Are generalized Lorentz “spaces” really spaces?

Abstract

Abstract. We show that the Lorentz space Λ p (w) need not be a linear set for certain "non-classical" weights w. We establish necessary and sufficient conditions on p and w for this situation to occur.

Optimality of function spaces for kernel integral operators

“…The ∆ 2 condition imposed on the function φ guarantees that • mϕ is a quasinorm (see also [13]), therefore we will consider m ϕ to be a quasinormed space. We say that a linear set M ⊂ M together with a functional F : M → [0, ∞) can be equivalently renormed with a rearrangementinvariant norm, if there exists a rearrangement-invariant space X and constants C 1 and C 2 such that X = M and…”

Note on linearity of rearrangement-invariant spaces

“…We also point out some applications of this result (including an alternative approach to the characterization of linearity of Orlicz-Lorentz) which enjoy the above-mentioned properties. In the case of linearity, the result is known (see [4]) but it assumes the ∆ 2 -condition of the function ϕ. In the present article, we will present a stronger version of this theorem without that restriction.…”

“…The following theorem is known in a weaker form (with the additional assumption of ϕ E 2 ϕ) (see [4,Theorem 4.1]). We will point an alternative proof based on Theorem 2.3 removing the assumption.…”

“…In 1990, Sawyer in [9] characterized the normability of classical Lorentz spaces. Many papers, with characterizations of linearity (see [4]) quasinormability (see [3]) and normability ([2]) followed (see also [5]).…”

Criterion for the coincidence of strong and weak Orlicz spaces

“…The following question is formulated in [6] by M. Cwikel, A. Kaminska, L. Maligranda and L. Pick: "Are the generalized Lorentz spaces really spaces? ", i.e., can these spaces be normed such that they are (complete) Banach functional rearrangement invariant spaces?…”