Plemelj–Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

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 $L^2$ anti-holomorphic one-forms on $\Sigma _1$ to the space of $L^2$ holomorphic one-forms 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 anti-holomorphic one-forms which are orthogonal to $L^2$ anti-holomorphic one-forms 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 one-forms on $\Sigma _2$. Using the relation between this 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 $\Gamma$ be a sufficiently regular curve separating a compact surface into two components $\Sigma _1$ and $\Sigma _2$. Given a sufficiently regular function h on that curve, it is well known that there are holomorphic functions $h_k$ on $\Sigma _k$, for $k=1,2$, such that

$$\begin{aligned} h = h_2 - h_1 \end{aligned}$$

if and only if $\int _\Gamma h \alpha =0$ 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 $\Sigma _1$ 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 non-rectifiable, 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 one-forms 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 Cauchy-type integral in higher genus (Theorem 4.2). The derivative of the Cauchy-type integral, when restricted to a finite co-dimension space of one-forms, 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 co-dimensional space is an isomorphism (Theorem 4.22).

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 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 non-rectifiable 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 doubly-connected regions in a Riemann surface. This is similar to a result of Askaripour and Barron [2] for $L^2$ k-differentials 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 Cauchy-type integral (in general genus), we relate it to the jump decomposition, and establish the isomorphism theorems for the Schiffer operator and the Cauchy-type 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.

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

$$\begin{aligned} *(a\, dx + b \, dy) = a \,dy - b \,dx \end{aligned}$$

in a local holomorphic coordinate $z=x+iy$. This is independent of the choice of coordinates. Harmonic functions f on R are those $C^2$ functions which satisfy $d *d f =0$, while harmonic one-forms $\alpha$ are those $C^1$ one-forms which satisfy both $d\alpha =0$ and $d *\alpha =0$. 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. $\overline{\alpha }$.

Harmonic one-forms $\alpha$ can always be decomposed as a sum of a holomorphic and anti-holomorphic 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

$$\begin{aligned} (\omega _1,\omega _2) = \frac{1}{2} \iint _R \omega _1 \wedge *\overline{\omega _2}; \end{aligned}$$

(2.1)

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

$$\begin{aligned} A(R) = \{ \alpha \in L^2(R) \,:\, \alpha \ \text {holomorphic} \} \end{aligned}$$

and the set of antiholomorphic $L^2$ one-forms will be denoted by $\overline{A(R)}$. This notation is of course consistent, because $\beta \in \overline{A(R)}$ if and only if $\beta = \overline{\alpha }$ for some $\alpha \in A(R)$. We will also denote

$$\begin{aligned} A_{\mathrm{harm}}(R) =\{ \alpha \in L^2(R) \,:\, \alpha \ \text {harmonic} \}. \end{aligned}$$

If $\alpha , \beta \in A(R)$ then $*\overline{\beta } = i \beta$, from which we see that

$$\begin{aligned} (\alpha ,\beta ) = \frac{i}{2} \iint _R \alpha \wedge \overline{\beta }. \end{aligned}$$

Observe that A(R) and $\overline{A(R)}$ are orthogonal with respect to the aforementioned inner product.

If $F: R_1 \rightarrow R_2$ is a conformal map, then we denote the pull-back of $\alpha \in A_{\mathrm{harm}}(R_2)$ under F by $F^*\alpha .$

We also define the Dirichlet spaces by

$$\begin{aligned} \mathcal {D}_{\mathrm{harm}} (R)&= \{ f:R \rightarrow \mathbb {C}, f \in C^2(R), \,:\, df\in L^2 (R)\quad \mathrm {and}\quad d*df =0 \},\\ \mathcal {D}(R)&= \{ f:R \rightarrow \mathbb {C} \,:\, df \in A(R) \}, \ \text {and} \\ \overline{\mathcal {D}(R)}&= \{ f:R \rightarrow \mathbb {C} \,:\, df \in \overline{A(R)} \}. \\ \end{aligned}$$

We can define a degenerate inner product on $\mathcal {D}_{\mathrm{harm}}(R)$ by

$$\begin{aligned} (f,g)_{\mathcal {D}_{\mathrm{harm}}(R)} = (df,dg)_{A_{\mathrm{harm}}(R)} \end{aligned}$$

where the right hand side is the inner product (2.1) restricted to elements of $A_{\mathrm{harm}}(R)$.

If we denote

$$\begin{aligned} \mathcal {D}_{\mathrm{harm}}(R)_q = \{ f \in \mathcal {D}_{\mathrm{harm}}(R) \, :\, f(q)=0 \} \end{aligned}$$

for some $q \in R$, then the scalar product defined above is a genuine inner product on $\mathcal {D}_{\mathrm{harm}}(R)_{q}$ 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

$$\begin{aligned} \partial f = \frac{\partial f}{\partial z}\, dz, \ \ \ \overline{\partial } f = \frac{\partial f}{\partial \bar{z}}\, d \bar{z}, \end{aligned}$$

then the aforementioned inner product can be written as

$$\begin{aligned} (f,g)_{\mathcal {D}_{\mathrm{harm}}(R)} = \frac{i}{2} \iint _{R} \left[ \partial f \wedge \overline{\partial g} - \overline{\partial } f \wedge \partial \overline{g} \right] . \end{aligned}$$

(2.2)

One can easily see from (2.2) that $\mathcal {D}(R)$ and $\overline{\mathcal {D}(R)}$ are orthogonal with respect to the inner product. We also note that if R is a planar domain and $f \in \mathcal {D}(R)$, then $(f,f)_{\mathcal {D}(R)} = \iint _R |f'(z)|^2 dA$ where dA denotes Lebesgue measure in the plane.

Finally, we will repeatedly use the following elementary fact.

Lemma 2.1

Let $U \subset \mathbb {C}$ be an open set. For any compact subset K of U, there is a constant $M_K$ such that

$$\begin{aligned} \sup _{z\in K}|\alpha (z)| \le M_K \Vert \alpha (z)\, dz \Vert _{A_{\mathrm{harm}}(U)} \end{aligned}$$

for all $\alpha (z)\,dz \in A_{\mathrm{harm}}(U)$.

For any Riemann surface R, compact subset K of R, and fixed $q \in R$, there is a constant $M_K$ such that

$$\begin{aligned} \sup _{z\in K} |h(z)| \le M_K \Vert h \Vert _{\mathcal {D}_{\mathrm{harm}}(R)_q} \end{aligned}$$

for all $h \in \mathcal {D}_{\mathrm{harm}}(R)_q$.

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 doubly-connected neighbourhood of $\Gamma$ if U is an open set containing $\Gamma$, which is bounded by two non-intersecting Jordan curves each of which is homotopic to $\Gamma$ within the closure of U. We say that a Jordan curve $\Gamma$ is strip-cutting if there is a doubly-connected 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 strip-cutting Jordan curve. A closed analytic curve is strip-cutting 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].

Now let $\Sigma$ be one of the connected components in R of the complement of $\Gamma$. Fix a point $q \in \Sigma$ and let $g_q$ be Green’s function of $\Sigma$ with singularity at q. We associate to $g_q$ a biholomorphism from a doubly-connected region in $\Sigma$, one of whose borders is $\Gamma$, onto an annulus as follows. Let $\gamma$ be a smooth curve in $\Sigma$ which is homotopic to $\Gamma$, and let $m = \int _\gamma *dg_q$. If $\tilde{g}$ denotes the multi-valued harmonic conjugate of $g_q$, then the function

$$\begin{aligned} \phi = \exp {[-2 \pi (g_q + i \tilde{g})/m]} \end{aligned}$$

is holomorphic and single-valued on some region $A_r$ bounded by $\Gamma$ and a level curve $\Gamma ^{q}_r = \{ z\,:\, g_q(z) = r \}$ of $g_q$ for some $r>0$. A standard use of the argument principle shows that $\phi$ is one-to-one and onto the annulus $\{ z : e^{-2\pi r/m}<|z|<1 \}$. It can be shown that $\phi$ has a continuous extension to $\Gamma$ which is a homeomorphism of $\Gamma$ onto $\mathbb {S}^1$. By decreasing r, one can also arrange that $\phi$ extends analytically to a neighbourhood of $\Gamma ^{q}_r$.

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

Let R be a compact Riemann surface and $\Gamma$ be a strip-cutting Jordan-curve separating R into two connected components $\Sigma _1$ and $\Sigma _2$. Let I be a closed set in $\Gamma$.

• (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 strip-cutting 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 non-tangentially at $p \in \Gamma$ with respect to $(\Sigma ,q)$ if $f\circ \phi ^{-1}$ has non-tangential limits at $\phi (p)$ where $\phi$ is the canonical collar chart induced by Green’s function $g_q$ of $\Sigma$. The conformal non-tangential limit of f at p is defined to be the non-tangential limit of $f \circ \phi ^{-1}$.

We will abbreviate “conformally non-tangential” as $\mathrm {CNT}$ throughout the paper.

Theorem 2.4

Let R be a compact Riemann surface and let $\Gamma$ be a strip-cutting 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 Dirichlet-bounded 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 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 $\Sigma _1$ and $\Sigma _2$ agree.

To make the first statement precise we define a kind of one-sided neighbourhood of $\Gamma$ which we call a collar neighourhood. Let $\Gamma$ be a strip-cutting 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 doubly-connected 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

Let R be a compact Riemann surface and let $\Gamma$ be a strip-cutting Jordan curve separating R into connected components $\Sigma _1$ and $\Sigma _2$. Let h be a function defined on $\Gamma$, except possibly on a null set in $\Gamma$. The following are equivalent.

• (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 $\Sigma _1$.

If $\Gamma$ is a quasicircle, then the following may be added to the list of equivalences above.

• (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

Let R be a compact Riemann surface and $\Gamma$ be a quasicircle separating R into components $\Sigma _1$ and $\Sigma _2$. The map

$$\begin{aligned} \mathfrak {O}(\Sigma _1,\Sigma _2):\mathcal {D}_{\mathrm{harm}}(\Sigma _1) \rightarrow \mathcal {D}_{\mathrm{harm}}(\Sigma _2) \end{aligned}$$

induced by Theorem 2.5 is bounded with respect to the Dirichlet semi-norm.

3 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;

• $\Gamma$ is a strip-cutting Jordan-like 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.

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 [12], 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 $R \backslash \{z,q\}$;

• (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 $z,q,w_0$.

It can be shown that g exists, is uniquely determined by these properties, and furthermore satisfies the symmetry properties

$$\begin{aligned} g(w,w_1;z,q)&= g(w,w_0;z,q) - g(w_1,w_0;z,q) \end{aligned}$$

(3.1)

$$\begin{aligned} g(w_0,w;z,q)&= - g(w,w_0;z,q) \end{aligned}$$

(3.2)

$$\begin{aligned} g(z,q;w,w_0)&= g(w,w_0;z,q). \end{aligned}$$

(3.3)

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 $\partial _w g$ 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

Let $\Gamma$ be a closed analytic curve separating R, enclosing $\Sigma$, which is positively oriented with respect to $\Sigma$. If h is holomorphic on $\Sigma$, and $z,q \notin \Gamma$, then for any fixed $p \in \Sigma$

$$\begin{aligned} - \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma _\epsilon ^p} h(w)\, \partial _w g(w,w_0;z,q) = \chi _\Sigma (z) h(z) - \chi _\Sigma (q) h(q) \end{aligned}$$

where $\chi _\Sigma$ is the characteristic function of $\Sigma$.

We will also need the following well-known reproducing formula for Green’s function of $\Sigma$.

Theorem 3.2

Let R be a compact Riemann surface and $\Gamma$ be a strip-cutting Jordan curve separating R. Let $\Sigma$ be one of the components of the complement of $\Gamma$. For any $h \in \mathcal {D}_{\mathrm{harm}}(\Sigma )$, we have

$$\begin{aligned} h(z) = \lim _{\epsilon \searrow 0} - \frac{1}{\pi i} \int _{\Gamma ^p_\epsilon } \partial _w g_{\Sigma }(w,z) \, h(w). \end{aligned}$$

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. Following [16], we define the Schiffer kernel to be the bi-differential

$$\begin{aligned} L_R(z,w) = \frac{1}{\pi i} \partial _z \partial _w g(w,w_0;z,q). \end{aligned}$$

and the Bergman kernel to be the bi-differential

$$\begin{aligned} K_R (z,w) = - \frac{1}{\pi i} \partial _z \overline{\partial }_{{w}} g(w,w_0;z,q). \end{aligned}$$

For non-compact surfaces $\Sigma$ with border, with Green’s function g, we define

$$\begin{aligned} L_\Sigma (z,w) = \frac{1}{\pi i} \partial _z \partial _w g(w,z). \end{aligned}$$

and

$$\begin{aligned} K_\Sigma (z,w) = - \frac{1}{\pi i} \partial _z \overline{\partial }_{{w}} g(w,z). \end{aligned}$$

Then the following identity holds. For any vector v tangent to $\Gamma ^{w}_{\epsilon }$ at a point z, we have

$$\begin{aligned} \overline{K_\Sigma (z,w)}(\cdot ,v) = -L_\Sigma (z,w)(\cdot ,v) \end{aligned}$$

(3.4)

This follows directly from the fact that the one form $\partial _z g (z,w) + \overline{\partial }_{{z}} g(z,w)$ vanishes on tangent vectors to the level curve $\Gamma ^{w}_{\epsilon }$.

It is well known that for all $h \in A(\Sigma )$

$$\begin{aligned} \iint _\Sigma K_\Sigma (z,w) \wedge h(w) = h(z). \end{aligned}$$

(3.5)

For compact surfaces, the reproducing property of the Bergman kernel is established in [12].

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 $w=z$.

• (4) $L_R(z,w)=L_R(w,z)$.

• (5) $K_R(w,z)= - \overline{K_R(z,w)}$.

For non-compact Riemann surfaces $\Sigma$ with Green’s function, $(2)-(5)$ hold with $L_R$ and $K_R$ replaced by $L_\Sigma$ and $K_\Sigma$.

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

It follows immediately from (3.1) that

$$\begin{aligned} \partial _w g(w,w_1;z,q) = \partial _w g(w,w_0;z,q) \ \ \text {and} \ \ \partial _{\bar{w}} g(w,w_1;z,q) = \partial _{\bar{w}} g(w,w_0;z,q), \end{aligned}$$

so $L_R$ and $K_R$ are independent of $w_0$. Applying (3.3) shows that similarly $\partial _w g$ and $\partial _{\bar{w}}$ 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, $\partial _{w} \overline{\partial }_{{w}} g(w,w_0;z,q) =0$ 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 Eq. (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 $g_\Sigma$, and the symmetry $g_\Sigma (z,w) = g_\Sigma (w,z)$ $\square$

Remark 3.5

The reader might find the negative sign in (5) surprising, since the Bergman kernel should be skew-symmetric. However this is in agreement with the usual convention when one takes into account that one usually integrates against a measure, whereas the kernel $K_R$ is a bi-differential to be integrated against one-forms. For example, if R is a region in the plane and $\alpha = h(w) dw$ is a one-form, then we have

$$\begin{aligned} \iint _R K_R(z,w) \wedge _w \alpha (w)&= - \frac{1}{\pi i} \iint _R \frac{\partial ^2 g}{\partial \bar{w} \partial z} (w,z) d\bar{w} \, dz \wedge _w h(w) dw \\&= - \frac{2}{\pi } \iint _R \frac{\partial ^2 g}{\partial \bar{w} \partial z} (w,z) h(w)\, dA_w \, dz, \end{aligned}$$

where dA is Euclidean Lebesgue measure. Observe that the kernel of the final integral is in fact skew-symmetric.

One can find the constant at the pole of L from the definition. Expressed in a local holomorphic coordinates $\eta = \phi (w)$ near a fixed point $\zeta = \phi (z)$,

$$\begin{aligned} (\phi ^{-1} \times \phi ^{-1})^* L(z,w) = \left( -\frac{1}{2\pi i} \frac{1}{(\zeta -\eta )^2} + H(\eta ) \right) d\zeta d\eta \end{aligned}$$

(3.6)

where $H(\eta )$ is holomorphic in a neighbourhood of $\zeta$. In most sources [3, 4, 8, 13] the integral kernel is expressed as a function (rather than a form) to be integrated against the Euclidean area form $dA_{\eta } = d\bar{\eta } \wedge d{\eta }/{2i}$. For example, if $\overline{\alpha (w)}$ is a holomorphic one-form given in local coordinates by $(\phi ^{-1})^{*} \alpha (\eta )= \overline{f(\eta )} d{\overline{\eta }}$ then we obtain the local expression

$$\begin{aligned} (\phi ^{-1} \times \phi ^{-1})^{*} L(z,w) \wedge _{\eta } \phi ^{-1*}\alpha (w) = \left( \frac{1}{\pi } \frac{1}{(\zeta -\eta )^2} + H(\eta ) \right) \overline{f(\eta )} \, d\zeta \, dA_{\eta } \end{aligned}$$

which agrees with the classical normalization [3].

Now let R be a compact Riemann surface and let $\Gamma$ be a strip-cutting 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.

Let $A(\Sigma _1 \cup \Sigma _2)$ denote the set of one-forms $\alpha$ on $\Sigma _1 \cup \Sigma _2$ which are holomorphic and square integrable, in the sense that their restrictions to $\Sigma _k$ is in $A(\Sigma _k)$ for $k=1,2$; that is, $\left\| \left. \alpha \right| _{\Sigma _1} \right\| ^2_{\Sigma _1} + \left\| \left. \alpha \right| _{\Sigma _2} \right\| ^2_{\Sigma _2} < \infty$. Note that we do not require the existence of a holomorphic or continuous extension to the closure of $\Sigma _1 \cup \Sigma _2$. For $k=1,2$ define the restriction operators

$$\begin{aligned} \mathrm {Res}(\Sigma _k):A(R)&\rightarrow A(\Sigma _k) \\ \alpha&\mapsto \left. \alpha \right| _{\Sigma _k} \end{aligned}$$

and

$$\begin{aligned} \text {Res}_0(\Sigma _k): A(\Sigma _1 \cup \Sigma _2)&\rightarrow A(\Sigma _k) \\ \alpha&\mapsto \left. \alpha \right| _{\Sigma _k}. \end{aligned}$$

It is obvious that these are bounded operators.

Definition 3.6

For $k=1,2$, we define the Schiffer comparison operators

$$\begin{aligned} T(\Sigma _k): \overline{A(\Sigma _k)}&\rightarrow A(\Sigma _1 \cup \Sigma _2) \\ \overline{\alpha }&\mapsto \iint _{\Sigma _k} L_R(\cdot ,w) \wedge \overline{\alpha (w)}. \end{aligned}$$

and

$$\begin{aligned} S(\Sigma _k): A(\Sigma _k)&\rightarrow A(R) \\ \alpha&\mapsto \iint _{\Sigma _k} K_R(\cdot ,w) \wedge \alpha (w). \end{aligned}$$

Also, we define for $j,k \in \{1,2\}$

$$\begin{aligned} T(\Sigma _j,\Sigma _k) = \text {Res}_0(\Sigma _k) T(\Sigma _j): \overline{A(\Sigma _j)} \rightarrow A(\Sigma _k). \end{aligned}$$

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

Let R be the Riemann sphere $\bar{\mathbb {C}}$, and let $\Gamma$ be a Jordan curve in $\mathbb {C}$ dividing $\bar{\mathbb {C}}$ into two Jordan domains $\Sigma _1$ and $\Sigma _2$; assume that $\Sigma _1$ is the bounded domain. With the normalization $w_0 = \infty$, we have

$$\begin{aligned} g(w,\infty ;z,q) = - \log { \frac{|w-z|}{|w-q|}}. \end{aligned}$$

From this, it can be calculated that

$$\begin{aligned} K_{\mathbb {C}}(z,w) =0. \end{aligned}$$

Thus, $S(\Sigma _1) = 0$, as is expected as a consequence of the non-existence of non-trivial holomorphic one-forms on $\bar{\mathbb {C}}$. We can also calculate that

$$\begin{aligned} L_{\bar{\mathbb {C}}}(z,w) = - \frac{1}{2 \pi i} \frac{dw \,dz}{(w-z)^2}. \end{aligned}$$

Thus for $\overline{\alpha (w)} = \overline{h(w)} d\bar{w} \in \overline{A(\Sigma _1)}$, we have

$$\begin{aligned} \left[ T(\Sigma _1,\Sigma _1 \cup \Sigma _2) \, \overline{\alpha } \right] (z) = \frac{1}{\pi } \iint _{\Sigma _1} \frac{\overline{h(w)}}{(w-z)^2} \frac{d\bar{w} \wedge d w}{2i} \cdot dz. \end{aligned}$$

If we choose for example $\Sigma _1 = \mathbb {D}$, we see that

$$\begin{aligned} g(z,w) = - \log {\frac{|z-w|}{|1-\bar{w}z|}}. \end{aligned}$$

So

$$\begin{aligned} L_{\mathbb {D} } = \frac{-1}{2\pi i} \frac{dw \, dz}{(w-z)^2} \end{aligned}$$

and

$$\begin{aligned} K_{\mathbb {D} } = \frac{1}{2\pi i} \frac{d\overline{w} \, dz}{(1-\bar{w}z)^2}. \end{aligned}$$

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

For all $\overline{\alpha } \in \overline{A(\Sigma )}$

$$\begin{aligned} \iint _{\Sigma ,w} L_\Sigma (z,w) \wedge \overline{\alpha (w)}=0. \end{aligned}$$

Proof

We assume momentarily that $\alpha$ has a holomorphic extension to the closure of $\Sigma$ and that $\Gamma$ is an analytic curve. Let $z \in \Sigma$ be fixed but arbitrary, and choose a chart $\zeta$ near z such that $\zeta (z)=0$. Write $\overline{\alpha }$ locally as $\overline{f(\zeta )} d\overline{\zeta }$ for some holomorphic function f. Let $C_r$ be the curve $|\zeta |=r$, and denote its image in $\Sigma$ by $\gamma _r$. Fixing $p \in \Sigma$ and using Stokes’ theorem yield

$$\begin{aligned} \iint _{\Sigma ,w} L_{\Sigma }(z,w) \wedge \overline{\alpha (w)} = \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma ^p_\epsilon } \partial _z g(w,z) \overline{\alpha (w)} - \lim _{r \searrow 0} \frac{1}{\pi i} \int _{C_r} \partial _z g(w,z) \overline{\alpha (w)}. \end{aligned}$$

The first term goes to zero uniformly as $\epsilon \rightarrow 0$. Writing the second term in coordinates $\eta = \phi (w)$ in a neighbourhood of $\zeta$ for fixed $\zeta$ (see Eq. (3.6)) we obtain

$$\begin{aligned} \iint _{\Sigma ,w} L_{\Sigma }(z,w) \wedge \overline{\alpha (w)}&= \lim _{r \searrow 0} -\frac{1}{2 \pi i} \int _{C_r} \left( \frac{1}{\eta } + h(\zeta ) \right) \overline{f(\eta )} d\bar{\eta } \end{aligned}$$

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 $\Sigma$ is a subset of its double and $\Gamma$ is analytic. By [2, Proposition 2.2], the set of holomorphic one-forms 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. $\square$

This implies that for R, $\Gamma$, and $\Sigma$ as in Theorem 3.8, we can write

$$\begin{aligned}{}[T(\Sigma ,\Sigma ) \alpha ](z)= \iint _{\Sigma ,w} \left( L_R(z,w) - L_\Sigma (z,w) \right) \wedge \overline{\alpha (w)}, \end{aligned}$$

(3.7)

which has the advantage that the integral kernel is non-singular.

Remark 3.9

The above expression shows that the operator $T(\Sigma ,\Sigma )$ 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 $\Sigma$ 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 strip-cutting 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

Fix j and let $k \in \{ 1,2 \}$ be such that $k \ne j$. By (3.7) we observe that

$$\begin{aligned} T(\Sigma _j) \overline{\alpha } (z) = \left\{ \begin{array}{cc} \iint _{\Sigma _{j},w} L_{R}(z,w) \wedge \overline{\alpha (w)} &{}\quad z \in \Sigma _k \\ \iint _{\Sigma _{j},w} \left( L_{R}(z,w) - L_{\Sigma _{j}}(z,w) \right) \wedge \overline{\alpha (w)} &{}\quad z \in \Sigma _j \end{array} \right. \end{aligned}$$

(3.8)

The integrand in both terms (3.8) is non-singular and holomorphic in z for each $w\in \Sigma _j$, and furthermore both integrals are locally bounded in z. Therefore the holomorphicity of $T(\Sigma _j)\overline{\alpha }$ follows by moving the $\overline{\partial }$ inside (3.7), and using the holomorphicity of the integrand. This also implies the holomorphicity of $T(\Sigma _j,\Sigma _k).$

Regarding the boundedness, the operator $T(\Sigma _j)$ is defined by integration against the L-Kernel 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 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 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.

Here are a few words on terminology. The Beurling transform in the plane is defined by

$$\begin{aligned} B_\mathbb {C} f(z)=\frac{-1}{\pi }\mathrm {PV}\iint _{\mathbb {C}}\frac{f(\zeta )}{(z-\zeta )^2}\, dA(\zeta ). \end{aligned}$$

Schiffer refers to this operator as the Hilbert transform, due to the fact that the operator in question behaves like the actual Hilbert transform

$$\begin{aligned} \mathcal {H}f(x):= \mathrm {PV}\int _{\mathbb {R}}\frac{f(y)}{x-y}\,dy. \end{aligned}$$

The term “Hilbert transform” is also the one used in Lehto’s classical book on Teichmüller theory [7]. 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 a Schiffer operator. Here, of course, we express this equivalently as an operator on anti-holomorphic one-forms.

3.4 Identities for comparison operators

Theorem 3.11

Let R be a compact surface and let $\Gamma$ be a strip-cutting Jordan curve separating R into two components, one of which is $\Sigma$. Then $S(\Sigma )= \mathrm {Res}(\Sigma )^*$, where $^*$ denotes the adjoint operator.

Proof

Let $\alpha \in A(\Sigma )$ and $\beta \in A(R)$. Then, using the reproducing property of $K_R$ and Proposition 3.3 we have

$$\begin{aligned} (S(\Sigma ) \alpha , \beta )_R&=\frac{i}{2} \iint _{R,z} \iint _{\Sigma ,\zeta } K_R(z,\zeta ) \wedge _\zeta \alpha (\zeta ) \wedge _z \overline{\beta (z)} \\&=\frac{-i}{2} \iint _{\Sigma _,\zeta } \iint _{R,z} \overline{K_R(\zeta ,z)} \wedge _z \overline{\beta (z)} \wedge {\alpha (\zeta )} \\&=\frac{-i}{2} \iint _{\Sigma ,\zeta } \overline{\beta (\zeta )} \wedge \alpha (\zeta ) =(\alpha ,\mathrm {Res}\,\beta )_{\Sigma }. \end{aligned}$$

Note that interchange of order of integration is legitimate by Fubini’s theorem, due to the analyticity and boundedness of the Bergman kernel. $\square$

Define

$$\begin{aligned} \overline{T}(\Sigma _j,\Sigma _k):A(\Sigma _j)&\rightarrow \overline{A(\Sigma _k)} \\ h&\mapsto \overline{T(\Sigma _j,\Sigma _k) \overline{h}}. \end{aligned}$$

and similarly for $\overline{S}(\Sigma _k)$.

Theorem 3.12

Let R be a compact surface. Let $\Gamma$ be a strip-cutting Jordan curve with measure zero, and assume that the complement of $\Gamma$ consists of two connected components $\Sigma _1$ and $\Sigma _2$. Then

$$\begin{aligned} T(\Sigma _j,\Sigma _k)^*= \overline{T}(\Sigma _k,\Sigma _j). \end{aligned}$$

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

$$\begin{aligned} \mathcal {L}_\mathbb {C}(z,w) = \frac{1}{\pi } \frac{1}{(z-w)^2}. \end{aligned}$$

We first show that for $G, H\in L^2 (\mathbb {C})$ one has

$$\begin{aligned}&\iint _{\mathbb {C}} \left( \iint _{\mathbb {C}} \mathcal {L}_{\mathbb {C}}(z, w) \overline{H(z)} \, dA(z) \right) \,\overline{G(w)}\, dA(w) \nonumber \\&\quad = \iint _{\mathbb {C}} \left( \iint _{\mathbb {C}} \mathcal {L}_{\mathbb {C}}(z, w) \overline{G(w)} \, dA(w) \right) \,\overline{H(z)}\, dA(z) \end{aligned}$$

(3.9)

where the inside integral is understood as a principle value integral in both cases.

Now, for $f \in L^2(\mathbb {C})$, the Beurling transform is given by

$$\begin{aligned} B_\mathbb {C} f(z)= \mathrm {PV}\iint _{\mathbb {C}}{\mathcal {L}}_{\mathbb {C}}(z, \zeta )\,f(\zeta )\, dA(\zeta )= \frac{-1}{\pi }\mathrm {PV}\iint _{\mathbb {C}}\frac{f(\zeta )}{(z-\zeta )^2}\, dA(\zeta ), \end{aligned}$$

(3.10)

With this notation, and denoting $\overline{H}(w) = \overline{H(w)}$, (3.9) amounts to

$$\begin{aligned} \iint _{\mathbb {C}} B_{\mathbb {C}} \overline{H}(w)\,\overline{G}(w)\, dA(w) = \iint _{\mathbb {C}} B_{\mathbb {C}} \overline{G}(z)\,\overline{H}(z)\, dA(z). \end{aligned}$$

(3.11)

If one defines the Fourier transform through

$$\begin{aligned} \widehat{f}(\xi ,\eta )=\iint _{\mathbb {R}^2} e^{-2\pi i(x\xi +y\eta )}\, f(x+iy)\, dx\, dy, \end{aligned}$$

then one has that $\widehat{B_\mathbb {C} f}(\xi ,\eta )=\frac{\xi -i\eta }{\xi +i\eta }\,\widehat{f}(\xi ,\eta ).$

Using Parseval’s formula and the above Fourier multiplier representation of the Beurling transform, one has that

$$\begin{aligned} \iint _{\mathbb {C}} B_\mathbb {C} \overline{H}(w)\,\overline{G}(w)\, dA(w)= & {} \iint _{\mathbb {C}} \widehat{B_\mathbb {C} \overline{H}}(\xi ,\eta )\,\widehat{\overline{G}}(\xi ,\eta )\, d\xi \,d\eta \\= & {} \iint _{\mathbb {C}} \frac{\xi -i\eta }{\xi +i\eta }\, \widehat{\overline{H}}(\xi ,\eta )\,\widehat{\overline{G}}(\xi ,\eta )\, d\xi \,d\eta , \end{aligned}$$

and

$$\begin{aligned} \iint _{\mathbb {C}} B_\mathbb {C} \overline{G}(z)\,\overline{H}(z)\, dA(z)= & {} \iint _{\mathbb {C}} \widehat{B_\mathbb {C} \overline{G}}(\xi ,\eta )\,\widehat{\overline{H}}(\xi ,\eta )\, d\xi \,d\eta \\= & {} \iint _{\mathbb {C}} \frac{\xi -i\eta }{\xi +i\eta }\, \widehat{\overline{G}}(\xi ,\eta )\,\widehat{\overline{H}}(\xi ,\eta )\, d\xi \,d\eta . \end{aligned}$$

This proves (3.11) and hence (3.9).

Now let B be a doubly-connected neighbourhood of $\Gamma$ and $\phi :B \rightarrow U \subseteq \mathbb {C}$ be a doubly-connected chart. Let $E= B \cap \Sigma _1$ and $E'=B \cap \Sigma _2$. Then $\Sigma _1 = D \cup E$ and $\Sigma _2 = D' \cup E'$ for some compact sets $D \subset \Sigma _1$ and $D' \subseteq \Sigma _2$ 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 $\alpha \in A(\Sigma _2)$ and $\beta \in A(\Sigma _1)$

$$\begin{aligned}&\iint _{\Sigma _1} \iint _{\Sigma _2}L (\zeta ,\eta ) \wedge _\zeta \overline{\alpha (\zeta )} \wedge _\eta \overline{\beta (\eta )}\nonumber \\&\quad = \left( \iint _{D} \iint _{D'} + \iint _{D} \iint _{E'} + \iint _{E} \iint _{D'} + \iint _E \iint _{E'} \right) L (\zeta ,\eta ) \wedge _\zeta \overline{\alpha (\zeta )} \wedge _\eta \overline{\beta (\eta )}.\nonumber \\ \end{aligned}$$

(3.12)

and

$$\begin{aligned}&\iint _{\Sigma _2} \iint _{\Sigma _1} L (\zeta ,\eta ) \wedge _\eta \overline{\beta (\eta )} \wedge _\zeta \overline{\alpha (\zeta )} \nonumber \\&\quad = \left( \iint _{D'} \iint _{D} + \iint _{D'} \iint _{E} + \iint _{E'} \iint _{D} + \iint _{E'} \iint _{E} \right) L (\zeta ,\eta ) \wedge _\eta \overline{\beta (\eta )} \wedge _\zeta \overline{\alpha (\zeta )}.\nonumber \\ \end{aligned}$$

(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 $L(z,\zeta )$ 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

$$\begin{aligned} \iint _E \iint _{E'} L (\zeta ,\eta ) \wedge _\zeta \overline{\alpha (\zeta )} \wedge _\eta \overline{\beta (\eta )} = \iint _{E'}\iint _E L (\zeta ,\eta )\wedge _\eta \overline{\beta (\eta )} \wedge _\zeta \overline{\alpha (\zeta )} . \end{aligned}$$

To show this, let $\phi$ be a local coordinate with $\eta =\phi (w)$ and $\zeta = \phi (z)$. We pull back the integral to the plane under $\psi =\phi ^{-1}$ so that we reduce the problem to showing that

$$\begin{aligned}&\iint _{\phi (E)} \iint _{\phi (E')} (\psi \times \psi )^* L (\zeta ,\eta ) \wedge _\zeta \psi ^* \overline{\alpha (\zeta )} \wedge _\eta \psi ^* \overline{\beta (\eta )} \nonumber \\&\quad = \iint _{\phi (E')}\iint _{\phi (E)} (\psi \times \psi )^*L (\zeta ,\eta )\wedge _\eta \psi ^* \overline{\beta (\eta )} \wedge _\zeta \psi ^* \overline{\alpha (\zeta )} . \end{aligned}$$

(3.14)

Recall that in local coordinates by Eq. (3.6)

$$\begin{aligned} (\psi \times \psi ^*) L(\zeta ,\eta )= \left( -\frac{1}{2\pi i} \frac{1}{(\zeta -\eta )^2} + H(\eta ) \right) d\zeta \, d\eta , \end{aligned}$$

where $H(\eta )$ is holomorphic near $\zeta$. 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 $\mathcal {L}_\mathbb {C}(\zeta ,\eta )$ which contains the singularity. We may write $\psi ^*\alpha (z) = h(z) dz$ and $\psi ^*\beta (w) = g(w) dw$ for some $L^2$ holomorphic functions g on E and h on $E'$. So the problem is reduced to showing that

$$\begin{aligned}&\iint _{\phi (E')} \iint _{\phi (E)} \mathcal {L}_{\mathbb {C}}(z,w) \wedge _\zeta \overline{h(z)} \wedge _\eta \overline{g(w)} dA(z) dA(w)\\&\quad := \iint _{\phi (E')} \iint _{\phi (E)} \mathcal {L}_{\mathbb {C}}(z,w)\overline{h(z)} \overline{g(w)} dA(w) dA(z). \end{aligned}$$

Letting

$$\begin{aligned} G(z)= \left\{ \begin{array}{ll} g(z), &{}\quad z\in E\\ 0, &{}\quad z\in \mathbb {C}{\setminus } E \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} H(z)= \left\{ \begin{array}{ll}h(z), &{}\quad z\in E'\\ 0, &{}\quad z\in \mathbb {C}{\setminus } E' \end{array}\right. \end{aligned}$$

then G and H are $L^2$ on $\mathbb {C}$ and the claim now follows directly from (3.9). $\square$

We also have the following identity.

Theorem 3.13

If $\Gamma$ is a quasicircle then

$$\begin{aligned} T(\Sigma _1,\Sigma _1)^*T(\Sigma _1,\Sigma _1) + T(\Sigma _1,\Sigma _2)^*T(\Sigma _1,\Sigma _2) + \overline{S}(\Sigma _1)^* \overline{S}(\Sigma _1) = I. \end{aligned}$$

Proof

By Theorem 3.12, and interchange of order of integration (which can be justified as in the proof of Theorem 3.12) we have that

$$\begin{aligned}{}[T(\Sigma _1,\Sigma _2)^* T(\Sigma _1,\Sigma _2) \alpha ](z)&= \iint _{\Sigma _2,\zeta } \overline{L_R(z,\zeta )} \wedge _\zeta \iint _{\Sigma _1,w} L_R(\zeta ,w) \wedge _w \alpha (w) \\&= \iint _{\Sigma _1,w} \left( \iint _{\Sigma _2,\zeta } \overline{L_R(z,\zeta )} \wedge _\zeta L_R(\zeta ,w) \right) \wedge _w \alpha (w) \end{aligned}$$

so the integral kernel of $T(\Sigma _1,\Sigma _2)^* T(\Sigma _1,\Sigma _2)$ is

$$\begin{aligned} \iint _{\Sigma _2,\zeta } \overline{L_R(z,\zeta )} \wedge _\zeta L_R(\zeta ,w). \end{aligned}$$

Similarly, by Eq. (3.7) and Theorem 3.12, the integral kernel of $T(\Sigma _1,\Sigma _1)^* T(\Sigma _1,\Sigma _1)$ is

$$\begin{aligned} \iint _{\Sigma _1,\zeta } \left( \overline{L_R(z,\zeta )} - \overline{L_{\Sigma _1}(z,\zeta )}\right) \wedge \left( L_R(\zeta ,w) - L_{\Sigma _1}(\zeta ,w) \right) . \end{aligned}$$

Finally, by Theorem 3.11, the integral kernel of $\overline{S}(\Sigma _1)^* \overline{S}(\Sigma _1)$ is $\overline{K_R(z,w)}$.

Using this and the reproducing property of $K_\Sigma$ we need only demonstrate the following identity:

$$\begin{aligned}&\iint _{\Sigma _1,\zeta } \left( \overline{L_R(z,\zeta )} - \overline{L_{\Sigma _1}(z,\zeta )}\right) \wedge \left( L_R(\zeta ,w) - L_{\Sigma _1}(\zeta ,w) \right) \nonumber \\&\quad + \iint _{\Sigma _2,\zeta } \overline{L_R(z,\zeta )} \wedge L_R(\zeta ,w) = \overline{K_{\Sigma _1}(z,w)} - \overline{K_R(z,w)}. \end{aligned}$$

(3.15)

Fix $w \in \Sigma _1$ and orient $\Gamma ^w_\epsilon$ positively with respect to $\Sigma _1$. For fixed w, $\partial _w g_{\Sigma _1}(\zeta ,w)$ goes to zero uniformly as $\epsilon \rightarrow 0$. We then have that, applying Theorem 3.8,

$$\begin{aligned}&\iint _{\Sigma _1,\zeta } \left( \overline{L_R(z,\zeta )} - \overline{L_{\Sigma _1}(z,\zeta )}\right) \wedge \left( L_R(\zeta ,w) - L_{\Sigma _1}(\zeta ,w) \right) \\&\quad = \iint _{\Sigma _1,\zeta } \left( \overline{L_R(z,\zeta )} - \overline{L_{\Sigma _1}(z,\zeta )}\right) \wedge L_R(\zeta ,w) \\&\quad = \lim _{\epsilon \rightarrow 0} \frac{1}{\pi i} \int _{\Gamma ^w_\epsilon } \left( \overline{L_R(z,\zeta )} - \overline{L_{\Sigma _1}(z,\zeta )}\right) \partial _w g(\zeta ,w) \\&\quad = \frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma ^{w}_\epsilon } \overline{L_R(z,\zeta )}\, \partial _w g(\zeta ,w) + \frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma ^{w}_\epsilon } K_{\Sigma _1}(z,\zeta ) \, \partial _w g(\zeta ,w) \end{aligned}$$

where we have applied Eq. (3.4) in the last step.

Applying Stokes’ theorem to the first term, we see that

$$\begin{aligned} \frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma ^w_\epsilon } \overline{L_R(z,\zeta )}\, \partial _w g(\zeta ,w) = -\iint _{\Sigma _2,\zeta } \overline{L_R(z,\zeta )} \wedge L_R(\zeta ,w). \end{aligned}$$

Here we used the fact that quasicircles have measure zero. Note that $\Gamma ^w_\epsilon$ is negatively oriented with respect to $\Sigma _2$. For the second term, we have

$$\begin{aligned} \frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma ^w_\epsilon } K_{\Sigma _1}(z,\zeta ) \, \partial _w g(\zeta ,w)&= \frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma ^w_\epsilon } K_{\Sigma _1}(z,\zeta ) \left( \partial _w g(\zeta ,w) - \partial _w g_{\Sigma _1}(\zeta ,w) \right) \\&= - \iint _{\Sigma _1,\zeta } K_{\Sigma _1}(z,\zeta ) \wedge \left( \overline{K_R(\zeta ,w)} - \overline{K_{\Sigma _1}(\zeta ,w)} \right) \\&= - \overline{K_R(z,w)} + \overline{K_{\Sigma _1}(z,w)} \end{aligned}$$

where in the last term we have used part (5) of Proposition 3.3 and the reproducing property of Bergman kernel on $\Sigma _1$. $\square$

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 Cauchy-type 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 $\Sigma _1$.

Let $h \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$. Fix $q \in R \backslash \Gamma$ and define

$$\begin{aligned} J_q(\Gamma )h (z) = - \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma ^{p_1}_\epsilon } \partial _w g(w;z,q) h(w) \end{aligned}$$

(4.1)

for $z \in R \backslash \Gamma$. Observe that, by definition, the curve $\Gamma ^{p_1}_\epsilon$ depends on a fixed point $p_1 \in \Sigma _1$. However, we shall show that $J_q(\Gamma )$ 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 $q \in \Sigma _2$, then for $z \in \Sigma _2$, we have by Stokes’ theorem that

$$\begin{aligned} J_q(\Gamma ) h(z) = - \frac{1}{\pi i} \iint _{\Sigma _1} \partial _w g(w;z,q) \wedge \overline{\partial } h(w) \end{aligned}$$

(4.2)

so the limit exists and is independent of $p_1$. For $z \in \Sigma _1$ we proceed as follows; let $\gamma _r$ 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

$$\begin{aligned} J_q(\Gamma ) h (z)&= - \frac{1}{\pi i} \iint _{\Sigma _1} \partial _w g(w;z,q) \wedge \overline{\partial } h(w) - \lim _{r \searrow 0} \frac{1}{\pi i} \int _{\gamma _r} \partial _w g(w;z,q) h(w) \nonumber \\&= - \frac{1}{\pi i} \iint _{\Sigma _1} \partial _w g(w;z,q) \wedge \overline{\partial } h(w) + h(z). \end{aligned}$$

(4.3)

This shows that the limit exists for $z \in R \backslash \Gamma$ and $q \in \Sigma _2$ and is independent of p. In the case that $q \in \Sigma _1$, we obtain similar expressions, but with the term h(q) added to both integrals.

This also shows that

Lemma 4.1

For strip-cutting 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

Let $\Gamma$ be a strip-cutting Jordan curve in R. For all $h \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ and any $q \in R \backslash \Gamma$,

Proof

Assume first that $q \in \Sigma _2$. The first claim follows from (4.2) and the fact that the integrand is non-singular. Similarly for $z \in \Sigma _2$, the third claim follows from (4.2).

The second claim follows from Stokes theorem:

$$\begin{aligned} \partial J_q(\Gamma ) h(z)&= \partial _z \left( - \frac{1}{\pi i} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } (\partial _w g(w;z,q) - \partial _w g_{\Sigma }(w,z) )\,h(w) \right) \nonumber \\&\quad - \partial _z \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma _\epsilon } \partial _w g_{\Sigma } (w,z) \,h(w) \nonumber \\&= \partial _z \left( - \frac{1}{\pi i} \iint _{\Sigma _1} (\partial _w g(w;z,q) - \partial _w g_{\Sigma }(w,z) )\wedge _w \overline{\partial } h(w) \right) \nonumber \\&\quad - \partial _z \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma _\epsilon } \partial _w g_{\Sigma } (w,z) \,h(w) \nonumber \\&= - \frac{1}{\pi i} \iint _{\Sigma _1} (\partial _z \partial _w g(w;z,q) - \partial _z \partial _w g_{\Sigma }(w,z) ) \wedge _w \overline{\partial } h(w) + \partial h(z) \end{aligned}$$

(4.4)

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 $w\in \Sigma _1$, and that

$$\begin{aligned} \iint _{\Sigma _1, w}|( \partial _w g(w;z,q) - \partial _w g_{\Sigma }(w,z) ) \wedge _w \overline{\partial }_{{w}} h(w)| \end{aligned}$$

is locally bounded in z, yield that derivation under the integral sign in the first term is legitimate.

Similarly removing the singularity using $\partial _w g_{\Sigma }$, and then applying Theorem 3.2 and Stokes’ theorem yield that

$$\begin{aligned} \overline{\partial } J(\Gamma ) h(z)&= - \overline{\partial }_{{z}} \frac{1}{\pi i} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } ( \partial _w g(w;z,q) - \partial _w g_{\Sigma }(w,z) )\,h(w) + \overline{\partial } h(z) \\&= - \frac{1}{\pi i} \iint _{\Sigma _1}(\overline{\partial }_{{z}} \partial _w g(w;z,q) - \overline{\partial }_{{z}} \partial _w g_{\Sigma }(w,z) ) \wedge _w \overline{\partial }_{{w}} h(w) + \overline{\partial } h(z). \end{aligned}$$

The third claim now follows by observing that the second term in the integral is just $- \overline{\partial } h$ because the integrand is just the complex conjugate of the Bergman kernel.

Now assume that $q \in \Sigma _1$. We show the second claim in the theorem. We argue as in Eq. (4.4), except that we must also add a term $\partial _w g_{\Sigma _1}(w;q) h(w)$. We obtain instead

$$\begin{aligned} \partial J(\Gamma ) h= & {} \partial _z J(\Gamma ) h = - \frac{1}{\pi i} \iint _{\Sigma _1} (\partial _z \partial _w g(w;z,q) - \partial _z \partial _w g_{\Sigma }(w;z) ) \wedge _w \overline{\partial }_{{w}} h(w)\\&+\, \partial _z \left( h(z) + h(q) \right) \end{aligned}$$

and the claim follows from $\partial _z h(q) =0$. The remaining claims follow similarly. $\square$

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

Let $\Gamma$ be a strip-cutting Jordan curve and assume that $q \in R \backslash \Gamma$.

• (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 $J_q(\Gamma ) h \in \mathcal {D}(\Sigma _1 \cup \Sigma _2)$.

Proof

The first claim follows immediately from Theorems 3.10 and 4.2. The second claim follows from Theorem 4.2 together with the fact that for fixed z $\overline{\partial }_{{z}} \partial _w g \in {A(R)}$. $\square$

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 one-forms (in fact, more generally differentials) on a region in a Riemann surface can be approximated by holomorphic one-forms on a larger domain. We need a result of this type for the Dirichlet space, for doubly-connected regions.

Theorem 4.4

Let R be a compact Riemann surface and $\Gamma$ be a strip-cutting Jordan curve. Let U be a doubly-connected 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

The proof proceeds in two steps. First, let $A'$ be any doubly-connected domain in $\mathbb {C}$, bounded by two Jordan curves $\Gamma _1$ and $\Gamma _2$. Let $B_1$ and $B_2$ be connected components of the complements of $\Gamma _1$ and $\Gamma _2$ respectively, chosen so that $B_1$ and $B_2$ both contain $A'$; thus $A' = B_1 \cap B_2$. We claim that every $h \in \mathcal {D}(A')$ can be written $h=h_1 + h_2$ where $h_1 \in \mathcal {D}(B_1)$ and $h_2 \in \mathcal {D}(B_2)$. To see this, one may take level curves $\Gamma ^k_{\epsilon ,p_k}$ of Green’s function of $B_k$ for $k=1,2$, and define

$$\begin{aligned} h_k(z) = \frac{1}{2 \pi i} \lim _{\epsilon \searrow 0} \int _{\Gamma ^k_{\epsilon ,p_k}} \frac{h(\zeta )}{ \zeta -z} \,d\zeta \end{aligned}$$

(where we assume that $\Gamma _k$ are positively oriented with respect to $B_k$ for $k=1,2$, and therefore also with respect to $A'$). Then $h_1$ and $h_2$ are clearly holomorphic and $h= h_1 + h_2$.

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. $\square$

We will also need a density result of another kind. Let $\Gamma$ be a strip-cutting Jordan curve in a compact Riemann surface R, which separates R into two components $\Sigma _1$ and $\Sigma _2$. Let A be a collar neighbourhood of $\Gamma$ in $\Sigma _1$. By Theorem 2.5 the boundary values of $\mathcal {D}_{\mathrm{harm}}(A)$ exist conformally non-tangentially in $\Sigma _1$ and are themselves CNT boundary values of an element of $\mathcal {D}_{\mathrm{harm}}(\Sigma _1)$. We then define

$$\begin{aligned} \mathfrak {G}: \mathcal {D}_{\mathrm{harm}}(A)&\rightarrow \mathcal {D}_{\mathrm{harm}}(\Sigma _1) \nonumber \\ h&\mapsto \tilde{h} \end{aligned}$$

(4.5)

where $\tilde{h}$ is the unique element of $\mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ with CNT boundary values equal to those of h. We have the following result:

Theorem 4.5

[20] Let $\Gamma$ be a strip-cutting 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

First, we prove this in the case that $A = \mathbb {A}$ is an annulus with outer boundary $\mathbb {S}^1$ and $\Sigma _1 = \mathbb {D}$, and $\mathfrak {G}$ is

$$\begin{aligned} \mathfrak {G}(\mathbb {A},\mathbb {D}):\mathcal {D}_{\mathrm{harm}}(\mathbb {A}) \rightarrow \mathcal {D}_{\mathrm{harm}}(\mathbb {D} ). \end{aligned}$$

Now the set of Laurent polynomials $\mathbb {C}[z,z^{-1}]$ are contained in $\mathbb {D}_{\mathrm{harm}}(\mathbb {A})$, and

$$\begin{aligned} \mathfrak {G}(\mathbb {A},\mathbb {D} ) z^n = z^n \quad \text {and} \quad \mathfrak {G}(\mathbb {A},\mathbb {D} ) z^{-n} = \bar{z}^n. \end{aligned}$$

Since the set $\mathbb {C}[z,\bar{z}]$ of polynomials in z, $\bar{z}$ is dense in $\mathcal {D}_{\mathrm{harm}}(\mathbb {D} )$, this proves the claim.

Next, let $F:\mathbb {A} \rightarrow A$ be a conformal map. Define the composition map

$$\begin{aligned} \mathcal {C}_F :\mathcal {D}_{\mathrm{harm}}(A)&\rightarrow \mathcal {D}_{\mathrm{harm}}(\mathbb {A}) \\ h&\mapsto h \circ F, \end{aligned}$$

which is bounded by conformal invariance of the Dirichlet norm, and furthermore is a bijection with bounded inverse $\mathcal {C}_{F^{-1}}$. Similarly the restriction of $\mathcal {C}_F$ to $\mathcal {D}(A)$ is a bounded bijection onto $\mathcal {D}(\mathbb {A})$. Thus, the image of $\mathcal {D}(A)$ under $\mathfrak {G}(\mathbb {A},\mathbb {D}) \mathcal {C}_F$ is dense in $\mathcal {D}_{\mathrm{harm}}(\mathbb {D} )$.

Now denote the restriction map from $\mathcal {D}_{\mathrm{harm}}(\mathbb {D} )$ to $\mathcal {D}_{\mathrm{harm}}(\mathbb {A})$ by $\text {Res}(\mathbb {D} ,\mathbb {A})$ and similarly for $\text {Res}(\Sigma _1,A)$. Define the linear map

$$\begin{aligned} \rho = \mathfrak {G}(A,\Sigma _1) \, \mathcal {C}_{F^{-1}} \, \text {Res}(\mathbb {D} ,\mathbb {A}) : \mathcal {D}_{\mathrm{harm}}(\mathbb {D} ) \rightarrow \mathcal {D}_{\mathrm{harm}}(\Sigma _1). \end{aligned}$$

This is obviously bounded, with bounded inverse

$$\begin{aligned} \rho ^{-1} = \mathfrak {G}(\mathbb {A},\mathbb {D})\, \mathcal {C}_F\, \text {Res}(\Sigma _1,A) \end{aligned}$$

by uniqueness of Dirichlet bounded harmonic extensions of elements of $\mathcal {H}(\mathbb {S}^1)$ and $\mathcal {H}(\Gamma )$ to $\mathcal {D}_{\mathrm{harm}}(\mathbb {D} )$ and $\mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ respectively.

Now by definition of $\mathfrak {G}(\mathbb {A},\mathbb {D} )$, for any $h \in \mathcal {D}_{\mathrm{harm}}(A)$, the $\mathrm {CNT}$ boundary values of

$$\begin{aligned} \mathcal {C}_{F^{-1}} \, \text {Res}(\mathbb {D} ,\mathbb {A})\,\mathfrak {G}(\mathbb {A},\mathbb {D} )\, \mathcal {C}_F h \end{aligned}$$

equal those of h. Thus we obtain the following factorization of $\mathfrak {G}(A,\Sigma _1)$:

$$\begin{aligned} \rho \, \mathfrak {G}(\mathbb {A},\mathbb {D}) \mathcal {C}_F = \mathfrak {G}(A,\Sigma _1) \mathcal {C}_{F^{-1}}\, \text {Res}(\mathbb {D} ,\mathbb {A}) \,\mathfrak {G}(\mathbb {A},\mathbb {D} ) \, \mathcal {C}_F = \mathfrak {G}(A,\Sigma _1). \end{aligned}$$

Since the image of $\mathcal {D}(A)$ under $\mathfrak {G}(\mathbb {A},\mathbb {D}) \mathcal {C}_F$ is dense in $\mathcal {D}_{\mathrm{harm}}(\mathbb {D} )$, and $\rho$ is a bounded bijection with bounded inverse, this completes the proof. $\square$

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.

For $h \in \mathcal {D}(A)$, letting $\Gamma _\epsilon$ be level curves of Green’s function of $\Sigma _1$ with respect to some fixed point $p \in \Sigma _1$, denote

$$\begin{aligned} J_q(\Gamma )'h(z) = - \frac{1}{\pi i} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } \partial _w g(w;z,q) h(w) \end{aligned}$$

(4.6)

for q fixed in $\Sigma _2$. We use the notation $J_q(\Gamma )'$ to distinguish it from the operator $J_q(\Gamma )$, which applies only to elements of $\mathcal {D}_{\mathrm{harm}}(\Sigma _1)$. For $\epsilon$ in some interval (0, R) the curve $\Gamma _\epsilon$ lies entirely in A, so this makes sense. Because the integrand is holomorphic, the integral is independent of $\epsilon$ for $\epsilon \in (0,R)$.

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

Let $\Gamma$ be a quasicircle and let $\beta \in A(B)$ where B is a collar neighbourhood of $\Gamma$ in $\Sigma _1$. Let $\Gamma _\epsilon$ be level curves of Green’s function in $\Sigma _1$. If $h \in \mathcal {D}_{\mathrm{harm}}(B)$ has CNT boundary values zero, then

$$\begin{aligned} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } \beta (w) h(w) = 0. \end{aligned}$$

Proof

Since B contains a canonical collar neighbourhood, it is enough to prove this for the case that B is a canonical collar neighbourhood. Let $\phi :B \rightarrow \mathbb {A}$ be a canonical collar chart onto an annulus $\mathbb {A} = \{ z : R< |z| < 1 \}$ for some $R \in (0,1)$. The level curves $\Gamma _\epsilon$ map onto circles $|z|=r=e^{-\epsilon }$ for all $\epsilon$ sufficiently close to zero. A change of variables reduces the problem to showing that

$$\begin{aligned} \lim _{r \nearrow 1} \int _{|z|=r} \alpha H = 0 \end{aligned}$$

for $\alpha = (\phi ^{-1})^* \beta \in A(\mathbb {A})$ and $H = h \circ \phi ^{-1} \in \mathcal {D}_{\mathrm{harm}}(\mathbb {A})$.

We demonstrate this first for $\alpha$ of the form a(z)dz for Laurent polynomials $a(z) \in \mathbb {C}[z,z^{-1}]$. By Corollary 2.14 in [20] there is a $c \in \mathbb {C}$, a $H_{1} \in \mathcal {D}_{\mathrm{harm}}(\mathbb {D} )$ and a $H_{2} \in \mathcal {D}(\Omega )$ where $\Omega = \{z : R < |z| \} \cup \{ \infty \}$, such that

$$\begin{aligned} H = c g_0 + H_1 + H_2 \end{aligned}$$

where $g_0$ is Green’s function with singularity at 0 (that is, $g_0(z) = -\log |z|$). Now since $H_2$ and $g_0$ are continuous up to $\mathbb {S}^1$, and $\alpha$ is continuous on $\mathbb {S}^1$, we have that

$$\begin{aligned} \lim _{r \rightarrow 1} \int _{|z|=r} \alpha H_2 = \lim _{r \rightarrow 1} \int _{|z|=r} \alpha \mathfrak {G}(\mathbb {A},\mathbb {D}) H_2 \end{aligned}$$

and

$$\begin{aligned} \lim _{r \rightarrow 1} \int _{|z|=r} \alpha cg_0 = \lim _{r \rightarrow 1} \int _{|z|=r} \alpha \mathfrak {G}(\mathbb {A},\mathbb {D}) cg_0. \end{aligned}$$

But we also have that $\mathfrak {G}(\mathbb {A},\mathbb {D}) H_1 = H_1$. Therefore

$$\begin{aligned} \lim _{r \rightarrow 1} \int _{|z|=r} \alpha \mathfrak {G}(\mathbb {A},\mathbb {D}) H&= \lim _{r \rightarrow 1} \int _{|z|=r} \alpha \mathfrak {G}(\mathbb {A},\mathbb {D}) (H_1 + cg_0 + H_2) \\&= \lim _{r \rightarrow 1} \int _{|z|=r} \alpha (H_1 + c \mathfrak {G}(\mathbb {A},\mathbb {D}) g_0 + \mathfrak {G}(\mathbb {A},\mathbb {D}) H_2) \\&= \lim _{r \rightarrow 1} \int _{|z|=r} \alpha (H_1 + c g_0 + H_2) = \lim _{r \rightarrow 1} \int _{|z|=r} \alpha H. \end{aligned}$$

Since if $H = 0$ then $\mathfrak {G}(\mathbb {A},\mathbb {D}) H = 0$, this proves the claim for the special case of $\alpha$ of the above form.

Next we show that $\alpha$’s of this form are dense in $A(\mathbb {A})$. To see this observe that $\mathbb {C}[z,z^{-1}]$ is dense in $\mathcal {D}(\mathbb {A})$. Thus, Laurent polynomials of the form

$$\begin{aligned} \frac{a_{-n}}{z^{n}} + \cdots + \frac{a_{-2}}{z^2} + a_0 + a_1 z + \cdots a_m z^m \end{aligned}$$

(4.7)

(that is, the set of derivatives of Laurent polynomials) are dense in the set of exact one-forms on $A(\mathbb {A})$.

Now let $\alpha \in A(\mathbb {A})$ be arbitrary and let $c = \int _{|z|=r} \alpha$ where $r\in (R,1)$. Then

$$\begin{aligned} \alpha _0 = \alpha - \frac{c}{2 \pi i z} \end{aligned}$$

is exact, and hence for any $\epsilon >0$ it can be approximated within $\epsilon$ in $A(\mathbb {A})$ by a Laurent polynomial p of the form (4.7). Then the Laurent polynomial $p(z) + c/(2 \pi i z)$ approximates $\alpha$ within $\epsilon$ in $A(\mathbb {A})$.

The proof will be complete if it can be shown that for fixed $H \in \mathcal {D}_{\mathrm{harm}}(\mathbb {A})$ the functional

$$\begin{aligned} \alpha \mapsto \lim _{r \nearrow 1} \int _{|z|=r} H \alpha \end{aligned}$$

on $A(\mathbb {A})$ is bounded (where we fix the orientation in the integral to be positive with respect to zero). Denote $\alpha (z) = a(z)dz$ as above (but now with no extra assumptions on a(z)). Fix an $s \in (R,1)$ and let $M = \sup _{|z|=s} |H(z)|$. Then denoting $B_s = \{ z : s< |z| < 1\}$, we have by Stokes’ theorem that

$$\begin{aligned} \left| \lim _{r \nearrow 1} \int _{|z|=r} H \alpha \right|&= \left| \int _{|z|=s} \alpha H + \iint _{B_s} \alpha \wedge \overline{\partial } H \right| \\&\le 2 \pi s \cdot M \cdot \sup _{|z|=s} |{ a(z)}| + \Vert \alpha \Vert _{A(B_s)} \Vert \overline{\partial } H \Vert _{A(B_s)}. \\ \end{aligned}$$

Now since $|z|=s$ is compact, by Lemma 2.1 there is a constant C which is independent of $\alpha$ such that

$$\begin{aligned} \sup _{|z|=s} |{ a(z)}| \le C \Vert \alpha \Vert _{A(\mathbb {A})}. \end{aligned}$$

Inserting this estimate in the line above we obtain

$$\begin{aligned} \left| \lim _{r \nearrow 1} \int _{|z|=r} H \alpha \right|&\le 2 \pi s \cdot M \cdot C \cdot \Vert \alpha (z) \Vert _{A(\mathbb {A})} + \Vert \alpha \Vert _{A(B_s)} \Vert H \Vert _{\mathcal {D}(B_s)} \\&\le \left( 2 \pi s \cdot M \cdot C+ \Vert H \Vert _{\mathcal {D}(\mathbb {A})} \right) \Vert \alpha \Vert _{A(\mathbb {A})}. \end{aligned}$$

Thus for fixed H the integral is a bounded functional on $A(\mathbb {A})$, which completes the proof. $\square$

We then have the following immediate consequence.

Theorem 4.8

Let $\Gamma$ be a quasicircle, $\alpha$ be a holomorphic one-form on a collar neighbourhood B of $\Gamma$ in $\Sigma _1$. If $\Gamma _\epsilon$ are the level curves of Green’s function in $\Sigma _1$ and if $h_1,h_2 \in \mathcal {D}_{\mathrm{harm}}(B)$ have the same CNT boundary values on $\Gamma$, then

$$\begin{aligned} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } \alpha (w) h_1(w) = \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } \alpha (w) h_2. \end{aligned}$$

In particular, if $\mathfrak {G}$ is given by (4.5) then

$$\begin{aligned} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } \alpha (w) h(w) = \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon } \alpha (w) \mathfrak {G} h(w). \end{aligned}$$

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

Let $\Gamma$ be a quasicircle and A be a collar neighbourhood of $\Gamma$ in $\Sigma _1$. Then for fixed $q \in R \backslash \Gamma$ and all $h \in \mathcal {D}_{\mathrm{harm}}(B)$ and $z \in R \backslash \Gamma$

$$\begin{aligned} J_q(\Gamma )'h(z) = J_q(\Gamma ) \mathfrak {G} h(z) \end{aligned}$$

(4.8)

where $\mathfrak {G}$ is as in (4.5) and $J_q(\Gamma )'$ is as in (4.6).

Proof

By restricting to a smaller canonical collar neighbourhood, we can assume that B does not contain z or q in its closure. For fixed z and q set

$$\begin{aligned} \alpha (w) = \left. - \frac{1}{\pi i} \partial _w g(w;z,q) \right| _B. \end{aligned}$$

Since the right hand side is holomorphic on an open neighbourhood of the closure of B, $\alpha \in A(B)$. Applying Theorem 4.8 proves the theorem. $\square$

We now show that for quasicircles, one can define the jump operator $J(\Gamma )$ using either limiting integrals from within $\Sigma _1$ or from within $\Sigma _2$ with the same result. We use the following temporary notation. For $q \in R \backslash \Gamma$ let $J_q(\Gamma ,\Sigma _i):\mathcal {D}_{\mathrm{harm}}(\Sigma _1) \rightarrow \mathcal {D}_{\mathrm{harm}}(\Sigma _1 \cup \Sigma _2)$ be defined by

$$\begin{aligned} J_q(\Gamma ,\Sigma _i)h (z) = - \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma ^{p_i}_\epsilon } \partial _w g(w;z,q) h(w). \end{aligned}$$

For definiteness, we assume that all curves $\Gamma ^{p_i}_\epsilon$ are oriented positively with respect to $\Sigma _1$. Aside from this change of sign, all previous theorems apply equally to $J_q(\Gamma ,\Sigma _1)$ and $J_q(\Gamma ,\Sigma _2)$.

Theorem 4.10

Let $\Gamma$ be a quasicircle. Then for all $q \in R \backslash \Gamma$

$$\begin{aligned} J_q(\Gamma ,\Sigma _1) = J_q(\Gamma ,\Sigma _2) \mathfrak {O}(\Sigma _1,\Sigma _2). \end{aligned}$$

Proof

Let U be a doubly-connected 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 $i=1,2$.

For any $h \in \mathcal {D}(U)$, let $\text {Res}_i\, h = \left. h \right| _{A_i}$. It follows immediately from the definition of $\mathfrak {G}_i$ that

$$\begin{aligned} \mathfrak {G}_2 \text {Res}_2\, h = \mathfrak {O}(\Sigma _1,\Sigma _2) \mathfrak {G}_1 \text {Res}_1\, h. \end{aligned}$$

(4.9)

Therefore

$$\begin{aligned} \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon ^{p_1}} \partial _w g(w;z,q) \mathfrak {G}_1 h(w)&= \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon ^{p_1}} \partial _w g(w;z,q) h(w) \\&= \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon ^{p_2}} \partial _w g(w;z,q) h(w) \\&= \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon ^{p_2}} \partial _w g(w;z,q) \mathfrak {G}_2 h(w), \end{aligned}$$

where we have used holomorphicity of the integrand in the second equality, and Proposition 4.9 to obtain the first and the third equalities. Thus for all $h \in \mathfrak {G}_i \text {Res}_i\, \mathcal {D}(U)$,

$$\begin{aligned} J(\Gamma ,\Sigma _1) h = J(\Gamma ,\Sigma _2) \mathfrak {O} (\Sigma _1,\Sigma _2) h. \end{aligned}$$

Now by Theorem 4.4$\text {Res}_i\, \mathcal {D}(U)$ is dense in $\mathcal {D}(A_i)$ for $i=1,2$ , and therefore by Theorem 4.6 part (2) $\mathfrak {G}_i \text {Res}_i\, \mathcal {D}(U)$ is dense in $\mathcal {D}_{\mathrm{harm}}(\Sigma _i)$. Since $\text {Res}_i$, $\mathfrak {G}_i$ and $J(\Gamma ,\Sigma _i)$ are all bounded, this completes the proof. $\square$

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

Let $\Gamma$ be a quasicircle. Let $\alpha$ be a holomorphic one-form in an open neighbourhood of $\Gamma$. For any $h \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$

$$\begin{aligned} \lim _{\epsilon \searrow 0} \int _{\Gamma ^{p_1}_\epsilon } h(w) \alpha (w) = \lim _{\epsilon \searrow 0} \int _{\Gamma ^{p_2}_\epsilon } [\mathfrak {O}(\Sigma _1,\Sigma _2) h](w) \alpha (w) \end{aligned}$$

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

We denote by $W_k$ the linear subspace of $\mathcal {D}_{\mathrm{harm}}(\Sigma _i)$ given by

$$\begin{aligned} W_k = \left\{ h \in \mathcal {D}_{\mathrm{harm}}(\Sigma _k) \,:\, \lim _{\epsilon \searrow 0} \int _{\Gamma ^{{p_k}}_\epsilon } h(w) \alpha (w) =0 \right\} \end{aligned}$$

for all $\alpha \in A(R)$ and for $k=1,2$. The elements of $W_k$ are the admissible functions for the jump problem.

Let

$$\begin{aligned} J(\Gamma )_{\Sigma _k} h = \left. J(\Gamma ) h \right| _{\Sigma _k} \end{aligned}$$

for $k=1,2$. We have the following result:

Theorem 4.13

Let R be a compact surface and $\Gamma$ be a quasicircle separating R into components $\Sigma _1$ and $\Sigma _2$. Let $q \in R \backslash \Gamma$. If $h \in W_1$ then

$$\begin{aligned} - \mathfrak {O}(\Sigma _2,\Sigma _1) J_q(\Gamma )_{\Sigma _2} h = h - J_q(\Gamma )_{\Sigma _1} h. \end{aligned}$$

To prove this theorem we need a lemma.

Lemma 4.14

Let $\Gamma$ be a quasicircle and let A be a collar neighbourhood of $\Gamma$ in $\Sigma _1$. Fix a smooth curve $\Gamma '$ in A homotopic to $\Gamma$, and assume that $h \in \mathcal {D}(A)$ satisfies

$$\begin{aligned} \int _{\Gamma '} h \alpha =0 \end{aligned}$$

(4.10)

for all $\alpha \in A(R)$. Then $\mathfrak {G} h \in W_1$ and

$$\begin{aligned} - \mathfrak {O}(\Sigma _2,\Sigma _1) J_q(\Gamma )_{\Sigma _2} \mathfrak {G} h = \mathfrak {G} h - J_q(\Gamma )_{\Sigma _1} \mathfrak {G} h. \end{aligned}$$

Proof

The fact that $\mathfrak {G} h \in W_1$ follows immediately from Theorem 4.8. By Royden [12, Theorem 4] and the explicit formula on the following page, there are holomorphic functions $H_1$ on $\Sigma _1$ and $H_2$ on $\text {cl}\, \Sigma _2 \cup A$ such that $H_1 - H_2 = h$ on A. Furthermore, these functions are given by the restrictions of $J_q(\Gamma )' h$ to $\Sigma _1$ and $\Sigma _2$. Thus, by Proposition 4.9, we have that

$$\begin{aligned} H_k = \left. J_q(\Gamma ) \right| _{\Sigma _k} \mathfrak {G} h \end{aligned}$$

(4.11)

for $k=1,2$ (where $H_2$ has a holomorphic extension to $\text {cl}\Sigma _2 \cup A$).

Since $H_1$, $H_2$ and h are all in $\mathcal {D}(A)$, they have conformally non-tangential boundary values in $\mathcal {H}(\Gamma )$ with respect to $\Sigma _1$. Since $H_1 - H_2 = h$ on A, then the boundary values also satisfy this equation. Thus

$$\begin{aligned} H_1 - \mathfrak {O}(\Sigma _2,\Sigma _1) H_2 = \mathfrak {G} h \end{aligned}$$

by definition of $\mathfrak {G}$ and $\mathfrak {O}(\Sigma _2,\Sigma _1)$. Finally Eq. (4.11) completes the proof. $\square$

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 $\mathcal {D}(A)$.

For $u \in \mathcal {D}(A)$ define

$$\begin{aligned} Q(u) = \left( \int _{\Gamma '}u \alpha _1, \ldots , \int _{\Gamma '} u \alpha _g \right) . \end{aligned}$$

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 $\mathcal {D}(A)$. Once again, a simple argument based on Riesz representation theorem and the Gram-Schmidt process yields that there is a C such that

$$\begin{aligned} \Vert \mathcal {P} u - u \Vert _{\mathcal {D}(A)} \le C \Vert Q(u) \Vert _{\mathbb {C}^g}. \end{aligned}$$

(4.12)

For $H \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ define now

$$\begin{aligned} Q_1(H) = \lim _{\epsilon \searrow 0} \left( \int _{\Gamma ^{p_1}_\epsilon }H \alpha _1, \ldots , \int _{\Gamma ^{p_1}_\epsilon } H \alpha _g \right) . \end{aligned}$$

We have that there is a $C'$ such that

$$\begin{aligned} \Vert Q_1(H) \Vert _{\mathbb {C}^g} \le C' \Vert H \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)}. \end{aligned}$$

This follows by applying Stokes’ theorem to each component:

$$\begin{aligned} \lim _{\epsilon \searrow 0} \int _{\Gamma ^{p_1}_\epsilon } H \alpha _k = \iint _{\Sigma _1} \overline{\partial } H \wedge \alpha _k \end{aligned}$$

which is proportional to $(\overline{\partial } H, \overline{\alpha _k})_{A(\Sigma _1)}$. Observe also that $Q_1( \mathfrak {G} u) = Q(u)$ for all $u \in \mathcal {D}(A)$ by Proposition 4.9.

Let $h \in W_1 \subseteq \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ be arbitrary. By density of $\mathfrak {G} \mathcal {D}(A)$, there is a $u \in \mathcal {D}(A)$ such that

$$\begin{aligned} \Vert \mathfrak {G} u - h \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)}< \varepsilon . \end{aligned}$$

We then have

$$\begin{aligned} \Vert \mathfrak {G} \mathcal {P} u - h \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)}&\le \Vert \mathfrak {G} \mathcal {P} u - \mathfrak {G} u \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)} + \Vert \mathfrak {G} u - h \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)} \\&\le \Vert \mathfrak {G} \Vert \Vert \mathcal {P} u - u \Vert _{\mathcal {D}(A)} + \Vert \mathfrak {G} u - h \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)}. \end{aligned}$$

Now

$$\begin{aligned} \Vert Q(u) \Vert = \Vert Q_1(\mathfrak {G} u) \Vert = \Vert Q_1(\mathfrak {G} u - h) \Vert \le C' \Vert \mathfrak {G} u - h \Vert < C' \varepsilon \end{aligned}$$

so (4.12) yields that

$$\begin{aligned} \Vert \mathcal {P} u - u \Vert _{\mathcal {D}(A)} \le C C' \varepsilon . \end{aligned}$$

Thus

$$\begin{aligned} \Vert \mathfrak {G} \mathcal {P} u - h \Vert _{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)} \le (C C'\Vert \mathfrak {G} \Vert +1 ) \varepsilon . \end{aligned}$$

$\square$

We also define a transmission operator for exact one-forms as follows. For spaces $A(\Sigma )$, $A_{\mathrm{harm}}(\Sigma )$, etc., denote the subset of exact one-forms with a subscript e, i.e. $A_e(\Sigma )$, $A_{\mathrm{harm}}(\Sigma )_e$, etc.

Definition 4.15

For an exact one-form $\alpha \in A_{\mathrm{harm}}(\Sigma _2)_e$ let $h_2$ be a harmonic function on $\Sigma _2$ such that $dh_2= \alpha$. Let $h_1$ be the unique element of $\mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ with boundary values agreeing with $h_2$. Then we define

$$\begin{aligned} \mathfrak {O}_e(\Sigma _2,\Sigma _1):A_{\mathrm{harm}}(\Sigma _2)_e&\rightarrow A_{\mathrm{harm}}(\Sigma _1)_e \\ \alpha&\mapsto d h_1. \end{aligned}$$

The transmission from $A_{\mathrm{harm}}(\Sigma _1)_e$ to $A_{\mathrm{harm}}(\Sigma _2)_e$ is defined similarly.

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

Let R be the double of $\Sigma$; so A(R) has dimension 2g where g is the genus of $\Sigma$. Let $a_1,\ldots ,a_{2g}$ be a collection of smooth curves which generate the fundamental group of $\Sigma$. Let

$$\begin{aligned} c_k = \int _{a_k} \overline{\alpha } \end{aligned}$$

for $k=1,\ldots ,2g$. Since A(R) has dimension 2g, there is a $\beta \in A(R)$ such that

$$\begin{aligned} \int _{a_k} \beta = -c_k \end{aligned}$$

for $k=1,\ldots ,2g$. Thus $\overline{\alpha } + \beta$ is exact in $\Sigma$ and hence is equal to dh for some $h \in \mathcal {D}_{\mathrm{harm}}(\Sigma )$. But clearly $\overline{\partial } h =\overline{\alpha }$.

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

Given R and $\Sigma _i$ as above, let

$$\begin{aligned} V_k = \overline{A(\Sigma _k)} \cap \overline{A(R)}^\perp , \end{aligned}$$

and

$$\begin{aligned} V_k' = \{ \overline{\alpha } + \beta \in A_{\mathrm{harm}}(\Sigma _k)_e \,:\, \overline{\alpha } \in V_k\}, \end{aligned}$$

for $k=1,2$.

Theorem 4.18

Let R be a compact Riemann surface and let $\Gamma$ be a quasicircle separating R into components $\Sigma _1$ and $\Sigma _2$. If $\overline{\alpha } \in V_1$ then

$$\begin{aligned} \mathfrak {O}_e (\Sigma _2,\Sigma _1) T(\Sigma _1,\Sigma _2) \overline{\alpha } = \overline{\alpha } + T(\Sigma _1,\Sigma _1) \overline{\alpha }. \end{aligned}$$

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$\overline{\partial } J(\Gamma )h =0$.

Applying d to both sides of Theorem 4.13 and using this fact yields

$$\begin{aligned} - \mathfrak {O}_e(\Sigma _2,\Sigma _1) {\partial } J(\Gamma )_{\Sigma _2} h = dh - \partial J(\Gamma )_{\Sigma _1} h. \end{aligned}$$

The Theorem now follows from the remaining relations in Theorem 4.2. $\square$

For $k=1,2$ denote by

$$\begin{aligned} P(\Sigma _k): A_{\mathrm{harm}}(\Sigma _k)&\rightarrow A(\Sigma _k) \\ \overline{P}(\Sigma _k):A_{\mathrm{harm}}(\Sigma _k)&\rightarrow \overline{A(\Sigma _k)} \end{aligned}$$

the orthogonal projections onto the holomorphic and anti-holomorphic parts of a given harmonic one-form.

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

Since $d:\mathcal {D}_{\mathrm{harm}}(\Sigma _k) \rightarrow A_{\mathrm{harm}}(\Sigma _k)$ is norm-preserving (with respect to the Dirichlet semi-norm), it follows from Theorem 2.7 that there is a $c \in (0,1)$ which is independent of $\overline{\alpha }$ such that

$$\begin{aligned} \Vert \mathfrak {O}_e (\Sigma _2,\Sigma _1) \, T(\Sigma _1,\Sigma _2)\overline{\alpha } \Vert ^2 \le \frac{1+c}{1-c} \, \Vert T(\Sigma _1,\Sigma _2) \overline{\alpha } \Vert ^2. \end{aligned}$$

(4.13)

We will insert the identity

$$\begin{aligned} \mathfrak {O}_e (\Sigma _2,\Sigma _1) T(\Sigma _1,\Sigma _2) \overline{\alpha } = \overline{\alpha } + T(\Sigma _1,\Sigma _1) \overline{\alpha } \end{aligned}$$

(4.14)

of Theorem 4.18 into (4.13).

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 $\text {Re}(H_i)$.

Since $\alpha + T(\Sigma _1,\Sigma _1) \overline{\alpha }$ 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

$$\begin{aligned} \frac{1+c}{1-c} \, \Vert T(\Sigma _1,\Sigma _2) \overline{\alpha } \Vert ^2&= \frac{1 + c}{1-c} \, 2 \Vert \text {Re}( H_2) \Vert ^2_{\mathcal {D}_{\mathrm{harm}}(\Sigma _2)} \nonumber \\&\ge 2 \Vert \text {Re} (H_1 )\Vert ^2_{\mathcal {D}_{\mathrm{harm}}(\Sigma _1)} \nonumber \\&= 2 \Vert d \,\text {Re} (H_1) \Vert ^2_{A_{\mathrm{harm}}}(\Sigma _1) = 2 \Vert \text {Re} (d H_1) \Vert ^2_{A_{\mathrm{harm}(\Sigma _1)}} \nonumber \\&= 2 \Vert \text {Re} \left( \overline{\alpha } + T(\Sigma _1,\Sigma _1) \overline{\alpha }\right) \Vert ^2 = \nonumber 2 \Vert \text {Re} \left( \alpha + T(\Sigma _1,\Sigma _1) \overline{\alpha }\right) \Vert ^2 \\&= \Vert \overline{\alpha } \Vert ^2 + 2 \text {Re}\,(T(\Sigma _1,\Sigma _1) \overline{\alpha }, \alpha ) + \Vert T(\Sigma _1,\Sigma _1) \overline{\alpha }\Vert ^2. \end{aligned}$$

(4.15)

where we have used the fact that $\alpha + T(\Sigma _1,\Sigma _1) \overline{\alpha }$ is holomorphic. By Theorem 3.13 we have that

$$\begin{aligned} \Vert \overline{\alpha }\Vert ^2 = (\overline{\alpha },\overline{\alpha })&= \left( \overline{\alpha }, T(\Sigma _1,\Sigma _1)^* T(1,1) \overline{\alpha } + T(\Sigma _1,\Sigma _2)^* T(\Sigma _1,\Sigma _2) \overline{\alpha } \right) \\&= \left( \overline{\alpha }, T(\Sigma _1,\Sigma _1)^* T(\Sigma _1,\Sigma _1) \overline{\alpha } \right) + \left( T(\Sigma _1,\Sigma _2)^* T(\Sigma _1,\Sigma _2) \overline{\alpha } \right) \\&= \Vert T(\Sigma _1,\Sigma _1) \overline{\alpha } \Vert ^2 + \Vert T(\Sigma _1,\Sigma _2) \overline{\alpha } \Vert ^2. \end{aligned}$$

Combining this with (4.15) yields

$$\begin{aligned} \frac{1-c}{1+c} \text {Re}(\alpha , T(\Sigma _1,\Sigma _1) \overline{\alpha }) \le \frac{c}{1+c} \Vert \overline{\alpha }\Vert ^2 - \frac{1}{1+c} \Vert T(\Sigma _1,\Sigma _1) \overline{\alpha } \Vert ^2. \end{aligned}$$

Hence

$$\begin{aligned} \text {Re} (\alpha , T(\Sigma _1,\Sigma _1) \overline{\alpha })&\le \frac{c}{1+c} \Vert \overline{\alpha }\Vert ^2 + \frac{2c}{1+c} \text {Re} (\alpha , T(\Sigma _1,\Sigma _1) \overline{\alpha }) - \frac{1}{1+c} \Vert T(\Sigma _1,\Sigma _1) \overline{\alpha }\Vert ^2 \\&= c \Vert \overline{\alpha }\Vert ^2 - \frac{1}{1+c} \Vert T(\Sigma _1,\Sigma _1) \overline{\alpha } - c \alpha \Vert ^2. \end{aligned}$$

Applying this to $e^{i\theta } \alpha$, we see that the same inequality holds with the left hand side replaced by $e^{-2i\theta } \text {Re}(\alpha , T(\Sigma _1,\Sigma _1) \overline{\alpha })$ for any $\theta$. So

$$\begin{aligned} |\text {Re} (\alpha , T(\Sigma _1,\Sigma _1)\overline{\alpha })| \le c \Vert \overline{\alpha }\Vert ^2. \end{aligned}$$

Together with the fact that $T(\Sigma _1,\Sigma _1)^*=\overline{T(\Sigma _1,\Sigma _1)}$ this proves the theorem. $\square$

Remark 4.21

This gives another proof that $T(\Sigma _1,\Sigma _2)$ is injective. Let $\nu =\Vert T(\Sigma _1,\Sigma _1)\Vert <1$. Observe that if $\overline{\alpha } \in \overline{A(\Sigma _1)}$ is in $V_1$, then since the kernel of the operator $S(\Sigma _1)$ is holomorphic we have that $\overline{\alpha } \in \text {Ker}\, \overline{S}(\Sigma _1)$. Thus by Theorem 3.13

$$\begin{aligned} \Vert T(\Sigma _1,\Sigma _2) \overline{\alpha } \Vert _{A(\Sigma _2)}^2&= \Vert \overline{\alpha }\Vert ^2_{\overline{A(\Sigma _1)}} - \Vert T(\Sigma _1,\Sigma _1) \overline{\alpha } \Vert ^2_{\overline{A(\Sigma _1)}} \\&\ge (1-\nu ^2) \Vert \overline{\alpha } \Vert ^2_{\overline{A(\Sigma _1)}}. \end{aligned}$$

Since $1-\nu ^2 >0$ this completes the proof.

4.5 Isomorphism theorem for the Schiffer operator

In this section, we prove the isomorphism theorem for the Schiffer operators. 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 one-forms 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.

We show that $T(\Sigma _1,\Sigma _2)(V_1) \subseteq A(\Sigma _2)_e$. If we take $\overline{\alpha } \in V_1$, then since

$$\begin{aligned} \iint _{\Sigma _1,w} \partial _{\bar{z}} \partial _w g(w,w_0;z,q) \wedge \overline{\alpha (w)} = 0, \end{aligned}$$

for any fixed $q \in \Sigma _2$ we have (without loss of generality, because $T(\Sigma _1,\Sigma _2)$ is independent of q)

$$\begin{aligned} T(\Sigma _1,\Sigma _2) \overline{\alpha }(z)&= \frac{1}{\pi i} \iint _{\Sigma _1,z} \partial _z \partial _w g(w,w_0;z,q) \wedge \overline{\alpha (w)} \\&= d_z \frac{1}{\pi i} \iint _{\Sigma _1,z} \partial _w g(w,w_0;z,q) \wedge \overline{\alpha (w)} \in A(\Sigma _2)_e, \end{aligned}$$

and therefore $T(\Sigma _1,\Sigma _2)(V_1) \subseteq A(\Sigma _2)_e$.

To show that $T(\Sigma _1,\Sigma _2) (V_1)$ contains $A(\Sigma _2)_e$, let $\beta \in A(\Sigma _2)_e$, and let h be the unique element of $\mathcal {D}(\Sigma _2)_q$ such that $\partial h = \beta$. By Theorem 2.7 there is an $H \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ such that h and H have the same boundary values on $\Gamma$. Now $dH = \delta _1 + \overline{\delta _2}$ for $\delta _1, \delta _2 \in A(\Sigma _1)$ (specifically, $\delta _1 = \partial H$ and $\overline{\delta _2} = \overline{\partial } H$). Now by Theorem 4.10 we have

$$\begin{aligned} \beta (z)&= - { \partial _z} \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma ^{p_2}_\epsilon } \partial _wg(z,q;w) h(w) \\&= -{ \partial _z} \lim _{\epsilon \searrow 0} \frac{1}{\pi i} \int _{\Gamma ^{p_1}_\epsilon } \partial _wg(z,q;w) H(w) \\&= -{ \partial _z}\frac{1}{\pi i} \iint _{\Sigma _1} \partial _w g(z,q;w) \wedge \overline{\delta _2(w)} \\&= - \frac{1}{\pi i}\iint _{\Sigma _1} \partial _z \partial _w(z,q;w) \wedge \overline{\delta _2(w)} \end{aligned}$$

which proves that $A(\Sigma _2)_e \subseteq \text {Im}(T_R(\Sigma _1,\Sigma _2))$. Now we need to show that $\overline{\partial } H \in V_1$.

Since h is holomorphic by assumption, we have that $\partial h = dh$, hence

$$\begin{aligned}&- \frac{1}{\pi i} \iint _{\Sigma _1} \partial _z \partial _w g(w,w_0;z,q) \wedge \overline{\partial } H(w) = \partial h(z) = dh(z)\\&\quad = - \frac{1}{\pi i} \iint _{\Sigma _1} \partial _z \partial _w g(w,w_0;z,q) \wedge \overline{\partial } H(w) - \frac{1}{\pi i} \iint _{\Sigma _1} \overline{\partial }_{{z}} \partial _w g(w,w_0;z,q) \wedge \overline{\partial } H(w). \end{aligned}$$

Thus

$$\begin{aligned} - \frac{1}{\pi i} \iint _{\Sigma _1} \overline{\partial }_{{z}} \partial _w g(w,w_0;z,q) \wedge \overline{\partial } H(w) =0 \end{aligned}$$

(4.16)

for all $z \in \Sigma _2$. If we now let $\overline{\alpha } \in \overline{A(R)}$, then we have

$$\begin{aligned} (\overline{\partial } H, \overline{\alpha })_{\Sigma _1}&= - \frac{i}{2} \iint _{\Sigma _1} \overline{\partial } H(w) \wedge \alpha (w) \\&= - \frac{i}{2} \iint _{\Sigma _1,w} \overline{\partial } H(w) \wedge _w \iint _{R,z} K_R(w;z) \wedge _z \alpha (z) \\&= - \frac{i}{2} \iint _{R,z} \alpha (z) \wedge _z \iint _{\Sigma _1,w} \overline{K_R(z;w)} \wedge _w \overline{\partial } H(w) \end{aligned}$$

which is zero by (4.16). Thus $\overline{\partial } H \in V_1$ as claimed. $\square$

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

Let R be a compact Riemann surface and let $\Gamma$ be a quasicircle separating R into components $\Sigma _1$ and $\Sigma _2$. For any $h \in \mathcal {D}_{\mathrm{harm}}(\Sigma )$ such that $\overline{\partial } h \in V_1$

$$\begin{aligned} -T(\Sigma _1,\Sigma _1) \overline{\partial } h+\partial h \in A(\Sigma _1)_e. \end{aligned}$$

Proof

By Corollary 4.3 we need only show that $- T(\Sigma _1,\Sigma _1) \overline{\partial }h+ \partial h$ is exact. As usual let $\Gamma _\epsilon$ be level curves of $g_{\Sigma _1}$ for fixed z. Since $L_R$ and hence $T(\Sigma _1,\Sigma _1)$ is independent of q, we can assume that $q \in \Sigma _2$. By Stokes’ theorem

$$\begin{aligned}&- \frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma _{\epsilon }} \left( \partial _w g(w;z,q) - \partial _w g_{\Sigma _1}(w,z) \right) h(w) \\&\quad = - \frac{1}{\pi i} \iint _{\Sigma _1} \left( \partial _w g(w;z,q) - \partial _w g_{\Sigma _1}(w,z) \right) \wedge \overline{\partial } h(w) =: \omega (z). \end{aligned}$$

The integral on the left hand side exists by (4.1) and Theorem 3.2. Thus the right hand side is a well-defined function $\omega (z)$ on $\Sigma _1$.

Thus

$$\begin{aligned} - T(\Sigma _1,\Sigma _1) \overline{\partial } h (z)&= \partial \omega (z) \\&= d \omega (z) - \overline{\partial } \omega (z) \\&= d\omega (z) + \frac{1}{\pi i} \iint _{\Sigma _1} \partial _{\bar{z}} \partial _w g(w;z,q) \wedge \overline{\partial } h(w) \\&\quad - \frac{1}{\pi i} \iint _{\Sigma _1} \partial _{\bar{z}} \partial _w g_{\Sigma _1}(w,z) \wedge \overline{\partial } h(w) \\&= d\omega (z) + 0 + \overline{\partial } h \end{aligned}$$

where the middle term vanishes because $\overline{\partial } h \in V_1$, and we have observed that the last term is just the conjugate of the Bergman kernel applied to $\overline{\partial } h$. Thus $-T(\Sigma _1,\Sigma _1) \overline{\partial } h -\overline{\partial } h$ is exact. Since $dh = \partial h + \overline{\partial } h$ is exact, the claim follows. $\square$

The following theorem is in some sense a derivative of the jump decomposition.

Theorem 4.25

Let R be a compact Riemann surface and let $\Gamma$ be a quasicircle separating R into components $\Sigma _1$ and $\Sigma _2$ and $V_1'$ be given as in Definition 4.17.

$$\begin{aligned} \hat{\mathfrak {H}}:V_1'&\rightarrow A(\Sigma _1)_e \oplus A(\Sigma _2)_e \\ dh&\mapsto \left( \partial h - T(\Sigma _1,\Sigma _1) \overline{\partial } h, -T(\Sigma _1,\Sigma _2) \overline{\partial } h \right) \end{aligned}$$

is an isomorphism.

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 set $\mu = \alpha - \partial \tilde{h} + T(\Sigma _1,\Sigma _1) \overline{\partial } \tilde{h}$. By construction $\mu$ is holomorphic and it is exact by Proposition 4.24. Let $u \in \mathcal {D}(\Sigma _1)$ be such that $\partial u = \mu$. Setting $h = \tilde{h} + u$ we see that

$$\begin{aligned} \hat{\mathfrak {H}}(dh)&= \left( \partial h - T(\Sigma _1,\Sigma _1) \overline{\partial } h, -T(\Sigma _1,\Sigma _2) \overline{\partial } h \right) \\&= \left( \partial \tilde{h} + \mu - T(\Sigma _1,\Sigma _1) \overline{\partial } \tilde{h}, -T(\Sigma _1,\Sigma _2) \overline{\partial } \tilde{h} \right) \\&= (\alpha ,\beta ). \end{aligned}$$

Thus $\hat{\mathfrak {H}}$ is surjective.

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

Let R be a compact Riemann surface, and let $\Gamma$ be a quasicircle separating R into two connected components $\Sigma _1$ and $\Sigma _2$. Fix $q \in \Sigma _2$ and let $W_1$ be given as in Definition 4.12. Then the map

$$\begin{aligned} {\mathfrak {H}}: \mathcal {D}_{\mathrm{harm}}(\Sigma _1)&\rightarrow \mathcal {D}(\Sigma _1) \oplus \mathcal {D}(\Sigma _2)_q \\ h&\mapsto \left( \left. J_q(\Gamma ) h \right| _{\Sigma _1},\left. J_q(\Gamma ) h \right| _{\Sigma _2} \right) \end{aligned}$$

is a bounded isomorphism from $W_1$ to $\mathcal {D}(\Sigma _1) \oplus \mathcal {D}(\Sigma _2)_q$.

Proof

By Corollary 4.3 we have that the image of the map is in $\mathcal {D}(\Sigma _1) \oplus \mathcal {D}(\Sigma _2)$. Now since $g(w_0,w_0;z,q)=0$ by definition of g, (3.3) yields that $g(w,w_0;q,q)=0$. Therefore $\partial _w g(w;q,q)=0$ and so

$$\begin{aligned} J_q(\Gamma ) h(q) = -\frac{1}{\pi i} \lim _{\epsilon \rightarrow 0} \int _{\Gamma _\epsilon } \partial _w g(w,w_0;q,q) h(w) =0. \end{aligned}$$

Thus the image of the map is in $\mathcal {D}(\Sigma _1) \oplus \mathcal {D}_q(\Sigma _2)$.

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

Let R, $\Gamma$, $\Sigma _1$ and $\Sigma _2$ be as above. Let $H \in \mathcal {H}(\Gamma )$ be such that its extension h to $\mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ is in $W_1$. There are unique $h_k \in \mathcal {D}(\Sigma _k)$, $k=1,2$ so that if $H_k$ are their CNT boundary values then $H=- H_2 + H_1$. These unique $h_i$’s are given by

$$\begin{aligned} h_k = \left. J_q(\Gamma ) h \right| _{\Sigma _k} \end{aligned}$$

for $k=1,2$.

Proof

We claim that $h = - \mathfrak {O}(\Sigma _2,\Sigma _1) h_2 + h_1$, which would imply that $H = -H_2 + H_1$. Proposition 4.27 yields

$$\begin{aligned} {\mathfrak {H}} (- \mathfrak {O}(\Sigma _2,\Sigma _1) h_2 + h_1) = (h_1,h_2) = {\mathfrak {H}} h. \end{aligned}$$

Thus by Theorem 4.25 the claim follows.

We need only show that the solution is unique. Given any other solution $(\tilde{h}_1,\tilde{h}_2)$ we have that $-\mathfrak {O}(\Sigma _2,\Sigma _1)(\tilde{h}_2 - h_2) + (\tilde{h}_1 - h_1) \in \mathcal {D}_{\mathrm{harm}}(\Sigma _1)$ has boundary values zero, so by uniqueness of the extension it is zero. Thus

$$\begin{aligned} 0= {\mathfrak {H}}\left( -\mathfrak {O}(\Sigma _2,\Sigma _1)(\tilde{h}_2 - h_2) + (\tilde{h}_1 - h_1) \right) = (\tilde{h}_1- h_1,\tilde{h}_2 - h_2) \end{aligned}$$

which proves the claim. $\square$

Finally, we show that the condition for existence of a jump formula is independent of the choice of side of $\Gamma$.

Theorem 4.29

Let $\Gamma$ be a quasicircle and $V_k$, $V_k'$ be as in Definition 4.17 and $W_k$ as in Definition 4.12. Then

$$\begin{aligned} \mathfrak {O}(\Sigma _1,\Sigma _2) W_1&= W_2 \\ \mathfrak {O}_e (\Sigma _1,\Sigma _2) V_1'&= V_2'. \end{aligned}$$

Proof

The first claim follows immediately from Theorem 4.11. Assume that $\overline{\alpha _k} + \beta _k \in A(\Sigma _k)_e$ for $k=1,2$ are such that

$$\begin{aligned} \mathfrak {O}_e (\Sigma _1,\Sigma _2) (\overline{\alpha _1} + \beta _1) = \overline{\alpha _2} + \beta _2. \end{aligned}$$

In other words, there are $h_k \in \mathcal {D}_{\mathrm{harm}}(\Sigma _k)$ such that $dh_k = \overline{\alpha _k} + \beta _k$ and $\mathfrak {O}(\Sigma _1,\Sigma _2) h_1 = h_2$. By Stokes’ theorem, we have that for any $\overline{\alpha } \in \overline{A(R)}$

$$\begin{aligned} \left( \overline{\alpha _k}, \overline{\alpha } \right) _{A_{\mathrm{harm}}}(\Sigma _k)&= \frac{1}{2i} \iint _{\Sigma _k} \alpha \wedge \overline{\alpha _k} = \frac{1}{2i} \iint _{\Sigma _k} \alpha \wedge dh_k \\&= \lim _{\epsilon \searrow 0} \int _{\Gamma _\epsilon ^{p_k}} h_k(w)\, \alpha (w). \end{aligned}$$

Theorem 4.11 yields that $\overline{\alpha _1} \in V_1$ if and only if $\overline{\alpha _2} \in V_2$. $\square$