Jensen measures and boundary values of plurisubharmonic functions

Wikström, Frank

doi:10.1007/bf02388798

Cited by 28 publications

(35 citation statements)

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“…This means, for example, that a domain is strongly regular if and only if it is B-regular. It is shown in [S] that a domain D is B-regular if and only if J c z = {δ z } for any z ∈ ∂D (see also [W,Corollary 3.8]). We can now combine these results with Theorem 6.3 in one corollary:…”

Section: Continuity Of Plurisubharmonic Envelopesmentioning

confidence: 99%

“…In [W,Thm. 4.10], it is shown that on star-shaped domains, J c z = J b z for all z ∈ D. It follows from Theorems 3.2, 3.7 and 4.2 that if D is starshaped, then the plurisubharmonic envelopes of continuous functions on D are continuous on D. So it is natural to ask if the envelopes are continuous up to the boundary.…”

Section: Continuity Of Plurisubharmonic Envelopesmentioning

confidence: 99%

“…If v ∈ PSH b (D) and v * ≤ f on ∂D, then v * ≤ f k on D. Thus we get S(z) = s(z). It follows from Edwards' theorem (see for instance [W,Theorem 2.1] for a general version) that…”

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confidence: 99%

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Continuity of plurisubharmonic envelopes

Göğüş

2005

Ann. Polon. Math.

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Abstract. Let D be a domain in C n . The plurisubharmonic envelope of a function ϕ ∈ C(D) is the supremum of all plurisubharmonic functions which are not greater than ϕ on D. A bounded domain D is called c-regular if the envelope of every function ϕ ∈ C(D) is continuous on D and extends continuously to D. The purpose of this paper is to give a complete characterization of c-regular domains in terms of Jensen measures.1. Introduction. The plurisubharmonic envelopes of functions have been quite useful and found a lot of applications in pluripotential theory. In this paper we address the problem of their continuity. [W] are obtained by comparing these classes of measures. In §2 we introduce additional classes of Jensen measures which we consider as multifunctions in C * (D). As it happens the continuity of envelopes is a consequence of such geometric properties of these multifunctions as upper and lower semicontinuity. In §3 we establish these properties for our classes and after that in §4 we prove the main theorem. Roughly speaking, envelopes of continuous functions are continuous if and only if any limit point of Jensen measures with respect to bounded plurisubharmonic functions can be obtained as a limit of such Jensen measures from every direction. We call such domains c-regular.

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Section: Continuity Of Plurisubharmonic Envelopesmentioning

confidence: 99%

Section: Continuity Of Plurisubharmonic Envelopesmentioning

confidence: 99%

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Continuity of plurisubharmonic envelopes

Göğüş

2005

Ann. Polon. Math.

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“…By adapting an approximation result by Cegrell for plurisubharmonic functions on hyperconvex domains [4] (see also [11]), we will prove that the two classes of Jensen measures coincide, and from this fact, Theorem 1.7 will follow. 2.…”

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confidence: 93%

“…The measures in J F z are called Jensen measures for the cone F , and the main reason for introducing these measures is the following duality theorem by Edwards [7]. A more accessible proof can be found in [11]. Theorem 2.1 (Edwards' Theorem).…”

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confidence: 99%

Continuity of the relative extremal function on analytic varieties in Cⁿ

Wikström

2005

Ann. Polon. Math.

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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 .

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