The Hardy Lorentz space H^p,q( Rⁿ) consists of those distributions f whose radial maximal function belongs to the Lorentz space L ^p,q(Rⁿ). In this thesis, we study the Hardy Lorentz spaces H^p,q( Rⁿ), 0 < p ≤ 1, 0 < q ≤ ∞. We show that H^p,q( Rⁿ) can be characterized in a number of different ways including an atomic decomposition characterization of H ^p,q(Rⁿ). We study the dual spaces and the enveloping Banach spaces of H^p,q( Rⁿ). We provide an interpolation result for these spaces, including the case of Weak H^p. We study the behavior of singular integral operators acting on H ^p,q(Rⁿ).

In the second part of this thesis, we study the dyadic structure of the Hardy spaces H^p(Rⁿ) and their duals the Lipschitz spaces Λ_α and BMO, the space of functions with bounded mean oscillations. We describe the spaces H^p(Rⁿ), Λ _α and BMO in terms of their dyadic counterparts. We show that H^p(Rⁿ) can be written as the sum of quasi-Banach spaces formed using special atoms. The dual result is a characterization of Λ_α as the intersection of dyadic Λ_α and a space of Zygmund type. We give a complete description of the norms in Λ_α and BMO in terms of the dyadic grid.