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The celebrated Fefferman's theorems on the general form of linear functionalson the Hardy space $H^1$ over the circle group is generalized to the case of anarbitrary compact Abelian group with totally ordered dual. Several corollariesthat can be applied to multiple lacunary Fourier series and atomicdecomposition on the two dimensional torus are obtained.

On the general form of linear functionals on the Hardy spaces $H^1$ over compact Abelian groups and some of its applications