Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $

Abstract

We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space H 2 that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space C 2 :One of the key notions of many-dimensional complex analysis is the notion of linear convexity. A domain D C n ; n 2; is called locally linearly convex if, for every point z 0 of its boundary @D; there exists a complex hyperplane … C n 1 that passes through the point z 0 and does not intersect the domain D in a certain neighborhood of this point; if, in addition, … C n 1 \ D D ¿; then the domain D is called (globally) linearly convex.The notions of local linear convexity and global linear convexity were introduced by Behnke and Peschl [1], who investigated the domains D D fz D .z 1