Local monotonicity structure of symmetric spaces with applications
[ 1 ] Instytut Matematyki, Wydział Elektryczny, Politechnika Poznańska | [ P ] employee
2014
scientific article
english
- symmetric Banach function spaces
- Orlicz–Lorentz spaces
- Lorentz spaces of maximal functions
- best dominated approximation problems
- local (global) monotonicity structure
- order continuity
EN In this paper we discuss several monotonicity properties in Banach lattices. We start with several general results on local structure of symmetric Banach function spaces discussing in particular whether a point x ∈ E has some local property if and only if its nonincreasing rearrangement x∗ has the same property (Section 2). In that section we also prove some general facts for nonincreasing rearrangements which may be of independent interest. Next, we apply these results to find complete criteria for local monotonicity structure of Lorentz spaces Γp,ω and Orlicz–Lorentz spaces Λϕ,ω (Sections 3, 4.1 and 4.2). We conclude with the description of global monotonicity structure of Lorentz spaces Γp,ω (Section 4.3). We finish with the applications of upper monotonicity and lower monotonicity points to local best dominated approximation problems in Banach lattices (Section 5).
649 - 662
40
1,12