Integral estimates for the Laplace-Beltrami and Green's operators applied to differential forms on manifolds
The author studies the \(L^ p\) theory of the Laplace-Beltrami operator \(\Delta=d\,d^*+d^* d\) and the Green operator \(G\) acting upon differential forms on manifolds. Some norm inequalities for \(\Delta\), \(G\), and their composition \(\Delta\circ G\) that are applied to differential forms on a compact orientable \(C^\infty\)-smooth Riemannian manifold \(M\) without boundary are established. Also, \(A_ r(M)\)-weighted boundedness for the composition \(\Delta\circ G\) is obtained. As applications, the author proves \(A_ r(M)\)-weighted Sobolev-Poincare embedding theorems for the Green operator and norm comparison theorems for solutions of the \(A\)-harmonic equation on manifolds. These results can be used in developing the \(L^ p\) theory of differential forms and the Hodge decomposition.
scientific article; zbMATH DE number 2075238