Regularity of invariant distributions

TL;DR

This paper investigates the regularity properties of invariant distributions supported on the limit set of a geometrically finite discrete group of conformal transformations, relating regularity to geometric and group-theoretic parameters.

Contribution

It provides estimates of the regularity of invariant distributions based on conformal weight, Hausdorff dimension, and cusp rank, advancing understanding of their analytical behavior.

Findings

Regularity depends on conformal weight and Hausdorff dimension.

Established bounds relating distribution regularity to geometric parameters.

Enhanced understanding of invariant distributions in conformal geometry.

Abstract

We consider a geometrically finite discrete group of conformal transformations of the sphere. Further we consider distributions which are supported on the limit set and are invariant with conformal weight. We estimate their regularity in terms of the conformal weight, the Hausdorff dimension of the limit set, and the maximal rank of the cusps of the group.