Local boundary Morera theorems
The following boundary Morera theorem is known [\textit{J. Globevnik}, J. Geom. Anal. 3, No. 3, 269-277 (1993; Zbl 0785.32008)]: Let \(D\subset \mathbb{C}^N\), \(N\geq 2\) be a bounded open set with \(C^2\) boundary and \({\mathcal L}\) be an open connected set of affine complex hyperplanes in \(\mathbb{C}^N\) containing a hyperplane that misses \(\overline D\). Let \(E=\cup_{\Lambda \in{\mathcal L}}\), \(\Gamma= E\cap bD\). Suppose that \(f\in C(\Gamma)\) has the Morera property with respect to every \(\Lambda\in {\mathcal L}\) that meets \(bD\) transversely. Then \(f\) is a CR function on \(\Gamma\). It was also shown by an example that the condition that \({\mathcal L}\) contains a hyperplane that misses \(\overline D\) cannot be dropped in general. We show that there are quite general geometric situations in which the condition that \({\mathcal L}\) contains a hyperplane missing \(\overline D\) is not necessary. This note also treats the open question whether an open connected set \({\mathcal L}\) of affine complex hyperplanes can be replaced by an open connected set of complex lines with the Morera property suitably modified. Our second result shows that there are geometric situations in which one can give an affirmative answer to this question.
scientific article; zbMATH DE number 1419358