Duality by reproducing kernels - A. Shlapunov; N. Tarkhanov (International Journal of Mathematics and Mathematical Sciences)
Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write 𝒮A(𝒟) for the space of solutions of the system Au=0 in a domain 𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of 𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of 𝒟, 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 domains 𝒟 which possess certain convexity properties with respect to A.