Characterization of Hardy Spaces by Singular Integrals and ‘divergence-free’ Wavelets

The Hardy space H ρr(R ) consists of all divergence free r-form distributions f whose non-tangential maximal functions are in L(R). We say that a system of singular integrals characterizes H ρr(R ) 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 sufficient conditions are prescribed on the multipliers in order that the system characterize H ρr(R ). 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.