Multivalued analytic continuation of the Cauchy transform
The Cauchy transform \[ {\widehat{g}}(z) = -\frac{1}{\pi} \int \frac{g(w)}{w - z} \,dA \] well-known in potential theory and applications, is studied. The following notion of the multivalued analytic continuation of the Cauchy transform of a characteristic function \(\chi_{\Omega}\) of a bounded domain \(\Omega\) is introduced: the Cauchy transform \({\widehat{\chi}}_{\Omega}\) possesses a multivalued analytic continuation in a neighborhood of \(z_0\in \partial \Omega\) if for certain positive \(r\) there exists an analytic function \(f(z)\) in \(B_r(z_0)\) such that for any (connected) component \(K\) of \(B_r(z_0)\setminus \Omega\) \[ {\widehat{\chi}}_{\Omega} = f(z) + c_K,\quad z\in K, \] where \(c_K\) is a constant depending on \(K\). The author proves the following property of such a continuation: if \({\widehat{\chi}}_{\Omega}\) has a multivalued continuation in \(B_r(z_0)\), \(z_0\in \partial \Omega\), then (a) \(\partial \Omega \cap {B}_r(z_0)\) has Lebesgue measure zero; (b) for any \(a\in \partial \Omega \cap {B}_r(z_0)\) \[ \int_{B_{\rho}(a)} \frac{\chi_{\Omega}(z)}{| z - a| ^2} \,dA = \infty \] for certain \(\rho > 0\). Connection of the introduced notion with the quadrature domains and approximation by rational functions are discussed too. An open problem is formulated.
scientific article; zbMATH DE number 5014106
scientific article