On the strong \(L^p\)-Hodge decomposition over complete Riemannian manifolds (Q1039417)

Strong \(L^2\)-Hodge decomposition is known for complete Riemannian manifolds (R.m.), while strong \(L^p\)-Hodge decomposition has been proved for compact R.m. The author of the present article has established Weak \(L^p\)-Hodge decomposition for complete R.m. in previous papers. In the present work, he studies Strong \(L^p\)-Hodge decomposition for complete, non-compact R.m., with weighted measure. More precisely, the author proves a criterion for the validity of the Strong \(L^p\)-Hodge decomposition and he finds two classes of \(L\)-stochastically complete R.m. with additional geometric conditions, where the Strong \(L^p\)-Hodge decomposition holds. As applications, he obtains (i) vanishing theorems for \(L^p\)-cohomology and \(L^p\)-torsion, and (ii) global solutions to the equation \(d\alpha = \omega\), on the above mentioned classes of R.m.