Harmonic Analysis and Fundamental Solutions on Nilpotent Lie Groups

The purpose of this paper is to study the harmonic analysis of a broad class of (hypoelliptic) partial differential operators on arbitrary (simply-connected) nilpotent Lie groups. We shall find the asymptotic development of their fundamental solutions, both at the origin and at infinity, and study the corresponding "Riesz transforms" and the analogues of the Hardy-Littlewood-Sobolev theorems on ''fractional integration".