International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 6, Pages 327-395
doi:10.1155/S0161171203206037
Duality by reproducing kernels
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.
Abstract
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.