A bounded open set with boundary of class C¹ which is locally weakly lineally convex is weakly lineally convex, but, as shown by Yuriĭ Zelinskiĭ, this is not true for unbounded domains. The purpose here is to construct explicit examples, Hartogs domains, showing this. Their boundary can have regularity $C^{1,1}$ or $C^{\infty }$. Obstructions to constructing smoothly bounded domains with certain homogeneity properties will be discussed.

Let Ω be a bounded strictly pseudoconvex domain in ${ℂ}^n$. In this paper we find sufficient conditions on a function f defined on Ω in order that the weighted Bergman projection $P_{s}f$ belong to the Hardy-Sobolev space $H_k^p\left(\partial \Omega \right)$. The conditions on f we consider are formulated in terms of tent spaces and complex tangential vector fields. If f is holomorphic then these conditions are necessary and sufficient in order that f belong to the Hardy-Sobolev space $H_k^p\left(\partial \Omega \right)$.