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) 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.

An optimal endpoint trace embedding

“…as one would expect in the light of other known optimal Sobolev embeddings such as those treated in [13,15,3,10] (see [6,9] for the optimality), and trace embeddings ( [5]). …”

Boundary trace inequalities and rearrangements

“…In the proof of Theorems 1.1 and 1.2 we shall need the following auxiliary result in the spirit of [EKP,Theorem 4.5] and [KP,Theorem 3.2].…”

Compactness properties of Sobolev imbeddings for rearrangement invariant norms

“…To verify the Claim let f ∈ Y satisfy f Y ≤ 1 and f ≥ 0. We follow the construction in the proof of Theorem 3.8 in [11] to define a function u :…”

“…In [11], Edmunds (Ω), where u * and |∇u| * are, respectively, the decreasing rearrangements of u and of the norm of its gradient. Defining X(Ω) := {u : Ω → R : u * ∈ X} and setting u X(Ω) := u * X (which is a norm because X is r.i.), the above inequality becomes…”

“…Edmunds, Kerman and Pick studied the boundedness of (1.2) by showing that (1.1) is equivalent to boundedness, from Y to X, of the kernel operator T associated with Sobolev's inequality, namely T f(t) : [11,Theorem 6.1]. We will investigate compactness of the Sobolev imbedding (1.2) via the associated kernel operator T , typically acting on classical function spaces, the advantage being that the general theory of such operators is well understood.…”