On the general form of linear functionals on the Hardy spaces H1 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 H1 over the circle group are 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 theory on the two dimensional torus are obtained.