The edge of the wedge theorem for separately holomorphic functions with singularities (Q842342)

Let \(\varGamma_\varepsilon:=\{x\in\mathbb R^n: \exists_k:\; 0<x_k<\varepsilon,\;x_j=0,\;j\neq k\}\), \(\varGamma:=\{x\in\mathbb R^n: \exists_k:\; x_k>0,\;x_j=0,\;j\neq k\}\), \(\widehat\varGamma_\varepsilon:=\text{convex}(\varGamma_\varepsilon)\), \(\widehat\varGamma:=\{x\in\mathbb R^n: x_j>0,\;j=1,\dots,n\}\). Let \(\varOmega\subset\mathbb R^n\) be a \(\mathcal C^1\)--domain. For \(x\in\partial\varOmega\) let \(v_x\) be the normal vector to \(\partial\varOmega\) at \(x\). The main result of the paper is the following theorem. Assume that \(\varOmega\) seats on one side of its boundary. Let \(K\subset\partial\varOmega\) be a compact set such that \(v_x\in\pm\widehat\varGamma\), \(x\in K\). Then there exist constants \(\varepsilon_0, c_1, c_2>0\) such that for any \(0<\varepsilon<\varepsilon_0\) and \(x\in K\), every function \(f\) continuous on \(\varOmega\cap B(x,c_1\varepsilon)\) that extends as a separately holomorphic function to \(B(x,c_1\varepsilon)+i\varGamma_{c_2\varepsilon}\), continuous up to \(\varOmega\cap B(x,c_1\varepsilon)\), extends holomorphically to \((\varOmega\cap B(x,\varepsilon))+i\widehat\varGamma_\varepsilon\). Let \(\varOmega=\mathbb R^n\setminus V\), where \(V\) is a \(\mathcal C^1\)--hypersurface. Using the above result, the authors get the following theorem. Let \(K\subset V\) be a compact such that \(v_x\in\pm\widehat\varGamma\), \(x\in K\). Then there exist \(\varepsilon_0, c_1, c_2>0\) such that for any \(0<\varepsilon<\varepsilon_0\) and \(x\in K\), every function \(f\) continuous on \((\mathbb R^n\setminus V)\cap B(x,c_1\varepsilon)\) that extends as a separately holomorphic function to \(B(x,c_1\varepsilon)+i\varGamma_{c_2\varepsilon}\), continuous up to \((\mathbb R^n\setminus V)\cap B(x,c_1\varepsilon)\), extends holomorphically to \(B(x,\varepsilon)+i\widehat\varGamma_\varepsilon\). Moreover, if \(v_x\in\pm\widehat\varGamma\), \(x\in V\), then any function \(f\) separately holomorphic in \(\mathbb R^n+i\varGamma\), continuous up to \(\mathbb R^n\setminus V\), extends holomorphically to \(\mathbb R^n+i\widehat\varGamma\), continuous up to \(\mathbb R^n\setminus V\).