A note on the spaces of variable integrability and summability of Almeida and H\"ast\"o

We address an open problem posed recently by Almeida and H\"ast\"o in \cite{AlHa10}. They defined the spaces $\ellqp$ of variable integrability and summability and showed that $\|\cdot|\ellqp\|$ is a norm if $q$ is constant almost everywhere or if $\esssup_{x\in\R^n}1/p(x)+1/q(x)\le 1$. Nevertheless, the natural conjecture (expressed also in \cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\ge 1$ almost everywhere. We show, that $\|\cdot|\ellqp\|$ is a norm, if $1\le q(x)\le p(x)$ for almost every $x\in\R^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\min(p(x),q(x))\ge 1$ for every $x\in\R^n$ such that the triangle inequality does not hold for $\|\cdot|\ellqp\|$.


Introduction
For the definition of the spaces ℓ q(·) (L p(·) ) we follow closely [1]. Spaces of variable integrability L p(·) and variable sequence spaces ℓ q(·) have first been considered in 1931 by Orlicz [5] but the modern development started with the paper [4]. We refer to [3] for an excellent overview of the vastly growing literature on the subject.
First of all we recall the definition of the variable Lebesgue spaces L p(·) (Ω), where Ω is a measurable subset of R n . A measurable function p : Ω → (0, ∞] is called a variable exponent function if it is bounded away from zero. For a set A ⊂ Ω we denote p + A = ess-sup x∈A p(x) and p − A = ess-inf x∈A p(x); we use the abbreviations p + = p + Ω and p − = p − Ω . The variable exponent Lebesgue space L p(·) (Ω) consists of all measurable functions f such that there exist an λ > 0 such that the modular This definition is nowadays standard and was used also in [1, Section 2.2] and [3, If we define Ω ∞ = {x ∈ Ω : p(x) = ∞} and Ω 0 = Ω \ Ω ∞ , then the Luxemburg norm of a function f ∈ L p(·) (Ω) is given by If p(·) ≥ 1, then it is a norm, but it is always a quasi-norm if at least p − > 0, see [4] for details. We denote the class of all measurable functions p : R n → (0, ∞] such that p − > 0 by P(R n ) and the corresponding modular is denoted by To define the mixed spaces ℓ q(·) (L p(·) ) we have to define another modular. For p, q ∈ P(R n ) and a sequence (f ν ) ν∈N 0 of L p(·) (R n ) functions we define where we put λ 1/∞ := 1. The (quasi-) norm in the ℓ q(·) (L p(·) ) spaces is defined as usually by This (quasi-) norm was used in [1] to define the spaces of Besov type with variable integrability and summability. Spaces of Triebel-Lizorkin type with variable indices have been considered recently in [2]. The appropriate L p(·) (ℓ q(·) ) space is a normed space whenever ess-inf x∈R n min(p(x), q(x)) ≥ 1. This was the expected result and coincides with the case of constant exponents.
As pointed out in the remark after Theorem 3.8 in [1], the same question is still open for the ℓ q(·) (L p(·) ) spaces.
In Theorem 3.6 of [1] the authors proved that if the condition 1 p(x) + 1 q(x) ≤ 1 holds for almost every x ∈ R n , then ·| ℓ q(·) (L p(·) ) defines a norm. They also proved in Theorem 3.8 that ·| ℓ q(·) (L p(·) ) is a quasi-norm for all p, q ∈ P(R n ). Furthermore, the authors of [1] posed a question if the (rather natural) condition p(x), q(x) ≥ 1 for almost every x ∈ R n ensures that ·| ℓ q(·) (L p(·) ) is a norm.

Positive results
We summarize in the following theorem all the cases when the expression ·| ℓ q(·) (L p(·) ) is known to be a norm. We include the proof of the case discussed already in [1] for the sake of completeness.
Proof. If p(x) ≥ 1 and q ≥ 1 is constant almost everywhere, then the proof is trivial.
Then (1) may be reformulated as and Our aim is to prove (2), which reads dx ≤ 1 and ess-sup We first prove the second part of (5). First we observe that (3) and (4) imply ν holds for almost every x ∈ Ω ∞ . Using q(x) ≥ 1, and Hölder's inequality in the form If q(x) = ∞, only notational changes are necessary. Next we prove the first part of (5). Let 1 ≤ q(x) ≤ p(x) < ∞ for almost all x ∈ Ω 0 . Then we use Hölder's inequality in the form If 1/p(x) + 1/q(x) ≤ 1 for almost every x ∈ Ω 0 , then we replace (6) by Using (6), we may further continue where we used also (3) and (4). If we start with (7) instead, we proceed in the following way In both cases, this finishes the proof of (5).
Remark 1. (i) A simpler proof of Theorem 1 is possible (and was proposed to us by the referee) if 1 ≤ q(x) ≤ p(x) ≤ ∞. Namely, if 1 ≤ q ≤ p ≤ ∞, λ > 0 and t ≥ 0, then where we use the convention that p q = 1 if p = q = ∞. This allows to simplify the modular ̺ ℓ q(·) (L p(·) ) to . (9) This shows that ̺ ℓ q(·) (L p(·) ) (f ν ) is a composition of only convex functions. Hence, it is a convex modular and therefore it induces a norm. Unfortunately, we were not able to find such a simplification for the case 1/p(x) + 1/q(x) ≤ 1.
The advantage of our proof of Theorem 1 is that it proves both the cases in a unified way.
(ii) Let us observe that (8) loses its sense if p < q = ∞. This shows, why (9) (which was already used in [1] for q + < ∞) has to be applied with certain care.
(iii) The method of the proof of Theorem 1 can be actually used to show that under the conditions posed on p(·) and q(·) in Theorem 1, ̺ ℓ q(·) (L p(·) ) is a convex modular, which is a stronger result than the norm property.

Counterexample
Theorem 2. There exist functions p, q ∈ P(R n ) with inf x∈R n p(x) ≥ 1 and inf x∈R n q(x) ≥ 1 such that · |ℓ q(·) (L p(·) ) does not satisfy the triangle inequality.
Using the calculation inf λ > 0 : which holds for every L > 1 fixed, we get Remark 2. Let us observe that 1 ≤ q(x) ≤ p(x) ≤ ∞ holds for x ∈ Q 0 and 1/p(x) + 1/q(x) ≤ 1 is true for x ∈ Q 1 . It is therefore necessary to interpret the assumptions of Theorem 1 in a correct way, namely that one of the conditions of Theorem 1 holds for (almost) all x ∈ R n . This is not to be confused with the statement that for (almost) every x ∈ R n at least one of the conditions is satisfied, which is not sufficient.
Remark 3. A similar calculation (which we shall not repeat in detail) shows that one may also put q(x) := q 0 large enough for x ∈ Q 1 to obtain a counterexample. Hence there is nothing special about the infinite value of q and the same counterexample may be reproduced with uniformly bounded exponents p, q ∈ P(R n ).