Bergman interpolation on finite Riemann surfaces. I: Asymptotically flat case (Q1713989)

The author studies the Bergman space interpolation problem on open Riemann surfaces $X$ obtained from a compact Riemann surface by removing a finite number of points. Such a surface is equipped with what the author calls an asymptotically flat conformal metric, i.e., a complete metric $\omega$ with zero curvature outside a compact subset. A discrete set $\Gamma\subset X$ is said to be uniformly separated with respect to some distance function $\rho$ if for $\gamma_1, \gamma_2\in \Gamma$, $\gamma_1\not=\gamma_2$, one has $\inf\{\rho(\gamma_1,\gamma_2)\}>0$. \par Let $\psi$ be a weight function on $X$ with the values in $[-\infty, \infty)$. The following two Hilbert spaces \[ \begin{split} H^2(X, e^{-\psi}\omega) & =\left\{g\in O(X): \int_X |g|^2e^{-\psi}\omega <+\infty\right\}, \\ \ell^2(\Gamma, e^{-\psi}) & =\left\{f: \Gamma\to\mathbb{C}, \sum_{\gamma\in\Gamma} |f(\gamma)|^2\,e^{-\psi(\gamma)}<+\infty\right\}, \end{split} \] and the restriction map $R_\gamma: H^2(X, e^{-\psi}\omega) \to \ell^2(\Gamma, e^{-\psi})$ are the main objects under consideration in the paper. $\Gamma$ is called an interpolation sequence, if $R_\Gamma$ is surjective. \par Sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces are established. For certain weights these conditions are shown to be necessary.