Suppose that Ω is the open region in R n above a Lipschitz graph and let d denote the exterior derivative on R n . We construct a convolution operator T which preserves support in Ω, is smoothing of order 1 on the homogeneous function spaces, and is a potential map in the sense that dT is the identity on spaces of exact forms with support in Ω. Thus if f is exact and supported in Ω, then there is a potential u, given by u = T f , of optimal regularity and supported in Ω, such that du = f . This

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... has implications for the regularity in homogeneous function spaces of the de Rham complex on Ω with or without boundary conditions. The operator T is used to obtain an atomic characterisation of Hardy spaces H p of exact forms with support in Ω when n/(n + 1) < p ≤ 1. This is done via an atomic decomposition of functions in the tent spaces T p (R n × R + ) with support in a tent T (Ω) as a sum of atoms with support away from the boundary of Ω. This new decomposition of tent spaces is useful, even for scalar valued functions.