Potential maps, Hardy spaces, and tent spaces on special Lipschitz domains

Suppose that Ω is the open region in ℝⁿ above a Lipschitz graph and let d denote the exterior derivative on ℝⁿ. We construct a convolution operator T which preserves support in Ω̄, 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 Ω̄. Thus if f is exact and supported in Ω̄, then there is a potential u, given by u = Tf, of optimal regularity and supported in Ω̄, such that du = f. This 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 Ω̄ when n/(n + 1) < p ≤ 1. This is done via an atomic decomposition of functions in the tent spaces T^p(ℝⁿ × 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.