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Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms

journal contribution
posted on 2023-06-07, 20:28 authored by D E Edmunds, R Kerman, L Pick
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)?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 Analysis

ISSN

0022-1236

Publisher

Elsevier

Issue

2

Volume

170

Page range

307-355

ISBN

0022-1236

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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