A Stampacchia-type inequality for a fourth-order elliptic operator on Kähler manifolds and applications
The author studies the relation between Aeppli groups \(\Lambda^{p,q}(X)\) and the groups \(H^{p,q}(X)\) of all global harmonic forms of bidegree \((p,q)\) on a connected complete Kähler manifold \(X\). Under a topological condition (satisfied for compact and Stein manifolds) it is proved that \(\Lambda^{p,q}(X)\) is isomorphic to \(H^{p,q}(X)\). The main tool in the proof is a suitable Hodge-Kodaira decomposition of \(W^{p,q}(X)\) the completion of \(D^{p,q}(X)\) the space of forms with compact support in \(X\). The proof of this requires an integral inequality of Stampacchia-type for a certain fourth-order elliptic operator. Both this inequality and the Hodge-Kodaira decomposition hold independently of the topological assumption.
scientific article; zbMATH DE number 5714229