Global regularity on 3-dimensional solvmanifolds

Cygan, Jacek M.; Richardson, Leonard F.

doi:10.1090/S0002-9947-1992-1055806-5

Global regularity on $3$-dimensional solvmanifolds
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by Jacek M. Cygan and Leonard F. Richardson PDF

Trans. Amer. Math. Soc. 329 (1992), 473-488 Request permission

Abstract:

Let $M$ be any $3$-dimensional (nonabelian) compact solvmanifold. We apply the methods of representation theory to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in ${C^\infty }(M)$. We show 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.

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