Let $\Gamma$ be a lattice in SU(n, 1). For each loxodromic element $\gamma_0\in\Gamma$ we define a closed curve $\{\gamma_0\}$ on $\Gamma\backslash SU(n,1)$ that projects to the closed geodesic on the factor of the complex hyperbolic space $\Gamma\backslash{\mathbb H}^n_{{\mathbb C}}$ associated with $\gamma_0$. We prove that the cohomological equation $\mathfrak{D} F=f$ 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 $\{\gamma_0\}$ 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. Zametki10 (1971), 555–564) where the curves are the closed geodesics themselves and the vanishing of the integrals of f 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.