Regular parameters and local pluriregularity

We study weighted extremal functions on a compact subset of C n when the weight is fixed and
the logarithmic growth parameter varies. Concavity and two-sided estimates show that the regular param-
eters at a point form either the empty set, the smallest contact parameter alone, or the whole contact ray.
At a regular parameter, we estimate the right derivative and characterize both persistence of regularity and
equality of this derivative with the unweighted Green function. A local estimate involving dd c shows that
a global admissible extension that is strictly plurisubharmonic near the point makes weighted regularity
equivalent to local pluriregularity. We also describe failure of local pluriregularity by removing a negligible
subset. This gives a characterization of the points at which a compact set is non-plurithin but not locally
pluriregular, and identifies its largest relatively plurifine open negligible subset. The paper presents a focused
account of these results from the author’s longer preprint.