×

Invariant subspaces in Bergman spaces and the biharmonic equation. (English) Zbl 0833.46044

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 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.
Reviewer: R.Mortini (Metz)

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI