Reproducing kernels for Hardy spaces on multiply connected domains (Q1921407)

Let \(D\) be a bounded planar domain with \(g\) holes \((g\geq 1)\). Specifically, \(D\) is a domain in \(\mathbb{C}\) bounded by a positively oriented boundary \(\Gamma= \partial D\), \(\Gamma= \Gamma_0\cup \Gamma_1 \cup\cdots \cup\Gamma_g\) where \(\Gamma_0, \Gamma_1, \dots, \Gamma_g\) are pairwise disjoint analytic Jordan curves with \(\Gamma_1, \dots, \Gamma_g\) bounding the \(g\) holes of \(D\) and \(\Gamma_0\) is the boundary of the unbounded component of \(\mathbb{C} \backslash \overline D\), \(\overline D=D \cup\Gamma\). There are two equivalent \(g\)-real dimensional tori of Hardy reproducing spaces of meromorphic functions on \(D\). One of these tori is the torus of least harmonic majorant Hardy spaces \(H^2_\lambda (D)\) parametrized by \(\lambda= (\lambda_1, \dots, \lambda_g)\) in \(\mathbb{T}^g\) as described in [\textit{D. Sarason}, Mem. Am. Math. Soc. 56, 78 p. (1965; Zbl 0127.07002)]. The second torus is the torus of Hardy spaces \(H^2_{\mathcal D} (\omega)\), \({\mathcal D} \in{\mathcal V}_a\), associated with divisors \(D\) of measures \(m\) on \(\Gamma\) representing evaluation at a point \(a\) of \({\mathcal D}\) of holomorphic functions on \(\overline D\). Here, every such representing measure is the restriction to \(\Gamma\) on a nonnegative symmetric meromorphic representing differential \(\omega\) on the Schottky double \(X\) of the bordered Riemann surface \(\overline D =D \cup \Gamma\). Recall that \(X\) is obtained topologically by gluing a second copy \(D'\) of \(D\) to \(\overline D\) along \(\Gamma\) and the complex atlas on \(X\) is determined by a reselection of the complex structure of \(D\) to \(D'\), i.e. the involution \(J:X\to X\) fixing \(\Gamma\) and interchanging a point \(p\in D\) with its twin point \(p'\in D'\) is conjugate-holomorphic. The explicit bijection between the two tori \(\{H^2_\lambda (D): \lambda\in \mathbb{T}^2\}\) and \(\{H_{\mathcal D}^2: {\mathcal D} \in{\mathcal V}_a\}\) is given in [Ill. J. Math. 35, No. 2, 286-311 (1991; Zbl 0806.46060)]. The main result of the paper is in providing explicit formulas for the reproducing kernels of these Hardy spaces. These formulas are expressed in terms of theta functions defined on the Jacobian variety of \(X\). Moreover, these results can be extended to the more general case where the planar domain \(D\) is replaced by a finite bordered Riemann surface. As an application, the authors use the above explicit formulas when \(g=1\) to settle a conjecture of \textit{M. B. Abrahamse} [Mich. Math. J. 26, 195-205 (1979; Zbl 0442.30034)] concerning the Pick-Nevanlinna interpolation problem on an annulus. As a second application, the authors also give an explicit formula, in terms of theta functions, for the curvature in the sense of \textit{M. J. Cowen} and \textit{R. G. Douglas} [Acta Math. 141, 187-261 (1978; Zbl 0427.47016)] of rank 1 bundle shift operators.