TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Upper envelopes of locally upper-bounded plurisubharmonic families

Statement

Let $F$ be a nonempty family of plurisubharmonic functions on a domain $Ω \subseteq C^{m}$ , and suppose that for every compact set $K \subseteq Ω$ there is a real number $M_{K}$ with $u \leq M_{K}$ on $K$ for every $u \in F$ . Define

$v (z) = \sup_{u \in F} u (z) .$

Assume also that $v$ is upper semicontinuous. Then $v$ is plurisubharmonic on $Ω$ .

Facts & Assumptions

Given: A locally bounded-above family $F$ of plurisubharmonic functions on a domain $Ω \subseteq C^{m}$ .

[L1]

Plurisubharmonicity is tested on affine complex lines (Plurisubharmonic functions).

[L2]

The upper-semicontinuous regularization of a locally bounded-above subharmonic supremum is subharmonic (The upper-semicontinuous regularization of a locally bounded-above subharmonic supremum is subharmonic).

Proof

technique · direct

1.1

The local upper bounds imply that $v$ never takes the value $+ \infty$ . By the added hypothesis, $v$ is upper semicontinuous. Because $F$ is nonempty and each member is not identically $- \infty$ on a component, the same is true of $v$ .

given

1.2

Fix an affine complex line in $Ω$ and one of the connected components of its line domain. Restricting the family $F$ there, [L1] gives a nonempty family of subharmonic functions that is still locally bounded above. Its restricted supremum is exactly the restriction of $v$ , and that restriction is upper semicontinuous because $v$ is. Therefore [L2] makes the restriction of $v$ subharmonic on that component. Thus the line test from [L1] is satisfied.

L1L2given

2.1

Step 1.1 gives the upper-semicontinuity and nontriviality conditions, and step 1.2 gives the line test. By [L1], $v$ is plurisubharmonic on $Ω$ .

L1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.4 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, Theorem 7 (standard reference, not scraped)