  In this paper, we generalize the CR Obata theorem to a compact strictly
pseudoconvex CR manifold with a weighted volume measure. More precisely, we
first derive the weighted CR Reilly's formula associated with the Witten
sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its
applications, we obtain the CR Obata theorem in a compact weighted Sasakian
manifold with or without boundary.
