Global solvability on compact nilmanifolds of three or more steps

Cygan, Jacek M.; Richardson, Leonard F.

doi:10.1090/S0002-9947-1987-0879578-8

Global solvability on compact nilmanifolds of three or more steps
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by Jacek M. Cygan and Leonard F. Richardson

Trans. Amer. Math. Soc. 301 (1987), 343-373

DOI: https://doi.org/10.1090/S0002-9947-1987-0879578-8

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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 $Df = g$ in ${C^\infty }$ of a compact nilmanifold of three or more steps. We investigate which algebraically well-defined conditions on $D$ 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.

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A symbolic calculus for

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