Restricted Perron envelopes and quasibounded functions

Abstract

Let $f$ be an upper semicontinuous function on a domain and consider the Perron envelope formed only with globally bounded-above subharmonic, or plurisubharmonic, minorants of $f$. Its upper semicontinuous regularization agrees with the envelope outside a polar, respectively pluripolar, set, but it is not immediate whether regularization is actually necessary. This question arises naturally when comparing quasiboundedness with quasiboundedness quasi-everywhere. We prove that in classical potential theory the restricted envelope is automatically upper semicontinuous, provided its regularization is subharmonic. The proof uses a local harmonic correction obtained from Brelot's resolutivity theorem. We then show that the corresponding pluripotential statement fails sharply by constructing a bounded B-regular complete Hartogs domain in $\mathbb{C}^2$ and a positive, continuous, unbounded pluriharmonic function $W$ whose envelope of bounded plurisubharmonic minorants is discontinuous along an analytic disc. The function $W$ nevertheless admits a positive plurisuperharmonic majorant growing faster than $W$, and it is an increasing limit of bounded plurisubharmonic functions outside a pluripolar set. Thus the exceptional pluripolar set cannot in general be removed, even under these strong growth and approximation properties and on a B-regular domain.