Characterisation of embeddings in Lorentz spaces

Some new integral conditions characterising the embedding Lambda(p) -> Gamma(q)(w), 0 < p, q <= infinity are presented, including proofs also for the cases (i) p = infinity, 0 < q <infinity, (ii) q = infinity, 1 < p < infinity and (iii) p = q = infinity. Only one condition is necessary for each case which means that our conditions are different from and simpler than other corresponding conditions in the literature. We even prove our results in a more general frame namely when the space Gamma(q)(w)(w) is replaced by the more general space Gamma(q)(u)(w)(w). In our proof we use a technique of discretisation and anti-discretisation developed by A. Gogatishvili and L. Pick, where they considered the opposite embedding.