Duality by reproducing kernels
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...
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| Language: | English |
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Hindawi Limited
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203206037 |
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doaj-2d6945675bdf4acf89cf3bf7659d3b012020-11-25T01:01:17ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003632739510.1155/S0161171203206037Duality by reproducing kernelsA. Shlapunov0N. Tarkhanov1Krasnoyarsk State University, pr. Svobodnyi 79, Krasnoyarsk 660041, RussiaUniversität Potsdam, Institut für Mathematik, Postfach 60 15 53, Potsdam 14415, GermanyLet 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.http://dx.doi.org/10.1155/S0161171203206037 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
A. Shlapunov N. Tarkhanov |
| spellingShingle |
A. Shlapunov N. Tarkhanov Duality by reproducing kernels International Journal of Mathematics and Mathematical Sciences |
| author_facet |
A. Shlapunov N. Tarkhanov |
| author_sort |
A. Shlapunov |
| title |
Duality by reproducing kernels |
| title_short |
Duality by reproducing kernels |
| title_full |
Duality by reproducing kernels |
| title_fullStr |
Duality by reproducing kernels |
| title_full_unstemmed |
Duality by reproducing kernels |
| title_sort |
duality by reproducing kernels |
| publisher |
Hindawi Limited |
| series |
International Journal of Mathematics and Mathematical Sciences |
| issn |
0161-1712 1687-0425 |
| publishDate |
2003-01-01 |
| description |
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. |
| url |
http://dx.doi.org/10.1155/S0161171203206037 |
| work_keys_str_mv |
AT ashlapunov dualitybyreproducingkernels AT ntarkhanov dualitybyreproducingkernels |
| _version_ |
1725209723670626304 |