Invariant subspaces in Bergman spaces and the biharmonic equation
The authors continue their research on contractive divisors in the Bergman space \(A^p (\Omega)\) of all \(p\)-integrable analytic functions on a bounded planar domain \(\Omega\). Results of \textit{H. Hedenmalm} [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)]\ for the case \(p=2\) and the authors [Pac. J. Math. 157, 37-56 (1993; Zbl 0782.30027)]for \(1\leq p\leq \infty\) on \(z\)-invariant subspaces of \(A^p\) defined by \(A^p\) zero sets are generalized to \(A^p\)-spaces with \(0< p< 1\) as well as to arbitrary \(z\)-invariant subspaces of \(A^p (\mathbb{D})\). It is also shown that parts of the theory can be extended to Bergman spaces on simply connected Jordan domains with analytic boundary. The key tool of this theory is an integral formula involving the biharmonic Green function. The paper concludes with several interesting questions.
scientific article; zbMATH DE number 682539