Theorems of Morera type for domains with weak cone condition (Q1902769)

The author gives two theorems of Morera type in the complex plane and three multidimensional theorems of Morera type. We consider two typical results. If \(G\) is an open set in the complex plane \(\mathbb{C}\), we write \(G\in A(\alpha, H)\) if for any \(z\in G\) there exists \(\theta\in [0, 2\pi)\) such that \(z+ T(\alpha, H) e^{i\theta}\in G\), where \[ T(\alpha, H):= \{z\in \mathbb{C}: 0\leq \text{Re } z\leq H, |\arg z|\leq \alpha\}. \] Theorem 1: Let \(\alpha\leq \pi/3\), \(G\in A(\alpha, H)\), \(f\in C(G)\), and \(\int_{\partial S} f(z) dz= 0\) for any circular sector \(S\subset G\) with a fixed opening \(\varphi\leq \alpha\) and a radius \(r< H\sin \alpha/2(1+ \sin \alpha)\). Then \(f\) is holomorphic in \(G\). Let \(\omega_\nu(z):= dz\wedge dz[\nu]\), \(1\leq \nu\leq n\), \(z= (z_1,\dots, z_n)\in \mathbb{C}^n\) and \(B^n_R\subset \mathbb{C}^n\) be the open ball \(|z|< R\). Theorem 4: Assume \(0< \varepsilon< R\), \(f\in C(B^n_R)\), and \(\int_{\partial D} f(z) \omega_\nu(z)= 0\) for \(1\leq \nu\leq n\) and for any ball \(D\subset B^n_R\) with a center in \(B^n_\varepsilon\). Then \(f\) is holomorphic in \(B^n_R\).