Global solvability on compact nilmanifolds of three or more steps

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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 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. © 1987 American Mathematical Society.

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Transactions of the American Mathematical Society

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