Integral conditions for the vanishing of the cohomology of open sets in \(\mathbb C^n\) (Q2882928)

``We develop and extend some techniques introduced in [the authors, C. R., Math., Acad. Sci. Paris 344, No. 11, 677--680 (2007; Zbl 1153.32022)] to find integral conditions for the vanishing of the cohomology of open bounded sets in \(\mathbb{C}^n\) with values in the sheaf of holomorphic functions.''NEWLINENEWLINENEWLINE Technically, an important step is the consideration of hyperholomorphic functions; see below an extract of the last part of introduction: NEWLINENEWLINENEWLINE``In a recent paper [loc. cit.], we used some ideas from quaternionic analysis to study the vanishing of the cohomology groups for open sets in \(\mathbb{C}^2\). Quaternions, however, are not helpful if one wants to use similar ideas to discuss the case of dimension higher than two. The use of several quaternionic variables, as well, does not seem to be of any help. But fortunately, some recent advances in Clifford analysis can help finding criteria for the vanishing of cohomology in higher dimensions. That this is indeed possible is due to the introduction of a more general definition of hyperholomorphicity.''