Lower Local Uniform Monotonicity Research Articles

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.

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.

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 ) .

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.

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.