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

Abstract

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.