Multidimensional version of the boundary Morera theorem
Let \(D\subset\mathbb C^n\), \(n\geq 2\), be a bounded domain with connected \(\mathcal C^2\)-smooth boundary and let \(f\in\mathcal C(\partial D)\). Consider a family \(L\) of holomorphic entire curves \(\ell:\mathbb C\to\mathbb C^n\) and assume that \(\int_{\ell\cap\partial D}f(\zeta) d\zeta_k=0\), \(k=1,\dots,n\), for any \(\ell\in L\). The problem is to characterize those families \(L\) which guarantee that \(f\) extends holomorphically to \(D\). The case of complex lines was solved by \textit{J. Globevnik} and \textit{E. L. Stout} [Duke Math. J. 64, 571-615 (1991; Zbl 0760.32002)]. The author presents a solution in the case where the family \(L\) consists of curves of the form \(\ell(t)=z+\Phi(b_1t^{k_1},\dots,b_nt^{k_n})\) with \(z, (b_1,\dots,b_n)\in\mathbb C^n\), \(k_1,\dots,k_n\in\mathbb N\), and \(\Phi:\mathbb C^n\rightarrow\mathbb C^n\) being an entire mapping for which \(\Phi(0)=0\) and there exists another entire mapping \(\Psi:\mathbb C^n\rightarrow\mathbb C^n\) such that \((\Psi\circ\Phi)(z_1,\dots,z_n)\equiv(z_1^{p_1},\dots,z_n^{p_n})\), where \(p_1,\dots,p_n\in\mathbb N\) and \(p_1k_1=\dots=p_nk_n\). The theorem generalizes also results by \textit{A. M. Kytmanov} and the author [J. Nat. Geom. 16, No. 1-2, 29-48 (1999; Zbl 0949.32004)].
scientific article; zbMATH DE number 1541509