$W_{0}^{1,p}(\Omega )$ that can be embedded in $L^{p^{\ast }}(
\Omega )?$ To make such questions sensible the class of competing spaces must be specified. If we restrict ourselves to Lebesgue spaces as targets and domain spaces that are Sobolev spaces based on Lebesgue spaces, then the embedding $W_{0}^{1,p}(\Omega )\hookrightarrow $ $L^{p^{\ast }}(\Omega )$ is optimal in the sense that neither the domain nor the target space can be improved. This leaves open the question of optimality in classes of spaces wider than those involving the Lebesgue scale. If the class of admissible target spaces is taken to be that of rearrangement-invariant (r.i.) spaces, then the optimal range space turns out to be the Lorentz space $L^{p^{\ast },p}(\Omega )$ there is a similar improvement of the domain space, involving a Sobolev space based on a Lorentz rather than a Lebesgue space.

The first results in this direction were obtained in [@EKP] in connection with rearrangement-invariant quasinorms. Further extensions concerning r.i. norms were added later in several papers, for instance [@T2; @T3]. A comprehensive treatment of optimal Sobolev embeddings on Euclidean domains equipped with general measures having specific isoperimetric properties was given in [@CPS].

Embeddings are not the only maps for which 