Let m and n be positive integers with n 2 and 1 m n&1. We study rearrangement-invariant quasinorms * R and * D on functions f: (0, 1) Ä R such that to each bounded domain 0 in R n , with Lebesgue measure |0|, there corresponds C=C( |0| )>0 for which one has the Sobolev imbedding inequality, involving the nonincreasing rearrangements of u and a certain m th order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which * D need not be rearrangementinvariant,In both cases we are especially interested in when the quasinorms are optimal, in the sense that * R cannot be replaced by an essentially larger quasinorm and * D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Bre zis, and Wainger.

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.1991 Mathematics Subject Classification: 46E35; 46E30. m f ðlÞ :¼ jfx A W : j f ðxÞj > lgj; l > 0:

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Abstract.Analogues of Young's Inequality and the Convolution Theorem are shown to hold when the Lp and L(p, q) spaces have underlying measure defined in terms of power weights.