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The concentration-compactness principle in the calculus of variations. The limit case, Part II.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

This paper is the second part of a work devoted to the study of variational problems (with constraints) in functional spaces defined on domains presenting some (local) form of invariance by a non-compact group of transformations like the dilations in RN. This contains for example the class of problems associated with the determination of extremal functions in inequalities like Sobolev inequalities, convolution or trace inequalities... We show how the concentration-compactness principle and method...

The concentration-compactness principle in the calculus of variations. The limit case, Part I.

Pierre-Louis Lions (1985)

Revista Matemática Iberoamericana

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in RN where the invariance of RN by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness...

The concept of boundedness and the Bohr compactification of a MAP Abelian group

Jorge Galindo, Salvador Hernández (1999)

Fundamenta Mathematicae

Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an ℒ ∞ -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This...

The Connes-Kasparov conjecture for almost connected groups and for linear p -adic groups

Jérôme Chabert, Siegfried Echterhoff, Ryszard Nest (2003)

Publications Mathématiques de l'IHÉS

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero.

The continuity of Lie homomorphisms

Bernard Aupetit, Martin Mathieu (2000)

Studia Mathematica

We prove that the separating space of a Lie homomorphism from a Banach algebra onto a Banach algebra is contained in the centre modulo the radical.

The continuity of superposition operators on some sequence spaces defined by moduli

Enno Kolk, Annemai Raidjõe (2007)

Czechoslovak Mathematical Journal

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = ( ϕ k ) let λ ( Φ ) = { x = ( x k ) ( ϕ k ( | x k | ) ) ∈ λ } . Given another sequence of modulus functions Ψ = ( ψ k ) , we characterize the continuity of the superposition operators P f from λ ( Φ ) into μ ( Ψ ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕ k ( k ∈ ...

The controlled separable projection property for Banach spaces

Jesús Ferrer, Marek Wójtowicz (2011)

Open Mathematics

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable...

The converse of the Hölder inequality and its generalizations

Janusz Matkowski (1994)

Studia Mathematica

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ʃ Ω x y d μ ≤ ϕ - 1 ( ʃ Ω ϕ ∘ x d μ ) ψ - 1 ( ʃ Ω ψ ∘ x d μ ) for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist...

The current situation in the linear problem of Molodenskii

Fausto Sacerdote, Fernando Sansò (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si prova l'esistenza di un'unica soluzione debole che dipende con continuità dai dati al contorno per il problema lineare di Molodenskii in approssimazione quasi sferica, nel caso che la superficie al contorno soddisfi una condizione di cono. Si segue un approccio costruttivo diretto, che generalizza una procedura precedentemente elaborata per il problema semplice di Molodenskii. Inoltre si prova che la soluzione ha derivate prime a quadrato integrabile al contorno, il che è essenziale per le applicazioni...

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