Perron envelopes with globally bounded minorants

Abstract

Let $f$ be an upper semicontinuous function on a domain $\Omega$, and let $V$ be the Perron envelope of $f$ formed only with subharmonic minorants that are bounded above on all of $\Omega$. The usual argument for upper semicontinuity of Perron envelopes breaks down, since the regularization $V^{\ast }$ need not be bounded above; one only knows that $V=V^{\ast }$ outside a polar set. We show that nevertheless $V=V^{\ast }$ 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 ${\mathbb{C}}^n$, $n\ge 2$, 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 $W\le \epsilon G+C_{\epsilon }$ for all $\epsilon >0$. 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.

arXiv comment9 pages; v2 contains a simplified proof of the main theorem as well as improvements to the exposition