\(L_p\)-cohomologies of warped products of Lipschitz Riemannian manifolds (Q1272001)

Let \(X\) and \(Y\) be Lipschitz Riemannian manifolds of dimensions \(m\) and \(n\) and \(p\in [1,\infty]\). The main result of the article is the demonstration of the Künneth formula in the case when either \(Y\) is compact or its de Rham \(L_p\)-complex is decomposable. More exactly, the result can be described as follows. Assume measurable functions \(\rho,f\: X\to \mathbb R_+\) and \(\sigma\: Y\to \mathbb R_+\) to be locally bounded with locally bounded inverses. If the function \(f\) is bounded and the Banach complex \(\mathbf{L}_p=( L_p^j(Y,\sigma;E),d^j_{W_p}\mid j\in \mathbb Z)\) \((\mathbf{W}_p=( W_p^i(Y,\sigma;E)\mid i\in \mathbb Z))\) is decomposable, then the Künneth formula \[ H_p^k(X \times_f Y,\rho \sigma;E) \cong\oplus_{i+j=k} H_p^i( X,\rho f^{n/p-j};H_p^j(Y,\sigma;E)) \] holds (\(E\) is a Banach space). Similarly, if the function \(f\) is bounded and \(Y\) is compact, then the formula \[ H_p^k(X \times_f Y,\rho;\mathbb R) \cong\oplus_{i+j=k} H_p^i( X,\rho f^{p/n-j};H^j(Y;\mathbb R)) \] holds.