Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms
✍ Scribed by D.E. Edmunds; R. Kerman; L. Pick
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 374 KB
- Volume
- 170
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
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 * R (u*(|0| t)) C* D (|{ m u|* (|0| t)), u # C m 0 (0), 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, * R (u*(|0| t)) C* D ((dÂdt) [x # R n : |u(x)| >u*( |0| t)] |({u)(x)| dx), u # C 1 0 (0). 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.