Some families of complex lines sufficient for holomorphic extension of functions (Q647860)

Let \(D\subset \mathbb C^n\) (\(n>1\)) be a bounded domain with connected \(\mathcal C^2\)-boundary and \(\Gamma\) be a germ of a real-analytic manifold of real dimension \(2n-2\). Denote by \(l_{z,b}\) the complex line passing through \(z\) in direction \(b\). The authors prove that, under certain regularity conditions on \(\partial D\) and \(\Gamma\), any function \(f\) that is real-analytic on \(\partial D\) and satisfies the generalized Morera condition \[ \int_{\partial D\cap l_{z,b}} f(z+tb) t^m dt =0 \] for any \(z\in \Gamma\), \(b\in\mathbb{CP}^{n-1} \), and some non-negative integer \(m\), is holomorphically extendable to the domain \(D\).