Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity
Bergur Snorrason
TL;DR
The paper develops regularization tools for plurisubharmonic functions with growth controlled by a compact convex set $S$, via generalized Lelong classes $\mathcal L^S(\mathbb{C}^n)$ and weighted Siciak-Zakharyuta functions $V^S_{K,q}$. It introduces infimal, supremal, and integral convolution operators $R^a_{\mu,\delta}$, $R^b_\delta$, and $R^c_\delta$, and a logarithmic-dilation type operator $R^d_\delta$, proving that these regularizations preserve $\mathcal L^S(\mathbb{C}^n)$ (under suitable conditions), yield smooth and convergent approximants, and in particular establish lower semicontinuity of $V^S_{K,q}$ on $\mathbb{C}^n$ when $S$ contains a neighborhood of $0$. The work also shows that Hölder continuity cannot hold in general unless $S$ is a lower set, correcting prior claims in the literature and clarifying the role of the geometry of $S$ in the regularity of weighted extremal functions. Overall, the results provide a framework for controlled regularization and regularity analysis of generalized extremal functions in pluripotential theory.
Abstract
We study various regularization operators on plurisubharmonic functions that preserve Lelong classes with growth given by certain compact convex sets. The purpose is to show that the weighted Siciak-Zakharyuta functions associated with these Lelong classes are lower semicontinuous. These operators are given by integral, infimal, and supremal convolutions. Continuity properties of the logarithmic supporting function are studied and a precise description is given of when it is uniformly continuous. This gives a contradiction to published results about the Hölder continuity of these Siciak-Zakharyuta functions.