Monotonicity properties of Lorentz spaces

Abstract

Abstract. Criteria for uniform monotonicity, local uniform monotonicity and strict monotonicity of Lorentz spaces are given.Let L° denote the space of all (equivalence classes of ) Lebesgue measurable real-valued functions defined on the interval [0, y), y < oc. In what follows, if fyg £ L°, then f < g means f(t) < g(t) almost everywhere (a.e.) with respect to the Lebesgue measure m on the real line. If f £ L° we denote by df the distribution function of \f\, that is, df(t) = m{s:\f(s)\>t}, and we denote by /* the decreasing rearrangement of \f\, that is, f(t) = inf{5 > 0 : df(s) < t}.

Non-squareness and local uniform non-squareness properties of Orlicz-Lorentz function spaces endowed with the Orlicz norm

“…where a i ≥ 0 and n i=1 a i = 1. Therefore we have Lemma 11,17]). The Lorentz function space L 1,ω is strictly monotone if and only if ω is positive on [0, γ) and 6]).…”

On a certain class of norms in semimodular spaces and their monotonicity properties

“…Recently, many authors have investigated intensively geometric properties of the Orlicz spaces L φ , Musielak-Orlicz spaces L φ and Lorentz spaces Λ φ,w for a Orlicz function φ and a weight function w (see e.g. [7,8,9,12]). It is worth mentioning that many papers were dedicated especially to rotundity and monotonicity properties which have a deep application to the approximation theory (see e.g.…”

Geometry of Orlicz spaces equipped with norms generated by some lattice norms in $$\mathbb {R}^{2}$$ R 2

“…For these reasons monotonicity properties were investigated in various classes of function spaces. Namely, in [13,24,26,32,33] for Musielak-Orlicz spaces, in [21] for Lorentz spaces, in [19] for Orlicz-Lorentz spaces, in [4,14,16,18,23,37] for Orlicz spaces, in [28,29] for Calderòn-Lozanovskiǐ spaces, in [12] for Cesàro-Orlicz sequence spaces, for Orlicz-Sobolev spaces in [8]. Relationships between monotonicity properties and rofundity properties as well as between monotonicity properties and orthogonal monotonicity properties in Kőthe spaces were given in [22].…”