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Marian Nowak - One of the best experts on this subject based on the ideXlab platform.
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Applications of the Theory of Orlicz Spaces to Vector Measures
Analysis Mathematica, 2018Co-Authors: Marian NowakAbstract:Let (Ω, Σ, λ) be a finite Complete Measure Space, (E, ξ) be a sequentially Complete locally convex Hausdorff Space and E′ be its topological dual. Let caλ (Σ, E) stand for the Space of all λ-absolutely continuous Measures m: Σ → E. We show that a uniformly bounded subset M of caλ (Σ, E) is uniformly λ-absolutely continuous if and only if for every equicontinuous subset D of E′, there exists a submultiplicative Young function φ such that the set \(\left\{ {\frac{{d\left( {e'om} \right)}}{{d\lambda }}:m \in M,e' \in D} \right\}\) is relatively weakly compact in the Orlicz Space Lφ(λ). As a consequence, we present a generalized Vitali–Hahn–Saks theorem on the setwise limit of a sequence of λ-absolutely continuous vector Measures in terms of Orlicz Spaces.
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operator Measures and integration operators
Indagationes Mathematicae, 2013Co-Authors: Marian NowakAbstract:Abstract Let ( Ω , Σ , μ ) be a finite Complete Measure Space and ( X , ‖ ⋅ ‖ X ) be a Banach Space with the Banach dual X ∗ . Let L ∞ ( μ , X ) denote the Space of all μ -measurable functions f : Ω → X such that ess sup ω ∈ Ω ‖ f ( ω ) ‖ X ∞ . We study the problem of the integral representation of some natural classes of linear operators from L ∞ ( μ , X ) to a Banach Space with respect to the corresponding operator Measures. We characterize relatively σ ( bvca μ ( Σ , X ∗ ) , L ∞ ( μ , X ) ) -sequentially compact sets in the Space bvca μ ( Σ , X ∗ ) of all countably additive Measures ν : Σ → X ∗ of bounded variation with ν ( A ) = 0 if μ ( A ) = 0 .
Jingshi Xu - One of the best experts on this subject based on the ideXlab platform.
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Proximinality in Banach Space-Valued Grand Bochner-Lebesgue Spaces with Variable Exponent
Mathematical Notes, 2019Co-Authors: Jingshi XuAbstract:Let (A, \(\mathscr{A}\), µ) be a σ-finite Complete Measure Space, and let p(·) be a µ-measurable function on A which takes values in (1, ∞). Let Y be a subSpace of a Banach Space X. By \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) and \({\tilde L^{p(\cdot),\varphi }}(A,X)\) we denote the grand Bochner-Lebesgue Spaces with variable exponent p(·) whose functions take values in Y and X, respectively. First, we estimate the distance of f from \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) when \(f \in {\tilde L^{p(\cdot),\varphi }}(A,X)\). Then we prove that \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) is proximinal in \({\tilde L^{p(\cdot),\varphi }}(A,X)\) if Y is weakly \(\mathcal{K}\)-analytic and proximinal in X. Finally, we establish a connection between the proximinality of \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) in \({\tilde L^{p(\cdot),\varphi }}(A,X)\) and the proximinality of L1(A, Y) in L1(A, X).
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Proximinality in Banach Space valued Musielak-Orlicz Spaces
Journal of Inequalities and Applications, 2014Co-Authors: Jingshi XuAbstract:Let (A, A, μ )b e aσ -finite Complete Measure Space and let Y be a subSpace of a Banach Space X.L etϕ be a generalized � -function on (A, A, μ). Denote by L ϕ (A, Y) and L ϕ (A, X) the Musielak-Orlicz Spaces whose functions take values in Y and X, respectively. Firstly, let f ∈ L ϕ (A, X), we characterize the distance of f from L ϕ (A, Y). Then, if Y is weakly K-analytic and proximinal in X, we show that L ϕ (A, Y) is proximinal in L ϕ (A, X). Finally, we give the connection between the proximinality of L ϕ (A, Y )i n L ϕ (A, X) and the proximinality of L 1 (A, Y )i nL 1 (A, X).
Henryk Hudzik - One of the best experts on this subject based on the ideXlab platform.
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In some symmetric Spaces monotonicity properties can be reduced to the cone of rearrangements
Aequationes mathematicae, 2016Co-Authors: Henryk Hudzik, Radosław Kaczmarek, Miroslav KrbecAbstract:Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric Spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric Spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric Space over a $${\sigma}$$ σ -finite Complete and non-atomic Measure Space. It is also showed that in the case of order continuous symmetric Spaces over a $${\sigma}$$ σ -finite and Complete Measure Space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM * coincide. As an application of this result, in the case of a non-atomic Complete finite Measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz Spaces, is presented. Finally, it is proved that every rotund and reflexive Space X such that both X and X * have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.
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Criteria for monotonicity properties of Musielak-Orlicz Spaces equipped with the Amemiya norm
Journal of Mathematical Analysis and Applications, 2005Co-Authors: Henryk Hudzik, L. Szymaszkiewicz, T. WangAbstract:Abstract Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity, and uniform monotonicity of Musielak–Orlicz Spaces over any σ-finite and Complete Measure Space, endowed with the Amemiya norm are given. The fact that the Spaces are considered over arbitrary σ-finite Measure Space is essential because, as it is shown in Example 3, the Musielak–Orlicz Spaces need not be strictly monotone even if their restrictions to the nonatomic part and the purely atomic part are strictly monotone.
M. Nowak - One of the best experts on this subject based on the ideXlab platform.
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Applications of the Theory of Orlicz Spaces to Vector Measures
Analysis Mathematica, 2019Co-Authors: M. NowakAbstract:Let (Ω, Σ, λ) be a finite Complete Measure Space, ( E , ξ ) be a sequentially Complete locally convex Hausdorff Space and E ′ be its topological dual. Let ca_λ (Σ, E ) stand for the Space of all λ-absolutely continuous Measures m : Σ → E . We show that a uniformly bounded subset M of ca_λ (Σ, E ) is uniformly λ-absolutely continuous if and only if for every equicontinuous subset D of E ′, there exists a submultiplicative Young function φ such that the set $$\left\{ {\frac{{d\left( {e'om} \right)}}{{d\lambda }}:m \in M,e' \in D} \right\}$$ { d ( e ′ o m ) d λ : m ∈ M , e ′ ∈ D } is relatively weakly compact in the Orlicz Space L ^ φ (λ). As a consequence, we present a generalized Vitali–Hahn–Saks theorem on the setwise limit of a sequence of λ-absolutely continuous vector Measures in terms of Orlicz Spaces.
Miroslav Krbec - One of the best experts on this subject based on the ideXlab platform.
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In some symmetric Spaces monotonicity properties can be reduced to the cone of rearrangements
Aequationes mathematicae, 2016Co-Authors: Henryk Hudzik, Radosław Kaczmarek, Miroslav KrbecAbstract:Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric Spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric Spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric Space over a $${\sigma}$$ σ -finite Complete and non-atomic Measure Space. It is also showed that in the case of order continuous symmetric Spaces over a $${\sigma}$$ σ -finite and Complete Measure Space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM * coincide. As an application of this result, in the case of a non-atomic Complete finite Measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz Spaces, is presented. Finally, it is proved that every rotund and reflexive Space X such that both X and X * have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.