Removable singularities for \(L^ p\) \(CR\) functions
Let \(\Omega\) be a bounded domain in \(\mathbb{C}^ n\) with \(C^ 2\)-smooth boundary \(\partial \Omega\). One says that a closed subset \(X \subset \partial \Omega\) is removable for \(L^ p\) CR-functions if, for each \(f \in L^ p (\partial \Omega,d \sigma)\) satisfying the tangential Cauchy- Riemann equations on \(\partial \Omega \backslash X\), there is \(F\) in the Hardy space \(H^ p (\Omega)\) with boundary values of \(\sigma\)-almost everywhere on \(\partial \Omega \backslash X\). Here \(\sigma\) is the induced \((2n-1)\)-dimensional Euclidean measure on \(\partial \Omega\). The main result of the paper is the following Theorem 2. Let \(\Omega\) be a smoothly bounded strongly starlike domain in \(\mathbb{C}^ n\), with strictly pseudoconvex boundary \(\partial \Omega\) of class \(C^ 2\). Let \(X\) be a closed subset of \(\partial \Omega\), and assume that there exists a smooth family \(\psi\) of analytic disks in \(\Omega\) with parameter space \(V \subset \mathbb{C}^{n-1}\), and set \(K<V\) of \(2n-2\) Lebesgue measure zero, such that \(X \subset \psi (K, \partial \Omega)\). If \(F\) is a \((\partial \Omega \backslash X)\)-local \(H^ p\) extension of \(f \in L^ p (\partial \Omega)\) to \(\Omega\), then \(F \in H^ p (\Omega)\).
scientific article; zbMATH DE number 575624