Livshitz theorem for the unitary frame flow

Let Γ be a lattice in SU(n, 1). For each loxodromic element γ₀ ∈ Γ we define a closed curve {γ ₀} on Γ\SU(n, 1) that projects to the closed geodesic on the factor of the complex hyperbolic space Γ\ℍ_ℂ ⁿ associated with γ₀. We prove that the cohomological equation Script D F = f has a solution if f is the lift of a holomorphic cusp form to SU(n, 1) under the following condition: for each restriction of f to {γ₀} a finite number of Fourier coefficients vanish, and this finite number grows linearly with the length of the curve. This is a generalization of the classical Livshitz theorem for SU (1, 1) (A. Livshitz. Mat. Zametki 10 (1971), 555-564) where the curves are the closed geodesies themselves and the vanishing of the integrals of / over them, i.e. the zeroth Fourier coefficients, is both necessary and sufficient. An application of our result to the construction of spanning sets for spaces of holomorphic cusp forms on complex hyperbolic spaces is given in Appendix A.