On normal solvability of the operator of exterior derivation of warped products
Given a Riemannian manifold \(M\), a number \(p\geq 1\), and a positive continuous function \(\sigma\) on \(M\), the de Rham \(L_p\)-complex \((L^k_p (M,\sigma), d)_{k\in \mathbb{Z}}\) is defined to consist of those differential forms on \(M\) whose moduli are integrable in power \(p\) with weight \(\sigma^p\). The differentials \(d\) of this complex are the exterior derivatives. These operators are densely defined and closed. The cohomology spaces of the de Rham complex are topological vector spaces, possibly nonseparated. These spaces are called the \(L_p\)-cohomologies of the manifold \(M\) with weight \(\sigma\) and denoted by \(H^k_p(M, \sigma)\), while the spaces \(\overline H^k_p (M,\sigma) =H^k_p (M,\sigma)/ \overline {\{0\}}\) are referred to as the reduced \(L_p\)-cohomologies of \(M\). If \(M\) is a compact manifold, then, for every \(p\geq 1\) and an arbitrary weight \(\sigma\), the \(L_p\)-cohomologies \(H^k_p (M, \sigma)\) coincide with the conventional real (singular) cohomologies of the manifold and hence are finite-dimensional. Thereby for every \(k\) the exterior derivative \(d:L^k_p (M,\sigma) \to L_p^{k+1} (M,\sigma)\) on a compact manifold \(M\) is normally solvable, i.e., it has closed range. In the present article, we put forward a new view on methods established earlier by the authors, V. M. Gol'dshtejn and J. Cheeger which allows us to improve the methods and, on their grounds, derive a series of necessary conditions for normal solvability of the operator \(d\) on a warped product. Given a Riemannian manifold \(X\) and two weight functions \(\tau\) and \(\xi\) on \(X\), we define the cochain complex \((R^i_p(X,\tau,\xi),\partial)\), where \[ R^i_p (X,\tau,\xi) = L_p^{i-1} (X,\tau) \times L^i_p (X,\xi), \quad \partial (\omega_1, \omega_2)= (\omega_2 -d \omega_1, d\omega_2). \] The complex \(R_p(X, \sigma_{j+1}, \sigma_j)\) with the weights \(\sigma_{j+1} =\sigma \cdot f^{n/p-j-1}\) and \(\sigma_j =\sigma \cdot f^{n/p-j}\) turns out to be equivalent (with the shift of dimension by \(j)\) to some direct summand of the complex \(L_p (X \times_fY, \sigma)\). Therefore, the normal solvability of the operator \(d:L^k_p (X\times_fY, \sigma) \to L_p^{k+1} (X\times_fY,\sigma)\) implies the normal solvability of the operators \(\partial: R_p^{k-j} (X,\sigma_{j+1}, \sigma_j) \to R_p^{k-j+1} (X,\sigma_{j+1}, \sigma_j)\). We find some conditions for normal solvability of the operator \(\partial: R^i_p(X, \tau, \xi) \to R_p^{i+1} (X,\tau,\xi)\); construct an exact sequence that binds the cohomologies of the complex \(R_p(X,\tau,\xi)\) to those of the de Rham complexes \(L_p(X, \tau)\) and \(L_p (X,\xi)\); and compute the cohomologies and reduced cohomologies of the complex \(R_p(X, \tau,\xi)\) for \(X=[a,b)\). Also, we reveal a connection between the question of normal solvability of the exterior derivative \(d\) on a warped cylinder \([a,b) \times_fY\) and the embedding theorems for weighted Sobolev spaces on the half-interval \([a,b)\).
scientific article; zbMATH DE number 1023967