Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $\Sigma$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $\mathcal{O}$ to the Bergman space of holomorphic forms on $\Sigma$ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $\Sigma$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $\Sigma$.

Let A(Σ) denote the set of holomorphic one-forms on Σ for which this pairing is finite. Similarly, let A(Σ) denote the set of anti-holomorphic one-forms for which (1.1) is finite. Denote the set of harmonic one-forms such that this pairing is finite by A harm (Σ). We have the orthogonal decomposition A harm (Σ) = A(Σ) ⊕ A(Σ).
The subscript e will denote the subset of exact forms; e.g. A(Σ) e , A harm (Σ) e etc. We also define the Dirichlet spaces. Let For a point q ∈ Σ, the subscript q will denote the subset of functions vanishing at q; e.g. D harm (Σ) q , etc. Denote complex conjugation of functions h and forms α by h and α. Of course, D(Σ) consists of the set of complex conjugates of elements of D(Σ), justifying the notation. The notation A(Σ) is similarly justified.
By a conformal map, we mean a holomorphic map which is a biholomorphism onto its image. To define a quasiconformal map we first need the notion of Beltrami differential on Σ, which is a (−1, 1)-differential ω on Σ, i.e., a differential given in a local biholomorphic coordinate z by µ(z)dz/dz, such that µ is Lebesgue-measurable in every choice of coordinate and ||µ|| ∞ < 1. Quasiconformal maps are by definition solutions to the Beltrami equation, 1 i.e. the differential equation given in local coordinates by ∂f = ω∂f where ω is a Beltrami differential on Σ. Let C denote the complex plane and C denote the Riemann sphere. By a quasicircle in the plane, we mean the image of the unit circle S 1 = {z ∈ C : |z| = 1} under a quasiconformal mapping of the plane. By a quasicircle Γ in Σ, we mean a simple closed curve such that there is a conformal map φ : U → C such that U is an open neighbourhood of Γ and φ(Γ) is a quasicircle in C in the sense above.
Let R be a compact surface. Fix points z, q, and w 0 ∈ R. Following for example H. Royden [10], we define Green's function of R to be the unique function g(w, w 0 ; z, q) such that (1) g is harmonic in w on R\{z, q}; (2) for a local coordinate φ on an open set U containing z, g(w, w 0 ; z, q)+log |φ(w)−φ(z)| is harmonic for w ∈ U; (3) for a local coordinate φ on an open set U containing q, g(w, w 0 ; z, q)−log |φ(w)−φ(q)| is harmonic for w ∈ U; (4) g(w 0 , w 0 ; z, q) = 0 for all z, q, w 0 . It can be shown that g exists, and is uniquely determined by these properties. The normalization at w 0 is inconsequential, because it can be shown that ∂ w g(w, w 0 ; z, q) and ∂ w g(w, w 0 ; z, q) are independent of w 0 . Thus we leave w 0 out of the notation for g. Also, g is harmonic in both w and z.
Let R be compact as above, and Σ ⊂ R be an open, proper and connected subset which we treat as a Riemann surface. For such surfaces we have a different notion of Green's function. We say that Σ has a Green's function if there is a harmonic function g(z, w) on Σ such that (1) for a local coordinate φ on an open set U ⊂ Σ containing w, g(z; w)+log |φ(z)−φ(w)| is harmonic in z on U; (2) lim z→p g(z, w) = 0 for all p ∈ ∂Σ and w ∈ Σ. It can be shown that g is also harmonic in w.
It is always understood that we use the first type of Green's function for compact surfaces, and the second type for open proper connected subsets. When necessary, we distinguish between Green's functions of different surfaces with a subscript, e.g. g R or g Σ .
1.2. Statement of results. Let R be a compact surface and O be an open subset. In this paper, we will always assume that O = Ω 1 ∪· · ·∪Ω n where Ω k are simply-connected domains for k = 1, . . . , n, each bounded by a quasicircle, whose closures are pairwise disjoint. Let Σ be the complement of the closure of O in R.
We define the integral operators where z ∈ Σ. We refer to the kernel of this integral operator L R (z, w) = ∂ z ∂ w g R (w; z, q)/(πi) as the Schiffer kernel. Note that the above case includes the possibility that the domain O is connected; we will frequently use this case, for example when restricting to one of the connected components Ω k of O. In that case we will denote the complement of the closure of Ω k by Ω * k . The fact that T (O, Σ) is bounded and has codomain A(Σ) will be justified ahead.
Our first result is the following: Theorem 1.1. Let R be a compact Riemann surface, and let O = Ω 1 ∪ · · · ∪ Ω n where Ω k are simply-connected domains in R bounded by quasicircles Γ k for k = 1, . . . , n. Assume that the closures of Ω k are pairwise disjoint. Let Σ be the complement of the closure of O in R.
This generalizes one direction of a result of V.V. Napalkov and R.S. Yulmukhametov [7], which says that in the case that n = 1 and R = C, the Schiffer operator is an isomorphism if and only if the domain Ω is bounded by a quasicircle. This is closely related to a result proven by Schippers and Staubach [14] which shows that the jump decomposition in the plane results in a bounded isomorphism if and only if the curve is a quasicircle, and also a result of Y. Shen [18] which shows that the boundary of Ω is a quasicircle if and only if a certain sequential Faber operator is a bounded isomorphism. These results motivate the particular interest in Schiffer operators for regions bounded by quasicircles. Schippers and Staubach proved this theorem [16] in the case of a single quasicircle dividing a compact Riemann surface into two disjoint connected components.
We also give three applications of Theorem 1.1. First, we prove a version of the Plemelj-Sokhotski jump formula for Γ = Γ 1 ∪ · · · ∪ Γ n where Γ k are as in Theorem 1.1. By H(Γ), we mean the set of complex functions on Γ whose restriction to Γ k is the boundary values of an element of D harm (Ω k ) in a sense which we refer to as "conformally non-tangential (CNT)" (see Section 2.2 for the precise definition). Let Theorem 1.2. Let R, Γ, Ω k and Σ be as in Theorem 1.1, and let H ∈ H(Γ) be such that its extension h to D harm (O) is in W . Fix q ∈ Σ. There are unique h k ∈ D(Ω k ), k = 1, . . . , n, and h Σ ∈ D(Σ) q so that if H k , H Σ are their CNT boundary values, then on each curve Γ k , H = −H Σ + H k . These are given by Here, J q (Γ) is an integral operator similar to the Cauchy integral, with integral kernel −∂ w g(w; z, q)/(πi). See Section 2.1 ahead for the precise definition, which involves approximations of the quasicircles by analytic curves.
It is classically known that there is such a jump decomposition for reasonably smooth curves and functions on Riemann surfaces; see [5,11]. This was generalized to the case of a single quasicircle separating a compact Riemann surface into two components, and data in H(Γ) as above, in [16]. A discussion of the literature can also be found there. The space W corresponds to the classical condition for existence of a jump; see Section 3.1 ahead.
The second application is an approximation theorem for Dirichlet spaces of holomorphic functions and Bergman spaces of holomorphic one-forms. Theorem 1.3. Let R be a compact Riemann surface and let Σ, Ω k , Γ k and Σ ′ , Ω ′ k , Γ ′ k each be as in Theorem 1.1. Assume further that Σ ⊂ Σ ′ and that the quasicircles Γ ′ k are isotopic to Γ k within the closure of Ω ′ k for each k = 1, . . . , n.
Since one may always view Σ ′ as embedded in its double, one can remove the mention of the outer surface and obtain Corollary 1.4. Let Σ ′ be a bordered Riemann surface whose boundary consists of n curves Γ ′ 1 , . . . , Γ ′ n homeomorphic to S 1 , whose double is compact. Assume that Σ ⊂ Σ ′ is an open set bordered by n quasicircles Γ 1 , . . . , Here, note that we mean that the boundaries are borders [1].
These results should be compared to a result of N. Askaripour and T. Barron [2], which says that if D 1 and D 2 are open subsets of a Riemann surface R such that D 1 ⊆ D 2 , and the lift to the universal cover (the disk D) of D 1 and D 2 are Carathéodory sets contained in a smaller disk, then restrictions of elements of the Bergman space A(D 2 ) to D 1 are dense in A(D 1 ). As far as we know, their result is the first general result for nested Riemann surfaces, for L 2 approximability, as opposed to uniform approximation, e.g. P. Gauthier and F. Sharifi [6] (see also F. Sharifi [17] for a literature review).
The approach of Askaripour and Barron uses a lift to the universal cover and application of Poincaré series. It would be of great interest to obtain our approximation theorems by applying their methods. Our approach here ultimately relies on sewing.
The third application involves another kind of operator which we now define. Let R be a compact Riemann surface, Σ and Σ ′ be Riemann surfaces such that Σ ⊂ Σ ′ ⊂ R and such that clΣ ⊂ Σ ′ (where the closure is with respect to the topology of R) and the inclusion maps from Σ to Σ ′ and Σ ′ to R are holomorphic.
The kernel of this integral operator is the Bergman kernel of Σ ′ . Note however that we integrate only over Σ and not all of Σ ′ . We then have M. Schiffer and others [3,4,12,13] have investigated these comparison operators in many cases. The Riemann surface R might be the Riemann sphere, or a subset of the plane bounded by analytic curves; while the subset O might be a multiply-connected planar domain or a subdomain of a compact surface R.

2.1.
A Cauchy-type operator on compact surfaces and a Schiffer comparison operator. In this section we bring together various identities for the integral operators, and generalize some of them to the case of several boundary curves. These include expressions for the integral operators in terms of a kind of Cauchy-integral.
We begin with the case of one boundary curve. Let R be a compact Riemann surface and let Γ be a quasicircle, whose complement we assume to consist of two connected components Ω and Σ. Let g Ω denote the Green's function of Ω, and for fixed p ∈ Ω and s > 0 let Γ p s be the level curves {w : g Ω (w, p) = s}. For s sufficiently small, these are in fact analytic simple closed curves, and we endow them with a positive orientation with respect to p. Fixing q ∈ Σ, we define for z ∈ R\Γ. This operator indeed takes D harm (Ω) into D harm (Ω ∪ Σ) by [16,Corollary 4.3], where by the latter we mean a function on the disjoint union which is harmonic on Ω and Σ. It was furthermore shown that the operator is bounded and independent of p. We may write the level curves Γ p s in terms of a conformal map f : D → Ω such that f (0) = p, as the images f ({z : |z| = e −s }) of circles centred at 0.
Recall the operator T (O, Σ) defined by (1.2) in the introduction. Specializing to the case that O consists of a single simply-connected domain Ω, yields an operator which we denote by T (Ω, Σ). It follows from [16,Theorem 3.9] that this operator is bounded. We also define where z ∈ Ω, which is also bounded by [16,Theorem 3.9]. 5 Finally define for z, q ∈ R, which is bounded because the kernel function is globally bounded [16]. The conjugate operator is defined by Conjugates of T operators are defined similarly.
The operators J q (Γ), T (Ω, Σ), T (Ω, Ω) and S(Ω, R) satisfy the identities [16, Theorem 4.2] We would like to generalize these identities to the case of many boundary curves. First, we make a general remark on notation.
Remark 2.1 (Direct sum notation). Let O be as in the introduction; that is, O = Ω 1 ∪· · ·∪Ω n for Ω k simply-connected, bordered by quasicircles, with pairwise disjoint closures. In that case, we have a natural isomorphism The inverse of this isomorphism is where χ k are the characteristic functions of Ω k for k = 1, . . . , n. To avoid needless insertion of this isomorphism into every formula, for α ∈ A(O) say, we use the notation α k = α| Ω k , and furthermore write without qualification Similarly, we have isomorphisms between A(O) e and n k=1 A(Ω k ) e ; D harm (O) and n k=1 D(Ω k ); and so on.
With this convention in mind, for (α 1 , . . . , α n ) ∈ n k=1 A(Ω k ), observe that T (O, Σ) can be written Here for a set A, by [·] A we mean the restriction to A. The above expression shows that T (O, Σ) is bounded as claimed in the introduction. For fixed j = 1, . . . , n, we now define As above, boundedness follows directly from boundedness in the case for one boundary curve. Finally define the bounded operator which again can be written as a sum of integrals over Ω k : We also set Γ = Γ 1 ∪ · · · ∪ Γ n and define for (h 1 , . . . , h n ) ∈ D harm (O) and z ∈ R\Γ The identities (2.3) can now be generalized as follows.
Proof. The first and third identities follow directly from (2.3), (2.5), and (2.7). Now let z ∈ Ω j for fixed j; in this case for every k = j, z ∈ Ω * k . Denoting by J q (Γ) Ω * k the operator obtained by restricting the output of J q (Γ) to Ω * k , and similarly for J q (Γ) Ω j , we Throughout the paper, we will denote and so on, as above. Thus for example the first two identities of the previous theorem can be expressed by

Transmission of harmonic Dirichlet-bounded functions.
In this section, we consider certain operators, which take Dirichlet bounded functions on one region to Dirichlet bounded functions on another region sharing a portion of the boundary with the first, in such a way that the functions have the same boundary values. These operators were studied in [15]; we briefly recall the necessary results. We now explain the sense of boundary values, which we call "conformally non-tangential boundary values", or CNT boundary values. All claims made here in the description of these boundary values are proven in [15]. For the purposes of this section we can assume that O = Ω is a single simply-connected domain and Ω * is the complement of its closure, as above.
For p ∈ Ω, let Ω p,ǫ = {w ∈ Ω : g Ω (w, p) < ǫ}. Similarly for p ′ ∈ Ω * , let where f extends to a homeomorphism from S 1 onto Γ. We can always assume that f has a conformal extension to a neighbourhood of |z| = e −ǫ so that the image of this curve is analytic. In the case of Ω p,ǫ , such a conformal map exists for all ǫ ∈ (0, 1) and is in fact the restriction of the conformal map from D to Ω taking 0 to p. In particular, Ω * p ′ ,ǫ and Ω p,ǫ are doubly-connected domains.
Given a u ∈ D harm (Ω p,ǫ ) or D harm (Ω * p ′ ,ǫ ), the non-tangential boundary values of u • f exist except perhaps on a Borel set of logarithmic capacity zero. The boundary value of u at a point ζ ∈ Γ is defined to be the boundary value of u • f at f −1 (ζ), where it exists. We call these boundary values the CNT boundary values.
We call the image of a Borel set of logarithmic capacity zero under f a "null set" on Γ. This definition can be shown to be independent on p or p ′ , and the particular choice of map f . Similarly, the CNT boundary values are independent of these choices. An important fact is that if u 1 , u 2 ∈ D harm (Ω) have the same CNT boundary values except possibly on a null set, then they are equal. Similarly for u 1 , u 2 ∈ D harm (Ω * ).
It is a much more subtle fact that a set on Γ is null with respect to the conformal map onto Ω p,ǫ if and only if it is null with respect to the conformal map onto Ω * p ′ ,ǫ , in the case that Γ is a quasicircle. More is true: if u is the CNT boundary values of an element of D harm (Ω) except possibly on a null set, then it is the CNT boundary values of a unique element of D harm (Ω * ) except on a null set, and the converse also holds.
This allows us to define an operator that we call the transmission operator. Given u ∈ D harm (Ω), one may obtain boundary values of u on Γ in the CNT sense (see Remark 2.3 below). There exists a unique element of D harm (Ω * ) with the same CNT boundary values, which we denote by O(Ω, Ω * )u. This defines a map and similarly a map By definition, these are inverses of each other. It was shown [15] that these are bounded with respect to the Dirichlet seminorm. We call these transmission operators, since in some sense they transmit a harmonic function through the quasicircle via a boundary value problem. Similarly, the CNT boundary values of any element of D harm (Ω p,ǫ ) are equal to the CNT boundary values of a unique element of D harm (Ω) up to a null set. The converse is obviously true by simply restricting from Ω to Ω p,ǫ ; however, one does not obtain a unique element of D harm (Ω p,ǫ ). The same claims are true for Ω * and Ω * p ′ ,ǫ . This allows us to define the following operator: to take u to the unique element of D harm (Ω) with the same CNT boundary values. Similarly we define . It was shown in [15] that these are bounded with respect to the Dirichlet seminorm. We call these bounce operators.
The integral (2.1) could equally be defined using level curves in Ω * . That is, let p ′ ∈ Ω * and let Γ p ′ ǫ denote the level curves of Green's function, but now give them a negative orientation with respect to Ω * . If we denote by J q (Γ, Ω * ) the new operator defined using negativelyoriented level curves in Ω * , and by J q (Γ, Ω) the original operator defined using positivelyoriented level curves in Ω, then [16, Theorem 4.10] The notation J q (Γ) will always refer to J q (Γ, Ω). The latter notation is used only when it is necessary to distinguish J q (Γ, Ω) from J q (Γ, Ω * ). We will also use the following notation for h ∈ D(Ω p,ǫ ).
Although the integral is the same, the prime is included to distinguish it from the operator J q (Γ), which has a different domain. For any one-form in A(Ω p,ǫ ) and h ∈ D harm (Ω p,ǫ ), we (2.14) Finally, we define H(Γ) to be the set of complex-valued functions defined on Γ, which are the CNT boundary values of an element h ∈ D harm (Ω), modulo the following equivalence relation. We say that h 1 ∼ h 2 if h 1 and h 2 are equal except possibly on a null set. We will continue to treat equivalence classes as functions in the customary way.
Remark 2.3. Except for Theorem 1.2, the precise meaning of "conformally non-tangential" is not directly relevant to the paper; the identities (2.11), (2.13), and (2.14) above are logically sufficient to obtain the results here. Of course, the meaning is helpful for an intuitive understanding of several theorems and proofs.

2.3.
Density theorems for the image of G. In this section we prove some preliminary density theorems.
Let R be a compact Riemann surface, and O, Ω k , and Γ k , for k = 1, . . . , n be as above. Now recall that where we have made use of the isomorphism (2.4) to write W in terms of the restrictions g k to Ω k . We also denote Because Ω k is simply connected, we may decompose any g as g = e + h where e has only holomorphic components and h has only anti-holomorphic components. Thus one may define W equivalently to be the set of elements g of ⊕ n k=1 D harm (Ω k ) whose anti-holomorphic component is in W ′ .
Note that by Stokes' theorem.
The last integral is zero since K R (z, ·) ∈ A(R) for each fixed z and (∂h 1 , . . . , ∂h n ) ∈ V .
For fixed choice of p k ∈ Ω k , k = 1, . . . , n, let Ω k,p k ,ǫ be the domains bounded by level curves of Green's function as in the previous section. Define now the spaces Here, for k = 1, . . . , n, γ k is any choice of simple closed analytic curve in Ω k,ǫ,p k which is isotopic to Γ p k ǫ within the closure of Ω k,p k ,ǫ . Since all theorems hold for any choice of p k , we will remove the points from the notation for the domains. That is, we will denote Ω k,p k ,ǫ by Ω k,ǫ .
Recall the definition (2.10) of the bounce operator. Denote We then have the following theorem.
We have the following two theorems.
Proof. Choose 0 < r < ǫ. Let Γ k,r denote the boundary of Ω k,r . Let B r be the region containing Σ and bounded by ∪ n k=1 Γ k,r . Applying Royden [10,Proposition 6], together with the explicit integral formula given there, we see that defines a holomorphic function on B r . But this integral is independent of r and thus equals the limiting integral (2.12). Since every point in cl Σ ∪ Ω 1,ǫ ∪ · · · ∪ Ω n,ǫ is contained in some B r this proves the theorem.
Proof. This follows immediately from Theorem 2.6 and (2.14).
Remark 2.8. Fix q ∈ Σ. Observe that if (g 1 , . . . , g n ) and (ĝ 1 , . . . ,ĝ n ) are in W , and g k −ĝ k are holomorphic in Ω k for all k, then To see this, by (2.3) we have that Since J q (Γ k )g k and J q (Γ k )ĝ k both vanish at q, this proves the claim. If q / ∈ Σ, then the claim is true up to a constant.
Proof. The proof follows the structure of that of Theorem 4.16 in [16], but generalizes it to the case of several boundary curves. Define P : n k=1 D(Ω k,ǫ ) → X ǫ the orthogonal projection to the subspace X ǫ . Fix a basis {α 1 , . . . , α g } for the vector space A(R), where g is the genus of R. Define the operator Q by Q : For any constants c k ∈ C, k = 1, . . . , n, Q(u 1 +c 1 , . . . , u n +c n ) = Q(u 1 , . . . , u n ). Furthermore, for any fixed k and fixed x k ∈ Γ ′ k there is a uniform C such that sup These two facts together imply that Q is bounded.

2.4.
Proof of Theorem 1.1. The proof proceeds in several steps.
We will show that the Schiffer operator is a bounded isomorphism on the subspace V of n k=1 A(Ω k ). Recall that V is given by 13 where we have made use of the isomorphism (2.4) to rewrite V in terms of the restrictions to Ω k . We record the following obvious fact.
Lemma 2.10. The operator is a surjective operator which preserves the norm.
For fixed j = 1, . . . , n we will define the transmission operator O(Σ, Ω j ) from Σ to Ω j . Recall that Ω * j is the complement of the closure of Ω j in R, which contains Σ. For fixed p ′ j ∈ Ω * j , let Ω * j,p ′ j ,ǫ be a doubly-connected domain bounded by Γ j in Σ as in (2.9). Let Res(Σ, Ω * j,p ′ j ,ǫ ) be the restriction operator from D(Σ) to D(Ω * j,p ′ j ,ǫ ). We then define . The interpretation of O(Σ, Ω j ) is that it takes elements of D harm (Σ) to elements of D harm (Ω j ) with the same CNT boundary values on Γ j ; in other words, it is a transmission operator from Σ to Ω j . This is independent of the choice of p ′ j [15]. The above expression establishes that the operator is bounded, since each operator on the right hand side is bounded. Finally, let Proof. Since every operator in the identity above is bounded, it suffices to prove this for ⊕ k G(Ω k,ǫ , Ω k )X ǫ , because this set is dense by Theorem 2.9.

Now since we have
applying G(Ω j,ǫ , Ω j ) to each term of (2.18) we obtain Finally, inserting the formulas for H 1 and H 2 yields that which completes the proof.
Remark 2.13. It is easily seen that this holds trivially for all holomorphic h ∈ W , since the left hand side vanishes by Remark 2.8. Thus the theorem holds for all h ∈ W .
Lemma 2.14. If h ∈ D(Σ) then Proof. We will distinguish J defined by limiting integrals from within Ω k and from within Ω * k in this proof. Recall that Since the function h is holomorphic on Σ it is equal to the sum of the limiting integrals from within Ω * k for the boundary curves Γ k . i.e.
Let Ω * k,ǫ be the doubly-connected domain bounded by Γ k in Σ. One can replace each integral in the sum by an integral over a fixed analytic curve Γ ′ k in Ω * k,ǫ . That defines an operator J q (Γ k , Ω * k,ǫ ) ′ . For every k = 1, . . . , n, [16,Theorem 4.9] yields that Now apply [16,Theorem 4.10] for each fixed curve Γ k and function G(Ω * k,ǫ , Ω * k ) h * | Ω k,ǫ to obtain that . Finally, taking a sum over all terms and using definition 2.11 yield that which completes the proof. Next we show that every element in A(Σ) e is in the image of T (O, Σ). Given β ∈ A(Σ) e , then there exists an h Σ ∈ D(Σ) q such that ∂ z h Σ = β. Let h k ∈ D harm (Ω k ) be such that O(Σ, Ω k )h Σ = h k , i.e. h Σ and h k have the same CNT boundary values on Γ k . Lemma 2.14 and Theorem 2.2 now imply that So we need only show that (∂h 1 , . . . , ∂h n ) is in V ; that is, for all α ∈ A(R), To see this we have Using Lemma 2.14 once again, we have On the other hand, h Σ is holomorphic so ∂ z h Σ = 0. Therefore n k=1 Ω k ,w ∂ z ∂ w g(w; z, q) ∧ w ∂h k (w) = 0 which inserted in (2.19) completes the proof.
We note that the transmission operator O(Σ, Ω) induces a transmission on the set of exact forms by conjugating by differentiation. Namely, for fixed k set Defining we then have the following version of Theorem 2.12 for one-forms.

Applications of the isomorphism theorem
3.1. Plemelj-Sokhtoski jump problem for finitely many quasicircles. In this section we establish a jump formula for n quasicircles. Setting aside analytic issues momentarily, the problem is as follows. Given a function U on Γ = Γ 1 ∪· · ·∪Γ n , find holomorphic functions u k on Ω k and u Σ on Σ such that on each curve Γ k the boundary valuesũ k andũ Σ respectively satisfyũ The solution to this problem is well-known for more regular curves, say for Γ and u smooth. Here, Γ k are of course quasicircles. We consider the class of functions H(Γ k ); recall that these functions are CNT boundary values of elements of D harm (Ω k ).
It is classically known [5], [11] that the topological condition for existence of a solution to the jump problem for functions U on Γ is that n k=1 Γ αU = 0 for all one-forms α ∈ A(R). On quasicircles, this integral condition would not make sense, because quasicircles need not be rectifiable. Thus, we replace this by the condition that U is the boundary values of an element of W , motivated by (2.15).
Our first theorem in some sense is the derivative of the jump isomorphism. Let Theorem 3.1. Let R be a compact Riemann surface, and Ω 1 , . . . , Ω n be simply connected regions in R, bounded by quasicircles Γ 1 , . . . , Γ n . Assume that the closures of Ω 1 , . . . , Ω n are pairwise disjoint. ThenĤ is an isomorphism.
Proof. Note thatĤ is well-defined, since the output is independent of the choice of constant in h. First we show that it is injective. Assume thatĤ(α + β) = 0, then α = 0 by Theorem Proof. By Lemma 2.4 the image of H is in D(O) ⊕ D(Σ). Since g(w; z, q) vanishes identically at z = q (see [16] proof of Theorem 4.26), so does ∂ w g(w; z, q). Thus the image of H is in By Theorem 2.2, ∂H =Ĥd. Assume that Hh = 0. ThenĤdh = 0, so dh = 0, so h is constant. But if h is a constant c then J q (Γ)c| Σ = c. Since c ∈ D(Γ) q it vanishes at q, so c = 0. So H is injective.
We also have that H(h + c) = Hh + (c, 0) for any constant c. This together with the fact thatĤ is surjective shows that H is surjective.
The proof of Theorem 3.2 also shows the following. Proof. If h ∈ D(O), then since ∂ w g(w; z, q) is holomorphic except for a simple pole of residue one at w = z, by the residue theorem J q (Γ)h = h. If h ∈ D(Σ) q then similarly J q (Γ, Σ)h = −h + h(q) = −h (note that Γ is negatively oriented with respect to Σ). By (2.11) and (2.5), J q (Γ)h = J q (Γ, Σ)h, which completes the proof.
We now prove Theorem 1.  We need only show that the solution is unique. Given any other solution ( has boundary values zero, so by uniqueness of the extension it is zero. Thus which proves the claim.
3.2. The approximation theorems. In this section, we prove some approximation theorems for Dirichlet and Bergman spaces of nested Riemann surfaces, including Theorem 1.3 and Corollary 1.4.
Since the Dirichlet semi-norm is not a norm, the meaning of density requires a clarification. Below, whenever we say that a linear subspace Y of a Dirichlet space is dense in a Dirichlet space, the space Y contains all constant functions. Thus, when we approximate in the Dirichlet semi-norm, we are still free to adjust any "approximating" function by a constant without leaving Y . Let denote orthogonal projection, where it is understood that for constants c P Ω k c = c. Let be the direct sum of these operators.
Corollary 3.5. The projection of ⊕ k G(Ω k,ǫ , Ω k )X ǫ onto the anti-holomorphic parts is dense in W ′ .
Proof. It is easily verified that the projection ⊕ k P Ω k takes W into W ′ . Since it is a bounded surjective operator, the claim follows immediately from Theorem 2.9.
This leads to the following density theorem.
Proof. Let q ∈ Σ. By Remark 2.8, the image of W under J q (Γ) Σ is equal to the image of W ′ under J q (Γ) Σ . By Theorem 3.3 J q (Γ) is a bounded isomorphism from W ′ to D(Σ) q . Thus by Corollary 3.5, is dense in D(Σ) q . Now Corollary 2.7 and Remark 2.8 yield that all of the functions in the set (3.1) have holomorphic extensions to cl Σ ∪ Ω 1,ǫ ∪ · · · ∪ Ω n,ǫ . Since constant functions automatically have such extensions, this completes the proof.
Corollary 3.7. Let R be a compact Riemann surface and Σ ⊂ R be a Riemann surface such that the inclusion map is holomorphic and the boundary of Σ consists of a finite number of pair-wise disjoint quasicircles Γ 1 , . . . , Γ n in R.
Assume that there is an open set Σ ′ ⊂ R which contains Σ, and is bounded by quasicircles Γ ′ k , k = 1, . . . , n, which are isotopic in the closure of Σ ′ \Σ to Γ k for k = 1, . . . , n respectively. Then the set of restrictions of elements of D(Σ ′ ) to Σ is dense in D(Σ).
Proof. Consider the compact Riemann surface R ′ obtained from Σ ′ by sewing disks D to the quasicircles Γ ′ k for k = 1, . . . , n using fixed quasisymmetric parametrizations τ k : S 1 → Γ ′ k , k = 1, . . . , n, say. It was shown in [8] that the topological space obtained from such a sewing has a unique complex structure compatible with that of Σ ′ and the sewn disks. Let Ω 1 , . . . , Ω n be the connected components of the complement of Σ in R ′ containing Γ ′ 1 , . . . , Γ ′ n respectively. It follows from the hypotheses that each Ω k is conformally equivalent to a disk bordered by Γ k . For each k = 1, . . . , n, fix a point p k ∈ Ω k \clΣ ′ , and let f k : D → Ω k be conformal maps such that f (0) = p k . We claim that for some ǫ > 0, Σ ′ is contained in clΣ ∪ Ω 1,ǫ ∪ · · · ∪ Ω n,ǫ . To see this, observe that the set f −1 k (Γ ′ k ) is compact and does not contain 0. Thus {|p|} > 0.
We now address the case of one-forms.
Theorem 3.8. Let R, Σ, and Σ ′ be as in Corollary 3.7. Assume that Σ ′ (and hence Σ) is a bordered Riemann surface of genus g and n borders with n ≥ 1. Then the set of restrictions of elements of A(Σ ′ ) to Σ is dense in A(Σ).
Proof. Let R ′ be the double of Σ ′ . It is a surface of genus 2g + n − 1, so the dimension of A(R ′ ) is 2g + n − 1. Let a 1 , . . . , a 2g+n−1 denote a set of generators for the fundamental group of Σ. Given any α ∈ A(Σ), there is a β ∈ A(R ′ ) with the same periods. Thus α − β is exact on Σ, with primitive H say.
This completes the proof.
The following example shows that the truth of Corollary 3.7 depends on the fact that every component of R\Σ contains a component of R\Σ ′ . Let R = C. Fix r ∈ (0, 1) and let Σ = {z : r < |z| < 1}. For 0 < r ′ < r and s ′ > 1, Theorem 3.6 says that for Σ ′ = {z : r ′ < |z| < s ′ }, D(Σ ′ ) is dense in D(Σ). However, setting instead Σ ′ = D, it is not true that D(D) is dense in D(Σ). To see this, fix z ∈ C such that |z| > 1 and observe that the functional on D Proof. (of Corollary 1.4). Observe that Σ ′ can be viewed as a subset of its double Σ D , and its boundary can be identified with n analytic curves in the double. Thus the claim follows from Theorem 1.3 applied with R = Σ D .
We indicate another possible approach to proving Theorem 3.8 (and therefore Theorem 1.3), using the result of Askaripour and Barron [2]. We assume that Γ k and Γ ′ k are analytic curves for k = 1, . . . , n. Assume also that the universal cover of R is the disk. Let π : D → R be the covering map. Choose a collection of curves γ j , j = 1, . . . , g dissecting the compact surface R, where g is the genus, to obtain a fundamental polygon F in the disk D. Choose the dissection such that every curve Γ k and Γ ′ k is crossed by at least one of the dissecting curves. In that case, the sets π −1 (Σ) ∩ F and π −1 (Σ ′ ) ∩ F will be Carathéodory sets in the plane, and one can apply [2, Proposition 2.1] to obtain the result in the case of analytic curves.
One would need to show that such a dissection exists in general, which should not pose much difficulty. However, if one attempts this argument in the case of quasicircles, then establishing that the dissecting curves can be made to have the intersection property might be a delicate problem. On the other hand, if the dissecting curves are chosen not to intersect Γ k and Γ ′ k , the lifted sets π −1 (Σ) ∩ F and π −1 (Σ ′ ) ∩ F would not be Carathéodory sets, and one could not apply their result directly.
It should be noted that [2, Proposition 2.1] does not require analytic conditions on the boundary of Σ and Σ ′ , as we do in Theorem 3.8. Although we were able to remove the restrictions on the outer domain Σ ′′ to some extent in Theorem 1.3, we did not do so for Σ itself. Thus their result suggests that the analytic conditions of Theorem 1.3 can be weakened.
3.3. The Schiffer comparison operator for open surfaces. Next we define a certain comparison operator, which generalizes an operator considered by Schiffer [4].
Let R be a compact Riemann surface. Let Σ and Σ ′ be Riemann surfaces such that Σ ⊆ Σ ′ ⊆ R, and such that the inclusion maps from Σ into Σ ′ and Σ ′ into R are holomorphic. Assume that Σ ′ has a Green's function g Σ ′ . We then define the Bergman kernel of Σ ′ to be The Schiffer comparison operator is then defined to be We then have the following result, which strangely seems to have been missed by Schiffer, even in the planar case. By a hyperbolic metric, we mean a complete, constant negative curvature metric. Theorem 3.9. Let R be a compact Riemann surface, Σ and Σ ′ be Riemann surfaces such that Σ ⊂ Σ ′ ⊂ R, clΣ ⊂ Σ ′ and the inclusion maps from Σ to Σ ′ and Σ ′ to R are holomorphic. Assume that Σ ′ has a Green's function g Σ ′ (w, z), and that Σ ′ possesses a hyperbolic metric. Denoting the adjoint of R(Σ ′ , Σ) by R(Σ ′ , Σ) * we have S(Σ, Σ ′ ) = R(Σ ′ , Σ) * .
Proof. Let α ∈ A(Σ) and β ∈ A(Σ ′ ). Then using the reproducing property of the Bergman kernel K Σ ′ and assuming that we are allowed to interchange the order of integration we have (S(Σ, Σ ′ )α, β) A(Σ ′ ) = i 2 Σ ′ ,z Σ,w K Σ ′ (z, w) ∧ w α(w) ∧ z β(z) To justify the change of the order of integration, let Σ be compactly included in Σ ′ and K n be a sequence of compact subsets of Σ that exhaust it (i.e. K n → Σ). Denote the L p norm over a set U with respect to the hyperbolic metric on Σ ′ by · p,U (see [9]). Note that the L 2 norm of a one-form (a one-differential in the terminology of [9]) with respect to the hyperbolic metric agrees with the L 2 norm used in this paper. Now for fixed w set c n (w) = K Σ ′ (·, w) 1,Kn .
Note that c n (w) is a one-form on Σ for every n (of the form a(w)dw in local coordinates). Then