Optimal Sobolev Imbeddings Involving Rearrangement-Invariant Quasinorms☆

By D. Edmunds, Ron Kerman, L. Pick. 2000

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Let m and n be positive integers with n⩾2 and 1⩽m⩽n−1. We study rearrangement-invariant quasinorms ϱR and ϱD on functions f: (0, 1)→View the MathML source such that to each bounded domain Ω in View the MathML sourcen, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality ϱR(u*(|Ω| t))⩽CϱD(|∇mu|* (|Ω| t)), u∈Cm0(Ω), 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 ϱD need not be rearrangement-invariant, ϱR(u*(|Ω| t))⩽CϱD((d/dt) ∫{x∈View the MathML sourcen : |u(x)|>u*(|Ω| t)} |(∇u)(x)| dx), u∈C10(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that ϱR cannot be replaced by an essentially larger quasinorm and ϱD 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, Brezis, and Wainger.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/92887ab67742641c5ec03364b0167c7d70da0419

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Citations in the corpus

Optimal Sobolev imbeddings (2006, methodology): “A brief history of such inequalities is given in the introduction of our paper with David Edmunds [5].”
Optimal Domains for Kernel Operators via Interpolation (2002, background): “Recently Edmunds, Kerman and Pick have studied this problem for the Sobolev embedding; see [12].”
Limiting reiteration for real interpolation with slowly varying functions (2005, background): “2 (ii)] and that in [30] or [12] one can also find results related to (7.”
Discretization and anti-discretization of rearrangement-invariant norms (2003, background): “Remark 6.3. Let us recall that the following special case of Theorem 6.2 was proved in [ 12, Theorem 2.7 ]:”
Hardy inequalities with non-standard remainder terms (2008, background): “13) goes back (apart from the constant) to [11,25,27], and has recently been shown to be optimal as far as the norm on the left-hand side is concerned [17,20].”
Optimal Orlicz-Sobolev embeddings (2004, result): “Let us mention that special instances of this example overlap with results scattered in various papers, including [2, 3, 8, 10, 15, 17, 18, 19, 20, 21, 25, 26] Case 1 ≤ p < n.”
On Lorentz spaces Γp,w (2004, background): “ACKNOWLEDGEMENT: We wish to thank the referee for valuable comments and for bringing reference [ 12 ] to our attention.”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “e inequality (6.3) kukY (Ω) ≤ C2 k∇ muk X(Ω) holds for some constant C2 and every u ∈ Vm ⊥ X(Ω). Forerunners of Theorem 6.1 are known. The ﬁrst order case (m = 1) on Lipschitz domains was obtained in [39]. Inthe case whenm = 2, and functionsvanishingon ∂Ωare considered, the equivalence of (6.1) and ”