On normal solvability of the exterior differentiation on a warped cylinder
The \(L_p\)-cohomology space \(H^k_p (M)\) of a manifold coincides with the reduced \(L_p\)-cohomology space \(\overline H^k_p (M)\) if and only if the exterior derivative \(d^{k - 1}_{p,M} : L^{k - 1}_p (M) \to L^k_p (M)\) is a normally solvable operator, i.e., it has a closed range. If \(\dim H^k_p (M) < \infty\) then \(d^{k - 1}_{p, M}\) is normally solvable. Due to J. Cheeger it is known that this holds for \(M\) obtained from a closed pseudomanifold by removing singularities. In the article under review the authors find necessary and sufficient normal solvability conditions for \(d^{k - 1}_{p, M}\) defined over a warped cylinder \(M = [a, b) \times_fY\). If \(\lim_{t \to b} f(t) = \infty\) then the normal solvability turns out to be equivalent to the validity of an embedding theorem for weighted Sobolev spaces on \([a,b)\).
scientific article; zbMATH DE number 825536