Displaying similar documents to “Representation of linear functionals on local L p -spaces”

Solvability of the functional equation f = (T-I)h for vector-valued functions

Ryotaro Sato (2004)

Colloquium Mathematicae

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Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and l i m n → ∞ f ₙ = f in measure for some f ∈ M(μ;X), then also l i m n → ∞ T f ₙ = T f in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X)...

The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

Janusz Brzdęk (1996)

Annales Polonici Mathematici

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Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation f ( x + f ( x ) n y ) = f ( x ) f ( y ) , then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.

Bargmann representation of q-commutation relations for q > 1 and associated measures

Ilona Królak (2007)

Banach Center Publications

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The classical Bargmann representation is given by operators acting on the space of holomorphic functions with the scalar product ⟨ z ⁿ | z k ⟩ q = δ n , k [ n ] q ! = F ( z ⁿ z ̅ k ) . We consider the problem of representing the functional F as a measure for q > 1. We prove the existence of such a measure and investigate some of its properties like uniqueness and radiality. The above problem is closely related to the indeterminate Stieltjes moment problem.

Invariance of the Gibbs measure for the Benjamin–Ono equation

Yu Deng (2015)

Journal of the European Mathematical Society

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In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than L 2 , extending the L 2 well-posedness result of Molinet [20].

Measurable envelopes, Hausdorff measures and Sierpiński sets

Márton Elekes (2003)

Colloquium Mathematicae

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We show that the existence of measurable envelopes of all subsets of ℝⁿ with respect to the d-dimensional Hausdorff measure (0 < d < n) is independent of ZFC. We also investigate the consistency of the existence of ℋ d -measurable Sierpiński sets.

Mean values and associated measures of δ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let u be a δ -subharmonic function with associated measure μ , and let v be a superharmonic function with associated measure ν , on an open set E . For any closed ball B ( x , r ) , of centre x and radius r , contained in E , let ℳ ( u , x , r ) denote the mean value of u over the surface of the ball. We prove that the upper and lower limits as s , t → 0 with 0 < s < t of the quotient ( ℳ ( u , x , s ) - ℳ ( u , x , t ) ) / ( ℳ ( v , x , s ) - ℳ ( v , x , t ) ) , lie between the upper and lower limits as r → 0 + of the quotient μ ( B ( x , r ) ) / ν ( B ( x , r ) ) . This enables us to use some well-known measure-theoretic results to prove new variants...

Metric transitivity and integer valued functions

Solomon Schwartzman (1960)

Annales de l'institut Fourier

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Soit X un espace mesurable de mesure μ finie ; φ : X → X une application vérifiant μ ( φ - 1 ( S ) ) = μ ( S ) pour chaque ensemble mesurable S ⊆ X . On donne des conditions nécessaires et suffisantes pour que X soit un ensemble ergodique.

On systems of imprimitivity on locally compact abelian groups with dense actions

J. Mathew, M. G. Nadkarni (1978)

Annales de l'institut Fourier

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Consider the four pairs of groups ( Γ , R ) , ( Γ / Γ 0 , R / Γ 0 ) , ( K ∩ S , P ) and ( S , B ) , where Γ , R are locally compact second countable abelian groups, Γ is a dense subgroup of R with inclusion map from Γ to R continuous; Γ 0 ⊆ Γ ⊆ R is a closed subgroup of R ; S , B are the duals of R and Γ respectively, and K is the annihilator of Γ 0 in B . Let the first co-ordinate of each pair act on the second by translation. We connect, by a commutative diagram, the systems of imprimitivity which arise in a natural fashion on each pair, starting...

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space L p ( μ , X ) (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of L p ( μ , X ) (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

Convex approximation of an inhomogeneous anisotropic functional

Giovanni Bellettini, Maurizio Paolini (1994)

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

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The numerical minimization of the functional F ⁢ u = ∫ Ω ϕ ⁢ x , ν u ⁢ D ⁢ u + ∫ ∂ ⁡ Ω μ ⁢ u ⁢ d ⁢ H n - 1 - ∫ Ω κ ⁢ u ⁢ d ⁢ x , u ∈ B ⁢ V ⁢ Ω ; - 1 , 1 is addressed. The function ϕ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that F can be equivalently minimized on the convex set B ⁢ V ⁢ Ω ; - 1 , 1 and then regularized with a sequence F ϵ ⁢ u ϵ , of stricdy convex functionals defined on B ⁢ V ⁢ Ω ; - 1 , 1 . Then both F and F ϵ , can be discretized by continuous linear finite elements. The convexity property of the functionals on B ⁢ V ⁢ Ω ; - 1 , 1 is useful in the numerical...