Separation of variables in the problems of normal and compact solvability of the exterior derivation operator (Q1586973)

The article continues previous publications of the authors devoted to the study of \(L_p\) and other de Rham complexes and their cohomologies of warped products of Riemannian manifolds [\textit{V. I. Kuz'minov} and \textit{I. A. Shvedov}, Sib. Mat. Zh. 40, No. 4, 893-904 (1999); translation in Sib. Math. J. 40, No. 4, 754-763 (1998; Zbl 0939.58001); ibid 39, No. 2, 354-368 (1998); translation in Sib. Math. J. 39, No. 2, 311-322 (1998; Zbl 0944.58001); ibid 38, No. 3, 573-590 (1997); translation in Sib. Math. J. 38, No. 3, 492-506 (1997; Zbl 0870.58003); ibid 37, No. 2, 324-337 (1996); translation in Sib. Math. J. 37, No. 2, 276-287 (1996; Zbl 0893.58004); \textit{V. I. Kuz'minov, I. A. Shvedov} and \textit{V. M. Gol'dshtejn}, ibid 30, No. 5(177), 49-59 (1989); translation in Sib. Math. J. 30, No. 5, 704-712 (1989; Zbl 0706.47005)]. The last article in this list was written in cooperation with V.~M.~Gol'dshtejn. In the article under review, the authors find necessary and sufficient conditions for normal and compact solvability of the exterior derivation operator \(\partial\) in the \(L_2\)-cohomologies of the warped product \(X\times_f Y \) of two Riemannian manifolds endowed by weight functions. If \(X\) is connected and has positive dimension, these conditions seem to be much simpler. If \(\dim X = 1\), some other necessary and sufficient conditions of the normal and compact solvability of \(\partial\) are obtained. Connections between embedding theorems and the results obtained are also discussed.