The Hardy space Hρr1(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
Hρr1(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
sufficient conditions are prescribed on the multipliers in order that the system
characterize Hρr1(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.
