Weighted Lorentz Spaces and the Hardy Operator

We find a new expression for the norm of a function in the weighted Loreritz space, with respect to the distribution funetion, and obtain as a simple consequence a generalization of the classical embeddings Lp, 1 ⊂ · · · ⊂ Lp ⊂ · · · ⊂ Lp, ∞ and a new definition of the weak space Λp, ∞u (w). We also give some applications to the boundedness of the Hardy operator Sƒ = ∫x0ƒ from Λp0u0(w0) into Λp1u1 (w1) with 0 < p0 ≤ p1.