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Transactions of the American Mathematical Society

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Cohomology of sheaves of holomorphic functions satisfying boundary conditions on product domains

Author: Alexander Nagel
Journal: Trans. Amer. Math. Soc. 172 (1972), 133-141
MSC: Primary 32C35; Secondary 32F15
MathSciNet review: 0320363
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Abstract: This paper considers sheaves of germs of holomorphic functions which satisfy certain boundary conditions on product domains in $ {{\mathbf{C}}^n}$. Very general axioms for boundary behavior are given. This includes as special cases $ {L^p}$ boundary behavior, $ 1 \leq p \leq \infty $; continuous boundary behavior; differentiable boundary behavior of order $ m,0 \leq m \leq \infty $, with an additional Hölder condition of order $ \alpha ,0 \leq \alpha \leq 1$, on the $ m$th derivatives. A fine resolution is constructed for those sheaves considered, and the main result of the paper is that all higher cohomology groups for these sheaves are zeŕo.

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Keywords: Sheaf cohomology, fine resolution, product domain, $ {L^p}$ boundary behavior, continuous boundary behavior, differentiable boundary behavior with Hölder condition, Poincaré lemma, Dolbeault lemma
Article copyright: © Copyright 1972 American Mathematical Society

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