zbMATH — the first resource for mathematics

Duality between harmonic and Bergman spaces. (English) Zbl 1239.32031
Barkatou, Y. (ed.) et al., Geometric analysis of several complex variables and related topics. Proceedings of the workshop, Marrakesh, Morocco, May 10–14, 2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5257-6/pbk). Contemporary Mathematics 550, 161-171 (2011).
Author’s abstract: In this paper we study the duality of the harmonic spaces on the annulus \(\Omega = \Omega_{1} \setminus \overline{\Omega^{-}}\) between two pseudoconvex domains with \(\Omega^{-} \subset\subset \Omega_{1}\) in \(\mathbb{C}^{n}\) and the Bergman spaces on \(\Omega^{-}\). We show that on the annulus \(\Omega\), the space of harmonic forms for the critical case of \((0,n-1)\)-forms is infinite dimensional and it is dual to the Bergman space on the pseudoconvex domain \(\Omega^{-}\). The duality is further identified explicitly by the Bochner-Martinelli transform, generalizing a result of L. Hörmander [Ann. Inst. Fourier 54, No. 5, 1305–1369 (2004; Zbl 1083.32033)].
For the entire collection see [Zbl 1221.32001].
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A36 Bergman spaces of functions in several complex variables