## Abstract

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 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}(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.

Original language | English |
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Pages (from-to) | 295-331 |

Number of pages | 37 |

Journal | Publicacions Matematiques |

Volume | 57 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 |