Global solvability on two-step compact nilmanifolds

Cygan, Jacek M.; Richardson, Leonard F.

doi:10.1090/S0002-9947-1983-0709567-1

Global solvability on two-step compact nilmanifolds
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by Jacek M. Cygan and Leonard F. Richardson PDF

Trans. Amer. Math. Soc. 279 (1983), 537-554 Request permission

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 two-step compact nilmanifold. We show that, under algebraically well-defined conditions on $D$ in the complexified Lie algebra, 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. Such strong estimates are not possible on multidimensional tori.

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