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The Hodge decomposition is well-known for compact manifolds. The result hasbeen extended by Kodaira to include non-compact manifolds and $L^2$ forms. Wefurther extend the Hodge decomposition to the Sobolev space $H^1$ for general$k$-forms on non-compact manifolds of nonpositive constant sectional curvature.As a result, we also obtain a decomposition on $\mathbb R^N$.

Hodge decomposition of the Sobolev space $H^1$ on a space form of nonpositive curvature