Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Curbera, Guillermo P.; Ricker, Werner J.

doi:10.1090/s0002-9947-06-04203-6

Cited by 27 publications

(22 citation statements)

References 14 publications

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“…In both [6] and [20] the function t → t m/n σ(χ (0,t) ) (in [6] m = 1) plays an important role. We observe that from our expression (3.5) for σ one readily obtains (6.2) σ (χ (0,t) ) ≈ t sup t≤s<1 s m/n−1 σ(χ (0,s) ), with t m/n σ(χ (0,t) ) equal to its quasiconcave majorant in (6.2) when the function t m/n−1 σ(χ (0,t) ) is nonincreasing.…”

Section: 2mentioning

confidence: 99%

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Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Kerman

Pick

2008

Studia Math.

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Abstract. We find necessary and sufficient conditions on a pair of rearrangementinvariant norms, and σ, in order that the Sobolev space W m, (Ω) be compactly imbedded into the rearrangement-invariant space Lσ(Ω), where Ω is a bounded domain in R n with Lipschitz boundary and 1 ≤ m ≤ n − 1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from L (0, |Ω|) into Lσ(0, |Ω|). The results are illustrated with examples in which and σ are both Orlicz norms or both Lorentz Gamma norms.1. Introduction. Sobolev spaces are one of the key elements of modern functional analysis. In applications their most important property is how they imbed into various function spaces. To be more specific, compactness of Sobolev imbeddings is useful in the theory of PDEs; indeed, it is quite indispensable when the methods of the calculus of variations are used.Among the function norms defining both the Sobolev and the imbedding spaces those of Lebesgue play a primary role, though a satisfactory description of all cases, especially the limiting ones, requires other, more delicate, norms.In this paper we characterize precisely when a Sobolev space defined by a rearrangement-invariant (r.i.) norm is compactly imbedded into a function space determined by another such norm. We will use the interpolation methods developed in our previous papers [8], [13] and [14], in which optimal (hence noncompact) imbeddings were studied.Suppose that Ω is a bounded domain in R n , n ≥ 2, having a Lipschitz boundary, written ∂Ω ∈ Lip 1 .Fix m ∈ Z + , 1 ≤ m ≤ n − 1 and let N = N (m, n) = 0≤|α|≤m 1 be the number of multiindices α = (α 1 , . . . , α k ) satisfying 0 ≤ |α|:= α 1 + · · ·+α k ≤ m.

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Section: 2mentioning

confidence: 99%

“…Recently, two papers, [6] and [20], on the compactness of Sobolev imbeddings involving r.i. norms have come to our attention. We discuss the relation of their results to ours in the final section.…”

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confidence: 99%

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Kerman

Pick

2008

Studia Math.

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“…This "optimal extension process" is known for kernel operators ( [2,7,46]), Sobolev imbeddings ( [8,9,14,25]), the Hardy operator, [10], and the Hausdorff-Young inequality, [35]. For convolutions with measures (in L p -spaces), which form a proper subclass of all p-multiplier operators, see [39,40], [41,Ch.7].…”

Section: Introductionmentioning

confidence: 99%

Optimal Extension of Fourier Multiplier Operators in L p (G)

Mockenhaupt

Okada²,

Ricker

2010

Integr. Equ. Oper. Theory

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“…For example, in order to find the largest space Y ( ) (i.e., having the smallest norm) for the right-hand side of (1), with X ( ) on the left-hand side being fixed, it suffices to find the largest domain for the 1-variable, X -valued kernel operator T in (2). In [5,7] we studied precisely this optimal domain for the extension of T , with T given by (2). It is a r.i. space, denoted by [T, X ] ri , and via the above mentioned results of [9,11], the inequality u X ( ) ≤ C |∇u| [T,X ] ri ( ) , u ∈ C 1 0 ( ),…”

Section: Introductionmentioning

confidence: 99%

Function spaces arising from kernel operators

Curbera

Ricker

2010

Positivity

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Given a probability space ( , μ) and a rearrangement invariant space X on [0,1], in certain situations inequalities for spaces of R-valued functions on are equivalent to the boundedness of an associated operator T K : L ∞ ([0, 1]) → X generated by a kernel K ≥ 0 on the unit square (e.g. Sobolev type inequalities or Riesz potentials on subsets ⊂ R n ). A natural class of spaces for treating such inequalities is given by [T K , X ]( ) := {u : → R : T K u * ∈ X } together with the functional u → T K u * X , where u * is the decreasing rearrangement of u. The investigation of these spaces is our main aim; the nature of the base space X and of K (via its monotonicity/growth properties) play a crucial role.

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Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Cited by 27 publications

References 14 publications

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Optimal Extension of Fourier Multiplier Operators in L p (G)

Function spaces arising from kernel operators

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