 2, we state necessary definitions and resultsregarding conformally non-tangential boundary values of Dirichlet bounded functions. Afterpreliminaries on the Dirichlet and Bergman space, and quasisymmetric mappings in Sections2.1 and 2.2, we define certain potential-theoretically negligible sets on a Jordan curve Γ withrespect to the enclosed domain in Section 2.3, which we call null sets, and derive their basicproperties. A particularly crucial fact is that, in the case that the Jordan curve is a quasi-circle, sets that are null with respect to one of the regions enclosed by Γ are also null withrespect to the other. In fact, for quasicircles not containing ∞ , null sets in Γ are preciselyBorel sets of capacity zero. After reviewing some basic results on boundary values of theDirichlet space of the disk in Section 2.4, we give the definition of CNT boundary values inSection 2.5 and basic properties.Section 3 contains the first of the main results, namely (1) ⇔ (2): a bounded transmissionexists on Dirichlet space if and only if Γ is a quasicircle. Section 3.1 reviews some knowntheorems which characterize quasisymmetries in terms of their action on the homogeneousSobolev space H / , and a reformulation in terms of CNT boundary values up to null sets.This refinement is necessary because sets of harmonic measure zero on a quasicircle withrespect to one side of a curve - which are the images of sets of Lebesgue measure zero onthe circle under a conformal map - need not be of harmonic measure zero with respect tothe other side of the curve. Thus, null sets are necessary. Section 3.2 contains the trans-mission result. In Section 3.3, we establish several useful results regarding boundary valuesand integrals. We prove that the so-called bounce operator described in the introduction isbounded. We also prove the “anchor lemma”, which shows that certain limiting integralstaken over curves approaching the non-rectifiable Jordan curve depend only on the CNTboundary values. Finally, we give a few useful dense subsets of Dirichlet spaces on simply- nd multiply-connected domains. These ultimately rely on density of polynomials.Section 4 contains the main results on Plemelj-Sokhotski jump isomorphism and Schifferisomorphism, that is (1) ⇔ (3) ⇔ (7) . Section 4.1 defines the Schiffer operator and provesbasic analytic results, M¨obius invariance, and an identity of Schiffer. Section 4.3 definesa Cauchy integral operator adapted to non-rectifiable curves using limits of integrals overcurves approaching the boundary. We show that for quasicircles the value of this operator isthe same for curves approaching Γ over either side, in a certain sense involving transmission.We also prove basic identities relating the Cauchy integral operator to the Schiffer opera-tors, and the M¨obius invariance of the operator. Section 4.3 contains the main results whichshow that the Plemelj-Sokhotski jump decomposition exists when Γ is a quasicircle, and ina certain sense this decomposition is an isomorphism if and only if Γ is a quasicircle. Wealso give a new proof of Napalkov and Yulmukhametov’s result that the Schiffer operator isan isomorphism if and only if Γ is a quasicircle.In Section 5 we prove that the Faber operator is an isomorphism if and only if Γ is aquasicircle, as well as the existence and uniqueness of Faber series; that is, (1) ⇔ (4) ⇔ (5) ⇔ (6). We also give a brief review of the equivalence with strict Grunsky inequalities.Finally, Section 6 contains notes on the literature, as well as some fine points which couldnot be put in the main text without interrupting the flow of the paper. Although the notesare fairly extensive for a paper of this size, we make no claims to completeness, and merelyindicate the tip of the literary iceberg.2. Function spaces and boundary values
Dirichlet and Bergman spaces.
We denote the complex plane by C and the Riemannsphere by C . We define D + = { z ∈ C : | z | < } and also D − = { z ∈ C : | z | > } ∪ {∞} .The circle bd( D ) = { z ∈ C : | z | = 1 } is denoted S . In this paper, a conformal map isalways assumed to be one-to-one (not just locally one-to-one). That is, a conformal map isa biholomorphism onto its image.The Riemann sphere C is endowed with the standard complex structure given by thecharts ψ : C → C ψ ( z ) = zψ ∞ : C \{ } → C ψ ∞ ( z ) = 1 /z z = ∞ ψ ∞ ( ∞ ) = 0 , and holomorphicity or harmonicity is defined with respect to these charts. That is, let Ωbe an open connected set in C . A function h is holomorphic on Ω if (1) it is holomorphicon C \{∞} and (2) if ∞ ∈ Ω then g ( z ) = f (1 /z ) is holomorphic in a neighbourhood of 0.Anti-holomorphic and harmonic functions on Ω are defined similarly.We will also consider smooth one-forms on subsets of C , where these are defined in theusual way in terms of the Riemann surface structure of C . Any one-form α is given in local oordinates by h ( z ) dz + h ( z ) d ¯ z for smooth functions h ( z ) and h ( z ). A one-form α on Ωis said to be holomorphic if it can be expressed in local coordinates z as h ( z ) dz where h ( z )is holomorphic. That is, α = a ( z ) dz on Ω \{∞} , and if ∞ ∈ Ω, then b ( w ) = − a (1 /w ) /w isholomorphic in an open set containing 0 (so that in a chart at ∞ , we may write α = b ( w ) dw ).A one-form is anti-holomorphic if it is the complex conjugate of a holomorphic one-form.We also define the ∗ -operator as follows. If α = h ( z ) dz + h ( z ) d ¯ z in local coordinateswe define ∗ α = ∗ ( h ( z ) dz + h ( z ) d ¯ z ) = − ih dz + ih d ¯ z. It is easily checked that this is well-defined with respect to the change of coordinates z = ψ ◦ ψ − ∞ ( w ) = 1 /w .Define(2.1) k α k = 12 π Z Z Ω α ∧ ∗ α which might of course diverge. Since any smooth one-form α on Ω can be written (uniquely)in z coordinates as(2.2) α = h ( z ) dz + h ( z ) d ¯ z for smooth functions h and h , then if in (2.2) z is the parameter in C (that is, in ψ coordinates), then (2.1) can be written as(2.3) k α k = 1 π Z Z Ω \{∞} (cid:0) | h ( z ) | + | h ( z ) | (cid:1) dA, where dA = ( d ¯ z ∧ dz ) / i is the Euclidean area element in C . This is justified as follows:when ∞ ∈ Ω, if k α k < ∞ then it is easily verified that there is an R such that Z Z | z | >R (cid:0) | h ( z ) | + | h ( z ) | (cid:1) dA < ∞ . Thus the point at ∞ can be removed from the domain of integration without changing theconvergence properties or value of the integral. Definition 2.1.
A smooth one-form α is said to be harmonic if dα = 0 and d ∗ α = 0;equivalently, for any point p ∈ Ω, α = dh for some harmonic function h on some openneighbourhood of p . Note that if ∞ ∈ Ω, this restricts the behaviour of α at ∞ since h (1 /z )must be harmonic at 0.We then define the space of L harmonic one-forms A harm (Ω) to consist of those harmonicone-forms α on Ω such that k α k Ω < ∞ . This is a Hilbert space with inner product(2.4) ( α, β ) = 12 π Z Z Ω α ∧ ∗ β, which is also consistent with (2.1). The Bergman space of one-forms is A (Ω) = { α ∈ A harm (Ω) : α is holomorphic } , and for α = h ( z ) dz, β = h ( z ) dz ∈ A (Ω) we have( α, β ) = 1 π Z Z Ω \{∞} h ( z ) h ( z ) dA. he anti-holomorphic Bergman space A (Ω) consists of complex conjugates of elements of A (Ω).Observe that A (Ω) and A (Ω) are orthogonal with respect to the inner product. We thenobtain the decomposition A harm (Ω) = A (Ω) ⊕ A (Ω)which induce the projection operators P (Ω) : A harm (Ω) → A (Ω) P (Ω) : A harm (Ω) → A (Ω)(2.5) Definition 2.2.
For an open connected set Ω and a smooth function h : Ω → C we definethe Dirichlet energy of h by(2.6) D Ω ( h ) = k dh k . The harmonic Dirichlet space D harm (Ω) consists of those harmonic functions h on Ω suchthat D Ω ( h ) < ∞ . If z is the coordinate in C then (2.6) can be written as(2.7) D Ω ( h ) = 1 π Z Z Ω \{∞} (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂z (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂ ¯ z (cid:12)(cid:12)(cid:12)(cid:12) ! dA. The holomorphic Dirichlet space D (Ω) is the set of holomorphic functions in D harm (Ω), andthe anti-holomorphic Dirichlet space D (Ω) is given by the set of complex conjugates ofelements of D (Ω). The Dirichlet energy on D (Ω) restricts to D Ω ( h ) = 1 π Z Z Ω \{∞} | h ′ ( z ) | dA and similarly for D (Ω). Observe that D harm (Ω) does not decompose into a sum of elementsof D (Ω) and D (Ω) unless Ω is simply connected (and even in that case, the decompositionis not unique because constants belong to both spaces).The Dirichlet energy is not a norm, since D ( c ) = 0 for constants c . It becomes a normif we restrict to normalized functions h ( p ) = 0 for some p ∈ Ω. When such a normalizationis imposed, we use the notations D p (Ω), D harm (Ω) p , and use k · k D p (Ω) and so on, for thecorresponding norms.We use the following notation for projection. Fix a simply-connected domain Ω in C . Fix p ∈ Ω. Define the decompositions D harm (Ω) = D (Ω) ⊕ D p (Ω)= D p (Ω) ⊕ D (Ω) . We then have projections P hp (Ω) : D harm (Ω) → D (Ω) P hp (Ω) : D harm (Ω) → D p (Ω)(2.8)induced by the first decomposition, and projections P ap (Ω) : D harm (Ω) → D p (Ω) P ap (Ω) : D harm (Ω) → D (Ω)(2.9) nduced by the second. Note that all four projections depend on the location of p . Thesuperscripts “ h ” and “ a ” stand for “holomorphic” and “anti-holomorphic” normalizations.Finally, for a domain G in the plane, with boundary Γ, we also define the Sobolev spaces H ( G ) and H / (Γ): Definition 2.3. H ( G ) consists of functions in L ( G ) such that(2.10) k h k H ( G ) := (cid:16) D G ( h ) + k h k L ( G ) (cid:17) / < ∞ . Moreover, if Γ is regular enough then one can also take the restriction (trace) of an H ( G )-function to Γ, which yields a function h | Γ ∈ H / (Γ) where H / (Γ) is the space of functionsin L (Γ) for which(2.11) k f k H / (Γ) := (cid:18)Z Γ Z Γ | f ( z ) − f ( ζ ) | | z − ζ | | dz | | dζ | + k f k L (Γ) (cid:19) / < ∞ , see Chapter 4 in [69] for all the details regarding Sobolev spaces.2.2. Quasisymmetries and quasiconformal maps.
In this section we review definitions and results about quasisymmetries and quasiconformalmaps.
Definition 2.4.
Let A and B be open connected subsets of the complex plane. An orientation-preserving homeomorphism Φ : A → B is a k - quasiconformal mapping if(1) for every rectangle [ a, b ] × [ c, d ] ⊂ A , Φ( x, · ) is absolutely continuous on [ c, d ] foralmost every x ∈ [ a, b ];(2) for every rectangle [ a, b ] × [ c, d ] ⊂ A , Φ( · , y ) is absolutely continuous on [ a, b ] foralmost every y ∈ [ c, d ];(3) there is a k ∈ (0 ,
1) such that | Φ ¯ z | ≤ k | Φ z | almost everywhere in A .We say that a map is quasiconformal if it is k -quasiconformal for some k ∈ (0 , ι : C \{ } → C \{ } .z /z. Let A and B be open connected subsets of C . We say that a homeomorphism Φ : A → B isa k -quasiconformal mapping ifΦ , ι ◦ Φ , Φ ◦ ι, and ι ◦ Φ ◦ ι are all k -quasiconformal on their maximal domains of definition; as above, we say that Φ isquasiconformal if it is k -quasiconformal for some k . If A, B ( C then Φ is quasiconformalif, given M¨obius transformations S and T such that S ( A ) ⊂ C and T ( B ) ⊂ C , T ◦ Φ ◦ S − is quasiconformal from S ( A ) to T ( B ). imilarly, for open connected sets A, B ⊂ C we say that a map f : A → B is conformal if f, ι ◦ f, f ◦ ι, and ι ◦ f ◦ ι are all conformal on their maximal domains of definition. Conformal maps are 0-quasiconformalonto their image, and it can be shown that 0-quasiconformal maps are conformal (see e.g.[2]). Furthermore, if Φ : A → B is quasiconformal and g : A ′ → A is conformal thenΦ ◦ g : A ′ → B is quasiconformal, and if f : B → B ′ is conformal, then f ◦ Φ is qua-siconformal. If C is any open connected subset of A , then the restriction of Φ to C is aquasiconformal map onto Φ( C ). Remark . Any quasiconformal map Φ : C → C extends to a quasiconformal map from C to C , which takes ∞ to ∞ [39, Theorem I.8.1]. Definition 2.6.
An orientation-preserving homeomorphism h of S is called a quasisymmet-ric mapping , iff there is a constant k >
0, such that for every α , and every β not equal to amultiple of 2 π , the inequality 1 k ≤ (cid:12)(cid:12)(cid:12)(cid:12) h ( e i ( α + β ) ) − h ( e iα ) h ( e iα ) − h ( e i ( α − β ) ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k holds.Let QS( S ) denote the set of quasisymmetric maps from S to S . QS( S ) consists preciselyof boundary values of quasiconformal maps, as the following two theorems show. Theorem 2.7.
Let
Φ : D + → D + be a quasiconformal map. Then Φ has a continuous exten-sion to S ∪ D + , and the restriction of this extension to S is a quasisymmetry. Conversely,if φ : S → S is a quasisymmetry, then φ is the restriction to S of the continuous extensionof a quasiconformal map Φ : D + → D + . In the above, one may replace D + everywhere by D − and the result still holds.Proof. By reduction of the problem to the upper-half plane using the conformal equivalenceof the unit disk and the former, this is just Ahlfors-Beurling’s result in [13]. (cid:3)
By a Jordan curve Γ in C , we mean the image of S under a continuous map into C whichis a homeomorphism onto its image. Equivalently, it is the image of a Jordan curve in theplane under a M¨obius transformation. Definition 2.8.
A Jordan curve Γ in C is a quasicircle if and only if it is the image of S under a quasiconformal map Φ : C → C . We say that a Jordan domain is a quasidisk if itsboundary is a quasicircle.Quasidisks have the following important property [26, Corollary 2.1.5]. Theorem 2.9.
Let Ω be a quasidisk. If f : D ± → Ω is a biholomorphism, then f extends toa quasiconformal map of C . One of the main tools in this paper is the conformal welding theorem.
Theorem 2.10 (Conformal welding theorem) . For any quasisymmetry φ : S → S , thereare conformal maps f : D + → C and g : D − → C , with the following properties. (1) f and g are quasiconformally extendible to C ( so that, in particular, Ω + = f ( D + ) and Ω − = g ( D − ) are quasidisks ) ;
2) bd( f ( D + )) = bd( g ( D − )) , where bd denotes the boundary; and (3) φ = ( g ◦ f − ) | S .If we specify the normalization f (0) = 0 , g ( ∞ ) = ∞ , and g ′ ( ∞ ) = 1 , then f and g areuniquely determined. The normalization above can be replaced with any three normalizations in the interior ofthe domains of f or g if desired.2.3. Null sets.
In this section, we define null sets, which are potential-theoretically negli-gible sets on the boundary of a Jordan domain. That is, in specifying a harmonic functionof bounded Dirichlet energy on a Jordan domain by its boundary values, changes to (ornon-existence of) the boundary values on null sets have no effect. We will see that in thespecial case that the Jordan domain is bounded by a quasicircle, null sets are those sets oflogarithmic capacity zero.We first recall the definition of logarithmic capacity [3, 51]; we follow [51].
Definition 2.11.
Let µ be a finite Borel measure in C with compact support. The potentialof µ is the function p µ ( z ) = Z Z C log | z − w | dµ ( w ) . The energy of µ is then defined to be I ( µ ) = Z Z p µ ( z ) dµ ( z ) . The equilibrium measure of a compact set K is the measure ν such that I ( ν ) = sup µ ∈P ( K ) I ( µ )where P ( K ) is the set of Borel probability measures on K . Every compact set posseses anequilibrium measure [51, Theorem 3.3.2]. Now the logarithmic capacity of a set E ⊆ C isdefined as c ( E ) = sup µ ∈P ( K ) K ⊆ E compact e I ( µ ) . For compact sets K we have c ( K ) = e I ( ν ) where ν is the equilibrium measure of K .We say a property holds quasieverywhere if it holds except possibly on a set of logarithmiccapacity zero. Remark . There are sets E which have Lebesgue measure zero but positive logarithmiccapacity. However, if a property holds quasi-everywhere, it holds almost everywhere.In what follows we will often drop the word “logarithmic” and use simply the word “ca-pacity”.The outer logarithmic capacity of a set E ⊆ C [21] is defined as c ∗ ( E ) = inf E ⊆ U ⊆ C U open c ( U ) . y Choquet’s theorem [18, 51], for any bounded Borel set E , c ( E ) = c ∗ ( E ). Lemma 2.13.
Every bounded set of outer logarithmic capacity zero in C is contained in aBorel set of logarithmic capacity zero.Proof. Let F be a set of outer capacity zero. Thus, there are open sets U n , n ∈ N , containing F such that c ( U n ) < /n . We can choose these sets such that U n +1 ⊆ U n for all n , byreplacing U n with U ′ n = ∩ nk =1 U k if necessary, and observing that by [51, Theorem 5.1.2(a)] c ( U ′ n ) ≤ c ( U n ) < /n since U ′ n ⊆ U n .The set V = ∩ ∞ n =1 U n is a Borel set containing F . Since V ⊆ U n for all n ∈ N , againapplying [51, Theorem 5.1.2(a)] we see that c ( V ) < /n for all n ∈ C , so c ( V ) = 0. (cid:3) Quasiconformal maps preserve compact sets of logarithmic capacity zero. We are gratefulto Malik Younsi for suggesting the following lemma and its proof.
Lemma 2.14.
Let K ⊆ C be compact. Let U be an open set containing K and let f : U → V be a homeomorphism onto the open set V ⊂ C , which is H¨older continuous of exponent α > .If K has capacity zero, then f ( K ) also has capacity zero.Proof. Let µ be a probability measure with support in f ( K ). If we define the Borel proba-bility measure ν = f ∗ ( µ ) by ν ( A ) = µ ( f ( A )) then I ( µ ) = Z Z V Z Z V log | z − w | dµ ( z ) dµ ( w ) = Z Z U Z Z U log | f ( z ) − f ( w ) | dν ( z ) dν ( w )= Z Z U Z Z U log | f ( z ) − f ( w ) || z − w | α dν ( z ) dν ( w ) + αI ( ν )Since the capacity of K is zero, I ( ν ) = −∞ . Moreover the H¨older-continuity of f means that | f ( z ) − f ( w ) | ≤ M | z − w | α , which therefore yields I ( µ ) = −∞ . Now since µ was arbitrary, f ( K ) has capacity zero. (cid:3) From this, it follows that
Lemma 2.15.
Let E ⊆ C be a bounded Borel set. Let f : C → C be a homeomorphism whichis H¨older continuous of exponent α > . If E has capacity zero, then f ( E ) has capacity zero.Proof. By [51, Theorem 5.1.2(b)],(2.12) c ( f ( E )) = sup K ⊆ f ( E ) K compact c ( K )(indeed, this follows directly from the definition of capacity). Thus, if f ( E ) does not havecapacity zero, there is a compact set K ⊆ f ( E ) such that c ( K ) >
0. Since f is a home-omorphism, f − ( K ) is a compact subset of E , so by the previous lemma c ( f − ( K )) > E in place of f ( E ) we see that c ( E ) >
0, a contradiction. (cid:3)
In particular, quasiconformal maps preserve bounded Borel sets of capacity zero, sincethey are uniformly H¨older on every compact subset ([39] p71).
Corollary 2.16.
Let φ : S → S be a quasisymmetry. Then I ⊆ S is a Borel set oflogarithmic capacity zero if and only if φ ( I ) is a Borel set of logarithmic capacity zero. roof. By the Beurling-Ahlfors extension theorem (Theorem 2.7), φ has a quasiconformalextension Ψ : D → D . In fact, this extends to a quasiconformal map of the plane viaΦ( z ) = (cid:26) Ψ( z ) z ∈ cl D / Ψ(1 / ¯ z ) z ∈ C \ cl D . Since a quasiconformal map is uniformly H¨older-continuous on every compact subset, theclaim follows from Lemma 2.15. (cid:3)
We now define null sets. Note that in the sphere, the boundary of a domain is taken withrespect to the sphere topology. So it might include ∞ . Definition 2.17.
Let Ω be a Jordan domain in C with boundary Γ. Let I ⊂ Γ. We say that I is null with respect to Ω if I is a Borel set, and there is a biholomorphism f : D + → Ω,such that f − ( I ) has logarithmic capacity zero.The meaning of f − ( I ) requires an application of Carath´eodory’s theorem, which says thatsince Ω is a Jordan domain, any biholomorphism f has a continuous extension which takes S homeomorphically to Γ. This is true even if Γ contains the point at ∞ , as can be seenby composing f by a M¨obius transformation taking Γ onto a bounded curve and applyingCarath´eodory’s theorem there, and then using the fact that T is a homeomorphism of thesphere. Thus f − ( I ) is defined using the extension of f . Note that I is a Borel set if andonly if f − ( I ) is Borel.If there is one biholomorphism f such that f − ( I ) has capacity zero, then g − ( I ) has ca-pacity zero for all biholomorphisms g : D + → Ω. This is because the M¨obius transformation T = g − ◦ f preserves Borel sets of capacity zero in S , for example by Corollary 2.16. Also,it is easily seen that one may replace D + with D − in the above definition.If Γ is a Jordan curve, bordering domains Ω and Ω , then I might be null with respect toΩ but not with respect to Ω , or vice versa. However, for quasicircles, the concept of nullset is independent of the choice of “side” of the curve. This is a key fact. Theorem 2.18.
Let Γ be a quasicircle in C , and let Ω and Ω be the connected componentsof C \ Γ . Then I ⊂ Γ is null with respect to Ω if and only if it is null with respect to Ω .Proof. Choose a M¨obius transformation T so that T (Γ) is a quasicircle in C (and in particularbounded). Clearly T ( I ) is null in T (Γ) with respect to T (Ω i ) if and only if it is null in Γwith respect to Ω i , for i = 1 ,
2. Thus it suffices to prove the claim for a quasicircle Γ in C .Let Ω + and Ω − be the bounded and unbounded components of the complement of Γrespectively. Let f ± : D ± → Ω ± be conformal maps. These have quasiconformal extensionsto C . Thus φ = f − − ◦ f + has a quasiconformal extension to C , and in particular is aquasisymmetry.By definition I is null with respect to Ω + if and only if f − ( I ) is a Borel set of logarithmiccapacity zero in S . By Corollary 2.16, this holds if and only if f − − ( I ) = φ ( f − ( I )) is a Borelset of logarithmic capacity zero in S , that is if and only if I is null with respect to Ω − . (cid:3) Remark . The proof can be modified to show that if Ω is a Jordan domain bounded by aquasicircle, and I ⊆ Γ is null with respect to Ω, then there is a M¨obius transformation T suchthat T ( I ) is a bounded Borel set of capacity zero (in fact, for any M¨obius transformationsuch that T ( I ) is bounded, it is a set of capacity zero). .4. Dirichlet space of the disk and boundary values.
If we write f ( z ) = P ∞ n =0 a n z n as a power series and setting z = re iθ , one can use polarcoordinates to see that(2.13) D D + ( f ) = ∞ X n =0 n | a n | . Another important fact about the Dirichlet space is that if f ∈ D ( D + ) then f has radialboundary values, i.e. for almost every z ∈ S , the limit lim r → − f ( rz ) =: ˜ f ( z ) exists, see e.g.[21]. Moreover a result of J. Douglas [19], one has that(2.14) D D + ( f ) = Z π Z π | ˜ f ( z ) − ˜ f ( ζ ) | | z − ζ | | dz | | dζ | . Now for ζ ∈ S , let(2.15) K ( ζ ) = 1 | − ζ | / , and define the convolution of two functions f, g defined on the unit circle via(2.16) ( f ∗ g )( z ) := Z π f ( zζ ) g ( ζ ) | dζ | . If z ∈ D + and ζ ∈ S then(2.17) P z ( ζ ) = 1 − | z | | z − ζ | , denotes the Poisson kernel of the disk, and we set(2.18) P ( u )( z ) = Z π P z ( ζ ) u ( ζ ) | dζ | . Regarding boundary values of harmonic functions with bounded Dirichlet energy, we willuse the following two results:
Theorem 2.20.
Let f be a harmonic function in D + and D ( f ) < ∞ . Then ˜ f ( z ) :=lim r → − f ( rz ) quasieverywhere.Proof. This is a classical result due to Beurling, see [11]. (cid:3)
Theorem 2.21.
Let f = P ( K ∗ ϕ ) for some ϕ ∈ L ( S ) . For fixed θ ∈ [0 , π ) , consider thefollowing four limits: (1) lim r → − f ( rz ) ( the radial limit of f )(2) lim N →∞ P Nn = − N \ ( K ∗ ϕ )( n ) e inθ ( the limit of the partial sums of the Fourier series for K ∗ ϕ )(3) lim h → + h R θ + hθ − h ( K ∗ ϕ )( e it ) dt ( the boundary trace of f )If one of them exists and is finite, they all do and they are equal. The equivalence of (1)and (2) is Abel’s theorem and a result of E. Landau [37] p. 65– 66. The equivalence of (2)and (3) is from Beurling in [11]. he boundary behaviour of elements of Dirichlet space is better than this result indicatesin two ways. Firstly, the limit exists not just rdially but non-tangentially. Secondly, thelimit exists not just almost everywhere, but up to a set of outer capacity zero.We now define non-tangential limit. A non-tangential wedge in D with vertex at p ∈ S isa set of the form(2.19) W ( p, M ) = { z ∈ D : | p − z | < M (1 − | z | ) } for M ∈ (1 , ∞ ). Definition 2.22.
We say that a function h : D → C has a non-tangential limit of ζ at p in S if lim z → pz ∈ W ( p,M ) h ( z ) = ζ for all M ∈ (1 , ∞ ).Equivalently, in the above definition one may replace non-tangential wedges with Stolzangles ∆( p, α, ρ ) = { z : | arg(1 − ¯ pz ) | < α and | z − p | < ρ } where α ∈ (0 , π/
2) and ρ ∈ (0 , α ).The following theorem of Beurling [21, Theorem 3.2.1] improves our understanding of theboundary behaviour, as promised. Theorem 2.23.
Let h ∈ D harm ( D ) . Then there is a set I ⊆ S of outer logarithmic capacityzero such that the non-tangential limit of h exists on S \ I .Remark . By Lemma 2.13, we may take I to be a Borel set of capacity zero.Since a wedge at p contains a radial segment terminating at p ∈ S , it is immediate thatif the non-tangential limit exists, then the radial limit exists and equals the non-tangentiallimit. Using Theorems 2.20 and 2.21 one then has: Theorem 2.25.
Let h ∈ D harm ( D ) . Let H be the non-tangential boundary values of h . TheFourier series of H converges, except possibly on a set of outer logarithmic capacity zero, to H . Finally, we have the following.
Theorem 2.26.
Let h , h ∈ D harm ( D ) . If the non-tangential limits of h and h are equalexcept on a Borel set of capacity zero, then h = h . To see this, it is enough to see that if the non-tangential limit of h ∈ D harm ( D ) is zero,then h is zero. This follows essentially from the equality of the radial and non-tangentiallimits and (2.14).2.5. Conformally non-tangential boundary values.
We now extend the notion of non-tangential limits to arbitrary Jordan domains. Thisextension is an immediate consequence of the Riemann mapping theorem, and is uniquelydetermined by the requirement that the definition be conformally invariant. Although thisextension is by itself trivial, substantial results arise when one considers boundary valuesfrom two sides of the curve, as we will see in Section 3. efinition 2.27. Let Ω be a Jordan domain in C with boundary Γ. Let h : Ω → C be afunction. We say that the conformally non-tangential (CNT) limit of h is ζ at p ∈ Γ if, fora biholomorphism f : D + → Ω, the non-tangential limit of h ◦ f is ζ at f − ( p ).The existence of the limit does not depend on the choice of biholomorphism, as the fol-lowing lemma shows. Lemma 2.28.
Let h : D → C , and let T : D → D be a disk automorphism. Then h has anon-tangential limit at p ∈ ∂ D if and only if h ◦ T has a non-tangential limit at T − ( p ) , andthese are equal.Proof. The claim follows from the easily verified fact that every Stolz angle at p is containedin the image under T of a Stolz angle at T − ( p ), and every Stolz angle at T − ( p ) is containedin the image under T − of a Stolz angle at p . (cid:3) If h ◦ f has non-tangential limit ζ at f − ( p ) for a biholomorphism f : D + → Ω, then h ◦ g has non-tangential limit ζ at g − ( p ) for any biholomorphism g : D + → Ω, by applying thelemma above to T = g − ◦ f . Remark . This notion of CNT limit is conformally invariant, in the following sense. If Ω and Ω are Jordan domains and f : Ω → Ω is a biholomorphism, then the CNT boundaryvalues of h : Ω → C exists and equals ζ at p ∈ ∂ Ω if and only if the CNT limit of h ◦ f exists and equals ζ at f − ( p ) ∈ Ω . The only role that the regularity of the boundary curvesplays in the definition, is that we use Carath´eodory’s theorem implicitly to uniquely associatepoints on the boundary of ∂ Ω with points on ∂ Ω . Therefore the boundary is required tobe a Jordan curve. However, even this condition can be removed, by replacing the boundaryof the domain in C with the ideal boundary [62]. Remark . An obviously equivalent definition is as follows. The CNT limit of h : Ω → C is ζ at p ∈ ∂ Ω if, given a conformal map f : D + → Ω, defining V ( p, M ) = f ( W ( f − ( p ) , M )),one has that lim z → pz ∈ V ( p,M ) h ( z ) = ζ . Note that, treating the ideal boundary of Ω as a border of Ω [4] (which can be done sinceΩ is biholomorphic to a disk), the angle of the wedge V ( p, M ) has a sensible geometricmeaning. That is, let φ be a border chart taking a neighbourhood U of p in Ω to a half-diskwhich takes a segment of the ideal boundary containing p to a segment of the real axis. Inthis neighbourhood, φ ( V ( p, M ) ∩ U ) is a wedge in the ordinary sense. The boundary of φ ( V ( p, M ) ∩ U ) meets the real axis at two angles which are independent of the choice ofchart.Using CNT limits, we can formulate a conformally invariant version of Beurling’s theoremon non-tangential limits. Theorem 2.31.
Let Ω be a Jordan domain with boundary Γ . For h ∈ D harm (Ω) , the CNT boundary values of h exist at every point in Γ except possibly on a null set I ⊂ Γ with respectto Ω . If h and h are CNT boundary values of some element of H and H in D harm (Ω) respectively, and h = h except possibly on a null set, then H = H . his follows directly from Theorem 2.23, Theorem 2.26, Lemma 2.13, and the conformalinvariance of CNT limits (Remark 2.29).We now define a particular class of boundary values. Let Ω be a Jordan domain in C withboundary Γ. We say that two functions h and h on Γ are equivalent if h = h exceptpossibly on a null set I with respect to Ω. Denote the set of such functions up to equivalenceby B (Γ , Ω). We say that h = h if they are equivalent. Definition 2.32.
The Osborn space of Γ with respect to Ω, denoted H (Γ , Ω), is the set offunctions h ∈ B (Γ , Ω) which arise as boundary values of elements of D harm (Ω).We then define the trace operator b Ω , Γ : D harm (Ω) → H (Γ , Ω)and the extension operator e Γ , Ω : H (Γ , Ω) → D harm (Ω)accordingly.In the case that Γ = S and Ω = D + , these maps have simple expressions in terms of theFourier series: b D + , S (cid:16) ∞ X n =0 a n z n + ∞ X n =1 a − n ¯ z n (cid:17) = ∞ X n = −∞ a n e inθ , and e S , D + ∞ X n = −∞ a n e inθ ! = ∞ X n =0 a n z n + ∞ X n =1 a − n ¯ z n . Similar expressions can be obtained for D − .By the Jordan curve theorem, a Jordan curve divides C into two connected componentsΩ and Ω . This leads to the following important question: Question: for which Jordan curves Γ is H (Γ , Ω ) = H (Γ , Ω )?That is the topic of the next section.3. Transmission of harmonic functions in quasicircles
Vodop’yanov-Nag-Sullivan theorem.
First we recall a result due to K. Vodop’yanov[72] regarding the boundedness of composition operators on fractional Sobolev spaces whichwill be useful in proving a characterization results for quasisymmetric homeomorphims of S . However the original result is formulated for Sobolev spaces on the real line. To thisend, one defines the homogeneous Sobolev (or Besov) space ˙ H / ( R ) as the closure of C ∞ c ( R )(smooth compactly supported functions) in the seminorm(3.1) k f k ˙ H / ( R ) = (cid:18)Z R Z R | f ( x ) − f ( y ) | | x − y | dx dy (cid:19) . heorem 3.1. The composition map C φ ( h ) := h ◦ φ is bounded from ˙ H ( R ) to ˙ H ( R ) , ifand only if φ is a quasisymmetric homeomorphism of R to R . See [72] Theorem 2.2.
Remark . As is customary in Sobolev space theory, the constructions of compositions,traces and so on, are done using dense subsets of Sobolev spaces, e.g. the set of smoothcompactly supported functions, where for example the composition C φ ( h ) is well-defined(i.e for h ∈ C ∞ c ( R )) . Thereafter one seeks boundedness estimates with bounds that areindependent of h and extends the results by density to the desired Sobolev space.As a side-note, using the existence of the solution of the Dirichlet’s problem and quasi-isometric extensions of quasisymmetries, the authors of the current exposition showed in[56] that C φ is bounded on H ( S ). To see this first we extend the quasisymmetry φ to aquasi-isometry Φ on D , which is possible thanks to a result of Z. Ibragimov, see [30, Theorem3.1 (5)], and the conformal equivalence of the half plane and the disk. Next we let F to bethe harmonic extension of f ∈ H ( S ) which according to Proposition 1.7 on page 360 in[69] belongs to H ( D ) and satisfies the estimate k F k H ( D ) ≤ C k f k H ( S ) .Now since the boundary value of C Φ ( F ) is C φ ( f ), by the continuity of the restriction to theboundary (see [69] Proposition 4.5), one has that k C φ ( f ) k H ( S ) ≤ C k C Φ ( F ) k H ( D ) . But itis well-known that k C Φ ( F ) k H ( D ) . k F k H ( D ) , for every quasi-isometric homeomorphism Φ,see e.g. [27, Theorem 4 . ′ ] and its Corollary 1. Thus k C φ ( f ) k H ( S ) . k C Φ ( F ) k H ( D ) . k F k H ( D ) . k f k H / ( S ) , which proves the claim.In [42] S. Nag and D. Sullivan showed that quasisymmetries of S are characterized by thefact that they are bounded maps of the Sobolev space H / ( S ) / R and in doing so reprovedTheorem 3.1. In what follows we give a presentation of their result adding also some morereferences for the sake of completeness. Theorem 3.3.
Let φ : S → S be a homeomorphism. Then the following are equivalent. (1) φ is a quasisymmetry; (2) φ has the following three properties: (a) φ takes Borel sets of capacity zero to Borel sets of capacity zero; (b) for every h ∈ H ( S ) , C φ ( h ) ∈ H ( S ) ; (c) the map h h ◦ φ obtained in ( b ) is bounded in the sense that there is a C suchthat (3.2) D D + ( e S , D + ( h ◦ φ )) ≤ CD D + ( e S , D + h ) . Proof.
That (1) implies 2(a) is Corollary 2.16.That (1) implies 2(b) are equivalent can be shown by transferring the problem to the realline. As a consequence of a much more general result for divergence-type elliptic operatorsdue to A. Barton and S. Mayboroda [9, Theorem 7.18], if H denotes the upper half-plane,then there exists a solution to the Dirichlet’s problem ( ∆ u = 0 on H ,u | ∂ H = f ∈ ˙ H / ( R ) , which is unique (up to additive constants) and the estimate(3.4) k u k ˙ H ( H ) ≤ C k f k ˙ H ( R ) , holds.Now the fact that for every h ∈ H ( R ), the composition C φ h ∈ H ( R ), is then a consequenceof Theorem 3.1.That (1) implies 2(c) can be shown as follows. Let H ∈ D harm ( D + ) be the function whoseCNT boundary values equal h quasieverywhere. Let Φ : D + → D + be a quasiconformalmap whose boundary values equal φ (which exists by the aforementioned Beurling-Ahlforsextension theorem). By quasi-invariance of Dirichlet energy (see e.g. [1]) we have D D + ( C Φ H ) ≤ C ′ D D + ( H ) = C ′ D D + ( e S , D + h )where C ′ is of course independent of H . Let F := C Φ H − e S , D + ( C φ h ) ∈ H ( D + ). Thenusing F | S = 0 , the harmonicity of e S , D + ( C φ h ) and the Sobolev space divergence theorem(see e.g. Theorem 4.3.1 page 133 in [22]) one can show that Z D + ∂ ( e S , D + ( C φ h )) ∂F dA = 0 . This yields that D D + ( e S , D + ( C φ h )) ≤ D D + ( e S , D + ( C φ h )) + D D + ( F )= D D + ( e S , D + ( C φ h )) + 2 Z D + ∂ ( e S , D + ( C φ h )) ∂F dA + D D + ( F )(3.5) = D D + ( C Φ H ) . Finally if (3.2) is valid for any homeomorphism φ , then transference of Douglas’s result to thereal line in equation (2.14) yields that k C φ u k ˙ H / ( R ) ≤ C k u k ˙ H / ( R ) , which by part 2(b) yieldsthat φ is a quasisymmetric homeomorphism of the real line. This completes the proof. (cid:3) In the remainder of the paper, we will say that an operator between Dirichlet spaces isbounded with respect to Dirichlet energy if it satisfies an estimate of the form given byequation (3.2).Conditions (2)(a) and (2)(b) of Theorem are not easy to verify, but of course the direction(2) → (1) can be stated in the following way. Theorem 3.4.
Let φ : S → S be a homeomorphism. Assume that there is a dense set L ⊆ ˙ H / ( S ) such that L ⊆ C ( S ) and there is an M such that k C φ h k ˙ H / ( S ) ≤ M k h k ˙ H / ( S ) .Then φ is a quasisymmetry.Proof. This is also a result whose proof is embedded in the proof of Theorem 3.1. See alsoCorollary 3.2 in [42] and Theorem 1.3 in [15]. (cid:3) .2. Transmission (Overfare).
We are now able to prove the transmission theorem in the simplest case.
Theorem 3.5.
Let Γ be a Jordan curve in C , and let Ω and Ω be the components of thecomplement. The statements (1), (2) , and (3) below are equivalent. (1) Γ is a quasicircle. (2) (a) If I ⊆ Γ is null with respect to Ω then it is null with respect to Ω , (b) H (Γ , Ω ) ⊆ H (Γ , Ω ) , and (c) the map e Γ , Ω b Γ , Ω : D harm (Ω ) → D harm (Ω ) is bounded with respect to Dirichletenergy. (3) (a) If I ⊆ Γ is null with respect to Ω then it is null with respect to Ω , (b) H (Γ , Ω ) ⊆ H (Γ , Ω ) , and (c) the map e Γ , Ω b Γ , Ω : D harm (Ω ) → D harm (Ω ) is bounded with respect to Dirichletenergy.Proof. We show that (2) implies (1). The truth of either (1) or (2) is unaffected by applyinga global M¨obius transformation, so we can assume that Γ is bounded. Let Ω ± be theconnected components of the complement in C ; assume for definiteness that Ω = Ω + (thiscan be arranged by composing by 1 /z ).Now let f ± : D ± → Ω ± be conformal maps. By Carath´eodory’s theorem, f ± each extendto homeomorphisms from S to Γ; denote the extensions also by f ± . The function φ = f − ◦ f − (cid:12)(cid:12) S : S → S is thus a homeomorphism. We will show that φ is a quasisymmetry.Once this is shown, it follows from the conformal welding theorem that Γ is a quasicircle.To do this, we show that φ has properties 2(a), 2(b), and 2(c) of Theorem 3.3. Let I be aBorel set of capacity zero in S . Then f − ( I ) is by definition null with respect to Ω + . So by2(a) of the present theorem, f − ( I ) is null with respect to Ω − . By definition φ ( I ) = f − ( f − ( I ))is a Borel set of capacity zero in S . This shows that φ has the property 2(a) of Theorem3.3.Given h ∈ H ( S ), there is an H ∈ D harm ( D + ) with CNT boundary values equal to h except possibly on a null set I . Also, H ◦ f − ∈ D harm (Ω + ). By definition, H ◦ f − has CNTboundary values except on the null set f + ( I ). By assumption 2(b) of the present theorem,there is a function u = e Γ , Ω − b Γ , Ω + ( H ◦ f − ) ∈ D harm (Ω − )whose CNT boundary values agree with those of H ◦ f − except on a null set K containing f + ( I ). Set I ′ = f − ( K ), which is a null set containing I .By definition, u ◦ f − ∈ D harm ( D − ) has CNT boundary values except on the null set f − − ( K ) = φ − ( I ′ ), which contains φ − ( I ). These CNT boundary values agree with h ◦ f − ◦ f − = h ◦ φ except on φ − ( I ′ ). Thus the function u ◦ f − (1 / ¯ z ) has CNT boundary values equalto h except on φ − ( I ′ ). That is, u ◦ f − (1 / ¯ z ) = e S , D + ( h ◦ φ ) , which shows that C φ h ∈ H ( S ). Since h is arbitrary, this shows that property 2(b) ofTheorem 3.3 holds.To show that 2(c) of Theorem 3.3 holds, by 2(c) of the present theorem and conformalinvariance of the Dirichlet norm, there is a constant C > D Ω + ( e Γ , Ω + b Γ , Ω − v ) ≤ CD Ω − ( v ) for all v ∈ D harm (Ω − ). Then for arbitrary h ∈ H ( S ), using the notation above we ave D D + ( e S , D + ( h ◦ φ )) = D D + ( u ◦ f − (1 / ¯ z )) = D D − ( u ◦ f − )= D Ω − ( u ) ≤ CD Ω + ( H ◦ f − )= CD D + ( H ) = CD D + ( e S , D + h ) . Thus φ is a quasisymmetry, completing the proof that (2) implies (1). It is easy to seethat this proof, with minor changes, shows that (3) implies (1).So we need only show that (1) implies (2). Again, we can assume that Γ is boundedand denote the bounded and unbounded components of the complement by Ω + and Ω − respectively. Let f ± : D ± → Ω ± be conformal maps, which have quasiconformal extensionsto C . Thus φ = f − ◦ f − is a quasisymmetry of S , and properties 2(a)–2(c) of Theorem 3.3hold.Given a Borel set I ⊆ Γ which is null with respect to Ω + , by definition f − ( I ) is aBorel set of capacity zero in S . Thus since φ − is a quasisymmetry, by Theorem 3.3 φ − ( f − ( I )) = f − − ( I ) is a Borel set of capacity zero. Thus by definition I is null withrespect to Ω − . This shows that 2(a) of the present theorem holds.Denoting R ( z ) = 1 / ¯ z , a proof similar to that given above for the reverse implication showsthat 2(b) of the present theorem holds with the extension to Ω − given by e Γ , Ω − b Γ , Ω + H = [ e S , D + ( b S , D + ( H ◦ f + ) ◦ φ )] ◦ R ◦ f − − , where H ∈ D harm (Ω + ) . To show that 2(c) of the present theorem holds, let C be the constant in Theorem 3.3 part2(c). Then we have D Ω − ( e Γ , Ω − b Γ , Ω + H ) = D D + ( e S , D + ( b S , D + ( H ◦ f + ) ◦ φ )) ≤ CD D + ( H ◦ f + ) = CD Ω + ( H )which completes the proof. (cid:3) Again, conditions (2/3)(a) and (2/3)(b) are difficult to verify in practice. So we give amore practical version of the (2/3) → (1) version of this theorem.First, we observe that harmonic functions which extend continuously to the boundaryhave a transmission. That is, let Γ be a Jordan curve separating C into two componentsΩ and Ω , and denote the set of functions continuous on the closure of Ω j by C (clΩ j ) andthe set of functions in C (clΩ j ) which are additionally harmonic in Ω j by C harm (Ω j ). Then bythe existence and uniqueness of solutions to the Dirichlet problem, given any h ∈ C harm (Ω )there is an h ∈ C harm (Ω ) whose boundary values agree with those of h everywhere. Wethus have the well-defined mapsˆ O Ω , Ω : C harm (Ω ) → C harm (Ω )ˆ O Ω , Ω : C harm (Ω ) → C harm (Ω ) . It follows immediately from the definition of CNT boundary values that if h extends contin-uously to a boundary point p ∈ Γ then the CNT boundary value exists and equals its CNTlimit. This motivates the definition of a transmission operator O Ω , Ω by restricting ˆ O Ω , Ω to D harm(Ω ) ∩ C harm(Ω ) , which we shall define momentarily. Before doing that we gather our bservations in the following theorem: Theorem 3.6.
Let Γ be a Jordan curve in C , and let Ω and Ω be the connected componentsof the complement of Γ . If there is a dense set L ⊆ D harm (Ω ) , such that L ⊂ C (cl Ω ) ,and the continuous transmission is bounded with respect to Dirichlet energy on L , then Γ isa quasicircle. Here, by dense set, we mean that for any h ∈ D harm (Ω ), for all ǫ > u ∈ L such that D Ω ( u − h ) < ǫ . Proof.
Let f : D + → Ω and g : D − → Ω be biholomorphisms. Then by Carath´eodory’stheorem φ := g − ◦ f is a well-defined homeomorphism of S .Given a L satisfying the hypotheses, observe that C g L is dense in D harm (Ω ) by con-formal invariance of Dirichlet energy, and by Carath´eodory’s theorem C g L ⊂ C ( S ). Also b D − , S C g L is dense. Now for h ∈ b D − , S C g L , defineˆ C φ h = b D + , S C f ˆ O Ω , Ω C g − e S , D − h and note that ˆ C φ h = C φ h. By conformal invariance of the Dirichlet spaces and the hypothesis, this is a bounded operatoron ˙ H / ( S ). Thus applying Theorem 3.4 we see that φ is a quasisymmetry which in turnyields that Γ is a quasicircle. (cid:3) Theorem 3.5 shows that if Γ is a quasicircle and Ω , Ω are the connected components ofthe complement, then H (Γ , Ω ) = H (Γ , Ω ). We thus define H (Γ) = H (Γ , Ω ) = H (Γ , Ω )in this special case. Now we are ready to define the transmission operators. Definition 3.7.
We have well-defined maps O Ω , Ω = e Γ , Ω b Γ , Ω : D harm (Ω ) → D harm (Ω ) O Ω , Ω = e Γ , Ω b Γ , Ω : D harm (Ω ) → D harm (Ω )which are bounded with respect to Dirichlet energy.We will also use the simplified notation O , = O Ω , Ω , O , = O Ω , Ω , wherever it can be done without ambiguity. Remark . The symbol “ O ” stands for old english “oferferian” meaning “to transmit”,which could be rendered as “overfare” in modern english.The overfare operators are inverses of each other by definition:Id D harm (Ω ) = O , O , Id D harm (Ω ) = O , O , where Id stands of course for the identity on the space indicated by the subscript. he overfare operators have a simple form in the case that Γ = S :[ O D + , D − h + ]( z ) = h + (1 / ¯ z ) , [ O D − , D + h − ]( z ) = h − (1 / ¯ z )for h ± ∈ D harm ( D ± ).We also observe that there is a transmission on Bergman space. Namely, if Γ is a quasicirclewe define O ′ , : A harm (Ω ) → A harm (Ω )to be the unique operator satisfying(3.6) O ′ , d = d O , and similarly for O ′ , . It is easily checked that this is well-defined using the fact that thetransmission of a constant is (the same) constant. Similarly, for arbitrary Jordan curves Γ,continuous transmission induces the transmission on harmonic one-formsˆ O ′ , : d C harm (Ω ) → d C harm (Ω ) . The formulation of CNT boundary values and limits was entirely conformally invariant.However, in the context of transmission, the existence of the overfare depends on the relativegeometry of the domain Ω and the sphere. That is, it depends on the regularity of theboundary. It is remarkable that complete symmetry between the boundary value problemsfor the inside and outside domains occurs precisely for quasicircles. To the authors, this isan indication of the principle that Teichm¨uller theory can be seen as a scattering theory forharmonic one-forms [62].Finally, we record the following result.
Corollary 3.9.
Let Γ be a Jordan curve in C and Ω and Ω be the connected components,and assume that f : D + → Ω and g : D − → Ω are biholomorphisms. If there is a dense set L ⊆ D harm (Ω ) such that L ⊂ C (cl Ω ) , on which C f ˆ O , is bounded, then Γ is a quasicircle.Conversely, if Γ is a quasicircle, then C f O , is bounded.Proof. If Γ is a quasicircle, then O , is bounded by Theorem 3.5, and C f is an isometry.Conversely, assume that there is a dense subset L with the stated properties. Then C g L is dense in D harm ( D − ) since C g preserves the Dirichlet energy, and furthermore C g L ⊂C (cl D − ). By assumption O D + , D − C f ˆ O , C g − is bounded on D harm ( D − ). Hence C g − ◦ f isbounded on ˙ H / ( S ) and therefore by Theorem 3.4 φ = g − ◦ f is a quasisymmetry. ThusΓ is a quasicircle. (cid:3) The bounce operator and density theorems.Definition 3.10.
Let Γ be a Jordan curve bounding a Jordan domain Ω in C . A collarneighbourhood of Γ in Ω is a set of the form A p,r = f ( A r )where A r = { z ∈ C : r < | z | < } and f : D + → Ω is a biholomorphism such that f (0) = p .In [60, 61] we used the term collar neighbourhood referred for more general domains, butthis special case suffices for our purposes here.We will show that functions in the Dirichlet space of a collar neighbourhood of Γ haveCNT boundary values. To prove this, we need a lemma. emma 3.11. Let A = { z ∈ C : r < | z | < R } B = { z ∈ C : | z | < R } and B = { z ∈ C : r < | z |} ∪ {∞} . For any h ∈ D harm ( A ) , there is a constant c ∈ C and functions h i ∈ D harm ( B i ) for i = 1 , ,such that h = h + h + c log ( | z | /R ) for all z ∈ A . If h is real, it is possible to choose h , h , and c real.Proof. We prove the claim for h real; the general case follows by separating h into real andimaginary parts.Choose s ∈ ( r, R ) and let γ be the curve | z | = s traced once counterclockwise. Set c = 12 π Z γ ∗ dh. Since Z γ ∗ d log ( | z | ) = 2 π we then have that Z γ ∗ d ( h − c log ( | z | )) = 0 . Set H = h − c log | z | . Since ∗ dH is exact, H has a single-valued harmonic anti-derivative G in A , which is the harmonic conjugate of H . Thus F = H + iG is a holomorphic function in A . Now define F ( z ) = lim s ր R πi Z γ F ( ζ ) ζ − z dζ , z ∈ B ;and define F by F ( z ) = lim s ց r πi Z γ F ( ζ ) ζ − z dζ , z ∈ B \{∞} and F ( ∞ ) = 0. Observe that F is holomorphic on B and F is holomorphic on B .Furthermore for z ∈ A clearly F ( z ) = F ( z ) − F ( z ). Now setting h = Re( F ) and h =Re( F ) we obtain the desired decomposition, where h , h , and c are real. It remains toshow that h i ∈ D harm ( B i ) for i = 1 , h ∈ D harm ( B ), it is enough to show that there is an annulus A ′ = { z ∈ C : r ′ < | z | < R } for r ′ ∈ ( r, R ) such that h is in D harm ( A ′ ), since h is holomorphic on an openneighbourhood of the closure of | z | < r ′ .Given any such r ′ ∈ ( r, R ), the closure of A ′ is in B , and thus the restriction of h to A ′ is in D harm ( A ′ ). Furthermore, the restriction of h to A ′ is in D harm ( A ′ ), and a directcomputation shows that log ( | z | ) is in D harm ( A ), and in particular in D harm ( A ′ ). Since h = h − h − c log ( | z | ), this proves that h ∈ D harm ( A ′ ) and hence in D harm ( B ).The same argument shows that h ∈ D harm ( B ). (cid:3) emark . It is easy to adapt this argument to any doubly-connected domain borderedby non-intersecting Jordan curves, even on Riemann surfaces [60]. It can be shown that thedecomposition is unique, up to the additive constant which can be transferred between h and h . Theorem 3.13.
Let Γ be a Jordan curve bounding a Jordan domain Ω in C . Let A be acollar neighbourhood of Γ in Ω . If h ∈ D harm ( A ) then h has CNT boundary values exceptpossibly on a null set with respect to Ω . Furthermore, there is an H ∈ D harm (Ω) whose CNT boundary values agree with those of h except possibly on a null set.Proof. By definition of collar neighbourhood, for some p ∈ Ω and r ∈ (0 , A = A p,r = f ( A r )where A r = { z ∈ C : r < | z | < } . By conformal invariance of Dirichlet spaces and CNTboundary values, it suffices to show this for Γ = S , Ω = D + , and A = A r .Let h ∈ D harm ( A r ). By Lemma 3.11, h = h + h + c log | z | for some functions h i ∈D harm ( B i ), i = 1 , B = D + and B = { z : | z | > r } ∪ {∞} . Now c log | z | extendscontinuously to 0 on S , and thus the non-tangential boundary values exist and are zeroeverywhere on S . Since h ∈ D ( D + ), it has non-tangential boundary values except possiblyon a null set by a direct application of Beurling’s theorem 2.23. Now h is continuous on anannular neighbourhood of S and thus the non-tangential boundary values exist with respectto D + everywhere. Thus the non-tangential boundary values of h exist except possibly on anull set.Furthermore, u ( z ) = h (1 / ¯ z ) ∈ D harm ( D + ) is continuous on an open neighbourhood of D + ,and its non-tangential boundary values exist everywhere with respect to D + and equal thoseof h with respect to D + . Thus the function H = h + u is in D harm ( D + ) has non-tangentialboundary values equal to h except possibly on a null set. (cid:3) Remark . The proof actually shows a slightly stronger statement: there is an H ∈D harm (Ω) whose CNT boundary values exist and equal those of h , precisely where those of h exist. Definition 3.15.
Since the function H ∈ D harm (Ω) is uniquely determined by its CNTboundary values on Γ, Theorem 3.13 induces a well-defined operator G A, Ω : D harm ( A ) → D harm (Ω)for any collar neighbourhood A of the boundary Γ of a Jordan domain. We call this the“bounce” operator.It follows immediately from the conformal invariance of the Dirichlet space and CNT limitsthat the bounce operator is conformally invariant. That is, if f : Ω ′ → Ω is a biholomorphismand A ′ is the domain such that f ( A ′ ) = A , then(3.7) G A ′ , Ω ′ ( h ◦ f ) = ( G A, Ω h ) ◦ f. We shall also need a result about the agreement of Sobolev and Osborn spaces.
Theorem 3.16.
Given a function f ∈ H / ( S ) there exits a unique harmonic function F ∈ D harm ( D ) whose CNT boundary values agree almost everywhere with values of f on S .Proof. By the existence and uniqueness of the solution to the Dirichlet problem (see e.g.Proposition 4.5 on page 334 in [69]), f has a unique harmonic extension F ∈ H ( D ), andthe CNT boundary values of F are equal to f almost everywhere. (cid:3) sing this we can prove an energy inequality for the bounce operator. Theorem 3.17.
Let Ω be a Jordan domain in C bounded by a Jordan curve Γ . For anycollar neighbourhood A of Γ in Ω , G A, Ω is bounded with respect to the Dirichlet energy. Thatis, there is a constant C such that D Ω ( G A, Ω h ) ≤ CD A ( h ) for all h ∈ D harm ( A ) .Proof. By conformal invariance of the Dirichlet semi-norm and CNT limits, it suffices toprove this for A = A r = { z : r < | z | < } and Ω = D + .Let h ∈ D harm ( A ). Then by Proposition 1.25.2 in [41], h is in H ( A ). By Theorem 3.16, G A, D + h is the unique Sobolev extension of the Sobolev trace of h in H / ( S ). Furthermoreby the result on the unique Sobolev extension, see e.g. Proposition 4.5 on page 334 in [69]and the fact that S ( ∂A , yields that k h | Γ k H / ( S ) ≤ k h | ∂ Ω k H / ( ∂A ) ≤ C k h k H ( A ) . Also, by the existence of the unique solution to the Dirichlet problem with boundarydata in Sobolev spaces (see e.g. Proposition 1.7 on page 360 in [69]), the harmonic Sobolevextension H of h | S satisfies k H k H ( D + ) ≤ C k h | S k H / ( S ) . This together with the estimate for k h | Γ k H / ( S ) above yields that(3.8) k G A, D + h k H ( D + ) ≤ k h k H ( A ) . Now if one applies (3.8) to the harmonic function h − h A where h A is the average of h givenby | A | R A h , then one has that k G A, D + h − G A, D + h A k H ( D + ) ≤ C k h − h A k H ( A ) . Moreover we know that D D + ( G A, D + h ) / = D D + ( G A, D + h − G A, D + h A ) / ≤ k G A, D + h − G A, D + h A k H ( D + ) and that k h − h A k H ( A ) = D A ( h − h A ) / + k h − h A k L ( A ) ≤ D A ( h ) / + D A ( h ) / = 2 D A ( h ) / , where the inequality k h − h A k L (Ω) ≤ CD A ( h ) / is the well-known Poincar´e-Wirtinger in-equality. Thus D D + ( G A, D + h ) ≤ CD A ( h ) , as desired. (cid:3) Theorem 3.18.
Let Ω be a Jordan domain in C bounded by Γ and let A be a collar neigh-bourhood of Γ in Ω . The set G A, Ω ( D ( A )) is dense in D harm (Ω) with respect to the Dirichletsemi-norm.Proof. By conformal invariance of the Dirichlet semi-norm and (3.7), we may assume that A = A r and Ω = D + as above.First, observe that the polynomials C [ z, z − ] are contained in D ( A r ). But for any integer n > G A r , D + z n = z n and G A r , D + z − n = ¯ z n so G A r , D + C [ z, z − ] = C [ z, ¯ z ]. Since C [ z, ¯ z ] is a dense subset of D harm ( D + ) this proves theclaim. (cid:3) heorem 3.19. Let A be any domain in C bounded by two non-intersecting Jordan curves,such that and ∞ are in distinct components of the complement of the closure of A . ThenLaurent polynomials C [ z, z − ] are dense in D ( A ) .Proof. Without loss of generality assume that the component of the complement of Γ con-taining A also contains ∞ , and let B denote this component. Let B then denote thecomponent of the complement of Γ containing A ; it must also contain 0. We have that A = B ∩ B .Now let f i : D + → B i be biholomorphisms for i = 1 ,
2. Let γ ri = f i ( | z | = r ) for r ∈ (0 , h ∈ D ( A ), setting h i ( z ) = lim r ր πi Z γ ri h ( ζ ) ζ − z dζ z ∈ B i , i = 1 , h i are holomorphic on B i , and h = h − h .We will show that h i are in D ( B i ) for i = 1 ,
2. Let C denote the open domain in B bounded by Γ and f ( γ s ) for s chosen close enough to 1 that it is entirely in A . This canbe done, because the function z
7→ | f − ( z ) | is continuous on B , and is strictly less thanone on B . This function has a maximum R < since Γ is compact, so we canchoose s ∈ ( R, h ∈ D ( B ) it suffices to show that h ∈ D ( C ), since h isholomorphic on an open neighbourhood of f ( | z | ≤ s ). Now h ∈ D ( C ) since C ⊆ A , and h ∈ D ( C ), since the closure of C is contained in B . Since h = h + h , this proves theclaim. A similar argument shows that h ∈ D ( B ).Now B is a Jordan domain and hence a Carath´eodory domain, so polynomials C [ z ] aredense in D ( B ) [40, v.3, Section 15]. Similarly C [ z ] is dense in D (1 /B ), so C [1 /z ] is densein D ( B ). So given any ǫ > p ∈ C [ z ] and p ∈ C [1 /z ] such that k h i − p i k D ( A ) ≤ k h i − p i k D ( B i ) < ǫ/ . Thus since h = h − h we see that k h − p + p k D ( A ) < ǫ. This proves the claim. (cid:3)
Corollary 3.20.
Let Γ be a Jordan curve in C and let Ω and Ω be the connected componentsof the complement. Let A and A be collar neighbourhoods of Γ in Ω and Ω respectively,and let U = A ∪ A ∪ Γ . Let R i : D ( U ) → D ( A i ) denote restriction from U to A i for i = 1 , . Then R i ( D ( U )) is dense in D ( A i ) for i = 1 , .Proof. Observe that U is open, so the statement of the theorem makes sense.Now A and A are each bounded by two non-intersecting Jordan curves in C . By applyinga M¨obius transformation and conformal invariance of the Dirichlet spaces and Dirichlet semi-norm, we can assume that ∞ and 0 are each contained in the interior of one of the connectedcomponents of the complement of U , and not both in the same one. In that case, the sameholds for A and A . Thus C [ z, z − ] is dense in D ( A ) and D ( A ) by Theorem 3.19. Since C [ z, z − ] ⊆ D ( U ), the theorem is proven. (cid:3) It is an indispensable fact that the limiting integral of harmonic functions against L one-forms is unaffected by application of the bounce operator. emma 3.21 (Anchor Lemma) . Let Γ be a Jordan curve in C bounding a Jordan domain Ω . Let A be a collar neighbouhood of Γ in Ω and let Γ ǫ = f ( | z | = e − ǫ ) for a biholomorphism f : D + → Ω and ǫ > . For any h ∈ D harm ( A ) and α ∈ A ( A )(3.9) lim ǫ ց Z Γ ǫ α ( w ) h ( w ) = lim ǫ ց Z Γ ǫ α ( w ) G A, Ω h ( w ) . In particular, if h has CNT boundary values equal to zero except possibly on a null set, thenfor any α ∈ A ( A ) lim ǫ ց Z Γ ǫ α ( w ) h ( w ) = 0 . Proof.
We assume that Γ ǫ are positively oriented with respect to 0. The fact that the leftintegral in (3.9) is finite follows from the fact that α and dh are L on A , since fixing ǫ suchthat Γ ǫ is in A , we have by Stokes’ theorem thatlim ǫ ց Z Γ ǫ α ( w ) h ( w ) = Z Γ ǫ α ( w ) h ( w ) + Z Z A ′ α ∧ dh where A ′ ⊂ A is the region bounded by Γ ǫ and Γ.Setting ˜ α ( w ) = α ( f ( w )) f ′ ( w ) and ˜ h ( w ) = h ( f ( w )), and denoting the circle | z | = e − ǫ traced counterclockwise by C ǫ , we have Z Γ ǫ α ( w ) h ( w ) = Z C ǫ ˜ α ( w )˜ h ( w )so it suffices to prove the claim for A = A = { z : e − ǫ < | z | < } , Ω = D + , and Γ ǫ = C ǫ .We first show (3.9) for α ( w ) = w n dw for some integer n . By Lemma 3.11 we can write h = h + h + c log | z | where h ∈ D harm ( D + ) and h ∈ D harm ( B ) where B = { z : | z | >e − ǫ } ∪ {∞} . Now α and h extend continuously to S ; thus so does G A , D + h and so triviallylim ǫ ց Z C ǫ α ( w ) h ( w ) = lim ǫ ց Z C ǫ α ( w ) G A , D + h ( w ) . Similarly lim ǫ ց Z C ǫ α ( w ) log | w | = lim ǫ ց Z C ǫ α ( w ) G A , D + log | w | ;in fact, both sides are zero. Finally, since G A , D + h = h , the claim follows.Thus the claim holds for any α ( w ) = p ( w ) dw for p ( w ) ∈ C [ z, /z ]. Now the set of such α is dense in A ( A ). This follows from the density of C [ z, /z ] in D ( A ) (which is a special caseof Theorem 3.19), and the fact that for some constant k , α − k/z is exact. So the proof ofthe claim will be complete if it can be shown that for h fixed, α lim ǫ ց Z C ǫ α ( w ) h ( w )is a continuous functional on A ( A ).With ǫ and A ′ the region bounded by Γ ǫ and S , let M = sup w ∈ Γ ǫ | h ( w ) | . Since Γ ǫ is a compact subset of A , by a standard result for Bergman spaces there is a constant C independent of α ( w ) = a ( w ) dw such thatsup w ∈ Γ ǫ | a ( w ) | ≤ C k α k A ( A ) . herefore Stokes’ theorem and Cauchy-Schwarz’s inequality yield that (cid:12)(cid:12)(cid:12)(cid:12) lim ǫ ց Z Γ ǫ α ( w ) h ( w ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ ǫ α ( w ) h ( w ) + Z Z A ′ α ∧ dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πe − ǫ M sup w ∈ Γ ǫ | α ( w ) | + k α k A ( A ′ ) k h k D harm ( A ′ ) ≤ (cid:0) πe − ǫ M + k h k D harm ( A ′ ) (cid:1) k α k A ( A ) which completes the proof of (3.9).The proof of the second claim follows immediately from the observation that if h has CNTboundary values zero except possibly on a null set, then G A, D + h = 0. (cid:3) Schiffer and Cauchy operator
Schiffer operators.
We will define certain operators on the Bergman space of anti-holomorphic one-forms which we call “Schiffer operators”. We require an identity to facilitatethe definition.Given a Jordan domain Ω ⊂ C , let g Ω ( z, w ) denote Green’s function of Ω, that is, theharmonic function in z on Ω \{ w } such that g Ω ( z, w ) + log | z − w | is harmonic near w andwhose limit as z → z is zero for any point z on the boundary of Ω. Schiffer considered thefollowing kernel function L Ω ( z, w ) dz dw = 1 πi ∂ g Ω ∂z∂w ( z, w ) dz dw. Note that L Ω is a meromorphic function in z on Ω with a pole of order two at z = w andno other poles. In fact, it is symmetric, so it is also holomorphic in w except for a pole at w = z . Theorem 4.1.
Let Γ be a Jordan curve, and let Ω be one of the connected components of thecomplement of Γ in C . Let g Ω ( z, w ) denote Green’s function of Ω . Then for any one-form ¯ α = h ( z ) d ¯ z ∈ A (Ω)(4.1) (cid:16) Z Z Ω L Ω ( z, w ) h ( w ) d ¯ w ∧ dw (cid:17) · dz = 0 as a principal value integral.Proof. Let f : D + → Ω be a biholomorphism, chosen so that f (0) = z and η be such that f ( η ) = w . Let Γ ǫ be the image of the circle with center at the origin and radius e − ǫ underthe biholomorphic map f , with positive orientation with respect to w . By Stokes’ theorem,if we denote by C r the circle of radius r centred at w with winding number one with respectto w , then Z Z Ω L Ω ( z, w ) h ( w ) d ¯ w ∧ dw · dz = lim r ց Z Z Ω \ B ( w ; r ) L Ω ( z, w ) h ( w ) d ¯ w ∧ dw · dz = lim ǫ ց πi Z Γ ǫ ∂g Ω ∂z ( z, w ) h ( w ) d ¯ w dz − lim r ց πi Z C r ∂g Ω ∂z ( z, w ) h ( w ) d ¯ w dz. ote that all integrals take place over the w variable while z is fixed. To say that the outputof the integral on the left hand side is zero as a form, is equivalent to demanding that forfixed z the coefficient of dz of the output is zero. Therefore it is enough to show that(4.2) lim ǫ ց πi Z Γ ǫ ∂g Ω ∂z ( z, w ) h ( w ) d ¯ w = 0 , and(4.3) lim r ց πi Z C r ∂g Ω ∂z ( z, w ) h ( w ) d ¯ w = 0 . Now Green’s function of the disk is given by g D + ( ζ , η ) = − log (cid:12)(cid:12)(cid:12)(cid:12) ζ − η − ¯ ηζ (cid:12)(cid:12)(cid:12)(cid:12) so by conformal invariance of Green’s function g Ω ( f ( ζ ) , f ( η )) = g D + ( ζ , η ) we have that(4.4) ∂g Ω ∂z ( z, f ( η )) = 1 f ′ (0) ∂∂ζ (cid:12)(cid:12)(cid:12)(cid:12) ζ =0 g D + ( ζ , η ) = 12 f ′ (0) (cid:18) η − ¯ η (cid:19) . Now let A r = { z : r < | z | < } for any r ∈ (0 ,
1) and set A = f ( A r ). One can see explicitlyfrom (4.4) that for fixed z , the function K ( η ) = ∂g Ω ∂z ( z, f ( η ))is in D harm ( A r ), so by conformal invariance of the Dirichlet space k ( w ) = K ( f − ( w )) = ∂g Ω ∂z ( z, w )is in D harm ( A ). Thus we can apply the anchor lemma Lemma 3.21 to ¯ k and α ( w ) = h ( w ) dw toconclude that the integral in (4.2) is zero. On the other hand, for w in an open neighbourhoodof z , by (4.4) (or directly from the definition of Green’s function) we can write ∂g Ω ∂z ( z, w ) = 12( w − z ) + H ( w )where H ( w ) is harmonic in w . Inserting this into the left side of (4.3) we obtain that theintegral is indeed zero. (cid:3) We may now define the Schiffer operator.
Definition 4.2.
Let Γ be a Jordan curve in C , and let Ω and Ω denote the connectedcomponents of the complement of Γ. For h ( w ) d ¯ w ∈ A (Ω ) we define for j = 1 , T Ω , Ω j h ( w ) d ¯ w = 1 π Z Z Ω h ( w )( w − z ) d ¯ w ∧ dw i · dz z ∈ Ω j . Note that the output is a one-form on Ω j . In the case that j = 1, we interpret (4.5) as aprincipal value integral. We will see that this is in general a bounded map into A (Ω j ), andin that role we refer to T Ω , Ω j as Schiffer operators. irst we establish the existence of this integral. Assume for the moment that Ω isbounded, that is, ∞ ∈ Ω . For fixed z ∈ Ω j , the integrand 1 / ( w − z ) is obviously in L (Ω ), so this is immediate. If z ∈ Ω , for a biholomorphism f : D + → Ω let Γ ǫ be theimage of the curve | z | = e − ǫ under f with the same orientation, and let C r be the circlecentred on z traced counterclockwise. Then(4.6) 1 π Z Z Ω h ( w )( w − z ) d ¯ w ∧ dw i · dz = lim ǫ ց πi Z Γ ǫ h ( w ) d ¯ w ( w − z ) dz − lim r ց πi Z C r h ( w ) d ¯ w ( w − z ) dz. Let A s = { z : s < | z | < } where s is fixed so that z is not in the closure of B s = f ( A s ).The first limit exists, by the fact that h ( w ) dw and dw/ ( w − z ) are in A ( B s ) andlim ǫ ց πi Z Γ ǫ h ( w ) d ¯ w ( w − z ) dz = 12 πi Z Γ − log s h ( w ) d ¯ w ( w − z ) dz + 1 π Z Z B s h ( w )( w − z ) d ¯ w ∧ dw i · dz. The limit of the second term in (4.6) can be shown to be zero by an explicit computation.Theorem 4.1 can now be applied to de-singularize the kernel function. We have for α ( w ) = h ( w ) dw ∈ A (Ω )(4.7) T Ω , Ω ¯ α ( z ) = Z Z Ω (cid:18) πi w − z ) − L Ω ( z, w ) (cid:19) h ( w ) d ¯ w ∧ dw · dz since this new term does not have an effect on the existence or value of the integral.We deal with the general case that Ω might be unbounded by establishing the invarianceof the integrals under M¨obius transformations, which is interesting on its own. To this enddefine the pull-back of ¯ α under w = M ( z ) by M ∗ ¯ α ( z ) = h ( M ( z )) M ′ ( z ) d ¯ z and similarly define the pull-back of β ( w ) = g ( w ) dw by M ∗ β ( z ) = g ( M ( z )) M ′ ( z ) dz. Theorem 4.3. If M : C → C is a M¨obius transformation taking Ω j bijectively to ˜Ω j for j = 1 , , then for all ¯ α = h ( w ) d ¯ w ∈ A ( ˜Ω ) we have (4.8) (cid:2) T Ω , Ω j M ∗ ¯ α (cid:3) = M ∗ h T ˜Ω , ˜Ω j ¯ α i . Proof.
Assume first that j = 2. Setting w = M ( z ), η = M ( ζ ), α ( η ) = h ( η ) d ¯ η and using theidentity(4.9) M ′ ( ζ ) M ′ ( z )( M ( ζ ) − M ( z )) = 1( ζ − z ) , hich holds for arbitrary M¨obius transformations, yield that M ∗ (cid:2) T ˜Ω , ˜Ω ¯ α (cid:3) ( z ) = 1 π Z Z ˜Ω h ( η )( η − M ( z )) d ¯ η ∧ dη i · M ′ ( z ) dz = 1 π Z Z Ω h ( M ( ζ ))( M ( ζ ) − M ( z )) M ′ ( ζ ) M ′ ( ζ ) d ¯ ζ ∧ dζ i · M ′ ( z ) dz = 1 π Z Z Ω h ( M ( ζ )) M ′ ( ζ )( ζ − z ) d ¯ ζ ∧ dζ i · dz = [ T Ω , Ω M ∗ ¯ α ] ( z ) . In the case that j = 1, we use the identity L M (Ω) ( M ( ζ ) , M ( z )) M ′ ( ζ ) M ′ ( z ) = L Ω ( ζ , z ) , which follows from g M (Ω) ( M ( ζ ) , M ( z )) = g Ω ( ζ , z ). When combined with (4.9), the argumentabove may be repeated using the expression (4.7). (cid:3) Note that the de-singularization of the integral allowed the application of change of vari-ables in the proof above. As a consequence, we see that the M¨obius transformation preservesthe original principal value integral. This can also be shown directly.
Remark . If one views the Schiffer operators as acting on a Bergman space of functions h ( z ) rather than on the L space of one-forms h ( z ) d ¯ z , their M¨obius invariance is obscured.As promised, Theorem 4.3 implies the existence of the integrals defining the Schiffer op-erator, since we may apply a M¨obius transformation to reduce the general case to the casethat Ω is bounded, which we dealt with above. Remark . If z ∈ Ω j , the meaning of this one-form at z = ∞ is obtained by applying thechange of coordinates z = 1 /ζ , dz = − dζ /ζ to express it in coordinates at ∞ :1 π Z Z Ω h ( w )(1 − wζ ) d ¯ w ∧ dw i · dζ ζ ∈ / Ω j . Alternatively one may transform both the input and output simultaneously using Theorem4.3 with M ( z ) = 1 /z .Finally, we have the following. Theorem 4.6.
Let Γ be a Jordan curve in C , and let Ω and Ω be the components of thecomplement of Γ in C . The Schiffer operators T Ω , Ω j are bounded from A (Ω ) to A (Ω j ) for j = 1 , .Proof. By Theorem 4.3, we may assume that ∞ ∈ Ω , so that Ω is bounded. The integrandsin the definitions of T Ω , Ω and T Ω , Ω given in (4.5) and (4.7) respecrively are non-singularand holomorphic in z for each w ∈ Ω and w ∈ Ω (in each case), and furthermore bothintegrals are locally bounded in z . Therefore the holomorphicity of T Ω , Ω j follows by movingthe ∂ inside (4.5) and (4.7), and using the holomorphicity of the integrands in each case.The L p -boundedness of these operators for 1 < p < ∞ , considered as singular integraloperators, is a consequence of the boundedness of singular integral operators of Calder´on-Zygmund type, see e.g. [38] page 26. (cid:3) s in the case of the overfare operator O , we will use the notation T j,k in place of T Ω k , Ω k wherever possible.4.2. Cauchy operator.
As usual, consider a Jordan curve Γ in C . For now we assumethat ∞ is not in Γ. Let Ω and Ω denote the components of the complement. Definition 4.7.
For h ∈ D (Ω ) we will consider a kind of Cauchy integral obtained asfollows. Let f : D → Ω be a biholomorphism. If we let Γ ǫ be the image of the closed curve | z | = e − ǫ under f , with the same orientation, and q / ∈ Γ, then we define J q Ω h ( z ) = 12 πi lim ǫ ց Z Γ ǫ h ( w ) (cid:18) w − z − w − q (cid:19) dw z ∈ C \ Γ(4.10) = 12 πi lim ǫ ց Z Γ ǫ h ( w ) z − q ( w − z )( w − q ) dw z ∈ C \ Γ . The term involving q amounts to an arbitrary choice of normalization. In the case that q = ∞ , this reduces to J q Ω h ( z ) = 12 πi lim ǫ ց Z Γ ǫ h ( w ) w − z dw z ∈ C \ Γ . This is almost a Cauchy integral, of course. We will motivate the definition of J q Ω afterfirst establishing some of its properties.First, we observe that J q is M¨obius invariant in a certain sense. The invariance followsfrom the identity(4.11) M ′ ( w )( M ( z ) − M ( q ))( M ( w ) − M ( z ))( M ( w ) − M ( q )) = z − q ( w − z )( w − q )which holds for any M¨obius transformation M . Observe that the usual normalization q = ∞ obscures the M¨obius invariance of the Cauchy kernel. Using this identity, together with achange of variables and conformal invariance of the Dirichlet space, we obtain the following. Theorem 4.8.
Let Γ be a curve in C and Ω , Ω be the connected components of thecomplement. Let M be a M¨obius transformation. Then for any h ∈ D ( M (Ω )) , we have h J M ( q ) M (Ω ) h i ◦ M = J q Ω ( h ◦ M ) . Observe that Theorem 4.8 extends the definition of the integral to the case that ∞ ∈
Γ.For the moment, this says only that if the limit exists on one side, then it exists on both,and the two sides are equal. We will show that the limit exists whenever h ∈ D harm (Ω ).In the remainder of the section we will: (1) provide identities relating J q to the Schifferoperators; (2) show that the output is in D harm (Ω ⊔ Ω ); and (3) show that for quasicircles,the limiting integral is in a certain sense independent of which side of Γ you choose to takethe limit in.Let ∂ and ∂ denote the Wirtinger operators on the Riemann sphere. heorem 4.9. Let Γ be a Jordan curve separating C into connected components Ω and Ω .Assume that q / ∈ Γ . Then ∂ J q Ω h ( z ) = − T Ω , Ω ∂h ( z ) z ∈ Ω (4.12) ∂ J q Ω h ( z ) = ∂h ( z ) − T Ω , Ω ∂h ( z ) z ∈ Ω (4.13) ∂ J q Ω h ( z ) = 0 z ∈ Ω ∪ Ω . (4.14) Proof. If q ∈ Σ , then the first claim follows by applying Stokes theorem and bringing ∂ underthe integral sign, as does the third in the case that z ∈ Ω . Denote the circle | z − q | = r traced counter-clockwise by C r . Using the fact thatlim r ց π Z C r ∂g Ω ∂w ( w ; z ) h ( w ) = h ( q )by Stokes’ theorem we have that for q ∈ Ω and z ∈ Ω , J q Ω h ( z ) = lim ǫ ց Z Γ ǫ (cid:20) πi (cid:18) w − z − w − q (cid:19) − π ∂g Ω ∂w ( w, q ) (cid:21) h ( w ) dw − h ( q )= Z Z Ω (cid:20) πi (cid:18) w − z − w − q (cid:19) − π ∂g Ω ∂w ( w, q ) (cid:21) h ( w ) d ¯ w ∧ dw − h ( q )(4.15)Noting that the integrand is non-singular, applying ∂ z to both sides using Theorem 4.1)proves the first claim. Applying ∂ z proves the third claim in the case that q ∈ Ω and z ∈ Ω .If z ∈ Ω and q ∈ Ω , we have similarly that(4.16) J q Ω h ( z ) = Z Z Ω (cid:20) πi (cid:18) w − z − w − q (cid:19) + 1 π ∂g Ω ∂w ( w, z ) (cid:21) h ( w ) d ¯ w ∧ dw + h ( z ) . Applying ∂ z completes the proof of the third claim, and applying ∂ z using (4.7) proves thesecond claim in the case that q ∈ Ω . To prove the second claim in the case that q ∈ Ω , weadd a further term to (4.16) which removes the singularity at q as in (4.15) and apply ∂ z . (cid:3) For j = 1 , J q Ω , Ω j h = J q Ω h (cid:12)(cid:12) Ω j . Then Theorem 4.9 immediately implies that these are bounded with respect to the Dirichletenergy.
Corollary 4.10.
Let Γ be a Jordan curve in C and choose q / ∈ Γ . Then J q Ω : D harm (Ω ) → D harm (Ω ∪ Ω ) q and J q Ω , Ω j : D (Ω ) → D (Ω j ) j = 1 , . If q ∈ Ω j , then the image of J q Ω , Ω j is D q (Ω j ) . Furthermore, each of the operators above arebounded with respect to Dirichlet energy.Proof. This follows immediately from Theorems 4.6 and 4.9. (cid:3) s in the case of the overfare and Schiffer operators, we will use the notation J qk in placeof J q Ω k and J qj,k in place of J q Ω j , Ω k wherever possible.The operator J q is motivated as follows. Setting aside the normalization at q , we wouldlike to define the Cauchy integral 12 πi Z Γ h ( w ) w − z dw of a function h ∈ H (Γ , Ω ), but there are two obstacles: the curve Γ is not rectifiable, andfunctions in h are not particularly regular. This problem is solved by considering instead J q Ω e Γ , Ω h .The question immediately arises: if one considers instead J q Ω e Γ , Ω h , is the result thesame? Of course, this requires that H (Γ , Ω ) ⊆ H (Γ , Ω ) at least, which we know is true forquasicircles. In fact, it is indeed sufficient that Γ is a quasicircle. Theorem 4.11.
Let Γ be a quasicircle in C , and let Ω and Ω be the connected componentsof the complement. Fix q / ∈ Γ . For any h ∈ D harm (Ω ) J q h = − J q O , h. The same result holds switching the roles of Ω and Ω .Proof. Let B and B be collar neighbourhoods of Γ in Ω and Ω respectively. Let U = B ∪ B ∪ Γ. This is an open set bordered by two analytic curves. By Corollary 3.20, theclass R D ( U ) of elements of D ( U ) to D ( U ) is dense in D ( B ). Furthermore, by Theorem3.18, G A , Ω D ( B ) is dense in D harm (Ω ). Thus since G B , Ω is bounded by Theorem 3.17, G B , Ω R D ( U ) is dense in D harm (Ω ). By Theorem 3.5 O , is bounded, so it is enough toprove the theorem for such functions.Let h ∈ D ( U ). Since h extends continuously to Γ, the CNT boundary values of R h and R h with respect to Ω and Ω both equal the continuous extension and hence each other.Thus(4.17) O , G B , Ω R h = G B , Ω R h. Fix z ∈ C \ Γ. Since B i are collar domains, by definition there are biholomorphisms f i : D + → Ω i so that f i ( A r i ) = B i for annuli A r i = { z : r i < | z | < } for i = 1 ,
2. Let Γ iǫ denote the limiting curves f i ( | z | = e − ǫ ) with orientations induced by f i . By Carath´eodory’stheorem, the maps f i extend homeomorphically to maps from S to Γ, so for any fixed ǫ ,the curves Γ iǫ are each homotopic to Γ and hence to each other. Thus, since z and q areeventually not in the domain bounded by Γ ǫ and Γ ǫ ,(4.18)lim ǫ ց πi Z Γ ǫ h ( w ) (cid:18) w − z − w − q (cid:19) dw = − lim ǫ ց πi Z Γ ǫ h ( w ) (cid:18) w − z − w − q (cid:19) dw where the negative sign arises from the change of orientation between the integrals.Finally, applying the anchor lemma 3.21 for fixed z with α ( w ) = (cid:18) w − z − w − q (cid:19) dw, e have that for i = 1 , J qi G B i , Ω i R i h ( z ) = lim ǫ ց πi Z Γ iǫ h ( w ) (cid:18) w − z − w − q (cid:19) dw. Here, we may have to shrink the domain B j so that neither z nor q are in the closure, toensure that α ∈ L ( B j ). This does not affect the validity of the argument, since given nestedcollar neighbourhoods B ′ j ⊂ B j , by definition G B ′ j , Ω j h | B ′ j = G B j , Ω j h | B j . Thus combining (4.17), (4.18), and (4.19) we have J q G B , Ω R h = − J q O , G B , Ω R h which completes the proof. (cid:3) Remark . The negative sign is an artifact of the change of orientation induced by theswitch from the domain Ω to Ω . In previous publications [58, 61] we chose the orientationsin such a way that the sign did not change.Finally, we record the following obvious fact. Theorem 4.13.
Let Γ be a Jordan curve in C and let Ω and Ω be the components of thecomplement. Fix q / ∈ Γ , and let h ∈ D (Ω j ) .If q ∈ Ω j , then J qj h ( z ) = (cid:26) h ( z ) − h ( q ) z ∈ Ω j − h ( q ) z / ∈ Ω j ∪ Γ whereas if q / ∈ Ω j then J qj h ( z ) = (cid:26) h ( z ) z ∈ Ω j z / ∈ Ω j ∪ Γ . Proof.
This follows from the ordinary Cauchy integral formula. (cid:3)
The Schiffer isomorphisms and the Plemelj-Sokhotski isomorphisms.
In thissection we show that the Schiffer operator T Ω , Ω and the jump decomposition induced bythe Cauchy operator J q are isomorphisms precisely for quasicircles.We refer to the isomorphism induced by the jump decomposition as the Plemelj-Sokhotskiisomorphism. The classical Plemelj-Sokhotski jump decomposition says the following, in thesmooth case. Let Γ be a smooth Jordan curve Γ separating C into components Ω and Ω ;assume that Ω is the unbounded component. For a smooth function u on Γ, define thefunctions h k ( z ) = 12 πi Z Γ h ( ζ ) ζ − z dζ z ∈ Ω k , k = 1 , . For any point w ∈ Γ, it is easily proven thatlim z → w h ( z ) − lim z → w h ( z ) = u ( w ) . In fact, this formula can be written in a stronger form involving the principal value integralof u on the boundary; see Section 6.4.The map taking u to ( h , h ) is what we call the Plemelj-Sokhotski isomorphism. Usingthe limiting integral in place of the Cauchy integral, we will prove that for u ∈ H (Γ), this is n isomorphism if and only if Γ is a quasicircle. We will also show the closely related resultof Napalkov and Yulmukhametov that T , is an isomorphism if and only if Γ is a quasicircle.To do this, we first require a lemma. Lemma 4.14.
Let Γ be a Jordan curve in C and let Ω and Ω be the connected componentsof the complement. Let B be a collar neighbourhood of Γ in Ω . Assume that q is not inthe closure of B . If H ∈ D ( B ) then J q , G B, Ω H extends to a holomorphic function H on Ω ∪ Γ ∪ B which satisfies H ( z ) = J q , G B, Ω H ( z ) − H ( z ) . z ∈ B. Furthermore, J q , G B, Ω H has a transmission in D harm (Ω ) , given explicitly by ˆ O , J q , G B, Ω H = G B, Ω H = J q , G B, Ω H − G B, Ω H. Recall that ˆ O , is the solution of the Dirichlet problem on Ω with continuous boundaryvalues H | Γ . Proof.
By Theorem 4.8, it is enough to prove this in the case that q = ∞ and Ω is theunbounded component of the complement of Γ.The first claim is just the ordinary Cauchy integral formula combined with the anchorlemma. Let f : D + → Ω be the biholomorphism such that f ( A ) = B for an annulus A = { z : r < | z | < } , and let Γ ǫ be the corresponding images under f of circles | z | = e − ǫ as usual, with orientation induced by f . Let γ be the analytic curve which is the innerboundary of B ; that is, the image of | z | = r under f .Define H ( z ) = 12 πi Z γ H ( w ) w − z dw which is holomorphic in the open set Ω ∪ Γ ∪ B . By the anchor lemma 3.21 and the factthat H is holomorphic, for all z ∈ Ω (4.20) J q , G B, Ω H ( z ) = lim ǫ ց πi Z Γ ǫ H ( w ) w − z dw = H ( z ) . By the ordinary Cauchy integral formula, for all z ∈ BH ( z ) = lim ǫ ց πi Z Γ ǫ H ( w ) w − z dw − H ( z ) . Applying the anchor lemma 3.21 again we see(4.21) H ( z ) = J q , G B, Ω H ( z ) − H ( z )for all z ∈ B .We now prove the second claim. Since H extends continuously to Γ, its CNT boundaryvalues with respect to Ω equal its CNT boundary values with respect to Ω , which are equalto those of J q , G B, Ω H by (4.21). Of course the CNT boundary values are all continuousextensions. Thus(4.22) G B, Ω H = ˆ O , J q , G B, Ω H. o see that G B, Ω H ∈ D harm (Ω ), let B = f ( A ′ ) be a collar neighbourhood of Γ in Ω where A ′ is chosen so that its inner boundary is contained in A . Since H is holomorphic onan open neighbourhood of the closure of B , its restriction to B is in D ( B ). Since G B, Ω is bounded by Theorem 3.17, the transmission G B, Ω H = G B , Ω H (where H is restrictedto B ) is in D harm (Ω ) as claimed.Finally, applying G B, Ω to both sides of (4.21), which leaves the first term of the righthand side unchanged, and using (4.22) we obtain G B, Ω H = J q , G B, Ω H ( z ) − ˆ O , J q , G ( B, Ω ) H. on Ω . This completes the proof. (cid:3) Theorem 4.15.
Let Γ be a quasicircle in C and let Ω and Ω be the connected componentsof the complement. For all h ∈ D harm (Ω ) h = J q , h − O , J q , h. Proof.
By Lemma 4.14 the claim holds for all h of the form G B, Ω H for H ∈ D ( B ). ByTheorem 3.18, G B, Ω D ( B ) is dense in D harm (Ω ). Thus the theorem follows from the factthat J q is bounded by Corollary 4.10. (cid:3) Remark . This can be thought of as the classical jump formula expressed in terms of thetransmission.Lemma 4.14 generates a large class of functions in the Dirichlet space with continuoustransmission. Namely, the bounce of any holomorphic Dirichlet-bounded function in thecollar has a continuous transmission. We show this now, as well as the corresponding factfor Bergman space. Recall that the overfare operator for one-forms O ′ used below wasdefined by equation (3.6). Lemma 4.17.
Let Γ be a Jordan curve separating C into components Ω and Ω . (1) For all h ∈ G B, Ω D ( B ) ∩ D (Ω ) , J q , h has a continuous transmission in D harm (Ω ) given by ˆ O , J q , h = J q , h − h. (2) For all α ∈ ∂ [ G B, Ω D ( B ) ∩D (Ω )] , T , h has a continuous transmission in A harm (Ω ) given by ˆ O ′ , T , α = α + T , α. Proof.
The first claim follows directly from Lemma 4.14. Now let α = ∂h . Applying nowTheorem 4.9 to the right hand side of (1), we see that ∂ ˆ O , T , α = − T , α and ∂ ˆ O , T , α = − α. Applying ∂ to the left hand side of (1) and using Theorem 4.9 again proves the claim. (cid:3) We can now prove that T , is one-to-one. Theorem 4.18.
Let Γ be a Jordan curve separating C into components Ω and Ω . (1) T , is injective. For any collar neighbourhood B of Γ in Ω , T (1 , restricted to ∂ [ G B, Ω D ( B ) ∩D (Ω )] has left inverse P (Ω ) ˆ O ′ , , where P (Ω ) is the projection defined in (2.5) .Proof. The second claim follows immediately from Lemma 4.17 part (2).Let B = f ( A ) be a collar neighbourhood of Γ in Ω induced by some biholomorphism f : D + → Ω and annulus A . Now for any n >
0, by conformal invariance of the bounceoperator (3.7), G B, Ω C f − w − n = C f − G A , D + w − n = C f − ¯ w n so ∂ C f − C [¯ z ] ⊆ ∂ [ G B, Ω D ( B ) ∩ D (Ω )] . Furthermore, ∂ C f − C [¯ z ] is dense in A (Ω ).By the second claim, for any α ∈ ∂ C f − C [¯ z ] if T Ω , Ω α = 0, then α = 0. On the otherhand, for z ∈ Ω fixed, dw/ ( w − z ) ∈ A (Ω ), and for any α ∈ A (Ω ) T , α = (cid:18) α ( w ) , dw ( w − z ) (cid:19) . This proves the first claim. (cid:3)
This implies that the Cauchy-type operator J q , is injective. It is convenient to recordthe two cases q ∈ Ω , Ω . In the following, see (2.8) and (2.9) for the definitions of theprojections . Corollary 4.19.
Let Γ be a Jordan curve Γ in C . (1) Fix q ∈ Ω . (a) For any p ∈ Ω , J q , is injective from D p (Ω ) to D q (Ω ) . (b) For any collar neighbourhood B of Γ in Ω , on D p (Ω ) ∩ G B, Ω D ( B ) , the leftinverse is given by − P ap (Ω ) ˆ O , . (2) Fix q ∈ Ω . (a) J q , is injective from D (Ω ) to D (Ω ) . (b) For any collar neighbourhood B of Γ in Ω , on D (Ω ) ∩ G B, Ω D ( B ) , the leftinverse is given by − P hq (Ω ) ˆ O , .Proof. We first prove the (b) claims. Lemma 4.14 tells us that for any h ∈ G B, Ω D (Ω ) − ˆ O , J q , h = − J q , h + h. (2) (b) follows by observing that the right hand side is the desired decomposition and applying P hq (Ω ) to both sides. (2) (a) follows similarly once one adds the assumption that h ( p ) = 0.To prove the (a) claims, by Theorem 4.18 part (1) and Theorem 4.9, if J q , h = 0 then h isa constant c . If h ∈ D p (Ω), then c = h ( p ) = 0 so h = 0. This proves (1)(a). If q ∈ Ω , then c = h ( q ) = 0. This proves (2)(a). (cid:3) Theorem 4.20.
Let Γ be a Jordan curve separating C into components Ω and Ω . If anyof the following three conditions hold, then Γ is a quasicircle. (1) T , is surjective. (2) The restriction of J q , to D p (Ω ) is surjective onto D q (Ω ) for some q ∈ Ω and p ∈ Ω . The restriction of J q , to D (Ω ) is surjective onto D (Ω ) for q ∈ Ω .Proof. The first claim follows from the second or third, since by Theorem 4.9, ∂ J q , = − T , ∂h for any h ∈ D (Ω ).Assume that the restriction of J q , to D (Ω ) is an isomorphism onto D (Ω ). Let K : D (Ω ) → D (Ω )be its inverse. Choose a collar neighbourhood B = f ( A ) of Γ in Ω , where f : D + → Ω is abiholomorphism and A = { z : r < | z | < } for some r ∈ (0 , G B, Ω C f − C [1 /z ] is dense in D (Ω ), since G B, Ω C f − C [1 /z ] = C f − G A , D + C [1 /z ] = C f − C [ z ]and polynomials are dense in D ( D + ). Since J q , is bounded and surjective, the set L = J q , G B, Ω C f − C [1 /z ]is dense in D (Ω ). Furthermore, Lemma 4.14 guarantees that L ⊂ C (clΩ ), and by Lemma4.17 for every element h ∈ L we haveˆ O , h = ˆ O , J q , K h = ( J q , K − K ) h. We can also conjugate to get transmission of elements h ∈ L ⊂ D (Ω ), that isˆ O , h = ( J q , K − K ) h. Since J q , K − K is bounded, Theorem 3.6 applies, and we can conclude that Γ is a quasicircle.This proves (3).If we assume that q ∈ Ω , then the argument above shows that we have bounded trans-mission on D p (Ω ) and D p (Ω ). Since constants are transmittable this proves (2). (cid:3) Theorem 4.21.
Let Γ be a Jordan curve separating C into components Ω and Ω . Thefollowing are equivalent. (1) Γ is a quasicircle. (2) T , is a bounded isomorphism. (3) For q ∈ Ω and p ∈ Ω , J q , is a bounded isomorphism from D p (Ω ) into D q (Ω ) . (4) For q ∈ Ω , J q , is a bounded isomorphism from D (Ω ) into D (Ω ) .In case (2), the inverse is − P (Ω ) O ′ , ; in case (3) the inverse is − P ap (Ω ) O , ; and in case(4), the inverse is − P hq (Ω ) O , .Proof. If (2), (3), or (4) holds, then by Theorem 4.20 Γ is a quasicircle.Conversely, assume that Γ is a quasicircle. By Theorem 4.18 and Corollary 4.19, we havethat the maps in (2), (3), and (4) are injective.By the inverse mapping theorem it is enough to show that the maps in (2), (3), and (4)are surjective. Assume that q ∈ Ω . To see that J q , is surjective from D p (Ω ) to D q (Ω ),let h ∈ D (Ω ). Let H = − O , h , where the bounded transmission O , exists by Theorem3.5. Now H = H + H where H ∈ D (Ω ) and H ∈ D p (Ω ). For all z ∈ Ω J q , H ( z ) = J q , H ( z ) = J q , h ( z ) = h ( z ) − h ( q ) = h ( z ) here the first equality is by part one of Theorem 4.13 with j = 2, the second equality isby Theorem 4.11, and the third equality is by Theorem 4.13 part two with j = 2. Thus (3)holds.A similar argument, after adjustment of the constants and decompositions, proves surjec-tivity in case (4). Finally, (2) follows from (3) or (4) and the fact that ∂ J q , h = T , ∂h .Thus (1) implies (2), (3), and (4), completing the proof. (cid:3) We now prove that the Plemelj-Sokhotski jump decomposition is an isomorphism preciselyfor quasicircles. For q ∈ Ω , define M q (Ω ) : D harm (Ω ) → D (Ω ) ⊕ D q (Ω ) h (cid:0) J q , h, J q , h (cid:1) . and for q ∈ Ω , define M q (Ω ) : D harm (Ω ) → D q (Ω ) ⊕ D (Ω ) . Similarly we have the following operator on harmonic Bergman space: M ′ (Ω ) : A harm (Ω ) → A (Ω ) ⊕ A (Ω )(4.23) α + β (cid:0) α − T , β, − T , β (cid:1) where α ∈ A (Ω ) and β ∈ A (Ω ).With this notation, we have the following theorem. Theorem 4.22.
Let Γ be a Jordan curve separating C into components Ω and Ω . Thefollowing are equivalent. (1) Γ is a quasicircle. (2) For any q ∈ C \ Γ , M q is an isomorphism. (3) M ′ (Ω ) is an isomorphism.It is enough that (2) holds for a single q .Proof. We first prove that (1) implies (2). Assuming that Γ is a quasicircle, by Theorem4.21 − T , is an isomorphism. Given τ = α + β ∈ A harm (Ω ), assume that M τ = 0. Then − T , β = 0 so β = 0. Since α = α − T , β = 0, we see that τ = 0 so M is injective. Givenany ( τ, σ ) ∈ A (Ω ) ⊕ A (Ω ), choose β such that − T , β = σ . Setting α = τ + T , β we have M ( α + β ) = ( τ, σ ). The converse is just the reversal of these arguments.A nearly identical argument using Theorem 4.21, as well as Theorem 4.13 to deal with theconstants, shows that (1) holds if and only if (2) holds. (cid:3) In the case that Γ is a quasicircle, we call M q the Plemelj-Sokhotski jump isomosphism.This establishes that the jump decomposition holds on quasicircles, with data in H (Γ). Theorem 4.23.
Let Γ be a quasicircle separating C into components Ω and Ω . For any u ∈ H (Γ) , there exist h j ∈ Ω j such that the CNT boundary values u j of h j satisfy u = u − u except possibly on a null set. Fixing q in one of the components Ω j , h and h are uniquelydetermined by the normalization h j ( q ) = 0 , and are given explicitly by ( h , h ) = M q e Γ , Ω u. roof. Fix q ∈ C \ Γ. Given u ∈ H (Γ), denote h = e Γ , Ω u so that ( h , h ) = M q h . To showthat u − u = u it suffices to show that h = h − O , h . But this is precisely Theorem 4.15.To see that the decomposition is unique, let H j be another suitably normalized pair offunctions such that u = b Γ , Ω H − b Γ , Ω H . In that case h = H − O , H so(4.24) h − H + O , ( H − h ) = 0 . In the case that q ∈ Ω , by Theorem 4.21 P hq (Ω ) O , is one-to-one on D (Ω ). Since h ( q ) − H ( q ) = 0, applying this to (4.24) we obtain that H − h = 0. Inserting this back into(4.24) yields h − H = 0. In the case that q ∈ Ω , h − H ∈ D q (Ω ). Fixing p ∈ Ω , byTheorem 4.21 again, P ap (Ω ) O , is injective on D q (Ω ). Applying this to (4.24) as aboveyields H − h = 0 and h − H = 0 . (cid:3) Faber and Grunsky operator
The Faber operator and Faber series.
The Faber operator [67] arises in the theoryof approximation by Faber series in domains in the plane or sphere, and has its origin inthe work of G. Faber [23]. The Faber operator is typically defined as follows. Let Γ be arectifiable Jordan curve separating C into components Ω and Ω . Assume for the momentthat Ω is bounded, that is ∞ ∈ Ω , and 0 ∈ Ω . Let f : D + → Ω be a conformal map, suchthat f (0) = 0. Let h be a holomorphic function on D − and assume that h ◦ f − extends to anintegrable function on Γ; that is, h has boundary values in some sense (e.g. non-tangential)and h ◦ f − ∈ L (Γ). Then the Faber operator is defined by F h ( z ) = − πi Z Γ h ◦ f − ( w ) w − z dw. For various choices of the regularity of Γ and the space of holomorphic functions on D + ,this is called the Faber operator. The Faber operator is closely related to the approximationby Faber polynomials of a holomorphic function on Ω in general. The n th Faber polynomialcorresponding to the domain Ω is defined by F n ( z ) = F (( · ) − n )( z ) = − πi Z Γ ( f − ( w )) − n w − z dw. The Faber operator produces a Faber series as follows. Let h ( z ) be a holomorphic functionin cl( D − ) which vanishes at the origin, and assume that Γ is an analytic Jordan curve, sothat we may focus on the heuristic idea. If h ( z ) = h z − + h z − + · · · , then it is easilyverified that the function H ( z ) = F h ( z ) is a holomorphic on the closure of Ω and vanishesat ∞ . Furthermore, one has the polynomial series H ( z ) = F h ( z ) = ∞ X n =1 h n F (( · ) − n ) = ∞ X n =1 h n F n ( z )called the Faber series of H . This series converges uniformly on the closure of Ω . One of theadvantages of Faber series over power series, is that sufficiently regular functions converge niformly on compact subsets of the domain. That is, unlike power series, they are adaptedto the geometry of the domain.If one refines the analytic setting, as we do below, then one can investigate differentkinds of convergence of the series. Existence and uniqueness of a Faber series correspond tosurjectivity and injectivity of the Faber operator respectively. Remark . Since h is defined on Ω and f − on Ω , the composition h ◦ f − is not necessarilydefined anywhere except on Γ. Thus the boundary behaviour of h and f − play a centralrole in the study of the Faber operator.The analytic properties of the Faber operator as they relate to the regularity of the curveand the function space, and approximability in various senses by series of Faber polynomialshas been extensively studied; see Section 5 for references. We will now choose specificconditions.We define a Faber operator with domain D ( D − ) for arbitrary Jordan curves using trans-mission on the circle. Since the boundary behaviour of a holomorphic function h ( z ) on D + isidentical in every sense to that of O D − , D + h ( z ) = h (1 / ¯ z ), we will replace the domain D ( D + )of the operator by D ( D + ). Definition 5.2.
For q ∈ Ω , we define a Faber operator by setting(5.1) I qf = − J q , C f − : D ( D + ) → D q (Ω ) . It follows immediately from Corollary 4.10 and conformal invariance of Dirichlet spacethat I qf is a bounded operator. The choice q = ∞ and p = f (0) corresponds to the casedescribed above. From here on, we refer to (5.1) as the Faber operator, and use the newnotation to distinguish it from the heuristic discussion above. Remark . It’s also possible to retain the constants in D ( D + ) and D (Ω ) by placing q ∈ Ω .Denote the set of polynomials vanishing at q by C q [ z ]. Assume that f (0) = p ∈ C , andlet Γ ′ be a fixed simple closed analytic curve in Ω with winding number zero with respectto p . By Lemma 3.21, for any ¯ h ∈ C [¯ z ] we have I qf ¯ h = I qf O D − , D + u = − πi Z Γ ′ u ◦ f − ( w ) (cid:18) w − z − w − q (cid:19) dw where u ( z ) = h (1 / ¯ z ). It is easily shown that the output is a polynomial in ( z − p ) − . Inparticular, we define the Faber polynomials as follows. Definition 5.4.
Let Γ be a Jordan curve separating C into Ω and Ω . Assume that q ∈ Ω and let p = f (0). Let f : D + → Ω be a conformal map. The n th Faber polynomial withrespect to f is Φ n ( z ) = I qf ( z − n ) ∈ C q [1 / ( z − p )] . If q = ∞ and p = 0 we have Φ n ( z ) ∈ C ∗ [1 /z ]. It is easily checked that Φ n has degree − n in ( z − p ). Remark . For a bounded domain D bounded by a Jordan curve, polynomials are densein A ( D ) [40], so polynomials vanishing at a fixed point q are dense in D q ( D ). So for p ∈ Ω ,setting M ( z ) = 1 / ( z − p ) and D = M (Ω ), we see that C q [1 / ( z − p )] is dense in D q (Ω ).Thus since Φ n has degree − n in ( z − p ) for each n , we see that the image of I qf is dense in D q (Ω ). y a Faber series we mean a series of the form ∞ X n =1 λ n Φ n ( z ) , whether or not it converges in any sense. We also define what we call the sequential Faberoperator : with notation as in Definition 5.4, I seq f : ℓ → D (Ω ) q ( λ , λ , . . . ) ∞ X k =1 λ k √ k Φ k . Theorem 5.6.
Let Γ be a Jordan curve separating C into components Ω and Ω . Let q ∈ Ω and fix a conformal map f : D + → Ω . The following are equivalent. (1) Γ is a quasicircle. (2) The Faber operator I qf is a bounded isomorphism. (3) The sequential Faber operator is a bounded isomorphism. (4)
Every element of D q (Ω ) is approximable in norm by a unique Faber series P ∞ n =1 h n Φ n satisfying ( h , h / √ , h / √ , . . . ) ∈ ℓ .If any of conditions (2)-(4) hold for a single q and single choice of f : D + → Ω , then theyhold for every q ∈ Ω and every choice of f .Proof. The equivalence of (1) and (2) follows immediately from Theorem 4.21 together withthe fact that C f − : D ( D + ) → D p (Ω ) is a bounded isomorphism, where p = f (0). Theequivalence of (2) and (3) follows from the fact that(5.2) ( λ , λ , λ , . . . ) ∞ X k =1 λ k √ k ¯ z k is a bounded isomorphism from ℓ to D ( D + ).To show that (2) and (4) are equivalent, first observe that for any Jordan curve I qf isinjective, since C f − : D ( D + ) → D p (Ω ) is an isomorphism and J q , is injective by Corollary4.19 part (1). Now assume that (2) holds. Given any H ( z ) ∈ D q (Ω ) let H = I qf h . Thisfunction h has a power series expression h ( z ) = h ¯ z + h ¯ z + · · · , which converges in D ( D + ) to ¯ h . Since I qf is bounded, applying it to both sides we see that H ( z ) = ∞ X n =1 h n Φ n ( z )is convergent in the norm. Uniqueness follows from injectivity of I qf .To see that (4) implies (2), observe that (4) implies that the sequential Faber operator issurjective. Since (5.2) is an isomorphism, I qf is surjective, and hence an isomorphism. (cid:3) The inverse can be given explicitly.
Theorem 5.7.
Let Γ be a quasicircle separating C into components Ω and Ω . Let q ∈ Ω ,and fix f : D + → Ω . The inverse of I qf is P a ( D + ) C f O , . roof. Let p = f (0). Observe that P a ( D + ) C f = C f P ap (Ω ). Thus by Theorem 4.21, − P a ( D + ) C f O Ω , Ω J q , C f − h = − C f P ap (Ω ) O , J q , C f − h = h for all h ∈ D ( D + ) . So this is a left inverse, which must also be the right inverse by Theorem5.6. (cid:3) Grunsky inequalities.
The Grunsky inequalities originally stem from H. Grunsky’sstudies in the context of univalent function theory [28]. The operators (or matrices) involvedin those studies have grown to become a powerful tool in many areas of mathematics.We shall first define the Grunsky operators acting on polynomials, and later extend themby Theorem 5.10 to Dirichlet spaces. In Theorem 5.12 we will define the Grunsky operatorsin the more general setting of quasicircles.
Definition 5.8.
Given a Jordan curve Γ separating C into Ω and Ω as above, let f : D + → Ω be a conformal map with f (0) = p and fix q ∈ Ω . The Grunsky operator on polynomialsis defined by(5.3) Gr f = P h ( D + ) C f ˆ O , I qf : C [¯ z ] → D ( D + ) . As we saw in the previous section, I qf takes polynomials to polynomials in C q [1 / ( z − p )],which have continuous transmission. It follows from Lemma 4.14 that the output of Gr f onpolynomials is in D ( D + ). Since for any h ∈ D ( D + ) I qf h − I q f h = [ J q , − J q , ] C f − h is constant, and the transmission and pull-back of constants are also constant, the Grunskyoperator is independent of q . Remark . By the anchor lemma 3.21, this agrees with the classical definition of theGrunsky coefficients. We choose q = ∞ ∈ Ω and p = 0 to be consistent with the usualconventions, though the reasoning works for arbitrary q and p . The classical definition (infact, one of several) is that the Grunsky coefficients b nk of a univalent map of the disk aregiven by(5.4) Φ n ( f ( z )) = z − n + ∞ X k =1 b nk z k where Φ n is the nth Faber polynomial. Recalling that I f ( z − n ) = Φ n , the fact that Φ n ( f ( z ))has this form follows from a simple contour integration argument (or from Corollary 4.19).By the Anchor Lemma 3.21 applied to Φ n , together with the fact that P h ( D + ) C f = C f P a ( D + )we have Gr f ( z n ) = ∞ X k =1 b nk z k . hus we see that the Grunsky coefficients are just the coefficients of the matrix representationof Gr f .We now extend Gr f to the full Dirichlet space. Theorem 5.10.
Let Γ be a Jordan curve separating C into components Ω and Ω . Fix q ∈ Ω . Let f : D + → Ω be a conformal map. Gr f extends to a bounded operator Gr f : D ( D + ) → D ( D + ) of norm less than or equal to one. For all h ∈ D ( D + ) the extended operator satisfies (5.5) k I qf h k D q (Ω ) ≤ k h k D ( D + ) − k Gr f h k D ( D + ) . If Γ has measure zero, then equality holds.Proof. First observe that the Grunsky operator satisfies the following invariance propertywhen restricted to polynomials. For any M¨obius transformation M (5.6) Gr M ◦ f = Gr f . This follows from the facts that C M ˆ O M (Ω ) ,M (Ω ) = ˆ O , C M and by Theorem 4.8 C M J M ( q ) M (Ω ) ,M (Ω ) = J q , C M . We will first show the inequality (5.5) for polynomials. By the above observation, it isenough to prove it when Ω contains ∞ , and p = 0 ∈ Ω . We can also assume that q = ∞ .For r ∈ (0 ,
1) let C r be the positively oriented curve | z | = r . For ¯ h = h ¯ z + · · · + h m ¯ z m ∈ C [¯ z ] we set H ( w ) = I qf ¯ h = m X n =1 h n Φ n . Observe that since H ( w ) vanishes at ∞ ,lim R ր∞ Z | z | = R H ( w ) H ′ ( w ) dw = 0 . Thus by (5.4) we have, using the fact that ¯ z = r /z on C r , k I qf ¯ h k D q (Ω ) ≤ − lim r ր πi Z f ( C r ) H ( w ) H ′ ( w ) dw = − lim r ր πi Z C r H ( f ( z ))( H ◦ f ) ′ ( z ) dz = lim r ր πi Z C r m X n =1 h n ¯ z − n + m X n =1 ∞ X k =1 b nk ¯ z k ! m X n =1 nh n z − n − − m X n =1 ∞ X k =1 kb nk z k − ! dz = lim r ր πi Z C r m X n =1 h n r z n + m X n =1 ∞ X k =1 b nk r z − k ! m X n =1 nh n z − n − − m X n =1 ∞ X k =1 kb nk z k − ! dz = k h k D ( D + ) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 m X n =1 b nk h n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) D ( D + ) . If Γ has measure zero, equality holds. The theorem now follow from density of C [¯ z ] in D ( D + ). (cid:3) rom now on, Gr f refers to this extended operator. Corollary 5.11.
For any M¨obius transformation M , Gr M ◦ f = Gr f .Proof. This follows from (5.6), since the extended operator must also satisfy this identity. (cid:3)
Theorem 5.12. If Γ is a quasicircle dividing C into Ω + and Ω − , and f : D + → Ω + is abiholomorphism, then Gr f = P h ( D + ) C f O , I qf . Proof.
The expression is a bounded extension of (5.3) by Theorem 3.5. (cid:3)
Remark . As is well-known, using the identity in Remark 5.9 one can show that onlyinjectivity of f is necessary in order to define the bounded Grunsky operator on Dirichletspace. This is usually formulated as an extension to sequences in ℓ . Theorem 5.14.
Let Γ be a Jordan curve in C . The following are equivalent. (1) Γ is a quasicircle. (2) The Grunsky operator has norm strictly less than one. (3)
The Grunsky operator has norm strictly less than one on polynomials. (4)
There is a κ such that or all ¯ h ∈ D ( D + ) , (5.7) Re (cid:10) O D + , D − ¯ h, Gr f ¯ h (cid:11) ≤ κ k h k . (5) The inequality (5.7) holds for polynomials.
Before giving the proof, we note the connection with the usual formulation of the Grunskyinequalities. Setting ¯ h ( z ) = λ z + · · · λ n z n , (3) says that there is some κ < λ , . . . , λ n ∈ C (5.8) n X k =1 k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m =1 b mk λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ n X k =1 k | λ k | . Item (5) says that there is some κ < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 n X m =1 b mk λ k λ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ n X k =1 k | λ k | . Proof.
By Theorem 5.10 and density of polynomials (2) and (3) are equivalent, and that (4)and (5) are equivalent. By the Cauchy-Schwarz inequality applied toRe (cid:10) ¯ h, O D − , D + Gr f ¯ h (cid:11) = Re (cid:10) O D + , D − ¯ h, Gr f ¯ h (cid:11) (2) implies (4), using the fact that O D − , D + is norm-preserving. Thus it is enough to show(1) ⇒ (2) and (5) ⇒ (1).(1) ⇒ (2). If Γ is a quasicircle, then by Theorem 5.6 I qf is an isomorphism, so there isa c > k I qf h k D q (Ω ) ≥ c k h k D ( D + ) for all h . Inserting this in (5.5) we see that k Gr f h k D ( D + ) ≤ √ − c k h k D ( D + ) .(5) ⇒ (1). This is [45, Theorem 9.12] applied to (5.7), applied to g ( z ) = 1 /f (1 /z ). Thedifferent convention for the mapping function does not alter the result; see (6.3) ahead. (cid:3) emark . A simple functional analytic proof that any of (2)-(5) implies (1) can be given,if we assume in addition that Γ is a measure zero Jordan curve. Assuming that k Gr f k ≤ k < k I qf h k D q (Ω ) ≥ √ − k k h k D ( D + ) .So the image of I qf is closed, and by Remark 5.5 it is D q (Ω ). Hence by the open mappingtheorem I qf is a bounded isomorphism, and therefore Theorem 5.6 yields that Γ is a quasi-circle. Remark . In order to define the Faber polynomials and the Grunsky coefficients b nk in(5.5), it is only required that f is defined in a neighbourhood of 0 and has non-vanishingderivative there. It is classical that (5.8) and (5.7) each hold for κ = 1 if and only if f extends to a one-to-one holomorphic function on D + [20]. Equation (5.8) (with κ = 1) isusually called the strong Grunsky inequalities, while (5.7) is usually called the weak Grunskyinequalities [20]. The computation in the proof of Theorem 5.10 is the usual proof of thestrong Grunsky inequalities. Remark . The proof of Theorem 5.10 is easily modified to show that for any one-to-one holomorphic function f on D , the Grunsky operator (expressed as a function of theparameters α k = λ k / √ k ) extends to a bounded operator on ℓ , see e.g. [20, 45].Finally, we include an integral expression for the Grunsky operator, due to Bergman andSchiffer [10]. It is most conveniently expressed in terms of the Bergman space of one-forms.If Ω is simply connected, then d : D (Ω) → A (Ω)is norm-preserving, and in fact if Ω is simply connected it becomes an isometry when re-stricted to D (Ω) q for any q ∈ Ω. We then defineˆ Gr f : A ( D + ) → A ( D + )by ∂ Gr f = ˆ Gr f ∂. With this definition, we have
Theorem 5.18.
For any Jordan curve Γ and conformal map f : D + → Ω , ˆ Gr f α = f ∗ T , ( f − ) ∗ α = Z Z D + πi (cid:18) f ′ ( w ) f ′ ( z )( f ( w ) − f ( z )) − w − z ) (cid:19) α ( w ) ∧ dw · dz. Proof.
Set p = f (0). Assume for the moment that Γ is a quasicircle. Then, fixing some q ∈ Ω , ˆ Gr f α = − ∂ P h ( D + ) C f O , J q , C f − ∂ − α = − ∂ C f P hp (Ω ) O , J q , ∂ − ( f − ) ∗ α = − f ∗ P (Ω ) ∂ O , J q , ∂ − ( f − ) ∗ α. ere, it is understood that ∂ − is a choice of inverse of ∂ : D ( D + ) → A ( D + ) . Now applyingTheorems 4.15 and 4.13 we see thatˆ Gr f α = − f ∗ P (Ω ) ∂ ( J q , − Id) ∂ − ( f − ) ∗ α = f ∗ P (Ω ) T , ( f − ) ∗ α which proves the first claim. The integral formula is obtained by substituting (4.7) into theright hand side and changing variables, using the fact that L Ω ( w, z ) = 12 πi ( f − ) ′ ( w )( f − ) ′ ( z )( f − ( w ) − f − ( z )) dw · dz. If Γ is a Jordan curve but not a quasicircle, we apply the same argument to polynomials h ∈ C [¯ z ], using Lemma 4.17 in place of Theorem 4.15. The result is then extended to D ( D + ) using Theorem 5.10. (cid:3) Bergman and Schiffer directly defined an operator using this integral formula, and observedthat it recovers the Grunsky operator when applied to polynomials, see [10, eq (9.7), (9.9)]and nearby text (in fact, their formulation is for germs of maps in arbitrarily simply connecteddomain, and the above is a special case). In particular, their integral formulation agreeswith the unique operator extension of the Grunsky matrix to ℓ , if we identify sequenceswith elements of A ( D + ) as in Theorem 5.6. Remark . It can be shown that the integral formula in Theorem 5.18 is a boundedoperator for arbitrary one-to-one f : D → C , and this agrees with the extension of theGrunsky operator of Theorem 5.10. Although Bergman and Schiffer [10] assume that theboundaries are analytic Jordan curves, they were certainly aware of this fact.6. Notes and literature
Notes on the Introduction.
We begin with some further remarks on attributions andproofs. Below, (n) refers to the claim that the statements (1) and (n) in the introductionare equivalent, unless the direction of the implication is specified.As mentioned in the introduction, the characterizations (5) and (6) of quasicircles are dueto Shen [63], where the implications (1) ⇒ (5,6) were obtained earlier by C¸ avu¸s [16]. Notethat C¸ avu¸s and Shen work with the conformal map on the outside of the disk, but this is onlya difference of convention. Similarly, they phrase their results in terms of Bergman spaces,but our formulation here is the same, after application of the isometry h h ′ betweenDirichlet and Bergman spaces.Characterization (4) is due to the authors [59]. The Faber operator typically involvesan integral over the Jordan curve, which must therefore be rectifiable in order to makesense. Rectifiability is added as an assumption in Wei, Wang and Hu [73], who showed (6)for rectifiable Jordan curves. By using the limiting integral, we were able to remove theassumption of rectifiability. A key result is the equality of limiting integrals from either side,stated in this paper as Theorem 4.11, which was originally proven in [58].It should be noted that although we have established the equivalences, the result of Shenis at face value stronger in the direction (5) ⇒ (1) in comparison to (4) ⇒ (1), and weakerthan (1) ⇒ (4) in the direction (1) ⇒ (5). Also, Shen’s result is stronger in that it does not equire assuming that the domain is a Jordan curve. It is not immediately clear what themeaning of transmission would be when the complement of f ( D + ) is more general than theclosure of a Jordan domain. We did not pursue this issue, since we could not shed new lighton his results. The issue seems to be of interest, in light of the fact that Faber polynomialshave meaning for degenerate domains (a classic example being the Chebyshev polynomialsfor an interval), among other things.The characterization (7) is due to Napalkov and Yulmukhametov [43]. It was provenindependently by the authors [59], using our characterization (4). Unfortunately none ofthe aforementioned authors, including us, were aware of the results of Napalkov and Yul-mukhametov. Both our proof and that of Wei, Wang and Hu use the result or approach ofShen. In some sense, transmission provides a bridge between the result (7) of Napalkov andYulmukhametov [43] and (5) of C¸ avu¸s [16] and Shen [63], by making it possible to replace thesequential Faber operator with the Faber operator for non-rectifiable curves. Theorem 4.11 isan essential ingredient of our approach (ultimately relying on both the bounded transmissiontheorem and the anchor lemma).The result (8) ⇒ (1), that the strict Grunsky inequality implies that Γ is a quasicircle, isused in every proof that (4), (5), (6), and (7) implies (1) given in the literature so far. Wegive an alternate proof in this paper which uses transmission only. Our present proofs inthe reverse direction (that (1) implies (2) through (7)) also differ from previous ones givenby the authors. We first applied this alternate approach in [61] in the case of Jordan curveson Riemann surfaces.6.2. Notes on Section 2.
The fact that quasiconformal maps and quasisymmetries preservecompact sets of capacity zero is well-known [13]. N. Arcozzi and R. Rochberg [6] gave acombinatorial proof that if φ : S → S is a quasisymmetry and I is a closed subset of S ,then there is a constant K > φ such that K c ( I ) ≤ c ( φ ( I )) ≤ Kc ( I ).This of course implies Corollary 2.16. Also, E. Villamor [71, Theorem 3] showed that if g : D − → C is a one-to-one holomorphic κ -quasiconformally extendible map satisfying g ( z ) = z + · · · near ∞ , then there is a κ depending only on the quasiconformal constantsuch that for any closed I ⊂ S , c ( I ) κ ≤ c ( g ( A )) ≤ c ( I ) − κ . This implies Remark 2.19 andhence Theorem 2.18.It is natural to ask whether the converse of Theorem 2.18 holds. That is, let Γ be a Jordancurve separating C into components Ω and Ω . Assume that any set I ⊂ Γ which is nullwith respect to Ω is null with respect to Ω is also null with respect to Ω , and vice versa.Must Γ be a quasicircle?In [58, 59] we used limits along hyperbolic geodesics (equivalently, orthogonal curves tolevel curves of Green’s function) in place of CNT limits, following H. Osborn [44]. If theCNT limit exists, then the radial limit exists. Besides being a stronger property, the CNTlimits we later defined [60, 61] seem to be more convenient. We use the name “Osborn space”for the set of boundary values of Dirichlet bounded harmonic functions, in order to drawattention to the paper [44].6.3. Notes on Section 3.
It is obvious that some notion of null set is necessary to formulatetransmission. As we saw, the fact that quasisymmetries or quasiconformal maps preserve apacity zero sets was central to defining a notion of null sets which allowed the formulationof transmission on quasicircles.On the other hand, quasisymmetries do not preserve sets of harmonic measure zero [13].In particular, even for a quasicircle Γ, sets of harmonic measure zero with respect to onecomponent of C \ Γ need not be of harmonic measure with respect to the other component.This can be seen immediately by a proof by contradiction using the conformal weldingtheorem. Thus harmonic measure is inadequate for our purposes.We have shown that a bounded transmission exists for quasicircles separating a compactRiemann surface into two components in [60]. The results of that paper develop a founda-tion for applying quasisymmetric sewing techniques to boundary value problems for generalRiemann surfaces, and ultimately to a “scattering theory” viewpoint of Teichm¨uller theory[62].Our proof of Theorem 4.11 given in [58] contains a gap, which is not hard to fill in a coupleof ways. Here it is filled by the proof of the anchor lemma, which we stated and proved forthe first time in [61].6.4.
Notes on Section 4.
The operators T j,k were first defined by Schiffer [53]. Schifferinvestigated these operators extensively with others; see e.g. Bergman and Schiffer [10],Schiffer and Spencer [55], Schiffer [17]. The connection to the jump problem was explicitfrom the beginning; see e.g. Bergman and Schiffer [10], and especially the survey [54] whichfocusses on the real jump problem and its relation to boundary layer potentials. The connec-tion to the complex jump theorem which we give here is more direct. The paper of Royden[52] connects the Schiffer kernel functions to the jump problem on Riemann surface. Hisresults are phrased somewhat differently in terms of topological conditions for extensionsof holomorphic and harmonic extensions on domains; indeed, the Plemelj-Sokhotski jumpformula is not mentioned explicitly. However it can be derived as a special case of his results,but with more restrictive analytic assumptions on the function on the curve; namely, thatit extend holomorphically to a neighbourhood of the curve. Our paper [61] considers jumpdecompositions and Schiffer kernels on Riemann surface, in the setting of Dirichlet spacesand quasicircles, extending some of the results given here to higher genus.The terminology surrounding the Schiffer operators is a bit confused. As a Calder´on-Zygmund singular integral operator acting on functions in the plane, the Schiffer operatoris bounded on L (more generally on L p , 1 < p < ∞ ). The integral operator on generalfunctions in L ( C ) is called the Beurling transform. It is also sometimes called the Hilberttransform, a term used more widely (in the harmonic analysis and integral equations con-text) for a principal value integral along the real line (the explicit formula is (6.1) ahead, ifone chooses there Γ = R ). Napalkov and Yulmukhametov use the term Hilbert transformto refer specifically to T , . Of course these integral operators are all closely related. Wereserve the term “Schiffer operator” for the restriction of the singular integral operator toanti-holomorphic functions on a subset of C .This “nesting” - that the kernel function is derived from the Green’s function of a largerdomain than the domain of integration - is a central feature of the Schiffer operators, whichhe explored at length in [17]. On general domains and Riemann surfaces, there is also arelated operator derived from integrating against the Bergman kernel obtained from a larger omain. (This does not appear in the present paper, because the Bergman kernel of thesphere is zero). Adding to the terminological confusion described above, some authors referto the Bergman kernel on Riemann surfaces as the Schiffer kernel, while at the same timethere is indeed a distinct Schiffer kernel (related to the so-called fundamental bi-differential).On the double, certain identities relate the Schiffer and Bergman kernels [55].The authors proved that a jump decomposition holds for the special case of Weil-Peterssonclass (WP-class) quasidisks in [46]. This class arises naturally in geometric function theory,Teichm¨uller theory and the theory of Schramm-Loewner evolution. As was demonstrated in[46] the rectifiability and Ahlfors-regularity of the WP-class quasicircles enables one to provea Plemelj-Sokhotzki-type jump decomposition. However the proof of [46, Theorem 2.8] (thechord-arc property of the WP-quasicircles) has a gap arising from our misinterpretation ofthe definition of a quasicircle given in the paper [24] by K. Falconer and D. Marsh, which infact only applies to weak quasicircles, see e.g. J-F. Lafont, B. Schmidt, W. van Limbeek [36].Therefore the theorem in [46] is not true as stated. Our claim that WP-class quasicirclesare chord-arc (and hence Ahlfors regular) was proven by C. Bishop [12]. As with the caseof quasicircles, there are an extraordinary number of characterizations of WP-class quasi-circles. Bishop [12] has listed over twenty, many of which are new (answering among manyothers a question raised by Takhtajan-Teo [68]). His paper also contains other far-reachinggeneralisations of the concept (to higher dimensions).In the case of WP-class quasidisks, not only is the curve rectifiable, but the boundary val-ues of Dirichlet bounded harmonic functions on such domains lie in a certain Besov space.So the contour integral could be defined directly. On the other hand, quasicircles are notin general rectifiable, which creates a hindrance to formulation of the jump decompositionin the setting of this paper. B. Kats studied Riemann-Hilbert problems on non-rectifiablecurves, see e.g. [34] for the case of H¨older continuous boundary values, and the survey ar-ticle [33] and references therein. The CNT boundary values of elements of D harm (Ω k ) arenot continuous, so this technology was not available. The jump decomposition was shown tohold for a range of Besov spaces of boundary values by the authors, for d -regular quasidisks[57], which are not necessarily rectifiable. This result did not include the case of boundaryvalues of the Dirichlet space, so it was also not available for use here.An interesting open question arises in association with the jump formula on quasidisks.The classical Plemelj-Sokhotski jump formula can be expressed using a principle value inte-gral on the curve. That is, if u is a smooth function and Γ is a smooth Jordan curve in C define, for z ∈ Γ,(6.1) H u ( z ) = P.V. 12 πi Z Γ u ( ζ ) ζ − z dζ . Of course, one could weaken the analytic assumptions. We have [14](6.2) lim z → z ± πi Z Γ u ( ζ ) ζ − z dζ = ± u ( z ) + H u ( z )where lim z → z ± respectively denotes the limits taken in the bounded and unbounded compo-nents Ω + and Ω − of the complement of Γ. This of course implies the jump formula. The uestion is: can a meaningful principal value integral H u be defined when Γ is a quasicircleand u ∈ H (Γ), and a corresponding formula (6.2) found? This would have many applicationsto the study of integral kernels.As mentioned above, Napalkov and Yulmukhametov were the first to recognize and provethat the Schiffer operator T , is an isomorphism for quasicircles; as far as we know this wasnot known to Schiffer even for stronger assumptions on the curve. We have generalized thisand the jump isomorphism to various settings (taking into account topological obstacles);namely to compact Riemann surfaces separated by a quasicircle [61]; and with M. Shirazi, tocompact Riemann surfaces with n quasicircles enclosing simply connected domains in [50].The converse result to that of Napalkov and Yulmukhametov, that if T , is an isomorphismthen Γ is a quasicircle, only exists in genus zero. It is an open question whether a suitablyformulated converse holds in genus g >
0, though it seems plausible once the topologicaldifferences are taken into account.The operator P a (Ω ) O , appears in conformal field theory (usually with stronger analyticassumptions). Theorem 5.7 generalizes to higher genus, that is this operator is inverse toa kind of Faber operator. This fact can be exploited to give an explicit description of thedeterminant line bundle of this operator; see [50] for the case of genus g surfaces with oneboundary curve. The general case of genus g with n boundary curves is work in progresswith D. Radnell.6.5. Notes on Section 5.
There is a vast literature on the Faber operator, Faber series,and their approximation properties. As mentioned in the main text, they are defined withvarious regularities. See the books of J. M. Anderson [5] and P. K. Suetin [67] (note thatthe 1998 English translation of the 1984 original has an extensively updated bibliography).Some more recent papers are Wei, Wang and Hu [73], D. Gaier [25], Y.E. Yıldırır and R.C¸ etinta¸s [74].The Grunsky operator has been explored by many authors, for example A. Baranov andH. Hedenmalm [8] and G. Jones [31]. L.A. Takhtajan and L.-P Teo showed that it providesan analogue of the classical period mapping of compact surfaces for the universal Teichm¨ullerspace [68]. See also V. L. Vasyunin and N. K. Nikol’ski˘ı for an exposition of its appearance inde Branges’ work on complementary spaces [70]. There are many interesting results relatingthe analytic properties of the Grunsky matrix Gr f to the geometric or analytic propertiesof the conformal map f and/or its image f ( D + ); see for example Jones [31], Shen [64], orTakhtajan and Teo [68].The treatment as an integral operator goes at least as far back as Bergman and Schiffer’sclassic paper [10], as described in the explanation following Theorem 5.18.The Grunsky inequalities have been generalized in many ways. J. A. Hummel [29] gen-eralized the inequalities to pairs of non-overlapping maps. The authors have extended thisto arbitrary numbers of non-overlapping maps in genus zero [48]; Grunsky inequalities wereproven for the case of n non-overlapping maps into a compact surface of genus g by M.Shirazi [65, 66]. This has applications to Teichm¨uller theory and is related to generalizationsof the classical period mapping to the infinite-dimensional Teichm¨uller space of borderedsurfaces of arbitrary genus and number of boundary curves [49], [62]. The thesis of Shirazi[65] also contains a historical survey of the Faber and Grunsky operator. he Faber polynomials, Faber operator, Grunsky operator, and Grunsky inequalities areformulated with an array of differing conventions and approaches. The existence of certainidentities also complicates matters. We will not attempt to untangle the conventions here,but rather content ourselves with a few remarks. For an early overview with a tidy descriptionof the algebraic identities and relation involved, see E. Jabotinsky [32]. See also the historicaloutline in Shirazi [65].Usually normalizations are imposed on the function classes, especially the derivative atthe origin or at ∞ . These normalizations obscure the M¨obius invariance of various objects,such as the Grunsky operator and Cauchy integral operator, and furthermore limit theapplicability of the stated theorems unnecessarily. So we have removed them as much aspossible throughout the paper.The Grunsky inequalities and Faber polynomials are often formulated for conformal mapsof the form g : D − → Ω , of the form g ( z ) = z + b + b /z + · · · , where Ω contains thepoint at ∞ . Choosing q = 0, the Faber polynomials are then defined by Φ n = I g ( z n ), andthe Grunsky coefficients by Φ n ( g ( z )) = z − n + − X n = −∞ b nk z k . The convention that g takes D − onto a domain containing ∞ seems to provide an advantagein some proofs [20, 45], in that the area of the complement of g ( D − ) is finite; this appears tobe the motivation for the choice. However, the advantage is illusory: the important fact isthat the functions to which the Grunsky operator is applied have finite Dirichlet energy. Thefollowing identity shows that either choice is as good as the other. Setting f ( z ) = 1 /g (1 /z ),then it is easily checked that for n > m > √ nm b − n, − m ( g ) = √ nm b nm ( f ) . Another approach to the definition of the Grunsky coefficients is through generating func-tions, e.g. log g ( z ) − g ( w ) z − w − log g ′ ( ∞ ) = − X n = −∞ ,m = −∞ b n,m z n w m where g ′ ( ∞ ) = lim z →∞ g ( z ) /z . for suitably chosen branches of logarithm. One can recognizeimmediately the relation to the integral kernel in Theorem 5.18. This also visibly demon-strates (6.3). The Faber polynomials can also be defined using generating functions relatedto the integral kernel of the Faber operator, see e.g. [20, 32, 67]. References [1] Ahlfors, L. V. Remarks on the Neumann-Poincare integral equation, Pacific J. Math., (1952), 271–280.[2] Ahlfors, L. V. Lectures on quasiconformal mappings. Second edition. With supplemental chapters by C.J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. University Lecture Series, 38. American MathematicalSociety, Providence, RI, 2006.[3] Ahlfors, L. V. Conformal invariants. Topics in geometric function theory. Reprint of the 1973 original.With a foreword by Peter Duren, F. W. Gehring and Brad Osgood. AMS Chelsea Publishing, Providence,RI, 2010.[4] Ahlfors, L. V.; Sario, L. Riemann surfaces. Princeton Mathematical Series, No. 26 Princeton UniversityPress, Princeton, N.J. 1960.
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A Model of the Teichmueller space of genus-zero borderedsurfaces by period maps.
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E-mail address : [email protected] Wolfgang StaubachDepartment of Mathematics, Uppsala University,S-751 06 Uppsala, Sweden
E-mail address : [email protected]@math.uu.se