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\(L^ p\) 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
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