Local monotonicity structure of symmetric spaces with applications

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 Γ p , ω and Orlicz–Lorentz spaces Λ φ , ω (Sections  3, 4.1 and 4.2). We conclude with the description of global monotonicity structure of Lorentz spaces Γ p , 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).