| Summary: | Abstract In this paper, we study the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. First of all, we give a general inequality for eigenvalues of the eigenvalue problem of elliptic operators in weighted divergence form on compact smooth metric measure space with boundary (possibly empty). Then applying this general inequality, we get some universal inequalities of Payne-Pólya-Weinberger-Yang type for the eigenvalues of elliptic operators in weighted divergence form on a connected bounded domain in the smooth metric measure spaces, the Gaussian shrinking solitons, and the general product solitons, respectively.
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