File(s) not publicly available
Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms
journal contribution
posted on 2023-06-07, 20:28 authored by D E Edmunds, R Kerman, L PickLet 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)?View the MathML source such that to each bounded domain O in View the MathML sourcen, with Lebesgue measure |O|, there corresponds C=C(|O|)>0 for which one has the Sobolev imbedding inequality ?R(u*(|O| t))?C?D(|?mu|* (|O| t)), u?Cm0(O), 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*(|O| t))?C?D((d/dt) ?{x?View the MathML sourcen : |u(x)|>u*(|O| t)} |(?u)(x)| dx), u?C10(O). 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.
History
Publication status
- Published
Journal
Journal of Functional AnalysisISSN
0022-1236Publisher
ElsevierExternal DOI
Issue
2Volume
170Page range
307-355ISBN
0022-1236Department affiliated with
- Mathematics Publications
Full text available
- No
Peer reviewed?
- Yes
Legacy Posted Date
2012-02-06Usage metrics
Categories
No categories selectedKeywords
Licence
Exports
RefWorks
BibTeX
Ref. manager
Endnote
DataCite
NLM
DC