Perron envelopes with globally bounded minorants
Abstract
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 ℝd, 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 ℂn, n≥2, 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≤ε G+Cε for all ε>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.
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BibTeX
@misc{wikstrom2026perron,
title = {{Perron envelopes with globally bounded minorants}},
author = {Frank Wikström},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.24139},
eprint = {2607.24139},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24139},
}
RIS
TY - UNPB
TI - Perron envelopes with globally bounded minorants
AU - Frank Wikström
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.24139
UR - https://arxiv.org/abs/2607.24139
AB - 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 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 ℂ^n, n≥2, 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≤ε G+C_ε for all ε>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.
ER -