Cohomology of Sheaves of Holomorphic Functions Satisfying Boundary Conditions on Product Domains

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.