# zbMATH — the first resource for mathematics

$$L^ p$$ theory of differential forms on manifolds. (English) Zbl 0849.58002
A Hodge-type decomposition for the $$L^p$$ space of differential forms on closed (i.e. compact, oriented and smooth) Riemannian manifolds is established. The author establishes an $$L^p$$ estimate which contains, as a special case, the Gaffney inequality. This is then used to show the equivalence of the usual definition of Sobolev space with a more geometric formulation, provided here in the case of differential forms on manifolds. The $$L^p$$ boundedness of Green’s operator is also established and then used to develop the $$L^p$$ theory of the Hodge decomposition. For the calculus of variations, the author rigorously verifies that the spaces of exact and coexact forms are closed in the $$L^p$$ norm. For nonlinear analysis, the existence and uniqueness of a solution to the $$A$$-harmonic equation is demonstrated.

##### MSC:
 58A10 Differential forms in global analysis 58A14 Hodge theory in global analysis 58J99 Partial differential equations on manifolds; differential operators 31C12 Potential theory on Riemannian manifolds and other spaces
Full Text: