On the general form of linear functionals on the Hardy spaces $H^1$ over   compact Abelian groups and some of its applications
  The celebrated Fefferman's theorems on the general form of linear functionals on the Hardy space $H^1$ over the circle group is generalized to the case of an arbitrary compact Abelian group with totally ordered dual. Several corollaries that can be applied to multiple lacunary Fourier series and atomic decomposition on the two dimensional torus are obtained. 
A. R. Mirotin
math.FA