Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms

Abstract

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)→R such that to each bounded domain Ω in Rn, with Lebesgue measure Ω, there corresponds C=C(Ω)>0 for which one has the Sobolev imbedding inequality R(u*(Ωt))≤CD(∇ mu*(Ωt)), 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 D need not be rearrangement-invariant, R(u*(Ωt))≤CD((d/dt)∫ {x∈Rn:u(x)>u*(Ωt)}(∇u)(x)dx), u∈C10(Ω). 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, Brézis, and Wainger. © 2000 Academic Press.