Gleason's problem and homogeneous interpolation in Hardy and Dirichlet-type spaces of the ball
The authors consider the reproducing kernel Hilbert spaces \({\mathcal H}_r\) with reproducing kernels \((1-\langle z,w\rangle)^{-r}\) for \(r>0\) and \(\log (1-\langle z,w\rangle)^{-1}\) for \(r=0\) on the unit ball of \({\mathbb C}^N\). These are spaces which are often called Dirichlet-type spaces or weighted Hardy (Bergman) spaces. The authors solve Gleason's problem for \({\mathcal H}_r\) by using the method of backward shift operators. Their proof actually works on any reproducing kernel Hilbert space with reproducing kernels which are holomorphic in \(\langle z,w\rangle\). By using these results, the authors obtain a description of the set of solutions of certain finite homogeneous interpolation problems in spaces of the form \({\mathcal H}_r^{n\times 1}\). These results extend the previous work of the authors for \({\mathcal H}_1\) [Integral Equations Oper. Theory 42, 1--21 (2001; Zbl 1010.47012)] to general \(r\).
scientific article; zbMATH DE number 1866770