The aim of this paper is to study Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. We establish the equivalence of a Sobolev imbedding to the boundedness of a certain weighted Hardy operator. This Hardy operator is then used to prove the existence of rearrangement-invariant norms that are optimal in the imbedding inequality. Our approach is to use the methods and principles of Interpolation Theory.
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Kerman et al. (2006) studied this question.
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