An embedding theorem for the Orlicz-Sobolev space W1,A₀(G) , G⊂ R^n , into a space of Orlicz-Lorentz type is established for any given Young function A . Such a space is shown to be the best possible among all rearrangement invariant spaces. A version of the theorem for anisotropic spaces is also exhibited. In particular, our results recover and provide a unified framework for various well-known Sobolev type embeddings, including the classical inequalities for the standard Sobolev space W1,p₀(G) by O'Neil and by Peetre ( 1≤ p< n ), and by Brezis-Wainger and by Hansson ( p=n ).
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Andrea Cianchi (2004) studied this question.