## Application of half-order differentials on Riemann surfaces to quadrature identities for arc-length.(English)Zbl 0652.30029

A region $$\Omega$$ in $${\mathbb{C}}$$ is called a quadrature domain for arc- length if there exist finitely many points $$z_ k\in \Omega$$ and complex numbers $$a_{kj}$$ (1$$\leq k\leq m$$, $$0\leq j\leq n_ k-1)$$ such that $\int_{\partial \Omega}f(z)| dz| =\sum^{m}_{k=1}\sum^{n_ k-1}_{j=0}a_{kj}f^{(j\quad)}(z_ k)$ for every function f in some suitable test class of holomorphic functions on $$\Omega$$. Here $$\partial \Omega$$ is assumed to consist of finitely many components each of which is a continuum of finite one-dimensional Hausdorff measure. In this paper the test class is typically the Hilbert space E 2($$\Omega)$$. The analogous situation of quadrature domains for area measure has been investigated earlier by a number of authors.
The main result of the paper is a characterization of quadrature domains for arc length in terms of conformal mappings from standard regions. Suppose W is a finitely connected region in $${\mathbb{C}}$$ such that each component of $$\partial W$$ is an analytic closed Jordan curve. Let $$\hat W$$ denote the Schottky double of W across $$\partial W$$. Assume g:W$$\to \Omega$$ is a conformal mapping. Then $$\Omega$$ is a quadrature domain for arc-length if and only if $$\sqrt{dg}$$ extends to a meromorphic half-order differential on $$\hat W.$$ In addition, the existence of quadrature domains for arc-length of all conformal types under consideration is established by showing that for a given W there always exist univalent functions g on W such that $$\sqrt{dg}$$ extends to a meromorphic half- order differential on $$\hat W.$$ The author also investigates quadrature domains containing $$\infty$$ and gives other characterizations of quadrature domains.
Reviewer: D.Minda

### MSC:

 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 30D55 $$H^p$$-classes (MSC2000)

### Keywords:

quadrature domain; Schottky double; half-order differential
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### References:

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