The spaces introduced by Sweezy are, in many respects, natural extensions of the real Hardy space \(H^1({\mathbb R}^d)\) . They are nested in full between \(H^1({\mathbb R}^d)\) and \(L_0^1({\mathbb R}^d)\) . Contrary to \(H^1({\mathbb R}^d)\) , they are subject only to atomic characterization. In this paper, the possibilities that atomic expansions allow one are used for proving analogs of the Fourier–Hardy inequality for the Sweezy spaces. The results obtained are used, in dimension one, for extending the scale of the spaces of functions with integrable Fourier transform. An application to trigonometric series is also given.
