On normal solvability of the operator of exterior derivation of warped products
type: Item
instance of: scholarly article
title: On normal solvability of the operator of exterior derivation of warped products
zbMATH Open document ID: 0893.58004
DOI: 10.1007/BF02104873
author: V. I. Kuz'minov
author: I. A. Shvedov
published in: Siberian Mathematical Journal
publication date: 1998-08-16T00:00:00Z
review text: 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 equi…
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 58A12
zbMATH DE Number: 1023967
zbMATH Keywords: operator of exterior derivation
zbMATH Keywords: warped products
zbMATH Keywords: normal solvability
zbMATH Keywords: warped cylinder
zbMATH Keywords: half-interval
MaRDI profile type: Publication
cites work: Q3911109
cites work: L//2 cohomology of warped products and arithmetic groups
cites work: Un théorème de dualité
cites work: Q5660209
cites work: Q3663911
cites work: On Weighted Norm Inequalities with Three Weights
full work available at URL: https://doi.org/10.1007/bf02104873
OpenAlex ID: W1964295550
description: scientific article; zbMATH DE number 1023967