Hardy factorization in terms of fractional commutators in Lorentz spaces

We provide a constructive proof of H^1\left({\text{ℝ}}^d\right) (the classical Hardy space) factorization in terms of fractional commutators in Lorentz spaces. As a direct application, we obtain a characterization of functions in BMO space. Furthermore, we also obtain a Lorentz compactness characterization of fractional commutators.