q-subharmonicity and q-convex domains in ℂn

Abstract

ABSTRACT. In this paper we study q-subharmonic and q-plurisubharmonic functions in C n . Next as an application, we give the notion of q-convex domains in C n which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.

The locally F-approximation property of bounded hyperconvex domains

“…Some elements of pluripotential theory that will be used throughout the paper can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

GEOMETRIC CHARACTERIZATION OF q-PSEUDOCONVEX DOMAINS IN ℂn

“…We will prove in Section 2 that, for q-pseudoconvex domains Ω ⊂ C n such that 2 ≤ q ≤ n, the function − log d(z, ∁Ω) is q-subharmonic. Note that by taking in account our convention about this notion, according to [5], this isn't generally the case for 1-pseudoconvex domains. By considering the function (1.1) δ Ω (z, E) = sup{r > 0, z + B E (r) ⊂ Ω}.…”

“…Further details about the notion of q-subharmonic functions and their properties can be obtained from [5] or [8].…”

“…In Section 3, we shall give a full rigorous proof of the local property of qpseudoconvexity, that differs of such given in [5]. Furthermore, we characterize the q-pseudoconvexity by the Levy form of the defining function and we prove that Ω is q-pseudoconvex, if and only if, for every boundary point, the Levi form of the boundary is semi-positive on the intersection of the holomorphic tangent space to the boundary with any (n − q + 1)-dimensional subspace E ⊂ C n .…”