On the Factor Opposing the Lebesgue Norm in Generalized Grand Lebesgue Spaces

We prove that if 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} and δ:]0,p-1]→]0,∞[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta :]0,p-1]\rightarrow ]0,\infty [$$\end{document} is continuous, nondecreasing, and satisfies the Δ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _2$$\end{document} condition near the origin, then This result permits to clarify the assumptions on the increasing function against the Lebesgue norm in the definition of generalized grand Lebesgue spaces and to sharpen and simplify the statements of some known results concerning these spaces. 


Introduction
The first appearence of generalized grand Lebesgue spaces-a class of Banach function spaces, see e.g. the monograph by Bennett and Sharpley [4]-is in the final remark in [5], where the factor against the L p−ε norm has been further generalized from a power into an increasing function. The growing interest in literature for this class of spaces has been motivated either by their utility in the theory of PDEs (see e.g. [1,8,11]), either in Function Spaces theory (see [6,9,10,[16][17][18]21]). We refer to [7] for a study of these spaces, to [13] for a survey, to [12] for a recent characterization of its norm in term of the decreasing rearrangement. The increasing interest on these spaces led to "maximize" the generalization of the original norm due to Iwaniec and Sbordone in 1992 (see [14]), into f L p),δ (0,1) = ess sup where δ is a nonnegative measurable bounded function on ]0, p − 1] (see [7,Theorem 2.1] for details; see [2,3,15,19] for a generalization also with respect to ε). Even without a deep knowledge of the literature on these spaces, already the expression of the norm · L p),δ (0, 1) suggests clearly that the most natural assumption on δ is the property to be nondecreasing of the function In [12, 3.8] it has been observed that if δ is nondecreasing, then δ is nondecreasing, and that the viceversa does not hold. Hence it is not a surprise that the main results in [7,12] have the "strong" assumption on the monotonicity of δ and not on the monotonicity of δ. The Δ 2 condition is a notion familiar for researchers working in Orlicz spaces (see e.g. [20]). When the so-called Δ 2 condition near the origin (we write δ ∈ Δ 2 if δ(2ε) ≤ cδ(ε) for ε small, for some c > 1) plays a role, again from [12, 3.8] we know that under the "weak" assumption that δ is nondecreasing, The The functionδ Proof. The left wing inequality in (2.3) is an immediate consequence of the continuity of δ: in fact, therefore the proof consists of showing the right wing inequality.
We begin observing thatδ is not affected, up to equivalence, multiplying δ by a positive constant k: assuming, without loss of generality, k > 1, we havē As a consequence, dividing δ by 2δ(p − 1), we may assume without loss of generality that δ(ε) ∈]0, 1] for every ε ∈]0, p − 1] and, since δ is nondecreasing, that δ 0 := δ(0+) ≤ 1/2. If δ 0 > 0, then for some ε 1 ∈]0, p − 1] we have The two relations above show that in the case δ 0 > 0, then In the following we may therefore assume that By continuity of δ, δ(0+) = 0 and, on the other hand, we recall that where the last inequality is due to: We go on with the estimate as follows: where the last inequality is due to the fact that 0 < δ(2 −m+1 ) < 1 and 2 −m p−2 −m > 0. Now fix any n 0 ∈ N such that note that it depends only on p and δ and that from (2.7) we have from which where ε 0 is from assumption (2.1).
Consider for the moment ε in the interval ]0, ε 2 [, so that from (2.5) this means that −n < −n 0 + 1, i.e. n ≥ n 0 , hence, from (2.8), we have This chain of inequalities gives us the possibility to establish that the exponent in the numerator of B is positive: in fact Hence, taking into account that m ≥ n and that δ is nondecreasing, By (2.9) and (2.1) and therefore we may estimate C as follows: Observe that, since m ≥ n, we have 2 −m ≤ 2 −n and therefore, by (2.9), 2 −m ≤ p − 1, from which p − 2 −m ≥ 1. On the other hand, by (2.8), iterating the argument in (2.10), we have We may therefore estimate Overall, we obtained that D is smaller than a term of a bounded sequence depending only on δ, c (from (2.1)), and n 0 (which in turn depends again on (2.1) and p), henceδ(ε)/δ(ε) is bounded for 0 < ε < ε 2 , where ε 2 depends on n 0 , too. Finally, we conclude arguing as in (2.4):

A Counterexample
The assumption δ ∈ Δ 2 in Theorem 1 cannot be dropped. We are going to exhibit an example of function δ continuous and nondecreasing, such thatδ ≈ δ (i.e., as usual, positive constants c 1 , c 2 such that c 1 δ(ε) ≤δ(ε) ≤ c 2 δ(ε) for every ε small cannot exist). We stress that the definition is well posed, because b n+1 < a n ⇔ 2 −n−1 which is true for n large. The function δ is continuous by definition and clearly nondecreasing (because the sequence (exp(−5 n )) n∈N is decreasing). Since a n < b n and δ(a n ) = δ(b n ), we havē and thereforeδ ≈ δ. We observe also that δ cannot be nondecreasing, otherwise it would bē δ = δ. The lost monotonicity of δ could be verified also directly, because of the constant behavior of δ in the intervals [a n , b n ].
Finally, we observe that δ / ∈ Δ 2 : this is a direct consequence of our Theorem 1, but it can be verified also directly. In fact, writing explicitly the values of a n and b n+1 , it is immediate to realize that a n < 2b n+1 , hence

On the Definition of Generalized Grand Lebesgue Spaces
After [7], the interesting class of spaces L p),δ is for functions δ defined pointwise, therefore the norm can be written with sup instead of esssup : (4.1) Again after [7], the whole class of spaces is covered by the assumption δ nondecreasing, and this is in agreement with the heart of these spaces, which is based on the monotonicity of the Lebesgue norm · p−ε , consequence of the Hölder's inequality (see [13] for details). With this assumption in order, one has that δ itself is nondecreasing, too. If one replaces the assumption " δ nondecreasing" with "δ nondecreasing", then one gets functions δ which are not nondecreasing (see Example 1): in principle, it gives a wider range of spaces. However, in [7] it is shown that replacing δ byδ one has an equivalent norm, and the corresponding δ is nondecreasing. The reader must be aware of the fact that in this case the norms are equivalent, but δ andδ are not necessarily equivalent. After our Theorem 1, they are equivalent if and only if δ ∈ Δ 2 . This essentially confirms that " δ nondecreasing" is the best option for the consideration of the whole class of spaces.
The point is that several results using techniques from Interpolation-Extrapolation theory involve the Δ 2 condition. With the addition of this assumption, after our Theorem 1, we know that the replacement of δ byδ gives not only the equivalence of the norms, but also the equivalenceδ ≈ δ.
The above considerations lead to state the following and the viceversa does not hold. Moreover, and the viceversa does not hold. If, moreover, δ ∈ Δ 2 , then We stress that the last sentence in Theorem 2 comes from the fact that when δ is nondecreasing and Δ 2 , by Theorem 1 we know thatδ ≈ δ without the use of L p),δ = L p),δ .
If one drops the assumption Δ 2 and assumes just δ nondecreasing, then Example 1 shows that (i), (iii) and (v) in general fail, while (ii) and (iv) remain true (because they have been proved without the Δ 2 assumption).
We stress that the problem of a complete characterization of the behavior of the fundamental function in generalized grand Lebesgue spaces remains still open.
Remark 1. Theorem 1 shows that the sentences in [12, 3.7] are correct. The equivalences stated therein use implicitly the stronger assumption that δ (called ϕ in [12]) must be such that δ(·) := δ(·) 1 p−· is nondecreasing, however, their correct (omitted and detailed) justification is a consequence of our main result.