Local Uniform Monotonicity Research Articles

Lower Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

Lower strict monotonicity points and lower local uniform monotonicity points are considered in the case of Musielak–Orlicz function spaces LΦ endowed with the Mazur–Orlicz F-norm. The findings outlined in this study extend the scope of geometric characteristics observed in F-normed Orlicz spaces, as well as monotonicity properties within specific F-normed lattices. They are suitable for the Orlicz spaces of ordered continuous elements, specifically in relation to the Mazur–Orlicz F-norm. In addition, in this paper presents results that can be used to derive certain monotonicity properties in F-normed Musielak–Orlicz spaces.

Upper Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

In this paper, the necessary and sufficient conditions for the upper strict monotonicity point and the upper local uniform monotonicity point are given in the case of Musielak–Orlicz spaces equipped with the Mazur–Orlicz F-norm. Moreover, strict monotonicity and upper local uniform monotonicity are easily deduced in the case of Musielak–Orlicz spaces endowed with the Mazur–Orlicz F-norm, and the work by Kaczmarek presented in the references is encompassed by the corollaries presented in this paper.

Some monotonicity properties in F-normed Musielak\u2013Orlicz spaces

Strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and their orthogonal counterparts are considered in the case of Musielak–Orlicz function spaces L^Phi (mu ) endowed with the Mazur–Orlicz F-norm as well as in the case of their subspaces E^Phi (mu ) with the F-norm induced from L^Phi (mu ). The presented results generalize some of the results from Cui et al. (Aequ Math 93:311–343, 2019) and Hudzik et al. (J Nonlinear Convex Anal 17(10):1985–2011, 2016), obtained only for Orlicz spaces as well as their subspaces of order continuous elements equipped with the Mazur–Orlicz F-norm.

Geometric properties of F-normed Orlicz spaces

The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on {mathbb {R}}_{+} and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Köthe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.

Lower and upper local uniform $K$ -monotonicity in symmetric spaces

Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform $K$-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$.

Rotundity and monotonicity properties of selected Ces\xe0ro function spaces

We study rotundity, strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some classes of Cesàro function spaces. We present full criteria of these properties in the Cesà ro–Orlicz function spaces Ces_{varphi } (under some mild assumptions on the Orlicz function varphi ). Next, we prove a characterization of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in the Cesàro–Lorentz function spaces CLambda _{phi }. We also show that the space CLambda _{phi } is never rotund. Finally, we will prove that Cesàro–Marcinkiewicz function space CM_{phi }^{(*)} is neither strictly monotone nor order continuous for any quasi-concave function phi .

Monotonicity in orlicz-lorentz sequence spaces equipped with the orlicz norm

In Orlicz-Lorentz sequence space λ°ϕ, w with the Orlicz norm, uniform monotonic ity, points of upper local uniform monotonicity and lower local uniform monotonicity are characterized. Moreover, the monotonicity coefficient in λ°ϕ, w are discussed.

In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements

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 \({\sigma}\)-finite complete and non-atomic measure space. It is also showed that in the case of order continuous symmetric spaces over a \({\sigma}\)-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.

Monotonicity and rotundity of Lorentz spaces [formula omitted

Criteria for rotundity, strict monotonicity, and lower local uniform monotonicity of the Lorentz spaces Γ p , w of maximal functions are given under arbitrary nonnegative weight function w . Necessary conditions are also established for uniform monotonicity of the spaces Γ p , w for 1 ≤ p < ∞ . Moreover, the spaces Γ 1 , w that are uniformly monotone are characterized.

Points of monotonicity in Orlicz–Lorentz function spaces

In Orlicz–Lorentz function space ⋀ M , ω [ 0 , γ ) , points of upper monotonicity, lower monotonicity, upper local uniform monotonicity ( γ < ∞ ) and lower local uniform monotonicity ( γ < ∞ ) are characterized. As the corollary, we can easily obtain the criteria for strict monotonicity, upper local uniform monotonicity ( γ < ∞ ) and lower local uniform monotonicity ( γ < ∞ ) of ⋀ M , ω [ 0 , γ ) .

Local Δ2E(x) condition as a crucial tool for local structure of Calderón–Lozanovskiĭ spaces

We shall give a full criterion for a point x∈Eφ to be a point of order continuity of Calderón–Lozanovskiĭ spaces Eφ. In order to get that result we introduce the new local Δ2E(x) condition which appears to be necessary and sufficient for this goal. Next we study the structure of points of lower local uniform monotonicity (LLUM for short). We give a result concerning general Köthe sequence spaces and we present a characterization for LLUM points in Calderón–Lozanovskiĭ spaces.

Criteria for monotonicity properties of Musielak–Orlicz spaces equipped with the Amemiya norm

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.

Points of monotonicity in Musielak-Orlicz function spaces endowed with the Luxemburg norm

Points of lower monotonicity, upper monotonicity, lower local uniform monotonicity and upper local uniform monotonicity of Musielak-Orlicz spaces L M and of their subspaces E M of order continuous elements from L M are characterized in the case of a nonatomic measure space $ (G, \Sigma, \mu) $ if L M and E M are endowed with the Luxemburg norm. Criteria for strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity are easily deduced.

Local monotonicity structure of Calderón-Lozanovskiǐ spaces

We first prove that if x is an element on the unit sphere of arbitrary Köthe space E, x is strickly positive μ-a.e. and x is an LM-point, then x is an UM-point. Criteria for lower and upper monotone points in Calderón-Lozanovskiǐ spaces E ϖ are presented. Points of lower local uniform monotonicity and upper local uniform monotonicity in E ϖ are also considered. Some sufficient conditions and necessary conditions for these properties of a given point x in S( E ϖ +) are given.

Monotonicity Properties of Musielak–Orlicz Spaces and Dominated Best Approximation in Banach Lattices

Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and uniform monotonicity of a Musielak–Orlicz space endowed with the Amemiya norm and its subspace of order continuous elements are given in the cases of nonatomic and the counting measure space. To complete the results of Kurc (J. Approx. Theory69(1992), 173–187), criteria for upper local uniform monotonicity of these spaces equipped with the Luxemburg norm are also given. Some applications to dominated best approximation are presented.

Monotonicity Properties of Lorentz Spaces

Criteria for uniform monotonicity, local uniform monotonicity and strict monotonicity of Lorentz spaces are given.

Monotonicity properties of Lorentz spaces

Criteria for uniform monotonicity, local uniform monotonicity and strict monotonicity of Lorentz spaces are given.