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In this paper, we generalize the CR Obata theorem for the Kohn Laplacian to aclosed strictly pseudoconvex CR manifold with a weighted volume measure. Moreprecisely, we first derive the weighted CR Reilly's formula associated with theweighted Kohn Laplacian and obtain the corresponding first eigenvalue estimate.With its application, we obtain the CR Obata theorem in a closed weightedSasakian manifold.

On the Obata theorem for the weighted Kohn Laplacian in a closed weighted Sasakian manifold