On the general form of linear functionals on the Hardy spaces H¹ over compact Abelian groups and some of its applications

Abstract

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.