Duality between Harmonic and Bergman spaces

In this paper we study the duality of the harmonic spaces on the annulus Ω = Ω1 \Ω − between two pseudoconvex domains with Ω− ⊂⊂ Ω1 in Cn and the Bergman spaces on Ω−. We show that on the annulus Ω, the space of harmonic forms for the critical case on (0, n−1)-forms is infinite dimensional and it is dual to the the Bergman space on the pseudoconvex domain Ω−. The duality is further identified explicitly by the Bochner-Martinelli transform, generalizing a result of Hörmander. Introduction Let Ω− and Ω1 be two bounded pseudoconvex domains in C with Ω− ⊂⊂ Ω1. In this paper we study the duality of the harmonic spaces on the annulus Ω = Ω1 \ Ω − and the Bergman spaces on Ω−. This paper is inspired by a recent paper of Hörmander [Hö 2] where the null space of the ∂̄-Neumann operator on a spherical shell as well as on an ellipsoid in C has been computed by explicit formula for the critical case for (0, n− 1)-forms. The ∂̄-Neumann problem on the annulus has been studied in [Sh 1] on an annulus between two pseudoconvex domains in C or in a hermitian Stein manifold. When the boundary is smooth, the closed range property and boundary regularity for ∂̄ were established in the earlier work (see [BS] or [Sh1]) for 0 < q ≤ n− 1 and n ≥ 2. In the case when 0 < q < n − 1, the space of harmonic forms is trivial. In this paper, we will study the critical case when q = n− 1 on the annulus Ω. In this case the space of harmonic forms is infinite dimensional. Our goal is to establish the duality between the harmonic forms in the critical degree with the Bergman spaces on the domain Ω−. In the first section, we recall the Hodge decomposition theorem on the annulus between two pseudoconvex domains. In the second section we establish the duality between the harmonic forms with coefficients in the Sobolev W (Ω) spaces with the Bergman spaces on Ω−. We then refine the duality to duality between L spaces in Section 3. 2010 Mathematics Subject Classification. Primary 32W05, 35N15, 58J32.