受限的佩龙包络与拟有界函数
Perron envelopes with globally bounded minorants
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中文总结 AI 辅助
研究上半连续函数\(f\)的受限佩龙包络，证明经典位势理论中若正则化是次调和的则受限包络自动上半连续，通过构造反例表明多重位势理论中类似结论不成立，即不能去掉例外多重极集。
中文摘要 AI 辅助
设\(f\)是区域上的上半连续函数，考虑仅由全局有上界的次调和或多重次调和函数构成的\(f\)的佩龙包络。其正则化与包络在极集或多重极集之外一致，但正则化是否必要并不明确。比较拟有界性和拟处处拟有界性时会自然产生这个问题。我们证明在经典位势理论中，若受限包络的正则化是次调和的，则它自动是上半连续的。证明使用了从布雷洛可解性定理得到的局部调和修正。然后通过在\(\mathbb{C}^2\)中构造一个有界的\(B\)正则完全哈托格斯域以及一个正的、连续的、无界的多重调和函数\(W\)，其有界多重次调和函数的包络沿解析圆盘不连续，表明相应的多重位势理论结论不成立。函数\(W\)有一个比它增长快的正多重上调和函数作为主函数，且在多重极集之外是有界多重次调和函数的递增极限。因此，即使在这些强增长和逼近性质下以及在\(B\)正则域上，一般也不能去掉例外的多重极集。
英文摘要
Let $f$ be an upper semicontinuous function on a domain $Ω$, and let $V$ be the Perron envelope of $f$ formed only with subharmonic minorants that are bounded above on all of $Ω$. The usual argument for upper semicontinuity of Perron envelopes breaks down, since the regularization $V^*$ need not be bounded above; one only knows that $V=V^*$ outside a polar set. We show that nevertheless $V=V^*$ everywhere in $\mathbb{R}^d$, using a local harmonic correction given by a Poisson integral on a ball. The plurisubharmonic analogue fails. We construct an explicit bounded B-regular Hartogs domain in $\mathbb{C}^n$, $n\geq2$, and a positive pluriharmonic function $W$ whose envelope of bounded-above plurisubharmonic minorants equals $W$ off an analytic disc and vanishes on it, although $W$ admits a positive plurisuperharmonic majorant $G$ with $W\leq\varepsilon G+C_\varepsilon$ for all $\varepsilon>0$. For pluriharmonic $W$, we show that the envelope recovers $W$ exactly on the union of the sets where such majorants are finite. The question arises in the study of quasibounded plurisubharmonic functions.
发表机构
- Lund University（隆德大学）
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