Duality by reproducing kernels

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.