The local limit theorem on nilpotent Lie groups

TL;DR

This paper proves a local limit theorem for certain measures on connected, simply connected nilpotent Lie groups, extending previous results to broader classes of measures and groups.

Contribution

It generalizes earlier local limit theorems to include measures satisfying specific moment and characteristic function conditions on nilpotent Lie groups.

Findings

Established a local limit theorem for a class of measures on nilpotent Lie groups.

Extended previous results to measures with less restrictive conditions.

Broadened the scope of local limit theorems beyond the Heisenberg group.

Abstract

A local limit theorem is proven on connected, simply connected nilpotent Lie groups, for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization. The result extends an earlier local limit theorems of Alexopoulos which treated absolutely continuous measures with a continuous density of compact support, and also extends local limit theorems of Breuillard and Diaconis-Hough which treated general measures on the Heisenberg group.