THE FORGOTTEN PARAMETER IN GRAND LEBESGUE SPACES

Let 1 < p < ∞, ε0 ∈]0, p − 1], Ω ⊂ Rn be a Lebesgue measurable set of positive, finite measure, and let δ : (0, p − 1] → (0, ∞) be such that δb(·):= δ(·) p−·1 is nondecreasing and bounded. We show that the linear set of functions 5 f Lebesgue measurable on Ω: 0<ε sup ≤ε0(δ(ε) k − |f(x)|p−εdx ) p−1 ε < ∞ 5 Ω does not depend on small values of ε0 if and only if δb ∈ ∆2(0+) (i.e., δb(2ε) ≤ cδb(ε) for ε small, for some c > 1), which is equivalent to say that δ ∈ ∆2(0+). This means that in the case δb ∈/ ∆2(0+), the parameter ε0 plays a crucial role in the definition of a generalized grand Lebesgue space, namely, different values of ε0 define different Banach function spaces.