Interpolating sequences in harmonically weighted Dirichlet spaces
Given a positive finite measure \(\mu\) defined on the unit circle, the weighted Dirichlet space \(D(\mu)\) is the Hilbert space of analytic functions on the unit disk \(D\) such that \(\int_D |f'(z)|^2 P_\mu(z) \,dA(z)\) is finite, where \(dA\) stands for the normalized area measure on the disk and \(P(\mu)\) for the Poisson integral of \(\mu\). The author first shows the following result for the measure \(\mu = a \delta_\lambda\) for \(\lambda \in\partial D\) and \(a >0\). Theorem. Suppose that a sequence \((z_j) \subset D\) is \(D(a\delta_\lambda)\)-separated and that \(\sum_j \|K_{z_j}\|^{-2}_{D(a\delta_1)} \delta_{z_j}\) is a \(D(a\delta_\lambda)\)-Carleson measure. Then \((z_j)\) is uniformly separated and the sequence \[ \frac{1 -|z_j|^2}{| 1-z_j|^2} \] belongs to \(l^1\). \ Combining a result due to \textit{A. Serra} [Proc. Am. Math. Soc. 131, No. 9, 2809--2817 (2003; Zbl 1039.30019)] with this theorem shows that interpolating sequences for the space of multipliers \(M\big(D(a\delta_\lambda)\big)\) can be characterized in terms of a separation condition and a Carleson-measure condition. This is the first example of a space with Nevanlinna-Pick kernel with non-radially symmetric weights in which this characterization remains true. Moreover, the author obtains a generalization of the above theorem to the spaces \(D(\mu)\) in the case where \(\mu\) is a finitely atomic measure. The proof is a combination of results of \textit{S. McCullough} and \textit{T. T. Trent} [J. Funct. Anal. 178, No. 1, 226--249 (2000; Zbl 0973.47003)], \textit{S. Richter} and \textit{C. Sundberg} [Mich. Math. J. 38, No. 3, 355--379 (1991; Zbl 0768.30040)], and \textit{A. Aleman} [The multiplication operators on Hilbert spaces of analytic functions. Habilitationsschrift. Fernuniversität Hagen (1993)] that reduce the problem to the above theorem from which the statement follows by another result due to Serra [loc. cit.].
scientific article; zbMATH DE number 5842770