theory of differential forms on manifolds

In this paper, we establish a Hodge-type decomposition for the LP space of differential forms on closed (i.e., compact, oriented, smooth) Rieman- nian manifolds. Critical to the proof of this result is establishing an LP es- timate which contains, as a special case, the L2 result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the LP boundedness of Green's operator which we use in developing the LP theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the LP norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the A-harmonic equation.

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