Global solvability on compact nilmanifolds of three or more steps

Abstract

We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations D f = g Df = g in C ∞ {C^\infty } of a compact nilmanifold of three or more steps. We investigate which algebraically well-defined conditions on D D in the complexified Lie algebra imply that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. This extends and improves the results of an earlier two step paper.