Abstract

The Hardy space H_ρr¹(Rⁿ) consists of all divergence free r-form distributions f whose non-tangential maximal functions are in L¹(Rⁿ). We say that a system of singular integrals characterizes H_ρr¹(Rⁿ) if this space consists precisely of those divergence-free r-form distributions f whose images under the singular integral operators are integrable. When the operators are determined by Fourier multipliers, necessary and suficient conditions are prescribed on the multipliers in order that the system characterize H_ρr¹(Rⁿ). The condition is analogous to the Janson–Uchiyama condition for the scalar-valued case and the characterization follows the lines of Uchiyama’s constructive decomposition of BMO. In particular, it is shown how to build divergence-free r-form wavelets which play the same role that the R. Fefferman–Chang elementary decomposition played in Uchiyama’s work.

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