Equivalent expressions for norms in classical Lorentz spaces

We characterize the weights w such that

?₀^? ?*(s)^pw(s) ds? ?₀^? (?**(s) – ?*(s))^pw(s) ds.

Our result generalizes a result due to Bennett–De Vore–Sharpley, where the usual Lorentz L^p,q norm is replaced by an equivalent expression involving the functional ? ** – ? *. Sufficient conditions for the boundedness of maximal Calderón–Zygmund singular integral operators between classical Lorentz spaces are also given.