International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 6, Pages 327-395

Duality by reproducing kernels

A. Shlapunov1 and N. Tarkhanov2

1Krasnoyarsk State University, pr. Svobodnyi 79, Krasnoyarsk 660041, Russia
2Universität Potsdam, Institut für Mathematik, Postfach 60 15 53, Potsdam 14415, Germany

Received 10 June 2002

Copyright © 2003 A. Shlapunov and N. Tarkhanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 𝒟X. 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.