Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity

TL;DR

This paper investigates regularization operators on plurisubharmonic functions linked to compact convex sets, demonstrating lower semicontinuity of associated extremal functions and challenging previous claims about their H"older continuity.

Contribution

It introduces new regularization techniques preserving growth conditions and provides a detailed analysis of their continuity properties, including a counterexample to earlier results.

Findings

Weighted extremal functions are lower semicontinuous.

Regularization operators preserve growth conditions.

H"older continuity of these functions is not generally valid.

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 H\"older continuity of these Siciak-Zakharyuta functions.