Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms

Let rn and n be positive integers with n greater than or equal to 2 and 1 l ess than or equal to m less than or equal to n-1. We study rearrangement-in variant quasinorms rho(R) and rho(D) on functions f :(0, 1) --> R such that to each bounded domain R in R-n, with Lebesgue measure \Omega\, there corr esponds C = C(\Omega\) > 0 for which one has the Sobolev imbedding inequali ty rho(R)(u*(\Omega\ t)) less than or equal to C rho(D)(\del(m)u\*(\Omega\ t)), u is an element of C-0(m)(Omega), involving the nonincreasing rearrang ements of u and a certain, mth order gradient of tl. When In = 1 we deal, i n fact, with a closely related imbedding inequality of Talenti, in which rh o(D) need not be rearrangement-invariant, rho R(u*(\Omega\ t)) less than or equal to C rho(D)((d/dt) integral ({x is an element of Rn:\u(x)\ > u*(\Ome ga\ t)})\(del u)(x)\dx), u is an element of C-0(1)(Omega). In both cases we are especially interested in when the quasinorms are optimal, in the sense that rho(R) cannot be replaced by an essentially larger quasinorm and rho( D) cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of T rudinger, Strichartz, Hansson, Brezis,and Wainger. (C) 2000 Academic Press.