Hyperfunction boundary values and a generalized Bochner-Hartogs theorem. (English) Zbl 0358.32014

Several complex Variables, Proc. Symp. Pure Math. 30, Part 1, Williamstown 1975, 187-193 (1977).
[This article was published in the book announced in this Zbl 0342.00011.]
This note is an announcement of several results which appear in full detail and with proofs in Abh. Math. Sem. Univ. Hamburg. Let \(X\) be a complex analytic manifold of dimension \(n\) which is countable at infinity. Let \(M\subset X\) be a real analytic hypersurface and let \(H^{p,q}(M)\) denote the cohomology of the hyperfunction \(\bar{\partial}_b\) complex
\[ \mathcal{B}^{p,0}(M) \overset{\bar{\partial}_b}{\rightarrow} \mathcal{B}^{p,1}(M) \rightarrow \ldots \overset{\bar{\partial}_b}{\rightarrow} \mathcal{B}^{p,n-1}(M) \]
where \(\mathcal{B}^{p,q}(M)\) is the space of hyperfunction \((p,q)\)-forms on \(M\). Then for \(u\in H^{p,q}(X-M)\) the “saltus” \(bu\in H^{p,q}(M)\) can be defined in a canonical way and a generalization of a theorem of A. Andreotti and C. D. Hill [Ann. Scuola norm. sup. Pisa, Sci. fis. mat., III. Ser. 26, 325-363 (1972; Zbl 0256.32007)] is obtained. For example, under the assumption \(H^{0,1}(X) = 0\) the generalization implies the exactness of the sequence
\[ 0\to \mathcal{O}(X)\to \mathcal{O}(X\setminus M)\to H^{0,0}(M)\to O. \]
The hyperfunction version of Bochner’s generalization of Hartogs’ theorem is as follows. Suppose \(H_{\mathrm{*}}^{1,0}(X) =0\) and \(n>1\). Let \(M=\partial \Omega\) be the real analytic boundary of an open, relatively compact set \(\Omega \subset X\) such that \(X\setminus\Omega\) is connected. Then \(\tilde{b} \colon \mathcal{O}(\Omega )\to H^{0,0}(M)\) is an isomorphism where \(\tilde{b}u=\tilde{b}u\) for \(u\in \mathcal{O}(\Omega )\) and \(\tilde{u} \in \mathcal{O} (X\setminus \partial \Omega)\) is the extension of \(u \in \mathcal{O}(\Omega )\) by zero to \(X\setminus \partial \Omega\). Finally, a local generalization of a result of H. Lewy is obtained and the H. Lewy example is discussed.
Reviewer: M. Schottenloher


32D15 Continuation of analytic objects in several complex variables
46F15 Hyperfunctions, analytic functionals