Peak sets in weakly pseudoconvex domains
The paper gives a necessary and a sufficient condition for a closed subset of the boundary of a pseudoconvex domain in \({\mathbb{C}}^ n\) to be locally a peak set for \(A^{\infty}(D)\). If E is locally a peak set for \(A^{\infty}(D)\) and \(p\in E\), then E is locally contained in the neighborhood of p in an \((n+q-1)\)-dimensional submanifold of \(\partial D\), complex-tangential at each point of E and \(\dim T{\mathbb{C}}_ z(M)\leq q\) for each \(z\in M\), where q represents the number of zero-eigenvalues of the Levi form at p and \(T{\mathbb{C}}_ z(M)\) is the maximal complex subspace of \(T_ z(M).\) An example in \({\mathbb{C}}^ 3\) shows that the dimension of M is varying with the q-pseudoconvexity and that generally M is not a CR-manifold. - If M is a complex-tangential CR-submanifold of type \((n+q-1,q)\) and CR(M) has certain additional properties, the above condition is also sufficient. An important step in the proof is the following result: the zero-set Z of a non-negative plurisubharmonic function \(\phi\) is locally contained in the neighborhood of \(z_ 0\in Z\) in a \((n+q)\)-dimensional generic submanifold, where q represents the number of zero-eigenvalues of the complex Hessian of \(\phi\) at \(z_ 0\). - The results and the methods used in the paper are generalizations of those of M. Hakim and N. Sibony in the case of strictly pseudoconvex domains, respectively F. R. Harvey and R. O. Wells jun. for strictly plurisubharmonic functions.
scientific article; zbMATH DE number 3847752