Plemelj–Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators
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Abstract
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 Calderón–Zygmund type operators \(T(\Sigma _1,\Sigma _k)\) taking the space of \(L^2\) antiholomorphic oneforms on \(\Sigma _1\) to the space of \(L^2\) holomorphic oneforms on \(\Sigma _k\) for \(k=1,2\), which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves \(\Gamma \), to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of antiholomorphic oneforms which are orthogonal to \(L^2\) antiholomorphic oneforms on R with respect to the inner product on \(\Sigma _1\). We show that the restriction of the Schiffer operator \(T(\Sigma _1,\Sigma _2)\) to V is an isomorphism onto the set of exact holomorphic oneforms on \(\Sigma _2\). Using the relation between this Schiffer operator and a Cauchytype 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 Dirichletbounded 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.
Mathematics Subject Classification
Primary 58C99 58A10 30F10 30F15 30F30 32W05 Secondary 58J401 Introduction
1.1 Results and literature
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 \(\Sigma _1\) or \(\Sigma _2\). In the case that \(\Gamma \) is analytic, this space agrees with the Sobolev \(H^{1/2}\) space on \(\Gamma \). We showed in an earlier paper [20] that the space of boundary values, for quasicircles, is the same for both \(\Sigma _1\) and \(\Sigma _2\), and the resulting map (which we call the transmission map) is bounded.
Since quasicircles are nonrectifiable, we replace the Cauchy integral by a limit of integrals along level curves of Green’s function in \(\Sigma _k\); for quasicircles, we show that this integral is the same whether one takes the limiting curves from within \(\Sigma _1\) or \(\Sigma _2\). This relies on our transmission result mentioned above. We show that the map from the harmonic Dirichlet space \(\mathcal {D}_{\mathrm{harm}}(\Sigma _k)\) to the direct sum of holomorphic Dirichlet spaces \(\mathcal {D}(\Sigma _1) \oplus \mathcal {D}(\Sigma _2)\) obtained from the jump integral is an isomorphism. We also consider a Calderón–Zygmund type integral operator on the space of oneforms which is one type of what we call a Schiffer operator. This was studied extensively by Schiffer and others in the plane and on Riemann surfaces (see Sect. 3.3 for a discussion of the literature). Schiffer discovered deep relations between these operators and inequalities in function theory, potential theory and Fredholm eigenvalues. We extend many known results from analytic boundary to quasicircles, and derive some new identities for the adjoints of the Schiffer operators (Theorems 3.11, 3.12, and 3.13), as well as a complete set of identities relating the Schiffer operator to the Cauchytype integral in higher genus (Theorem 4.2). The derivative of the Cauchytype integral, when restricted to a finite codimension space of oneforms, equals a Schiffer operator which we denote below by \(T(\Sigma _1,\Sigma _2)\). We prove that the restriction of this Schiffer operator to this finite codimensional space is an isomorphism (Theorem 4.22).
In the case of simplyconnected domains in the plane (where the finite codimensional space is the full space of oneforms), the fact that aforementioned Schiffer operator \(T(\Sigma _1,\Sigma _2)\) is an isomorphism is due to Napalkov and Yulmukhametov [8]. In fact, they showed that it is an isomorphism precisely for domains bounded by quasicircles. This is closely related to a result of Shen [22], 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 a jump operator and the Schiffer operator are isomorphisms precisely for quasicircles [18]. The isomorphism for the jump operator is what we call the Plemelj–Sokhotski isomorphism. As mentioned above, here we generalize the isomorphism theorem for \(T(\Sigma _1,\Sigma _2)\) and the Plemelj–Sokhotski isomorphism (Theorem 4.26) 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.
Let us 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 nonrectifiable curves have been studied extensively by Kats, see e.g. [6] for the case of Hölder continuous boundary values, and the survey article [5] 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 [20] (which, in the plane, also characterizes quasicircles [17]). Indeed our approach to proving surjectivity of \(T(\Sigma _1,\Sigma _2)\) 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 for Dirichlet space functions on nested doublyconnected regions in a Riemann surface. This is similar to a result of Askaripour and Barron [2] for \(L^2\)kdifferentials for nested surfaces satisfying certain conditions. Their result uses the density of polynomials in the Bergman space of a Carathéodory domain in the plane. The proof of our result is similar.
The results of this paper can be applied to families of operators over Teichmüller space, as we will pursue in future publications. Applications to a certain determinant line bundle occurring in conformal field theory appear in [11].
1.2 Outline of the paper
In Sect. 2 we establish notation and state preliminary results. We also outline previous results of the authors which are necessary here. In Sect. 3 we define the Schiffer operators, generalize known results to quasicircles, and establish some new identities for adjoints. In Sect. 4, we give identities relating one type of Schiffer operator to a Cauchytype integral (in general genus), we relate it to the jump decomposition, and establish the isomorphism theorems for the Schiffer operator and the Cauchytype integral. We call the latter isomorphism the Plemelj–Sokhotski isomorphism.
2 Notations and preliminaries
2.1 Forms and functions
We begin by establishing notation and terminology.
Harmonic oneforms \(\alpha \) can always be decomposed as a sum of a holomorphic and antiholomorphic oneform. The decomposition is unique. On the other hand, harmonic functions do not possess such a decomposition.
If \(F: R_1 \rightarrow R_2\) is a conformal map, then we denote the pullback of \(\alpha \in A_{\mathrm{harm}}(R_2)\) under F by \(F^*\alpha .\)
Finally, we will repeatedly use the following elementary fact.
Lemma 2.1
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 [20].
Let R be a compact Riemann surface. Let \(\Gamma \) be a Jordan curve in R, that is a homeomorphic image of \(\mathbb {S}^1\). We say that U is a doublyconnected neighbourhood of \(\Gamma \) if U is an open set containing \(\Gamma \), which is bounded by two nonintersecting Jordan curves each of which is homotopic to \(\Gamma \) within the closure of U. We say that a Jordan curve \(\Gamma \)is stripcutting if there is a doublyconnected neighbourhood U of \(\Gamma \) and a conformal map \(\phi :U \rightarrow \mathbb {A} \subseteq \mathbb {C}\) so that \(\phi (\Gamma )\) is a Jordan curve in \(\mathbb {C}\). We say that \(\Gamma \) is a quasicircle if \(\phi (\Gamma )\) is a quasicircle in \(\mathbb {C}\). By a quasicircle in \(\mathbb {C}\) we mean the image of the circle \(\mathbb {S}^1\) under a quasiconformal mapping of the plane. In particular a quasicircle is a stripcutting Jordan curve. A closed analytic curve is stripcutting by definition.
If R is a Riemann surface and \(\Sigma \subset R\) is a proper open connected subset of R which is itself a Riemann surface, in such a way that the inclusion map is holomorphic, then we say that g(w, z) is the Green’s function for \(\Sigma \) if \(g(w,\cdot )\) is harmonic on \(R \backslash \{w\}\), \(g(w,z) + \log {\phi (z)\phi (w)}\) is harmonic in z for a local parameter \(\phi :U \rightarrow \mathbb {C}\) in an open neighbourhood U of w, and \(\lim _{z \rightarrow z_0} g(w,z) =0\) for all \(z_0 \in \partial \Sigma \) and \(w \in \Sigma \). 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 \(\Sigma \) reduces to a point, so Green’s function of \(\Sigma \) exists; see for example Ahlfors and Sario [1, II.3 11H, III.1 4D].
We call this the canonical collar chart with respect to \((\Sigma ,q)\). It is uniquely determined up to a rotation and the choice of r in the definition of domain.
We say that a closed set \(I \subseteq \Gamma \) is null with respect to \((\Sigma ,q)\) if \(\phi (I)\) has logarithmic capacity zero in \(\mathbb {S}^1\). 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
 (1)
I is null with respect to \((\Sigma _1,q)\) for some \(q \in \Sigma _1\) if and only if it is null with respect to \((\Sigma _1,q)\) for all \(q \in \Sigma _1\).
 (2)
If \(\Gamma \) is a quasicircle, then I is null with respect to \((\Sigma _1,q)\) for some \(q \in \Sigma _1\) if and only if I is null with respect to \((\Sigma _2,p)\) for all \(p \in \Sigma _2\).
Thus for quasicircles we can say “I is null in \(\Gamma \)” without ambiguity. For stripcutting Jordan curves, we may say that “I is null in \(\Gamma \) with respect to \(\Sigma \)” without ambiguity.
Definition 2.3
Given a function f on an open neighbourhood of \(\Gamma \) in the closure of \(\Sigma \), we say that the limit of f exists conformally nontangentially at \(p \in \Gamma \) with respect to \((\Sigma ,q)\) if \(f\circ \phi ^{1}\) has nontangential limits at \(\phi (p)\) where \(\phi \) is the canonical collar chart induced by Green’s function \(g_q\) of \(\Sigma \). The conformal nontangential limit of f at p is defined to be the nontangential limit of \(f \circ \phi ^{1}\).
We will abbreviate “conformally nontangential” as \(\mathrm {CNT}\) throughout the paper.
Theorem 2.4
Let R be a compact Riemann surface and let \(\Gamma \) be a stripcutting Jordan curve separating R into two connected components. Let \(\Sigma \) be one of these components. For any \(H \in \mathcal {D}_{\mathrm{harm}}(\Sigma )\), the \(\mathrm {CNT}\) limit of H exists at every point in \(\Gamma \) except possibly on a null set with respect to \(\Sigma \). For any q and \(q'\) in \(\Sigma \), the boundary values so obtained agree except on a null set I in \(\Gamma \). If \(H_1,H_2 \in \mathcal {D}(\Sigma )\) have the same \(\mathrm {CNT}\) boundary values except on a null set then \(H_1 = H_2\).
From now on, the terms “\(\mathrm {CNT}\) boundary values” and “boundary values” of a Dirichletbounded harmonic function refer to the \(\mathrm {CNT}\) limits thus defined except possibly on a null set. Also, if \(\Gamma \) is a quasicircle, we say that two functions \(h_1\) and \(h_2\) agree on \(\Gamma \) (\(h_1=h_2\)) 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 Dirichletbounded harmonic functions in a certain sense determined only by a neighbourhood of the boundary. For quasicircles, it is sideindependent: that is, the set of boundary values of the Dirichlet spaces of \(\Sigma _1\) and \(\Sigma _2\) agree.
To make the first statement precise we define a kind of onesided neighbourhood of \(\Gamma \) which we call a collar neighourhood. Let \(\Gamma \) be a stripcutting Jordan curve in a Riemann surface R. By a collar neighbourhood of \(\Gamma \) we mean an open set A, bounded by two Jordan curves one of which is \(\Gamma \), and such that (1) the other Jordan curve \(\Gamma '\) is homotopic to \(\Gamma \) in the closure of A and (2) \(\Gamma ' \cap \Gamma \) is empty. For example, if U is a doublyconnected neighbourhood of \(\Gamma \), and \(\Gamma \) separates a compact Riemann surface R into two connected components, the intersection of U with one of 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
 (1)
There is some \(H \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) whose \(\mathrm {CNT}\) boundary values agree with h except possibly on a null set.
 (2)
There is a collar neighbourhood A of \(\Gamma \) in \(\Sigma _1\), one of whose boundary components is \(\Gamma \), and some \(H \in \mathcal {D}(A)\) whose \(\mathrm {CNT}\) boundary values agree with h except possibly on a null set with respect to \(\Sigma _1\).
 (3)
There is some \(H \in \mathcal {D}_{\mathrm{harm}}(\Sigma _2)\) whose \(\mathrm {CNT}\) boundary values agree with h except possibly on a null set.
 (4)
There is a collar neighbourhood A of \(\Gamma \) in \(\Sigma _2\), one of whose boundary components is \(\Gamma \), and some \(H \in \mathcal {D}(A)\) whose \(\mathrm {CNT}\) boundary values agree with h except possibly on a null set.
Thus, for a quasicircle \(\Gamma \) we may define \(\mathcal {H}(\Gamma )\) to be the set of equivalence classes of functions \(h:\Gamma \rightarrow \mathbb {C}\) which are boundary values of elements of \(\mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) 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 \(\mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) to \(\mathcal {D}_{\mathrm{harm}}(\Sigma _2)\) as follows:
Definition 2.6
Let \(\Gamma \) be a quasicircle in a compact Riemann surface R, separating it into two connected components \(\Sigma _1\) and \(\Sigma _2\). Given \(H \in \mathcal {D}(\Sigma _1)\), let h be the \(\mathrm {CNT}\) boundary values of H on \(\Gamma \). Define \(\mathfrak {O}(\Sigma _1,\Sigma _2) H\)^{1} to be the unique element of \(\mathcal {D}_{\mathrm{harm}}(\Sigma _2)\) with boundary values equal to h.
This operator enables one to transmit harmonic functions from one side of the Riemann surface to the other side through the quasicircle \(\Gamma \).
Theorem 2.7
3 Schiffer’s comparison operators
3.1 Assumptions

R is a compact Riemann surface;

\(\Gamma \) is a stripcutting Jordanlike curve separating R;

\(\Sigma _1\) and \(\Sigma _2\) are the connected components of \(R \backslash \Gamma \);

\(\Sigma \) stands for an unspecified component \(\Sigma _1\) or \(\Sigma _2\);

\(\Gamma \) is positively oriented with respect to \(\Sigma _1\);

\(\Gamma ^{p_k}_\epsilon \) the level curves of Green’s function \(g_{\Sigma _k}(\cdot ,p_k)\) with respect to some fixed points \(p_k \in \Sigma _k\);

when an integrand depends on two variables, we will use the notation \(\iint _{\Sigma ,w}\) to specify that the integration takes place over the variable w.
3.2 Schiffer’s comparison operators: definitions
 (1)
g is harmonic in w on \(R \backslash \{z,q\}\);
 (2)
for a local coordinate \(\phi \) on an open set U containing z, \(g(w,w_0;z,q) + \log  \phi (w) \phi (z) \) is harmonic for \(w \in U\);
 (3)
for a local coordinate \(\phi \) on an open set U containing q, \(g(w,w_0;z,q)  \log  \phi (w) \phi (z) \) is harmonic for \(w \in U\);
 (4)
\(g(w_0,w_0;z,q)=0\) for all \(z,q,w_0\).
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 \(\partial _w g\) 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
We will also need the following wellknown reproducing formula for Green’s function of \(\Sigma \).
Theorem 3.2
Proposition 3.3
 (1)
\(L_R\) and \(K_R\) are independent of q and \(w_0\).
 (2)
\(K_R\) is holomorphic in z for fixed w, and antiholomorphic in w for fixed z.
 (3)
\(L_R\) is holomorphic in w and z, except for a pole of order two when \(w=z\).
 (4)
\(L_R(z,w)=L_R(w,z)\).
 (5)
\(K_R(w,z)=  \overline{K_R(z,w)}\).
Remark 3.4
The symmetry statements (4) and (5) are formally expressed as follows. If \(D:R \times R \rightarrow R \times R\) is the map \(D(z,w)=(w,z)\) then \(D^*L =L \circ D\) and \(D^*K =\overline{K \circ D}\).
Proof
Since g is harmonic in w, \(\partial _{w} \overline{\partial }_{{w}} g(w,w_0;z,q) =0\) so \(K_R\) is antiholomorphic 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 Eq. (3.3) applied directly to the definitions of \(L_R\) and \(K_R\).
The noncompact case follows similarly from the harmonicity with logarithmic singularity of \(g_\Sigma \), and the symmetry \(g_\Sigma (z,w) = g_\Sigma (w,z)\)\(\square \)
Remark 3.5
Now let R be a compact Riemann surface and let \(\Gamma \) be a stripcutting Jordan curve. Assume that \(\Gamma \) separates R into two surfaces \(\Sigma _1\) and \(\Sigma _2\). We will mostly be concerned with the case that \(\Gamma \) is a quasicircle.
Definition 3.6
We will also call these Schiffer comparison operators.
Note that the operator S is bounded and the image is clearly in A(R). This can be seen from the fact that the kernel form is holomorphic in w and R is compact. On the other hand, for \(j \ne k\), the integral kernel of the operator \(T(\Sigma _j,\Sigma _k)\) is nonsingular, but if \(j =k\), then the kernel has a pole of order 2 when \(z=w\); thus the output of the operator \(T(\Sigma _j)\) need not have a holomorphic continuation across \(\Gamma \). In fact, the jump formula will show that it does not. We will show below that the image of \(T(\Sigma _j,\Sigma _k)\) is in fact in \(A(\Sigma _k)\), as the notation indicates.
Example 3.7
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.
Theorem 3.8
Proof
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 \(\Sigma \) is a subset of its double and \(\Gamma \) is analytic. By [2, Proposition 2.2], the set of holomorphic oneforms on an open neighbourhood of the closure of \(\Sigma \) is dense in \(A(\Sigma )\). The \(L^2\) boundedness of the \(L_\Sigma \) operator yields the desired result. \(\square \)
Remark 3.9
The above expression shows that the operator \(T(\Sigma ,\Sigma )\) is welldefined. 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 \(\Sigma \) might in principle lead to different values of the integral.
However the proof of Theorem 3.8 shows that the integral of \(L_\Sigma \) 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 3.8 shows that the integral in the definition of \(T(\Sigma ,\Sigma )\) is independent of the choice of exhaustion. One may also obtain this fact from the general theory of Calderón–Zygmund operators on manifolds, see [21].
Theorem 3.10
Let R be a compact Riemann surface, and \(\Gamma \) be a stripcutting Jordan curve in R. Assume that \(\Gamma \) separates R into two surfaces \(\Sigma _1\) and \(\Sigma _2\). Then \(T(\Sigma _j) \overline{\alpha } \in A(\Sigma _1 \cup \Sigma _2)\) for all \(\alpha \in A(\Sigma _j)\) for \(j=1,2\). Furthermore for all \(j,k \in \{1,2\}\), \(T(\Sigma _j)\) and \(T(\Sigma _j,\Sigma _k)\) are bounded operators.
Proof
Regarding the boundedness, the operator \(T(\Sigma _j)\) is defined by integration against the LKernel which in local coordinates is given by \(\frac{1}{\pi (\zeta \eta )^2}\), 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 Seeley in [21] to conclude that, the operators with kernels such as \(L_R(z,w)\) are bounded on \(L^p\) for \(1<p<\infty \). The boundedness of \(T(\Sigma _j, \Sigma _k)\) follows from this and the fact that \(R_0(\Sigma _j)\) is also bounded. \(\square \)
3.3 Attributions
The comparison operators \(T(\Sigma _j,\Sigma _k)\) were studied extensively by Schiffer [13, 14, 15], 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 [16]. See also our review paper [19].
In this section, we demonstrate some necessary identities for the Schiffer operators. Most of the identities were stated by for example Bergman and Schiffer [3], Schiffer [4], and Schiffer and Spencer [16] for the case of analytic boundaries. Versions can be found in different settings, for example multiplyconnected domains in the sphere, nested multiplyconnected 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 Theorems 3.11, 3.12, and 3.13. 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.
We shall refer to the restriction of the Beurling transform to antiholomorphic functions on fixed domain as a Schiffer operator. Here, of course, we express this equivalently as an operator on antiholomorphic oneforms.
3.4 Identities for comparison operators
Theorem 3.11
Let R be a compact surface and let \(\Gamma \) be a stripcutting Jordan curve separating R into two components, one of which is \(\Sigma \). Then \(S(\Sigma )= \mathrm {Res}(\Sigma )^*\), where \(^*\) denotes the adjoint operator.
Proof
Theorem 3.12
Proof
If \(j =k\), the claim follows from the nonsingular integral representation (3.7) and interchanging the order of integration.
We also have the following identity.
Theorem 3.13
Proof
Remark 3.14
Theorem 3.13 (in various settings) appears only as a norm equality in the literature.
4 Jump formula on quasicircles and related isomorphisms
4.1 The limiting integral in the jump formula
In this section, we show that the jump formula holds when \(\Gamma \) is a quasicircle. We also prove that in this case the Schiffer operator \(T(\Sigma _1,\Sigma _2)\) is an isomorphism, when restricted to a certain subclass of \(\overline{A(\Sigma _1)}\).
To establish a jump formula, we would like to define a Cauchytype integral for elements \(h \in \mathcal {H}(\Gamma )\). Since \(\Gamma \) is not necessarily rectifiable, instead we replace the integral over \(\Gamma \) with an integral over approximating curves \(\Gamma ^{p_1}_\epsilon \) (defined at the beginning of Sect. 3), and use the harmonic extensions \(\tilde{h} \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) of elements of \(\mathcal {H}(\Gamma )\).
It is an arbitrary choice whether to approximate the curve from within \(\Sigma _1\) or from within \(\Sigma _2\). Later, we will show that the result is the same in the case that \(\Gamma \) is a quasicircle. For now, we have chosen to approximate the curves from within \(\Sigma _1\).
This also shows that
Lemma 4.1
For stripcutting Jordan curves \(\Gamma \), the limit (4.1) exists and is independent of the choice of \(p_1\).
Therefore, in the following we will usually omit mention of the point \(p_1\) in defining the level curves, and write simply \(\Gamma _\epsilon \).
Theorem 4.2
Proof
Assume first that \(q \in \Sigma _2\). The first claim follows from (4.2) and the fact that the integrand is nonsingular. Similarly for \(z \in \Sigma _2\), the third claim follows from (4.2).
Below, let \(\overline{A(R)}^\perp \) denote the orthogonal complement in \(A_{\mathrm{harm}}(\Sigma _1)\) of the restrictions of \(\overline{A(R)}\) to \(\Sigma _1\).
Corollary 4.3
 (1)
\(J_q(\Gamma )\) is a bounded operator from \(\mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) to \(\mathcal {D}_{\mathrm{harm}}(\Sigma _1 \cup \Sigma _2)\).
 (2)
If \(\overline{\partial } h \in \overline{A(R)}^\perp \) then \(J_q(\Gamma ) h \in \mathcal {D}(\Sigma _1 \cup \Sigma _2)\).
4.2 Density theorems
In this section we show that certain subsets of the Dirichlet space are dense.
Our first density result parallels a general theorem of Askaripour and Barron [2], which asserts that \(L^2\) holomorphic oneforms (in fact, more generally differentials) on a region in a Riemann surface can be approximated by holomorphic oneforms on a larger domain. We need a result of this type for the Dirichlet space, for doublyconnected regions.
Theorem 4.4
Let R be a compact Riemann surface and \(\Gamma \) be a stripcutting Jordan curve. Let U be a doublyconnected neighbourhood of \(\Gamma \). Let \(A_i = U \cap \Sigma _i\) for \(i=1,2\), and let \(\mathrm {Res}_i:\mathcal {D}(U) \rightarrow \mathcal {D}(A_i)\) denote restriction for \(i=1,2\). Then \(\mathrm {Res}_i \mathcal {D}(U)\) is dense in \(\mathcal {D}(A_i)\) for \(i=1,2\).
Proof
We now show that they are in \(\mathcal {D}(B_k)\) for \(k=1,2\). Let \(C \subseteq B_1\) be a collar neighbourhood of \(\Gamma _1\), and let \(D \subset B_1\) be an open set whose closure is in \(B_1\), which furthermore contains the closure of \(B_1 \backslash C\). Since \(C \subset A'\), we have that \(h \in \mathcal {D}(C)\). Since the closure of C is contained in \(B_2\), we see that \(h_2 \in \mathcal {D}(C)\). Thus using \(h_1 = h  h_2\) we see that \(h_1 \in \mathcal {D}(C)\). Now since the closure of D is contained in \(B_1\), \(h_1 \in \mathcal {D}(D)\). This proves that \(h_1 \in \mathcal {D}(B_1)\). The proof that \(h_2 \in \mathcal {D}(B_2)\) is obtained by interchanging the indices 1 and 2 above.
Next we claim that the linear space \(\mathbb {C}[z,z^{1}]\) of polynomials in z and \(z^{1}\) is dense in \(\mathcal {D}(A')\). To see this, assume for definiteness that \(B_1\) is the bounded domain and \(B_2\) is the unbounded domain. Since polynomials in z are dense in \(\mathcal {D}(B_1)\) and polynomials in \(z^{1}\) are dense in \(\mathcal {D}(B_2)\), this proves the claim.
Returning to the statement of the theorem, observe that we can assume that U is an annulus \(\mathbb {A} = \{ z : r< z < 1/r \}\). This is because we can map U conformally onto \(\mathbb {A}\), and every space in the statement of the theorem is conformally invariant. But since \(\mathbb {C}[z,z^{1}]\) is dense in both \(\mathcal {D}(A_1)\) and \(\mathcal {D}(A_2)\), and \(\mathbb {C}[z,z^{1}] \subset \mathcal {D}(U)\), this completes the proof. \(\square \)
Theorem 4.5
[20] Let \(\Gamma \) be a stripcutting Jordan curve in a compact Riemann surface R. Assume that \(\Gamma \) separates R into two components, one of which is \(\Sigma \). Let A be a collar neighbourhood of \(\Gamma \) in \(\Sigma \). Then the associated map \(\mathfrak {G}:\mathcal {D}_{\mathrm{harm}}(A) \rightarrow \mathcal {D}_{\mathrm{harm}}(\Sigma )\) is bounded.
Theorem 4.6
Let \(\Gamma \), R, A and \(\Sigma \) be as above. The image of \(\mathcal {D}(A)\) under \(\mathfrak {G}\) is dense in \(\mathcal {D}_{\mathrm{harm}}(\Sigma _1)\).
Proof
4.3 Limiting integrals from two sides
In this section, we show that for quasicircles, the limiting integral defining \(J_q(\Gamma )\) can be taken from either side of \(\Gamma \), 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 \(J_q(\Gamma )\) is easier to work with when restricting to \(\mathcal {D}(A)\). To make use of this simplification, we must first show that the limiting integrals of \(\mathfrak {G} h\) and h are equal.
We first require a more general theorem, which shows that the limiting integral is the same for any functions with the same CNT boundary values.
Theorem 4.7
Proof
We then have the following immediate consequence.
Theorem 4.8
The following special case will allow us to make convenient use of the density of \(\mathfrak {G} \mathcal {D}(B)\) in \(\mathcal {D}_{\mathrm{harm}}(B)\), as was mentioned above.
Theorem 4.9
Proof
Theorem 4.10
Proof
Let U be a doublyconnected neighbourhood of \(\Gamma \), bounded by \(\Gamma _i \subset \Sigma _i\). Let \(A_i = U \cap \Sigma _i\). Then \(A_i\) are collar neighbourhoods of \(\Gamma \) in \(\Sigma _i\). Let \(\mathfrak {G}_i: \mathcal {D}(A_i) \rightarrow \mathcal {D}_{\mathrm{harm}}(\Sigma _i)\) be induced by these collar neighbourhoods for \(i=1,2\).
Thus one may think of \(J(\Gamma )\) as an operator on \(\mathcal {H}(\Gamma )\).
In the rest of the paper, we return to the convention that \(J_q(\Gamma )\) is an operator on \(\mathcal {D}_{\mathrm{harm}}(\Sigma _1)\). However, Theorem 4.10 plays an important role in the proof that \(T(\Sigma _1,\Sigma _2)\) is surjective.
Also, by using Theorem 4.8 and proceeding exactly as in the proof of Theorem 4.10 we obtain
Theorem 4.11
4.4 A transmission formula
In this section we prove an explicit formula for the transmission operator \(\mathfrak {O}\) on the image of the jump operator.
Definition 4.12
Theorem 4.13
To prove this theorem we need a lemma.
Lemma 4.14
Proof
We continue with the proof of Theorem 4.13.
Proof
Let E be the linear subspace of \(\mathcal {D}(A)\) consisting of those elements of \(\mathcal {D}(A)\) for which (4.10) is satisfied. It is enough to show that \(\mathfrak {G} E\) is dense in \(W_1\).
Fix a basis \(\alpha _1,\ldots \alpha _g\) for A(R). Let \(\mathcal {P}:\mathcal {D}(A) \rightarrow E\) denote the orthogonal projection in \(\mathcal {D}(A)\).
We also define a transmission operator for exact oneforms as follows. For spaces \(A(\Sigma )\), \(A_{\mathrm{harm}}(\Sigma )\), etc., denote the subset of exact oneforms with a subscript e, i.e. \(A_e(\Sigma )\), \(A_{\mathrm{harm}}(\Sigma )_e\), etc.
Definition 4.15
To prove the transmission formula for \(\mathfrak {O}_e\), we require the following elementary lemma.
Lemma 4.16
Let \(\Sigma \) be a Riemann surface of finite genus g bordered by a curve homeomorphic to a circle. Let \(\overline{\alpha } \in \overline{A(\Sigma )}\). There is an \(h \in \mathcal {D}_{\mathrm{harm}}(\Sigma )\) such that \(\overline{\partial } h = \overline{\alpha }\). If \(\tilde{h} \in \mathcal {D}_{\mathrm{harm}}(\Sigma )\) is any other such function, then \(\tilde{h}  h \in \mathcal {D}(\Sigma )\).
Proof
If \(\tilde{h}\) is any other such function then \(\overline{\partial } (\tilde{h}  h) =0\), which completes the proof. \(\square \)
Recall that \(\overline{A(R)}^\perp \) denotes the set of elements in \(A_{\mathrm{harm}}(\Sigma )\) which are orthogonal, with respect to \((\cdot ,\cdot )_{A_{\mathrm{harm}}}(\Sigma )\), to the restrictions to \(\Sigma \) of elements of \(\overline{A(R)}\).
Definition 4.17
Theorem 4.18
Proof
Let \(\overline{\alpha } \in V_1\), then by Lemma 4.16 there is an \(h \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) such that \(\overline{\partial } h = \overline{\alpha }\).
Since \(\overline{\partial } h = \overline{\alpha } \in \overline{A(R)}^\perp \), \(\overline{S}(\Sigma _1) \overline{\partial } h = 0\), so by Theorem 4.2\(\overline{\partial } J(\Gamma )h =0\).
Corollary 4.19
Let R be a compact Riemann surface and \(\Gamma \) be a quasicircle separating R into components \(\Sigma _1\) and \(\Sigma _2\). Then \(\overline{P}(\Sigma _1) \mathfrak {O}(\Sigma _2,\Sigma _1)\) is a left inverse of \(\left. T(\Sigma _1,\Sigma _2) \right _{V_1}\). In particular, the restriction of \(T(\Sigma _1,\Sigma _2)\) to \(V_1\) is injective.
Proof
This follows immediately from Theorem 4.18 and the fact that for \(\overline{\alpha } \in V_1\), \(T(\Sigma _1,\Sigma _1) \overline{\alpha }\) and \(T(\Sigma _1,\Sigma _2) \overline{\alpha }\) are holomorphic. \(\square \)
As another consequence of Theorem 4.18 we are able to prove an inequality analogous to the strengthened Grunsky inequality for quasicircles [10].
Theorem 4.20
Let R be a compact Riemann surface and \(\Gamma \) be a quasicircle separating R into disjoint components \(\Sigma _1\) and \(\Sigma _2\). Then \(\Vert \left. T(\Sigma _1,\Sigma _1) \right _{V_1} \Vert <1\).
Proof
In the following computation, we need two observations. First, if a function H is holomorphic on a domain \(\Omega \), then \(\Vert H \Vert ^2_{\mathcal {D}_{\mathrm{harm}}}(\Omega ) = 2\Vert \text {Re} (H) \Vert ^2_{\mathcal {D}(\Omega )}\). Second, if \(H_2\) is a primitive of \(T(\Sigma _1,\Sigma _2) \overline{\alpha }\) and if we let \(H_1 = \mathfrak {O}(\Sigma _2,\Sigma _1) H_2\) (so that \(H_1\) is a primitive of \(\overline{\alpha } + T(\Sigma _1,\Sigma _1) \overline{\alpha }\) by definition), then we observe that \(\mathfrak {O}(\Sigma _2,\Sigma _1) \text {Re}(H_2) = \text {Re}(H_1)\), and therefore the boundedness of transmission estimate applies to \(\text {Re}(H_i)\).
Remark 4.21
4.5 Isomorphism theorem for the Schiffer operator
In this section, we prove the isomorphism theorem for the Schiffer operators. Theorem 4.22 shows that \(T(\Sigma _1,\Sigma _2)\) is an isomorphism between \(V_1 \subset \overline{A(\Sigma _1)}\) and the space \(A(\Sigma _2)_e\) of exact oneforms on \(\Sigma _2\), thus generalizing Napalkov and Yulmukhametov’s theorem to compact Riemann surfaces. In Proposition 4.24 we establish that for harmonic Dirichlet space functions h on \(\Sigma _1\) such that \(\overline{\partial } h \in V_1\), \({\partial } h + T(\Sigma _1,\Sigma _1) \overline{\partial } h\) is exact. These two facts, combined with the identities of Theorem 4.2, allow us to give, in Theorem 4.25, an isomorphism between \(V_1' \subset A_{\mathrm{harm}}(\Sigma _1)_e\) and \(A(\Sigma _1)_e \oplus A(\Sigma _2)_e\). This last theorem is the “derivative” of the Plemelj–Sokhotski isomorphism, which will be given in the final section of the paper.
Theorem 4.22
Let \(\Gamma \) be a quasicircle. Then the restriction of \(T(\Sigma _1,\Sigma _2)\) to \(V_1\) is an isomorphism onto \(A(\Sigma _2)_e\).
Proof
Injectivity of \(T(\Sigma _1,\Sigma _2)\) is Corollary 4.19.
Remark 4.23
Although we have only proven that \(T(\Sigma _1,\Sigma _2)\) is injective for quasicircles, we conjecture that this is true in greater generality, as in Napalkov and Yulmukhametov [8] 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 3.12 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.
Proposition 4.24
Proof
The following theorem is in some sense a derivative of the jump decomposition.
Theorem 4.25
Proof
First we show surjectivity. Let \((\alpha ,\beta ) \in A(\Sigma _1)_e \oplus A(\Sigma _2)_e\). By Theorem 4.22, \(T(\Sigma _1,\Sigma _2)\) is surjective so there is a \(\overline{\delta } \in V_1\) such that \(T(\Sigma _1,\Sigma _2) \overline{\delta } =\beta \). By Lemma 4.16 there is a \(\tilde{h} \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)\) such that \(\overline{\partial } \tilde{h} = \overline{\delta }\).
Now assume that \(\hat{\mathfrak {H}}(dh) = 0\). The vanishing of the second component yields that \(T(\Sigma _1,\Sigma _2) \overline{\partial } h =0\), so by Theorem 4.22 we have that \(\overline{\partial } h =0\). Thus the vanishing of the first component of \(\hat{\mathfrak {H}}(dh)\) yields that \(\partial h =0\), hence \(dh = 0\). \(\square \)
4.6 The jump isomorphism
In this section we establish the existence of a jump decomposition for functions in \(\mathcal {H}(\Gamma )\). The first theorem phrases the decomposition in terms of an isomorphism, which we call the Plemelj–Sokhstki isomorphism.
Theorem 4.26
Proof
By Theorem 4.2\(\partial {\mathfrak {H}} h = \hat{\mathfrak {H}}\, dh\), so since \(\hat{\mathfrak {H}}\) is an isomorphism by Theorem 4.25, we only need to deal with constants. If \(J_q(\Gamma ) h =0\) then \(dh=0\) so h is constant on \(\Sigma _1\). Since the second component of \(\mathfrak {H} h\) vanishes at q we see that \(h=0\), so \({\mathfrak {H}}\) is injective. Now observe that \({\mathfrak {H}}(h+c) = {\mathfrak {H}}h +(c,0)\) for any constant c. Thus surjectivity follows from surjectivity of \(\hat{\mathfrak {H}}\). \(\square \)
Proposition 4.27
Let R be a compact Riemann surface, and let \(\Gamma \) be a quasicircle separating R into components \(\Sigma _1\) and \(\Sigma _2\). Assume that \(\Gamma \) is positively oriented with respect to \(\Sigma _1\). For \(q \in \Sigma _2\), let \(J_q(\Gamma )\) be defined using limiting integrals from within \(\Sigma _1\). If \(h \in \mathcal {D}(\Sigma _1)\) then \(J_q(\Gamma ) h = (h,0)\), and if \(h \in \mathcal {D}_q(\Sigma _2)\) then \(J_q(\Gamma ) \mathfrak {O}(\Sigma _2,\Sigma _1) h = (0,h)\).
Proof
The first claim follows immediately from Theorem 3.2. The second claim follows from Theorems 3.2 and 4.10 (note that \(\Gamma \) is negatively oriented with respect to \(\Sigma _2\)). \(\square \)
We then have a version of the Plemelj–Sokhotski jump formula.
Corollary 4.28
Proof
Finally, we show that the condition for existence of a jump formula is independent of the choice of side of \(\Gamma \).
Theorem 4.29
Proof
Footnotes
 1.
The notation \(\mathfrak {O}\) for this transmission operator stems from the first letter in the Old English word “oferferian” which means “to transmit” (or “to overfare”).
Notes
Acknowledgements
Open access funding provided by Uppsala University. The authors are grateful to the referee, whose insightful comments have improved the overall presentation of the paper.
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