Continuity of the relative extremal function on analytic varieties in Cⁿ

Abstract

Abstract. Let V be an analytic variety in a domain Ω ⊂ C n and let K ⊂⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V , we prove that the relative extremal function ω K is continuous on V if Ω is hyperconvex and K is regular.1. Introduction. Let P (D) be a linear partial differential operator with constant coefficients. Hörmander [9] gave a characterization of when P (D) is surjective on the space A(Ω) of real-analytic functions when Ω is a convex domain in R n . This characterization is stated in terms of Phragmén-Lindelöf type estimates for plurisubharmonic functions on the zero variety of the symbol of P (D). Hörmander's result has been the main inspiration for trying to find various kinds of geometric or algebraic criteria for recognizing varieties satisfying such Phragmén-Lindelöf estimates, and there are a large number of papers attacking this problem. One of the tools that have been used is the relative extremal function ω K . (See for example [1,2,3].) The main goal of this paper is to prove that ω K is continuous if K is regular. To state the result more precisely, we will require some preliminary definitions.there is an open neighborhood U of x such that V ∩ U is irreducible, we say that V is locally irreducible at x. We denote by V irr the set of locally irreducible points in V . If x ∈ V irr , we say that V is reducible at x; we denote the set of reducible points by V red . Note that V irr is an open dense subset of V containing all regular points of V .

Monge-Ampère operators on complex varieties in $\C^n

“…Lemma 3.11 (c) was proved in Theorem 1.7 of [Wik1] under the additional assumption that D is hyperconvex. Note that our method is intrinsic (working directly on V ) and so different from those in [Wik1].…”

“…A weaker version of the above result was established, using the same scheme as in the case of domains in C n (cf. Theorem 2.1 in [Ce1]), was established in Theorem 2.3 of [Wik1] where the ambient domain D is assumed to be hyperconvex. Notice also that if the continuity requirement on u j is dropped then we may simply take u j := max{u, jρ}.…”

Sublevel sets of certain extremal functions

“…For more background on relative extremal functions on complex varieties, the reader may consult [15]. The result below gives a sufficient condition for the regularity of a given compact set.…”