Weighted regularity and plurithinness in $\mathbb{C}^n$

Abstract

We study weighted pluricomplex extremal functions with a fixed weight and a variable logarithmic growth parameter. We show that the regular parameters at a point form the empty set, the smallest contact parameter alone, or the whole contact ray. For continuous weights, we represent the right parameter derivative by probability measures satisfying quasi-everywhere inequalities. This gives a criterion for equality with the unweighted Green function. We construct two counterexamples to preservation of weighted regularity, using smooth plurisubharmonic weights and polynomially convex compact sets that are non-pluripolar and non-plurithin at every point. For the product example, we compute the derivative on the entire exceptional polydisc through planar Green functions and determine exactly where equality holds. The second example is globally pluriregular but has a positive right derivative. We also characterize the points where local pluriregularity fails without plurithinness, using negligible subsets and the plurifine topology, and prove preservation of regularity under local strict plurisubharmonicity of the extending weight.