Plemelj-Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operator

Let $R$ be a compact Riemann surface and $\Gamma$ be a Jordan curve separating $R$ into connected components $\Sigma_1$ and $\Sigma_2$. We consider Calder\'on-Zygmund type operators $T(\Sigma_1,\Sigma_k)$ taking the space of $L^2$ anti-holomorphic one-forms on $\Sigma_1$ to the space of $L^2$ holomorphic one-forms on $\Sigma_k$, which we call the Schiffer operators. We extend results of Menahem M. Schiffer and others, which where confined to analytic Jordan curves $\Gamma$, to general quasicircles in a characterizing manner, and prove new identities for adjoints of the Schiffer operators. Furthermore, we show that if $V$ is the space of anti-holomorphic one-forms orthogonal to $L^2$ forms on $R$ with respect to the inner product on $\Sigma_1$, then the Schiffer operator $T(\Sigma_1,\Sigma_2)$ is an isomorphism onto the set of exact one-forms on $\Sigma_2$. Using the relation between the Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

1. Introduction 1.1. Results and literature. Let Γ be a sufficiently regular curve separating a compact surface into two components Σ 1 and Σ 2 . Given a sufficiently regular function h on that curve, it is well known that there are holomorphic functions h k on Σ k such that if and only if Γ hα = 0 for all holomorphic one forms on R. In the plane, this is a consequence of the Plemelj-Sokhotski jump formula (which is a more precise formula in terms of a principal value integral). The functions h k are obtained by integrating h against the Cauchy kernel.
Different regularities of the curve and the function are possible. In this paper, we show that the jump formula holds for quasicircles on compact Riemann surfaces, where the function h is taken to be the boundary values of a harmonic function of bounded Dirichlet energy on either Σ 1 or Σ 2 . In the case that Γ is analytic, this space agrees with the Sobolev H 1/2 space on Γ. We showed in an earlier paper that the space of boundary values, for quasicircles, is the same for both Σ 1 and Σ 2 , and the resulting map (which we call the transmission map) is bounded.
Since quasicircles are non-rectifiable, we replace the Cauchy integral by a limit of integrals along level curves of Green's function in Σ k ; for quasicircles, we show that this integral is the same whether one takes the limiting curves from within Σ 1 or Σ 2 . This relies on our transmission result mentioned above. We also show that the map from the harmonic 2010 Mathematics Subject Classification. 58J05, 30F15. 1 Dirichlet space D harm (Σ k ) to the direct sum of holomorphic Dirichlet spaces D(Σ 1 ) ⊕ D(Σ 2 ) obtained from the jump integral is an isomorphism.
We also consider a Calderón-Zygmund type integral operator on the space of one-forms which we call the Schiffer operator. This was studied extensively by M. Schiffer and others in the plane and on Riemann surfaces (see Section 3.3 for a discussion of the literature). Schiffer discovered deep relations between inequalities in function theory, potential theory and Fredholm eigenvalues, and properties of these operators. We extend many known results from analytic boundary to quasicircles. We also derive some new identities for the adjoint of the Schiffer operator, and a complete set of identities relating the Schiffer operator to the jump integral in higher genus. The derivative of the jump integral, when restricted to a finite co-dimension space of one-forms, equals the Schiffer operator. We show that the restriction of the Schiffer operator to this finite co-dimensional space is an isomorphism.
In the case of simply-connected domains in the plane (where the finite co-dimensional space is the full space of one-forms), the fact that the Schiffer operator is an isomorphism is due to V. V. Napalkov and R. S. Yulmukhametov [12]. In fact, they showed that it is an isomorphism precisely for domains bounded by quasicircles. This is closely related to a result of Y. Shen [21], who showed that the Faber operator of approximation theory is an isomorphism precisely for domains bounded by quasicircles. Indeed, using Shen's result, the authors (at the time unaware of Napalkov and Yulmukhametov's result) derived a proof that the jump isomorphism and the Schiffer operator are isomorphisms precisely for quasicircles [17]. As mentioned above, here we generalize the jump and Schiffer isomorphism to Riemann surfaces separated by quasicircles. We conjecture that the converse holds, as in the planar case; namely, if either of these is an isomorphism, then the separating curve is a quasicircle.
We conclude with a few remarks on technical issues and related literature. The main hindrance to the solution of the Riemann boundary problem and the establishment of the jump decomposition is that quasicircles are highly irregular, and are not in general rectifiable. Riemann-Hilbert problems on non-rectifiable curves have been studied extensively by B. Kats, see e.g. [8] for the case of Hölder continuous boundary values, and the survey article [7] and references therein. However the boundary values of Dirichlet bounded harmonic functions need not be Hölder continuous. For Dirichlet spaces boundary values exist for quasicircles and the jump formula can be expressed in terms of certain limiting integrals. A key tool here is our proof of the existence and boundedness of a transmission operator for harmonic functions in quasicircles [19] (which, in the plane, also characterizes quasicircles [16]). Indeed our approach to proving surjectivity of the Schiffer operator relies on the equality of the limiting integral from both sides. We have also found that the transmission operator has a clarifying effect on the theory as a whole.
In this paper, approximation by functions which are analytic or harmonic on a neighbourhood of the closure plays an important role. We rely on an approximation result of N. Askaripour and T. Barron [2] for L 2 k-differentials on nested surfaces. Their result replaces the density of polynomials in the Bergman space of a Carathéodory domain used in proving the results in the case of the sphere.

1.2.
Outline of the paper. In Section 2 we establish notation and state preliminary results. We also outline previous results of the authors which are necessary here. In Section 3 we define the Schiffer operators, generalize known results to quasicircles, and establish some new identities. In Section 4, we give identities relating the Schiffer operator to a Cauchytype integral (in general genus), we relate it to the jump decomposition, and establish the isomorphism theorems for the jump decomposition and the Schiffer operator.

Notations and Preliminaries
2.1. Forms and functions. We begin by establishing notation and terminology.
Let R be a Riemann surface, which we will always assume to be connected. For smooth real one-forms, define the dual of the almost complex structure * by * (a dx + b dy) = a dy − b dx in a local holomorphic coordinate z = x+iy. This is independent of the choice of coordinates. Harmonic functions f on R are those which satisfy d * df = 0, while harmonic one-forms α are those which satisfy both dα = 0 and * dα = 0. Equivalently, harmonic one-forms are those which can be expressed locally as df for some harmonic function f . We consider complex-valued functions and forms. Denote complex conjugation of functions and forms with an overline, e.g. α.
Harmonic one-forms α can always be decomposed as a sum of a holomorphic and antiholomorphic one-form. The decomposition is unique. On the other hand, harmonic functions do not possess such a decomposition.
The space of complex one-forms on R has the natural inner-product Denote by L 2 (R) the set of one-forms which are L 2 with respect to this inner product. The Bergman space of holomorphic one forms is A(R) = {α ∈ L 2 (R) : α holomorphic} and the set of antiholomorphic L 2 one-forms will be denoted by A(R). This notation is of course consistent, because β ∈ A(R) if and only if β = α for some α ∈ A(R). We will also denote A harm (R) = {α ∈ L 2 (R) : α harmonic}.
Observe that A(R) and A(R) are orthogonal with respect to the aforementioned inner product.
We also define the Dirichlet spaces by 3 We can define a degenerate inner product on D harm (R) by If we denote D harm (R) q = {f ∈ D harm (R) : f (q) = 0} for some q ∈ R, then the scalar product defined above is a genuine inner product on D harm (R) q and also makes it a Hilbert space. In what follows, a subscript q on a space of functions indicates the subspace of functions such that f (q) = 0.
If we now define the Wirtinger operators via their local coordinate expressions then the aforementioned inner product can be written as One can easily see from (2.1) that D(R) and D(R) are orthogonal with respect to the inner product. We also note that if R is a planar domain and f ∈ D(R), where dA denotes Lebesgue measure in the plane.
Finally, we will repeatedly use the following elementary fact.
for all α(z) dz ∈ A harm (U). For any Riemann surface R, compact subset K of R, and fixed q ∈ R, there is a constant The first claim is classical and the second claim is an elementary consequence of the first.

2.2.
Transmission of harmonic functions through quasicircles. In this section we summarize some necessary results of the authors. The proofs can be found in [19].
Let R be a compact Riemann surface. Let Γ be a Jordan curve in R, that is a homeomorphic image of S 1 . We say that U is a doubly-connected neighbourhood of Γ if U is an open set containing Γ, which is bounded by two non-intersecting Jordan curves each of which is homotopic to Γ within the closure of U. We say that a Jordan curve Γ is strip-cutting if there is a doubly-connected neighbourhood U of Γ and a conformal map φ : U → A ⊆ C so that φ(Γ) is a Jordan curve in C. We say that Γ is a quasicircle if φ(Γ) is a quasicircle. In particular a quasicircle is a strip-cutting Jordan curve. A closed analytic curve is strip-cutting by definition.
If R is a Riemann surface and Σ ⊂ R is a proper subset of R, then we say that g(w, z) is the Green's function for Σ if g(w, ·) is harmonic on R\{w}, g(w, z) + log |φ(z) − φ(w)| is harmonic in z for a local parameter φ : U → C in an open neighbourhood U of w, and lim z→z 0 g(w, z) = 0 for all z 0 ∈ ∂Σ and w ∈ Σ. Green's function is unique and symmetric, provided that it exists. In this paper, we will consider only the case where R is compact and no boundary component of Σ reduces to a point, so Green's function of Σ exists; see for example L. Ahlfors and L. Sario [1,II.3]. Now let Σ be one of the connected components in R of the complement of Γ. Fix a point q ∈ Σ and let g q be Green's function of Σ with singularity at q. We associate to g q a biholomorphism from a doubly-connected region in Σ, one of whose borders is Γ, onto an annulus as follows. Let γ be a smooth curve in Σ which is homotopic to Γ, and let m = γ * dg q . Ifg denotes the multi-valued harmonic conjugate of g q , then the function is holomorphic and single-valued on some region A r bounded by Γ and a level curve Γ q r = {z : g q (z) = r} of g q for some r > 0. A standard use of the argument principle shows that φ is one-to-one and onto the annulus {z : e −2πr/m < |z| < 1}. It can be shown that φ has a continuous extension to Γ which is a homeomorphism of Γ onto S 1 . By decreasing r, one can also arrange that φ extends analytically to a neighbourhood of Γ q r . We call this the canonical collar chart with respect to (Σ, q). It is uniquely determined up to a rotation and the choice of r in the definition of domain.
We say that a closed set I ⊆ Γ is null with respect to (Σ, q) if φ(I) has logarithmic capacity zero in S 1 . The notion of a null set does not depend on the position of the singularity q. For quasicircles, it is also independent of the side of the curve.
Theorem 2.2. Let R be a compact Riemann surface and Γ be a strip-cutting Jordan-curve separating R into two connected components Σ 1 and Σ 2 . Let I be a closed set in Γ.
(1) I is null with respect to (Σ 1 , q) for some q ∈ Σ 1 if and only if it is null with respect to (Σ 1 , q) for all q ∈ Σ 1 . (2) If Γ is a quasicircle, then I is null with respect to (Σ 1 , q) for some q ∈ Σ 1 if and only if I is null with respect to (Σ 2 , p) for all p ∈ Σ 2 .
Thus for quasicircles we can say "I is null in Γ" without ambiguity. For strip-cutting Jordan curves, we may say that "I is null in Γ with respect to Σ" without ambiguity. Definition 2.3. Given a function f on an open neighbourhood of Γ in the closure of Σ, we say that the limit of f exists conformally non-tangentially at p ∈ Γ with respect to (Σ, q) if f • φ −1 has non-tangential limits at φ(p) where φ is the canonical collar chart induced by Green's function g q of Σ. The conformal non-tangential limit of f at p is defined to be the non-tangential limit of f • φ −1 .
We will abbreviate "conformally non-tangential" as CNT throughout the paper.
Theorem 2.4. Let R be a compact Riemann surface and let Γ be a strip-cutting Jordan curve separating R into two connected components. Let Σ be one of these components. For any H ∈ D harm (Σ), the CNT limit of H exists at every point in Γ except possibly on a null set with respect to Σ. For any q and q ′ in Σ, the boundary values so obtained agree except on a null set I in Γ. If H 1 , H 2 ∈ D(Σ) have the same CNT boundary values except on a null set then H 1 = H 2 .
From now on, the terms "CNT boundary values" and "boundary values" of a Dirichletbounded harmonic function refer to the CNT limits thus defined except possibly on a null set. Also, if Γ is a quasicircle, we say that two functions h 1 and h 2 agree on Γ (h 1 = h 2 ) if they agree except on a null set. Outside of this section we will drop the phrase "except on a null set", although it is implicit wherever boundary values are considered.
The set of boundary values of Dirichlet-bounded harmonic functions in a certain sense determined only by a neighbourhood of the boundary. For quasicircles, it is side-independent: that is, the set of boundary values of the Dirichlet spaces of Σ 1 and Σ 2 agree.
To make the first statement precise we define a kind of one-sided neighbourhood of Γ which we clal a collar neighourhood. Let Γ be a strip-cutting Jordan curve in a Riemann surface R. By a collar neighbourhood of Γ we mean an open set A, bounded by two Jordan curves one of which is Γ, and such that (1) the other Jordan curve Γ ′ is homotopic to Γ in the closure of A and (2) Γ ′ ∩ Γ is empty. For example, if U is a doubly-connected neighbourhood of Γ, and Γ separates a compact Riemann surface R into two connected components, the intersection of U with one of the the components is a collar neighbourhood. Also, the domain of the canonical collar chart is a collar neighbourhood if the annulus r < |z| < 1 is chosen with r sufficiently close to one.
Theorem 2.5. Let R be a compact Riemann surface and let Γ be a strip-cutting Jordan curve separating R into connected components Σ 1 and Σ 2 . Let h be a function defined on Γ, except possibly on a null set in Γ. The following are equivalent.
(1) There is some H ∈ D harm (Σ 1 ) whose CNT boundary values agree with h except possibly on a null set. (2) There is a collar neighbourhood A of Γ in Σ 1 , one of whose boundary components is Γ, and some H ∈ D(A) whose CNT boundary values agree with h except possibly on a null set with respect to Σ 1 .
If Γ is a quasicircle, then the following may be added to the list of equivalences above.
(3) There is some H ∈ D harm (Σ 2 ) whose CNT boundary values agree with h except possibly on a null set. (4) There is a collar neighbourhood A of Γ in Σ 2 , one of whose boundary components is Γ, and some H ∈ D(A) whose CNT boundary values agree with h except possibly on a null set.
Thus, for a quasicircle Γ we may define H(Γ) to be the set of equivalence classes of functions h : Γ → C which are boundary values of elements of D harm (Σ 1 ) except possibly on a null set, where we define two such functions to be equivalent if they agree except possibly on a null set.
This theorem also induces a map from D harm (Σ 1 ) to D harm (Σ 2 ) as follows: induced by Theorem 2.5 is bounded with respect to the Dirichlet semi-norm.

Schiffer's comparison operators
3.1. Assumptions. The following notation and assumptions will be in place throughout the rest of the paper (see the relevant sections for further explanations): • R is a compact Riemann surface; • Γ is a strip-cutting Jordan-like curve separating R; • Σ 1 and Σ 2 are the connected components of R\Γ; • Σ stands for an unspecified component Σ 1 or Σ 2 ; • Γ is positively oriented with respect to Σ 1 ; • Γ p k ǫ the level curves of Green's function g Σ k (·, p k ) with respect to some fixed points p k ∈ Σ k ; • when an integrand depends on two variables, we will use the notation Σ,w to specify that the integration takes place over the variable w. We will sometimes alter the assumptions or repeat them for emphasis. When no assumptions are indicated at all, the above assumptions are in place.

3.2.
Schiffer's comparison operators: definitions. Following for example Royden [11], 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(z)| 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, is uniquely determined by these properties, and furthermore satisfies the symmetry properties In particular, g is also harmonic in z away from the poles.
We will treat w 0 as fixed throughout the paper, and notationally drop the dependence on w 0 as much as possible. In fact, it follows immediately from (3.1) that ∂ w g is independent of w 0 . All formulas of consequence in this paper are independent of w 0 for this reason.
The following is an immediate consequence of the residue theorem and the fact that g is harmonic in w.
Theorem 3.1. Let Γ be a closed analytic curve separating R, enclosing Σ, which is positively oriented with respect to Σ. If h is holomorphic on Σ, and z, q / ∈ Γ, then for any fixed p ∈ Σ − lim where χ Σ is the characteristic function of Σ.
We will also need the following well-known reproducing formula for Green's function of Σ.
Theorem 3.2. Let R be a compact Riemann surface and Γ be a strip-cutting Jordan curve separating R. Let Σ be one of the components of the complement of Γ. For any h ∈ D harm (Σ), we have Next we turn to the definitions of the relevant kernel forms. Let R be a compact Riemann surface, and let g(w, w 0 ; z, q) be the Green's function. We define the Schiffer kernel to be the bi-differential We also define the Bergman kernel function For non-compact surfaces Σ with border, with Green's function g, we define Then the following identity holds. For any vector v tangent to Γ w ǫ at a point z, we have This follows directly from the fact that the one form ∂ z g(z, w)+∂ z g(z, w) vanishes on tangent vectors to the level curve Γ w ǫ . It is well known that for all h ∈ A(Σ) For compact surfaces, the reproducing property of the Bergman kernel is established in [11]. Proposition 3.3. Let R be a compact Riemann surface with Green's function g(w, w 0 ; z, q).

Then
(1) L R and K R are independent of q and w 0 .
(2) K R is holomorphic in z for fixed w, and anti-holomorphic in w for fixed z.
(3) L R is holomorphic in w and z, except for a pole of order two when w = z.
For non-compact Riemann surfaces Σ with Green's function, (2) − (5) hold with L R and K R replaced by L Σ and K Σ .
Remark 3.4. The symmetry statements (4) and (5) are formally expressed as follows. If Proof. It follows immediately from (3.1) that ∂ w g(w, w 1 ; z, q) = ∂ w g(w, w 0 ; z, q) and ∂wg(w, w 1 ; z, q) = ∂wg(w, w 0 ; z, q), so L R and K R are independent of w 0 . Applying (3.3) shows that similarly ∂ w g and ∂w are independent of q, and hence the same holds for L R and K R . This demonstrates that property (1) holds.
Since g is harmonic in w, ∂ w ∂ w g(w, w 0 ; z, q) = 0 so K R is anti-holomorphic in w. As observed above, (3.2) shows that g is also harmonic in z, so we similarly have that K R is holomorphic in z. This demonstrates (2).
Similarly harmonicity of g in z and w implies that L R is holomorphic in z and w. The fact that L R has a pole of order two at z follows from the fact that g has a logarithmic singularity at w = z. This proves (3).
Properties (4) and (5) follow from equation (3.3) applied directly to the definitions of L R and K R .
The non-compact case follows similarly from the harmonicity with logarithmic singularity of g Σ , and the symmetry g Σ (z, w) = g Σ (w, z) One can find the constant at the pole of L from the definition. Expressed in a local holomorphic coordinates η = φ(w) near a fixed point ζ = φ(z), where H(η) is holomorphic in a neighbourhood of ζ. In most sources [3], the integral kernel is expressed as a function (rather than a form) to be integrated against the Euclidean area form dA η = dη ∧ dη/2i. E.g. if α(w is a holomorphic one-form given in local coordinates by (φ −1 ) * α(η) = f (η)dη then we obtain the local expression which agrees with the classical normalization [3]. Now let R be a compact Riemann surface and let Γ be a strip-cutting Jordan curve. Assume that Γ separates R into two surfaces Σ 1 and Σ 2 . We will mostly be concerned with the case that Γ is a quasicircle.
Let A(Σ 1 ∪Σ 2 ) denote the set of one-forms on Σ 1 ∪Σ 2 which are holomorphic and square integrable. Note that we do not require the existence of a holomorphic or continuous extension to the closure of Σ 1 ∪ Σ 2 . For k = 1, 2 define the restriction operators It is obvious that these are bounded operators.
Definition 3.5. For k = 1, 2, we define the Schiffer comparison operators Note that the operator S is bounded and the image is clearly in A(R). Moreover, for j = k, the integral kernel of this operator is nonsingular, but if j = k, then the kernel is singular. We will show below that the image is in fact in A(Σ k ).
First we require an identity of Schiffer. Although this identity was only stated for analytically bounded domains, it is easily seen to hold in greater generality.
Proof. We assume momentarily that α has a holomorphic extension to the closure of Σ and that Γ is an analytic curve. Let z ∈ Σ be fixed but arbitrary, and choose a chart ζ near z such that ζ(z) = 0. Write α locally as f (ζ)dζ for some holomorphic function f . Let C r be the curve |ζ| = r, and denote its image in Σ by γ r . Fixing p ∈ Σ and using Stokes' theorem yield The first term goes to zero uniformly as ǫ → 0. Writing the second term in coordinates η = φ(w) in a neighbourhood of ζ for fixed ζ (see equation (3.6)) we obtain where h is some harmonic function in a neighbourhood of 0. Now since both terms on the right hand side go to zero, we obtain the desired result. Note that this shows that the principal value integral can be taken with respect to any local coordinate with the same result. Furthermore, the integral is conformally invariant.
Thus, we may assume that Σ is a subset of its double and Γ is analytic. By [2, Proposition 2.2], the set of holomorphic one-forms on an open neighbourhood of the closure of Σ is dense in A(Σ). The L 2 boundedness of the L Σ operator yields the desired result.
This implies that for R, Γ, and Σ as in Theorem 3.6, we can write which has the advantage that the integral kernel is non-singular.
Remark 3.7. The above expression shows that the operator T (Σ, Σ) is well-defined. The subtlety is that the principal value integral might depend on the choice of coordinates, which determines the ball which one removes in the neighbourhood of the singularity. Since the integrand is not in L 2 , different exhaustions of Σ might in principle lead to different values of the integral. However the proof of Theorem 3.6 shows that the integral of L Σ is independent of the choice of coordinate near the singularity. Since the integrand of (3.7) is L 2 bounded, it is independent of the choice of exhaustion; combining this with Theorem 3.6 shows that the integral in the definition of T (Σ, Σ) is independent of the choice of exhaustion. One may also obtain this fact from the general theory of Calderón-Zygmund operators on manifolds, see [20].
Theorem 3.8. Let R be a compact Riemann surface, and Γ be a strip-cutting Jordan curve in R. Assume that Γ separates R into two surfaces Σ 1 and Σ 2 .
Proof. Fix j and let k ∈ {1, 2} be such that k = j. By (3.7) we observe that The integrand in both terms (3.8) is non-singular and holomorphic in z for each w ∈ Σ j , and furthermore both integrals are locally bounded in z. Therefore the holomorphicity of T (Σ j )α follows by moving the ∂ inside (3.7), and using the holomorphicity of the integrand. This also implies the holomorphicity of T (Σ j , Σ k ).
Regarding the boundedness, the operator T (Σ j ) is defined by integration against the L-Kernel which in local coordinates is given by 1 π(ζ−η) 2 , modulo a holomorphic function. Since the singular part of the kernel is a Calderón-Zygmund kernel we can use the theory of singular integral operators on general compact manifolds, developed by R. Seeley in [20] to conclude that that, the operators with kernels such as L R (z, w) are bounded on L p for 1 < p < ∞. The boundedness of T (Σ j , Σ k ) follows from this and the fact that R 0 (Σ j ) is also bounded.

3.3.
Attributions. The comparison operators T (Σ j , Σ k ) were studied extensively by Schiffer [14], [5,6], and also together with other authors, e.g. Bergman and Schiffer [3]. In the setting of planar domains, a comprehensive outline of the theory was developed in a chapter in [4]. The comparison theory for Riemann surfaces can be found in Schiffer and Spencer [15]. See also our review paper [18].
In this section, we demonstrate some necessary identities for the Schiffer operator. Most of the identities were stated by for example Bergman and Schiffer [3], Schiffer [4], and Schiffer and Spencer [15] for the case of analytic boundaries. Versions can be found in different settings, for example multiply-connected domains in the sphere, nested multiply-connected domains, and Riemann surfaces.
On the other hand, we introduce here several identities involving the adjoints of the operators, which Schiffer seems not to have been aware of. These are Theorem 3.9, Theorem 3.10, and Theorem 3.11. The introduction of the adjoint operators has significant clarifying power. Proofs of the remaining identities are included because it is necessary to show that they hold for regions bordered by quasicircles.
Here are a few words on terminology. The Beurling transform in the plane is defined by Schiffer refers to this operator as the Hilbert transform, due to the fact that the operator in question behaves like the actual Hilbert transform The term "Hilbert transform" is also the one used in O. Lehto's classical book on Teichmüller theory [9]. Indeed the integrands of both operators exhibit a similar type of singularity in their respective domains of integration and both fall into the general class of Calderón-Zygmund singular integral operators. For such operators, one has quite a complete and satisfactory theory, both in the plane and on differentiable manifolds. We shall refer to the restriction of the Beurling transform to anti-holomorphic functions on fixed domain as the Schiffer operator. Here, of course, we express this equivalently as an operator on anti-holomorphic one-forms.

Identities for comparison operators.
Theorem 3.9. Let R be a compact surface and let Γ be a strip-cutting Jordan curve separating R into two components, one of which is Σ. Then S(Σ) = Res(Σ) * , where * denotes the adjoint operator.
Proof. Let α ∈ A(Σ) and β ∈ A(R). Then, using the reproducing property of K R , Note that interchange of order of integration is legitimate by Fubini's theorem, due to the analyticity and boundedness of the the Bergman kernel. Define and similarly for S(Σ k ).
Theorem 3.10. Let R be a compact surface. Let Γ be a strip-cutting Jordan curve with measure zero, and assume that the complement of Γ consists of two connected components Σ 1 and Σ 2 . Then Proof. If j = k, the claim follows from the non-singular integral representation (3.7) and interchanging the order of integration. The claim essentially follows from the corresponding fact for planar domains, and we need only reduce the problem to this case using coordinates. Denote We first show that for G, H ∈ L 2 (C) one has (3.9) where the inside integral is understood as a principle value integral in both cases. Now, for f ∈ L 2 (C), the Beurling transform is given by With this notation, and denoting H(w) = H(w), (3.9) amounts to (3.11) If one defines the Fourier transform through . Using Parseval's formula and the above Fourier multiplier representation of the Beurling transform, one has that This proves (3.11) and hence (3.9). Now let B be a doubly-connected neighbourhood of Γ and φ : B → U ⊆ C be a doublyconnected chart. Let E = B ∩ Σ 1 and E ′ = B ∩ Σ 2 . Then Σ 1 = D ∪ E and Σ 2 = D ′ ∪ E ′ 13 for some compact sets D ⊂ Σ 1 and D ′ ⊆ Σ 2 whose shared boundaries with E and E ′ are strip-cutting Jordan curves. We may choose these as regular as desired (say, analytic Jordan curves, which in particular have measure zero). Observe that we then have, for any forms α ∈ A(Σ 2 ) and β ∈ A(Σ 1 ) (3.12) and (3.13) We only need to show that one can interchange integrals in each term. The first three integrals in the right hand side of (3.12) are equal to their interchanged counterparts in the first three terms of (3.13). This follows from Fubini's theorem, using the fact that L(z, ζ) is non-singular and in fact bounded on all of the six domains of integration involved in those integrals. Therefore it is enough to show that To show this, let φ be a local coordinate with η = φ(w) and ζ = φ(z). We pull back the integral to the plane under ψ = φ −1 so that we reduce the problem to showing that (3.14) Recall that in local coordinates by equation (3.6) where H(η) is holomorphic near ζ. For the holomorphic error term, we can just apply Fubini's theorem, so matters reduce to the demonstration of (3.14) for the principal term of L C (ζ, η) which contains the singularity. We may write ψ * α(z) = h(z)dz and ψ * β(w) = g(w)dw for some L 2 holomorphic functions g on E and h on E ′ . So the problem is reduced to showing that Letting then G and H are L 2 on C and the claim now follows directly from (3.9).
We also have the following identity.
Theorem 3.11. If Γ is a quasicircle then Proof. By Theorem 3.10, and interchange of order of integration (which can be justified as in the proof of Theorem 3.10) we have that Similarly, by equation (3.7) and Theorem 3.10, the integral kernel of T ( Finally, by Theorem 3.9, the integral kernel of S(Σ 1 ) * S(Σ 1 ) is K R (z, w).
Using this and the reproducing property of K Σ we need only demonstrate the following identity: Fix w ∈ Σ 1 and orient Γ w ǫ positively with respect to Σ 1 . For fixed w, ∂ w g Σ 1 (ζ, w) goes to zero uniformly as ǫ → 0. We then have that (applying (3.6) where we have applied equation (3.4) in the last step. Applying Stokes' theorem to the first term, we see that Here we used the fact that quasicircles have measure zero. Note that Γ w ǫ is negatively oriented with respect to Σ 2 . For the second term, we have where in the last term we have used part (5)   In this section, we show that the jump formula holds when Γ is a quasicircle. We also prove that in this case the Schiffer operator T (Σ 1 , Σ 2 ) is an isomorphism, when restricted to a certain subclass of A(Σ 1 ).
To establish a jump formula, we would like to define a Cauchy-type integral for elements h ∈ H(Γ). Since Γ is not necessarily rectifiable, instead we replace the integral over Γ with an integral over approximating curves Γ p 1 ǫ (defined at the beginning of Section 3), and use the harmonic extensionsh ∈ D harm (Σ 1 ) of elements of H(Γ).
It is an arbitrary choice whether to approximate the curve from within Σ 1 or from within Σ 2 . Later, we will show that the result is the same in the case that Γ is a quasicircle. For now, we choose to approximate from within Σ 1 . We thus define the operator on elements of D harm (Σ 1 ), which in the end will be chosen as the unique extension of a fixed element of H(Γ).
Let h ∈ D harm (Σ 1 ). Fix q ∈ R\Γ and define for z ∈ R\Γ. Observe that, by definition, the curve Γ p 1 ǫ depends on a fixed point p 1 ∈ Σ 1 . However, we shall show that J q (Γ) is independent of p 1 in a moment.
First we show that the limit exists. There are several cases depending on the locations of z and q. Assume that q ∈ Σ 2 , then for z ∈ Σ 2 , we have by Stokes' theorem that so the limit exists and is independent of p 1 . For z ∈ Σ 1 we proceed as follows; let γ r denote the circle of radius r centered at z, positively oriented with respect to z, in some fixed chart near z. By applying Stokes' theorem and the mean value property of harmonic functions we obtain This shows that the limit exists for z ∈ R\Γ and q ∈ Σ 2 and is independent of p. In the case that q ∈ Σ 1 , we obtain similar expressions, but with the term h(q) added to both integrals.
This also shows that Therefore, in the following we will usually omit mention of the point p 1 in defining the level curves, and write simply Γ ǫ .
Theorem 4.2. Let Γ be a strip-cutting Jordan curve in R. For all h ∈ D harm (Σ 1 ) and any q ∈ R\Γ, Proof. Assume first that q ∈ Σ 2 . The first claim follows from (4.2) and the fact that the integrand is non-singular. Similarly for z ∈ Σ 2 , the third claim follows from (4.2). The second claim follows from Stokes theorem: where we have used Theorem 3.2. Also observe that the fact that the integrand of the first term is non-singular and holomorphic in z for each w ∈ Σ 1 , and that is locally bounded in z, yield that derivation under the integral sign in the first term is legitimate.
Similarly removing the singularity using ∂ w g Σ , and then applying Theorem 3.2 and Stokes' theorem yield that The third claim now follows by observing that the second term in the integral is just −∂h because the integrand is just the complex conjugate of the Bergman kernel. Now assume that q ∈ Σ 1 . We show the second claim in the theorem. We argue as in equation (4.4), except that we must also add a term ∂ w g Σ 1 (w; q)h(w). We obtain instead and the claim follows from ∂ z h(q) = 0. The remaining claims follow similarly.
Below, let A(R) ⊥ denote the orthogonal complement in A harm (Σ 1 )of the restrictions of A(R) to Σ 1 Corollary 4.3. Let Γ be a strip-cutting Jordan curve and assume that q ∈ R\Γ.
Proof. The first claim follows immediately from Theorems 3.8 and 4.2. The second claim follows from Theorem 4.2 together with the fact that for fixed z ∂ z ∂ w g ∈ A(R).

Density theorems.
In this section we show that certain subsets of the Dirichlet space are dense. Our first density result follows from a theorem of N. Askaripour and T. Barron [2]. In brief, their result says that L 2 holomorphic one-forms (in fact, differentials) on a region in a Riemann surface can be approximated by holomorphic one-forms on a larger domain. More specifically they proved the following: We need a result of this form for the Dirichlet space. Equivalently, one must show that exact L 2 one-forms can be approximated by exact L 2 one-forms on the larger region. (1) Let R be a Riemann surface and Σ 1 be bounded by a strip-cutting Jordan curve in R which separates R. Let Σ ′ 1 contain Σ 1 and be such Σ ′ 1 \cl Σ 1 is a collar neighbourhod of Γ. Let Res : (2) Let R be a compact Riemann surface and Γ be a strip-cutting Jordan curve. Let U be a doubly-connected neighbourhood of Γ. Let A i = U ∩ Σ i for i = 1, 2, and let Proof. Assume that R is a hyperbolic Riemann surface. Applying Proposition 4.4 with k = 1, B 1 = Σ 1 , B 2 = Σ ′ 1 , and Σ = R, we see that Res A(Σ ′ 1 ) is dense in A(Σ 1 ). If on the other hand R is not hyperbolic, let Σ be obtained by removing a conformal disk from R whose closure is disjoint from the closure of Σ ′ 1 . Let k, B 1 and B 2 be as before. The conditions of Proposition 4.4 are then easily verified, and we have that Res A(Σ ′ 1 ) is dense in A(Σ 1 ). We need to show that the claim holds for exact forms. Choose a set of closed contours γ 1 , . . . , γ m in Σ 1 generating the fundamental group of Σ 1 . For α ∈ A(Σ 1 ) let Each component of Q • Res is a bounded linear functional. To see this, for fixed i let φ : B → U be a local holomorphic chart in a neighbourhood B of γ i . Now Γ i = φ • γ i is a compact subset of U, and has finite length l say. Let M Γ i be obtained by applying Lemma 2.1 to A(U) and Γ i . We then have that → Ker Q denote the orthogonal projection onto the kernel of Q • Res. By using the Riesz representation theorem and the Gram-Schmidt process, one can show that there is a C such that Again using Lemma 2.1 to control the uniform norm on γ 1 ∪ · · · ∪ γ m by the norm on A(Σ 1 ), there is also an M such that for all α ∈ A(Σ 1 ). Now let β ∈ A e (Σ 1 ) and let ε > 0. Choose α ∈ A(Σ ′ 1 ) such that β − Res α A(Σ 1 ) < ε. Using the fact that Q(β) = 0 together with the two preceding bounds, we have Since Res Pα ∈ A e (Σ 1 ), this completes the proof of the first claim.
To prove the second claim, fix i = 1 or 2 and set B 1 = A i , B 2 = U, in Proposition 4.4 and let Σ = R be as above (with disks removed if necessary as above). Choose a single curve γ in A i which is homotopic to Γ, and define Q(α) = γ α. The proof now proceeds identically, but with a single curve.
Remark 4.6. We will only use the second part of Theorem 4.5. However, part (a) is no more difficult than part (b), and will be useful in future applications.
Remark 4.7. Note that the result of Askaripour and Barron [2] will not in general apply to exact one-forms without restrictions on the relation between the homology of Σ 1 and Σ ′ 1 . It is clear that weaker conditions could be imposed, so that we have not used the full power of their result.
We will also need a density result of another kind. Let Γ be a strip-cutting Jordan curve in a compact Riemann surface R, which separates R into two components Σ 1 and Σ 2 . Let A be a collar neighbourhood of Γ in Σ 1 . By Theorem 2.5 the boundary values of D harm (A) exist conformally non-tangentially in Σ 1 and are themselves CNT boundary values of an element of D harm (Σ 1 ). We then define whereh is the unique element of D harm (Σ 1 ) with CNT boundary values equal to those of h. We have the following result:  Since the set C[z,z] of polynomials in z,z is dense in D harm (D), this proves the claim.
Next, let F : A → A be a conformal map. Define the composition map Since the image of D(A) under G(A, D)C F is dense in D harm (D), and ρ is a bounded bijection with bounded inverse, this completes the proof.

4.3.
Limiting integrals from two sides. In this section, we show that for quasicircles, the limiting integral defining J q (Γ) can be taken from either side of Γ, with the same result.
We will need to write the limiting integral in terms of holomorphic extensions to collar neighbourhoods. The integral in the definition of J q (Γ) is easier to work with when restricting to D(A). To make use of this simplification, we must first show that the limiting integrals of Gh and h are equal.
For h ∈ D(A), letting Γ ǫ be level curves of Green's function of Σ 1 with respect to some fixed point p ∈ Σ 1 , denote for q fixed in Σ 2 . We use the notation J q (Γ) ′ to distinguish it from the operator J q (Γ), which applies only to elements of D harm (Σ 1 ). For ǫ in some interval (0, R) the curve Γ ǫ lies entirely in A, so this makes sense. Because the integrand is holomorphic, the integral is independent of ǫ for ǫ ∈ (0, R).
We have the following.  We claim that (4.5) holds for all h ∈ D poly (A). To see this, observe that for any ǫ, f −1 takes the curve Γ ǫ to the curve |z| = e −cǫ for some c > 0 which is independent of ǫ. So for z ∈ Γ ǫ , f −1 (z) n = e −2ncǫ (f −1 (z)) −n for any n ∈ Z. Thus where we use the fact that the integrand on the right side of the first line is holomorphic in w and thus the integral is independent of ǫ. This proves the claim.
By Lemma 2.1, for compact K there is an M K such that for all g ∈ D harm (A) x (4.6) sup The theorem now follows from the boundedness of G and of J q (Γ). To see this let h ∈ D(A). Given u ∈ D poly (A) we have that Fixing any z ∈ Σ 1 ∪ Σ 2 , let K be a compact set containing z. By (4.6), the norm estimate above, and the normalization of J q (Γ) we have that for any ε > 0, there is a δ 1 > 0 so that Because the integrands of J q (Γ) ′ u and J q (Γ) ′ h are holomorphic, the limiting integral is equal to the integral over Γ ′ for some fixed level curve Γ ′ = Γ ǫ ⊂ A. Recall (see above) that we can assume that h and u are in D(A) x for some fixed x ∈ A. Again by (4.6), there . Using the fact that ∂ w g(w; z, q) is bounded on the compact set K for w ∈ Γ ′ , there is a δ 2 such that u − h D(A) < δ 2 implies that Using the fact that J q (Γ) ′ u = J q (Γ)Gu, we have that Combining this with (4.8) and (4.7), by the density of D(A) poly we see that for all ε > 0, which proves the theorem.
Since every collar neighbourhood of Γ contains a canonical collar neighbourhood Γ, we have Corollary 4.11. Proposition 4.10 holds for any collar neighbourhood A of Γ in Σ 1 .
Finally we observe that the same proof, replacing ∂ w g(w; z, q) with any holomorphic oneform α on a neighbourhood of Γ shows the following.  We now show that for quasicircles, one can define the jump operator J(Γ) using either limiting integrals from within Σ 1 or from within Σ 2 with the same result. We use the 22 following temporary notation. For q ∈ R\Γ let J q (Γ, Σ i ) : For definiteness, we assume that all curves Γ p i ǫ are oriented positively with respect to Σ i . Aside from this change of sign, all previous theorems apply equally to J q (Γ, Σ 1 ) and J q (Γ, Σ 2 ). Theorem 4.13. Let Γ be a quasicircle. Then for all q ∈ R\Γ induced by these collar neighbourhoods for i = 1, 2.
For any h ∈ D(U), let Res i h = h| A i . It follows immediately from the definition of G i that (4.9) In the rest of the paper, we return to the convention that J q (Γ) is an operator on D harm (Σ 1 ). However, Theorem 4.13 plays an important role in the proof that T (Σ 1 , Σ 2 ) is surjective.
Also, by using Theorem 4.12 and proceeding exactly as in the proof of Theorem 4.13 we obtain Definition 4.15. We denote by W k the linear subspace of D harm (Σ i ) given by for all α ∈ A(R) and for k = 1, 2. The elements of W k are the admissible functions for the jump problem. Let We have the following result: Theorem 4.16. Let R be a compact surface and Γ be a quasicircle separating R into com- To prove this theorem we need a lemma.
Proof. The fact that Gh ∈ W 1 follows immediately from Theorem 4.12. By Royden [11,Theorem 4] and the explicit formula on the following page, there are holomorphic functions H 1 on Σ 1 and H 2 on cl Σ 2 ∪ A such that H 1 − H 2 = h on A. Furthermore, these functions are given by the restrictions of J q (Γ) ′ h to Σ 1 and Σ 2 . Thus, by Proposition 4.10, we have that  By Lemma 2.1 and the fact that Q(u + c) = Q(u) for any constant c, it follows that each component of Q is a bounded linear functional on D(A). Once again, a simple argument based on Riesz representation theorem and the Gram-Schmidt process yields that there is a C such that (4.12) Pu For H ∈ D(Σ 1 ) harm define now We have that there is a C ′ such that This follows by applying Stokes' theorem to each component: We also define a transmission operator for exact one-forms as follows: Definition 4.18. For an exact one-form α ∈ A e (Σ 2 ) harm let h 2 be a harmonic function on Σ 2 such that dh 2 = α. Let h 1 be the unique element of D(Σ 1 ) harm with boundary values agreeing with h 2 . Then we define The transmission from A e (Σ 1 ) harm to A e (Σ 2 ) harm is defined similarly.
To prove the transmission formula for O e , we require the following elementary lemma.
Lemma 4.19. Let Σ be a Riemann surface of finite genus g bordered by a curve homeomorphic to a circle. Let α ∈ A(Σ). There is an h ∈ D harm (Σ) such that ∂h = α. Ifh ∈ D harm (Σ) is any other such function, thenh − h ∈ D(Σ).
Proof. Let R be the double of Σ; so A(R) has dimension 2g where g is the genus of Σ. Let a 1 , . . . , a 2g be a collection of smooth curves which generate the fundamental group of Σ. Let for k = 1, . . . , 2g. Thus α + β is exact in Σ and hence is equal to dh for some h ∈ D harm (Σ). But clearly ∂h = α.
Ifh is any other such function then ∂(h − h) = 0, which completes the proof.
Recall that A(R) ⊥ denotes the set of elements in A harm (Σ) which are orthogonal to the restrictions of elements of A(R) with respect to (·, ·) A harm (Σ).
Definition 4.20. Given R and Σ i as above, let and Theorem 4.21. Let R be a compact Riemann surface and let Γ be a quasicircle separating R into components Σ 1 and Σ 2 . If α ∈ V 1 then Proof. Let α ∈ V 1 , then by Lemma 4.19 there is an h ∈ D harm (Σ 1 ) such that ∂h = α.
Applying d to both sides of (4.16) and using this fact yields The Theorem now follows from the remaining relations in Theorem 4.2.
For k = 1, 2 denote by the orthogonal projections onto the holomorphic and anti-holomorphic parts of a given harmonic one-form.
As another consequence of Theorem 4.21 we are able to prove an inequality analogous to the strengthened Grunsky inequality for quasicircles [13].
Proof. Since d : D harm (Σ k ) → A harm (Σ k ) is norm-preserving (with respect to the Dirichlet semi-norm), it follows from Theorem 2.7 that there is a c ∈ (0, 1) which is independent of α such that We will insert the identity of Theorem 4.21 into (4.13).
In the following computation, we need two observations. First, if a function H is holomorphic on a domain Ω, then , then we observe that O(Σ 2 , Σ 1 )Re(H 2 ) = Re(H 1 ), and therefore the boundedness of transmission estimate applies to Re(H i ).
Since α − T (Σ 1 , Σ 1 )α has the same real part as the right hand side of (4.14), combining with (4.13) (applied to the real part of the primitives) we obtain where we have used the fact that α − T (Σ 1 , Σ 1 )α is holomorphic. By Theorem 3.11 we have that Combining this with (4.15) yields Applying this to e iθ α, we see that the same inequality holds with the left hand side replaced by −e −2iθ Re(α, T (Σ 1 , Σ 1 )α) for any θ. So Together with the fact that T (Σ 1 , Σ 1 ) * = T (Σ 1 , Σ 1 ) this proves the theorem.
Remark 4.24. This gives another proof that Observe that if α ∈ A(Σ 1 ) is in V 1 , then since the kernel of the operator S(Σ 1 ) is holomorphic we have that α ∈ Ker S(Σ 1 ). Thus by Theorem 3.11 Since 1 − ν 2 > 0 this completes the proof.

4.5.
Isomorphism theorem for the Schiffer operator. In this section, we prove the isomorphism theorem for the Schiffer operators. We require two facts.
Remark 4.26. Although we have only proven that T (Σ 1 , Σ 2 ) is injective for quasicircles, we conjecture that this is true in greater generality, as in Napalkov and Yulmakhumetov [12] in the planar case. It would also be of interest to give a proof of surjectivity using their approach. One would use the adjoint identity of Theorem 3.10 in place of the symmetry of the L kernel, which is used implicitly in their proof. One would also need to take into account the topological obstacles as we did above.
Since dh = ∂h + ∂h is exact, the claim follows.
The following theorem is in some sense a derivative of the jump decomposition.
ThusĤ is surjective. Now assume thatĤ(dh) = 0. The vanishing of the second component yields that −T (Σ 1 , Σ 2 )∂h = 0, so by Theorem 4.25 we have that ∂h = 0. Thus the vanishing of the first component of H(dh) yields that ∂h = 0, hence dh = 0. 4.6. The jump isomorphism. In this section we establish the existence of a jump decomposition for functions in H(Γ).
Theorem 4.29. Let R be a compact Riemann surface, and let Γ be a quasicircle separating R into two connected components Σ 1 and Σ 2 . Fix q ∈ Σ 2 and let W 1 be given as in Definition 4.15. Then the map is a bounded isomorphism from W 1 to D(Σ 1 ) ⊕ D(Σ 2 ) q .
Thus the image of the map is in D(Σ 1 ) ⊕ D q (Σ 2 ). By Theorem 4.2 ∂Hh =Ĥ dh, so sinceĤ is an isomorphism by Theorem 4.28, we only need to deal with constants. If J q (Γ)h = 0 then dh = 0 so h is constant on Σ 1 . Since the second component of Hh vanishes at q we see that h = 0, so H is injective. Now observe that H(h + c) = Hh + (c, 0) for any constant c. Thus surjectivity follows from surjectivity of H.
Proof. The first claim follows immediately from Theorem 3.2. The second claim follows from Theorems 3.2 and 4.13 (note that Γ is negatively oriented with respect to Σ 2 ).
We then have a version of the Plemelj-Sokhotski jump formula.
Corollary 4.31. Let R, Γ, Σ 1 and Σ 2 be as above. Let H ∈ H(Γ) be such that its extension h to D harm (Σ 1 ) is in W 1 . There are unique h k ∈ D(Σ k ), k = 1, 2 so that if H k are their CNT boundary values then H = −H 2 + H 1 . These unique h i 's are given by h k = J q (Γ)h| Σ k for k = 1, 2.
Proof. We claim that h = −O(Σ 2 , Σ 1 )h 2 + h 1 , which would imply that H = −H 2 + H 1 . Finally, we show that the condition for existence of a jump formula is independent of the choice of side of Γ.