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We apply the methods of primary and irreducible Fourier series on compact nilmanifolds to determine the ranges of all first order invariant operators on the compact Heisenberg manifolds. We show that the sums of primary solutions behave better on these manifolds than on any multidimensional torus.doi:10.2307/1999207 fatcat:lhnblo2juzeyjaknakksegwf5a