Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms

Let m and n be positive integers with n⩾2 and 1⩽m⩽n−1. We study rearrangement-invariant quasinorms ϱR and ϱD on functions f: (0, 1)→View the MathML source such that to each bounded domain Ω in View the MathML sourcen, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality ϱR(u*(|Ω| t))⩽CϱD(|∇mu|* (|Ω| t)), u∈Cm0(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which ϱD need not be rearrangement-invariant, ϱR(u*(|Ω| t))⩽CϱD((d/dt) ∫{x∈View the MathML sourcen : |u(x)|>u*(|Ω| t)} |(∇u)(x)| dx), u∈C10(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that ϱR cannot be replaced by an essentially larger quasinorm and ϱD 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, Brézis, and Wainger