Traces of pluriharmonic functions on the boundaries of analytic varieties (Q1319350)

Let \(\Omega\) be a bounded strictly pseudoconvex domain of class \({\mathcal C}^ \infty\) in \(\mathbb{C}^ 2\) and \(E\) be an analytic variety properly embedded into \(\Omega\) such that \(E\) is of class \({\mathcal C}^ \infty\) near \(\partial\Omega\) and transversal to \(\partial\Omega\). Let \(PH^ \infty(\Omega)\) be the space of all real-valued functions of class \({\mathcal C}^ \infty\) on \(\overline\Omega\), pluriharmonic on \(\Omega\). Denote by \(PH^ \infty_{\partial E}(\Omega)\) the space of all traces on \(\partial E\) of functions of \(PH^ \infty(\Omega)\). We prove that \(PH^ \infty_{\partial E}(\Omega)\) is a closed subspace of finite codimension in the space \({\mathcal C}^ \infty(\partial E)\) of all smooth real-valued functions on \(\partial E\) and characterize this codimension in terms of geometrical properties of the variety \(E\). As a consequence of this result, we answer a question of J. Bruna and J. Ortega.