Some monotonicity properties in F-normed Musielak–Orlicz spaces

Abstract

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 Φ (µ) endowed with the Mazur-Orlicz F-norm as well as in the case of their subspaces E Φ (µ) with the F-norm induced from L Φ (µ). The presented results generalize some of the results from Cui et al.

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

“…It is worth noting that quasi-Banach spaces have been extensively studied over the last century (see [1][2][3][4][5][6]). As we know, in the realm of quasi-Banach spaces, the geometry is heavily influenced by the significant role played by monotonicity properties.…”

“…), as mentioned above, are Σ−measurable functions. The methods used to prove this statement are similar to [7] or [5]. Definition 6 (see [5]).…”

“…The methods used to prove this statement are similar to [7] or [5]. Definition 6 (see [5]). If there exists a set T 1 ⊂ Σ with measure m(T 1 ) = 0, a positive constant K and a non-negative function h(t) in the Lebesgue space L 1 (T, Σ, m) for which the inequality Φ(t, 2u) ≤ KΦ(t, u) + h(t) holds for all t ∈ T\T 1 , then we say that the monotone Musielak-Orlicz function Φ is said to satisfy the ∆ 2 −condition (Φ ∈ ∆ 2 for short).…”

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

“…are Σ measurable and the proofs are similar to the proof of [5]. The function Φ(t, u) is continuous on [0, b Φ (t)) in regard to u for almost every t ∈ T. Definition 2 (see [6]). We say that a monotone Musielak-Orlicz function Φ satisfies the ∆ 2 − condition (for brevity, we write Φ ∈ ∆ 2 ) if there exists a set T 1 ∈ Σ with m(T 1 ) = 0, a constant K > 0, and a function 0…”

Uniform monotonicity of Orlicz spaces equipped with the Mazur–Orlicz F‐norm and dominated best approximation in F‐normed Köthe spaces

“…Clearly, the theory of normed Orlicz spaces equipped with the Luxemburg-Nakano norm is very well known. Recently, some authors studied 𝐹-normed Orlicz and Musielak-Orlicz spaces with the Mazur-Orlicz 𝐹-norm (see [10,13,14,18,31,32,34,35]). Moreover, the quasi-normed Calderón-Lozanovskiȋ spaces (in particular Orlicz spaces) have been studied in [19].…”