Growth Envelopes of Some Variable and Mixed Function Spaces

We study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{i}$$\end{document} is the smallest, is crucial. More precisely, the growth envelope is given by EG(Lp→(Ω))=(t-1/min{p1,…,pd},min{p1,…,pd})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$\end{document} for mixed Lebesgue and EG(Lp→,q(Ω))=(t-1/min{p1,…,pd},q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$\end{document} for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain EG(Lp(·)(Ω))=(t-1/p-,p-)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$\end{document}, where p-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{-}$$\end{document} is the essential infimum of p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document}, subject to some further assumptions. Similarly, for the variable Lorentz space, it holdsEG(Lp(·),q(Ω))=(t-1/p-,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$\end{document}. The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.


Introduction
Using Sobolev embeddings, the integrability properties of a real function can be deduced from those of its derivatives. Sobolev's famous embedding theorem [31] says, that for 1 ≤ p < ∞ and k ∈ N, the embedding W k p (Ω) → L r (Ω) holds for all 1 ≤ r ≤ ∞ such that k < d/ p and k/d − 1/ p ≥ −1/r , where Ω ⊂ R d is a bounded domain with sufficiently smooth boundary. In the limiting case, when k = d/ p, we have the embedding W d/ p p (Ω) → L r (Ω) only for finite r . It can be understood as the impossibility of specifying integrability conditions of a function f ∈ W d/ p p (Ω) merely by means of L r conditions. Refinements of the Sobolev embeddings in the limiting case were investigated in [25,26,32,35] and the embedding W d/ p p (Ω) → L ∞, p (log L) −1 (Ω) was obtained (see [6,18] [15] proved that the space H d/ p p can be characterized by sharp inequalities and the non-increasing rearrangement function f * of f : let κ be a continuous, decreasing function on (0, 1] and 1 < p < ∞. Then the inequality p , if and only if κ is bounded. The idea of the growth envelopes come from Edmunds and Triebel [15] and appears first in Triebel's monograph [33]. The concept was studied in detail by Haroske [19][20][21]. Starting from the previous characterization of H d/ p p , to investigate the unboundedness of functions on R d belonging to the quasi-normed function space X , the growth envelope function and the additional index u X G ∈ (0, ∞] have been introduced. The growth envelope function E X G is monotonically decreasing, see Haroske [20,Prop. 3.4]. The latter index gives a finer description of unboundedness and is defined as the infimum of those v > 0, for which the inequality holds for all f ∈ X . Here μ G is the Borel measure associated with 1/E X G . In particular, μ G ( dt) ∼ f p(·) := inf λ > 0 : dx ≤ 1 is finite and Ω ⊂ R d . These spaces were introduced by Kováčik and Rákosník [24] in 1991, where some of their properties were investigated. From this starting point, a lot of research has been undertaken regarding this topic. We refer, in particular, to the monographs by Diening et al. [13] and Cruz-Uribe and Fiorenza [10]. The variable Lebesgue spaces are used for variational integrals with non-standard growth conditions [1,36,37], which are related to modeling of so-called electrorheological fluids [27][28][29]. These spaces are widely used in the theory of harmonic analysis, partial differential equations [7,8,12,14], moreover in fluid dynamics and image processing [2,3,11,16,30], as well.
The variable Lorentz space L p(·),q will be defined in this paper, where p(·) is an exponent function and q is a number. The measurable function f : Ω → R belongs to the space L p(·),q , if In this paper, we will study the growth envelope of the spaces L p(·) and L p(·),q . We will show in Corollaries 5.7 and 6.3, that subject to some restrictions for small t > 0, If, additionally, the so-called locally log-Hölder continuity for the exponent function p(·) is assumed, the growth envelope function of L p(·) and L p(·),q can be written in the form where p − denotes the essential infimum of the exponent function p(·), see Corollaries 5.18 and 6.10 below. Here and in what follows the symbol f ∼ g means for positive functions f and g, that there are positive constants A and B such that for all t, A g(t) ≤ f (t) ≤ B g (t). Moreover, if Ω is bounded and p(·) is locally log-Hölder continuous with p − > 1, then the growth envelope of the variable Lebesgue space is E G (L p(·) (Ω)) = t −1/ p − , p − , see Corollary 5.21. For the variable Lorentz spaces when additionally 1 < q ≤ ∞, we obtain in Corollary 6.12, E G (L p(·),q (Ω)) = t −1/ p − , q .
All in all, it will turn out, that the unboundedness is determined by p − , which "extends" our observation from the mixed Lebesgue and Lorentz spaces in a natural way: the "minimal" integrability is the crucial one.
In [23], Kempka and Vybíral defined for exponent functions p(·) and q(·), the space L p(·),q (·) . It would be a natural conjecture, that this space has a growth envelope function of the form E L p(·),q(·) G (t) = sup{ χ A −1 L p(·),q(·) : measure of A is equal to t}. However, this space is technically so complicated that we have to postpone an answer to this question.
The paper is organized as follows. In Sect. 2, we recall the concept of the growth envelopes, collect some of its properties, and recall classical examples.
In Sects. 3 and 4, we concentrate on the mixed Lebesgue and mixed Lorentz spaces, respectively, and determine their growth envelopes.
We will consider the variable Lebesgue spaces in Sect. 5. In Sect. 6, we will prove similar theorems for the variable Lorentz space L p(·),q . Finally, in Sect. 7, we present some applications of our new results.

Growth Envelope
First, we need the concept of the rearrangement function. Let (Ω, A, μ) be a totally σfinite measure space. For simplicity, we shall restrict ourselves to the setting Ω ⊆ R d in what follows, where μ stands for the Lebesgue measure. For a measurable function It is easy to see that μ f is non-negative and non-increasing. The non-increasing rearrangement function f * : As usual, the convention inf ∅ = ∞ is assumed. In particular, for a measurable set A, we have The growth envelope function was first introduced and studied in [33,Chap. 2] and [19]; see also [20].
Strictly speaking, we obtain equivalence classes of growth envelope functions when working with equivalent quasi-norms in X : if · 1 ∼ · 2 , then E For 0 < p, q ≤ ∞, the classical Lorentz spaces contain all measurable functions for which the quasi-norm  [20,Sect. 4.2], that for 0 < p < ∞ and 0 < q ≤ ∞, More precisely, taking care of the (usually hidden) constants, the growth envelope function of L p,q (see, for example, in Haroske [20,Rem. 3.13]) is

The Mixed Lebesgue Space
Let d ∈ N and (Ω i , A i , μ i ) be measure spaces for i = 1, . . . , d, and − → p : with the usual modification if p j = ∞ for some j ∈ {1, . . . , d}. In general, the mixed Lebesgue space will be denoted by L− → p , but if the domain is important, for example, if it is bounded, we write L− → p (Ω).
As mentioned in the introduction, these spaces are not invariant for the permutations of the coordinates of the vector − → p . This is illustrated in the following example. Example 3.1 For simplicity, we deal with the two-dimensional case. Let − → p := ( p 1 , p 2 ) and − → q : Consider the function where we have used that −1 < α p 1 < 0. Using (3.1), we obtain that and therefore On the other hand since in (3.1), we choose Hence If for some 0 < p ≤ ∞, − → p = ( p, . . . , p), we get back the classical Lebesgue space, i.e., L− → p = L p in this case. This means that the mixed Lebesgue spaces are generalizations of the classical Lebesgue spaces. Throughout the paper, 0 < − → p ≤ ∞ will mean that the coordinates of − → p satisfy the previous condition, e.g., for all i = 1, . . . , d, 0 < p i ≤ ∞. When μ(Ω) < ∞, Benedek and Panzone [4] showed, that if In the next theorem, we show that the space L min{ p 1 ,..., p d } (Ω) is indeed the smallest classical Lebesgue space, which contains the mixed Lebesgue space L− → p (Ω).
Then 0 < p l < ∞. We assume that ε > 0 is sufficiently small, that is, ε satisfies p l + ε < p j for all p j for which p j > p l . Now, for p j < ∞, let us consider the numbers For those j = 1, . . . , d, for which p j < ∞, we consider the functions f j (x j ) = x −α j j and if p j = ∞, we put f j (x j ) := 1, where x j ∈ (0, 1]. Let us define the function By (3.3) and (3.4), if p j < ∞, then α j p j < 1 and therefore that is, f ∈ L− → p (Ω). By the construction (see (3.3)), if p l < p j < ∞, then ε > 0 was chosen such that p l + ε < p j , that is α j ( p l + ε) ≤ α j p j < 1, and if p j = p l , then α j ( p l + ε) ≥ 1. Hence In the first product, α j ( p l + ε) < 1, therefore the first term is finite. But, the second term is infinite since α j ( p l + ε) ≥ 1, which means that f / ∈ L p l +ε (Ω) implying From (3.2) and Theorem 3.2, we obtained that if and only if min{ p 1 , . . . , p d } < ∞.

Lemma 3.3 Let
. . , d) and consider their Cartesian product A := A 1 × · · · × A d . Then which proves the lemma.
We have the following lower estimate for E Proof Suppose that p k = min{ p 1 , . . . , p d } and for a fixed t > 0, let s > t. Consider the following function and by (2.1) which finishes the proof.
In conclusion, for the growth envelope function E , we obtain the following result.

The Mixed Lorentz Space
It is known that L ∞,∞ = L ∞ , and for all 0 < q < ∞, the space L ∞,q contains the zero function only. Therefore, if p = ∞, then it is supposed that q = ∞, too. Moreover, for 0 < p < ∞, it follows from [17, Prop. 1.4.9.] (see [5,34]), that the quasi-norm of the classical Lorentz space can be written as Therefore the quasi-norm is equivalent with the previous one. This approach allows for a generalization to mixed Lorentz spaces and later on to variable Lorentz spaces. For a vector 0 < − → p ≤ ∞ and for a number 0 < q ≤ ∞, the mixed Lorentz space L− → p ,q contains all measurable functions for which the quasi-norm is finite. If it does not cause misunderstanding, the mixed Lorentz space is denoted by L− → p ,q , but if the domain is important, for example, if it is bounded, then we denote Proof Let us start with the case q 2 = ∞. Then for all s > 0, And if 0 < q 1 < q 2 < ∞, then by the previous inequality, which means that L− → p ,q 1 → L− → p ,q 2 and the proof is complete.
In terms of the growth envelope function, our previous results yield the following.
Concerning the additional index u of the mixed Lorentz space L− → p ,q , we can state the following.
Proof Using Theorem 4.3, we obtain that there exists ε > 0, such that We put again p l := min{ p 1 , . . . , p d } and suppose that q < ∞. We will show that if v < q, then the inequality does not hold for all f ∈ L− → p ,q . Let γ ∈ R again, such that 1/q < γ < 1/v and consider the same function as in the proof of Theorem 3.7: where 0 < s < 1 and A (l) Therefore, This means that f ∈ L− → p ,q (Ω). At the same time, in the proof of Theorem 3.7, we have seen that the left-hand side of (4.6) is not finite, since γ v > 1.
Hence, it follows that v ≥ q, which implies u Together with the first part of the proof, we have that u Let q = ∞. Now, for an arbitrary 0 < v < ∞, let us choose a number γ > 0, such that γ v < 1. Then by the same extremal function f , we have again that χ {| f |>t} − → p = χ {|g s,γ |>t} p l and therefore by the definition of the · L p l ,∞ quasi-norm and (4.1), we obtain that Since γ > 0, recall the proof of Theorem 3.7, we have seen, that the integral on the left-hand side of (4.6) is infinite and the proof is complete.
Altogether, in terms of growth envelopes for mixed Lorentz spaces, we have obtained the following.

The Variable Lebesgue Space
We can generalize the classical Lebesgue space L p in another way. In this case, the exponent will not be a vector, but a function of x. Let Ω ⊆ R d , p(·) : Ω → (0, ∞) be a measurable function and denote Similarly, for a measurable set A, If p − > 0, then we say that p(·) is an exponent function. Moreover, the set of all exponent functions is denoted by P. For p(·) ∈ P and for a measurable function f , the p(·)-modular is defined by we get a quasi-normed space (L p(·) , · p(·) ). In general, we denote the variable Lebesgue space by L p(·) , except the domain is important. In particular, if μ(Ω) < ∞, then the variable Lebesgue space on Ω is denoted by L p(·) (Ω). If the function p(·) = p is constant, we get back the classical Lebesgue space L p . If μ(Ω) < ∞ and r (·) ≤ p(·) pointwise, then (see, e.g., Diening [13,Cor. 3.3.4.]) We have the following inequalities. If p + < ∞, then for any |λ| ≤ 1 and | λ| > 1 where the set supp( f ) denotes the support of f . From this, it follows that for all f ∈ L p(·) , the map α → p(·) (α f ) is increasing. Indeed, suppose that α 1 < α 2 , then α 2 /α 1 > 1, and therefore (α 2 /α 1 ) p − > 1, too. Thus From this, we get as well, that for all f ∈ L p(·) , the function λ → p(·) ( f /λ) is non-increasing (moreover, decreasing). Besides that, the · p(·) -quasi-norm of the function f can be estimated by (see [9]) (5.2) and (5.3) for a characteristic function χ A , we get and the quasi-norm of a characteristic function χ A can be estimated (see (5.4) and (5.5)) by The proof of the following theorem for p(·) ∈ P with p − ≥ 1 can be found in [13, Lemma 3.2.4.]. If p − < 1, the proof is similar using inequality (5.2).
The following lemma can be proved easily.

Lemma 5.2
If p(·) ∈ P, then the following holds: 2. if f ∈ L p(·) , g is measurable and |g| ≤ | f | almost everywhere, then g ∈ L p(·) and g p(·) ≤ f p(·) ; We will also need the result that if the sequence of functions ( f n ) n tends to f in the · p(·) -norm, i.e., f n − f p(·) → 0, then the sequence of the norms ( f n p(·) ) n tends to f p(·) . This is very easy, if we have the triangle inequality. Indeed, in this case, is not a Banach space, just a quasi-Banach space, and the triangle inequality does not hold. We circumvent this problem by using the following lemma. The proof can be found in [9].
If p − > 1, then Lemma 5.3 is not true. But in this case, the triangle inequality holds. Using these observations, we get the following result. Proof If p − ≥ 1, then by the triangle inequality, If p − < 1, then by Lemma 5.3 we have thus, which means that lim k→∞ f k Now, for fixed t > 0, let us consider the sets A t ⊂ A s where s > t, μ(A t ) = t, μ(A s ) = s, and χ A t and χ A s denote their characteristic functions. We may suppose that s ≤ t + 1. Then μ(A s \ A t ) ≤ 1 and by (5.6), (5.8) After these preparations, we now study the growth envelopes of variable Lebesgue spaces. We proceed as follows: We obtain the lower estimate of the growth envelope function of the space L p(·) under some mild condition on the exponent function p(·), namely that the exponent function p(·) is bounded. For the upper estimate, we need the condition, that p(·) is constant p − on a (small) set. Assuming that the exponent function p(·) is additionally locally log-Hölder continuous and p − is attained, we show that the growth envelope function is actually equivalent to t −1/ p − near to the origin. Moreover, in case Ω is bounded, then is proved, that the additional index of the function space L p(·) (Ω) is p − .

Lower Estimate for E L p(·) G
We recall the following very simple result which follows immediately from the definition of f * . If A ⊂ R d is measurable, then Proposition 5.5 Let p(·) ∈ P and p + < ∞. Then We have seen in (5.8), that this supremum is where the set A t was an arbitrary set with measure t. Thus and the proof is complete.

Upper Estimate for E L p(·) G
For the upper estimate, we need to assume more conditions on the exponent function p(·).
Theorem 5.6 Let p(·) ∈ P with p + < ∞ and suppose that there exists a set A t 0 , with Proof Let 0 < t < min{1, t 0 } be fixed and let us denote Then our claim E We prove it by contradiction. Assume on the contrary, that there exists a function f ∈ L p(·) , f p(·) ≤ 1 such that μ ({| f | > α}) > t. We can suppose w.l.o.g. that f ∈ L p(·) such that It is easy to see, that From this, we have that By our general assumption, there exists a set A t 0 with measure t 0 , such that for all x ∈ A t 0 , p(x) = p − . It can be assumed that t 0 ≤ 1. By (5.6), for this set A t 0 , we compute It is clear that for all t < t 0 , and a set A t ⊂ A t 0 we have for (5.13) By (5.12) and (5.13), we obtain for all t < t 0 , Using (5.11), Lemma 5.2, (5.6) (with the condition (5.10)) and (5.14), it follows so we have that 1 > 1, which is a contradiction. Hence, which proves the theorem.
By Proposition 5.5 and Theorem 5.6, the following corollary is obtained.

Remark 5.8
If p(·) = p, then for any set A, p| A = p = p − and for all sets A t with measure t, A t p = μ(A t ) 1/ p = t 1/ p , hence in this case that is, we get back the classical result.

Remark 5.9
Obviously, the space L p(·) is not rearrangement-invariant for arbitrary p(·) ∈ P satisfying the assumptions of Corollary 5.7. So this can be seen now as the extension of our result connecting the growth envelope function E X G and fundamental function ϕ X in rearrangement-invariant spaces, see Remark 2.3, to more general spaces.

Remark 5.10
The condition for the upper estimate that the exponent function p(·) is constant p − on a set, may be too strong. This condition can be omitted, if we suppose that μ(Ω) < ∞. Indeed, in this case, we have the embedding L p(·) (Ω) → L p − (Ω) and therefore, see Proposition 2.2 and (2.3), we obtain that Using this together with Theorem 5.5, we see that if μ(Ω) < ∞ and p + < ∞, then In what follows we show that if we additionally assume the exponent function p(·) to be locally log-Hölder continuous at a point x 0 , where p(x 0 ) = p − , then the lower estimate in Remark 5.10 can be replaced by c t −1/ p − . The function r (·) is locally log-Hölder continuous at the point x 0 , if there exists a constant C 0 > 0, such that for all x ∈ Ω, |x − x 0 | < 1/2, We will denote this by r (·) ∈ L H 0 {x 0 }. If the previous condition holds for all x 0 ∈ Ω, then r (·) is locally log-Hölder continuous (not only in x 0 ), in notation r (·) ∈ L H 0 . The ball with radius r > 0 and center x 0 is denoted by B r (x 0 ) := {y ∈ Ω : y −x 0 2 < r }.

Lemma 5.11
Let p(·) ∈ P and suppose that x 0 ∈ Ω, such that p − = p(x 0 ). Then the function p(·) is locally log-Hölder continuous at x 0 if, and only if, there exists C > 0, such that for all r > 0, Proof If r ≥ 1/2, then by the positivity of the exponent p Now, suppose that r < 1/2. It is enough to show that for some constant C > 0, It is known that for every ball B r ⊂ R d with radius r > 0 we have that μ(B r ) = c d r d , where the constant c d depends only on d. By our assumptions where c 1 is the log-Hölder constant from (5.15), which proves (5.17). Now let us see the other direction and suppose that (5.16) holds. Let x, x 0 ∈ Ω with |x − x 0 | < 1/2. We distinguish between c d < 1 and c d ≥ 1.
If c d < 1, then there exists ε > 0, such that c d (1 + ε) d < 1. Let us choose such an ε and consider the ball B r (x 0 ) with radius r := (1 + ε)|x − x 0 |. Then x ∈ B r (x 0 ) and by (5.16), Since c d (1 + ε) d < 1, the first factor can be estimated from below by 1. At the same time, p + (B r (x 0 )) − p − (B r (x 0 )) ≥ |p(x) − p(x 0 )| and therefore (5.18) can be estimated from below by The above considerations lead us to Taking the logarithm on both sides, after ordering we obtain . (5.19) Now, c d 2 d > 1, and therefore (5.19) can be estimated from below by and obtain Taking the logarithm on both sides, after ordering, we have the form that is, p(·) ∈ L H 0 {x 0 }, which proves the lemma.
Using the previous lemma, we get the following result.
Then for j ≥ j 0 , we define the functions where a j = μ(B 2 − j (x 0 )) −1/ p − . Then by Lemma 5.12, that is, by the norm-modular unit ball property (see Theorem 5.1), the · p(·) -norm of the function We have for any 0 < h < 1, Since the function E is decreasing (see Proposition 2.2), we obtain Summing up our previous results, we obtain the following.
Corollary 5.14 Let μ(Ω) < ∞, p(·) ∈ P with p + < ∞. If there exists x 0 ∈ Ω, such that p(x 0 ) = p − and p(·) ∈ L H 0 {x 0 }, then there exists ε > 0, such that Moreover, under the condition that p(·) ∈ L H 0 , a similar upper estimate can be reached without assuming that Ω is bounded. In order to show this, we need the following lemmas. The proofs can be found in [

Lemma 5.15 Let p(·)
Then the following assertions are equivalent:

then for all cubes Q ⊂ R d with μ(Q) ≤ 1 and for all x ∈ Q,
Again, instead of cubes, it is also possible to use balls.

Theorem 5.17
Let p(·) ∈ P with p + < ∞, p(·) ∈ L H 0 and suppose that there exists x 0 ∈ Ω, such that p(x 0 ) = p − . Then Proof Let t ∈ (0, 1] be fixed and let us denote where the constant c 2 is equal with the constant in (5.21). Similarly, as in the proof of Theorem 5.6, we use argument by contradiction. By the definition of α and (5.21), As in the proof of Theorem 5.6, using (5.23), we have that which is a contradiction. From this, by (5.22), we have that which proves the theorem.
Using Theorems 5.13 and 5.17, we get the following corollary. If we take the weaker condition, that the exponent function is locally log-Hölder continuous and there is a

Additional Index
We study the index u L p(·) G now. Recall Proposition 2.4.
Proof From Corollary 5.14, we obtain that for all 0 < t < ε, E On the other hand, for any 1 ≤ v < p − , there exists α > 1, such that 1 ≤ v < αv ≤ p − . Let us choose a j 0 ∈ N, such that for all j ≥ j 0 , For k > j 0 and j = j 0 , . . . , k, let us consider Since b j ≥ 1, we get that Let us denote β := C 1/ p − ∞ j=1 1 j α . Naturally, β > 1, so by (5.2), By the norm-modular unit ball property (see Theorem 5.1), β −1/ p − s k p(·) ≤ 1, that is, s k p(·) ≤ β 1/ p − . Moreover, the right-hand side does not depend on k, therefore sup k> j 0 We will show that Since s k is a step function, we have that Since v ≥ 1, we obtain that . After simplifying, we have that which means that v ≥ p − , and therefore u

Remark 5.20
During the proof, we need the following lower estimate . This lower estimate holds for v ≥ 1. We have supposed that v < p − (we wanted to show that for all v < p − , the inequality in the definition of the additional index is not satisfied), that is, 1 ≤ v < p − . From this follows that p − > 1 must hold.
We get immediately the following corollary.

Lower Estimate for E
L p(·),q G Proposition 6.1 If p + < ∞, then for all 0 < q ≤ ∞, Proof The proof is similar to the proof of Proposition 5.5. Let t > 0, s > t be fixed, and choose A s ⊂ R d with μ(A s ) = s. We consider the functions ϕ s,A s := χ A s −1 L p(·),q χ A s . Then ϕ s,A s L p(·),q = 1 and ϕ * s,A s (t) = χ A s −1 L p(·),q . Using (6.1) and (5.8), we get the following lower estimate for 0 < q < ∞ (the case q = ∞ follows analogously): for all sets A t with measure t. This implies that which proves the proposition.

Upper Estimate for E
L p(·),q G Theorem 6.2 Let p(·) ∈ P with p + < ∞ and 0 < q ≤ ∞. If there exists t 0 > 0 and a set A t 0 with measure t 0 , such that p( For convenience, we may assume that q < ∞, but the necessary modifications otherwise are obvious. We proceed by contradiction and suppose that there exists a function Since μ({| f | > α}) < 1, we get from (5.6), that By the monotonicity of the · p(·) -norm (see Lemma 5.2), for all 0 < u < α, χ {| f |>u} p(·) > t 1/ p − . Similarly, as before, because of p(x) = p − (x ∈ A t 0 ), we have that χ A t 0 p(·) ≤ t 1/ p − 0 and by (6.1), Moreover, for all t < t 0 and sets A t ⊂ A t 0 , so we have that 1 > 1, which is a contradiction.
Using Proposition 6.1 and Theorem 6.2, we have Corollary 6.3 Let p(·) ∈ P with p + < ∞ and 0 < q ≤ ∞. If there exists t 0 > 0 and a set A t 0 with measure t 0 , such that p(

Remark 6.4
If p(·) = p, where 0 < p < ∞ is a constant and 0 < q < ∞, then by (4.3), which is not the classical result. But if p(·) = p, then · L p(·),q = · L p,q , which was only an equivalent norm with the norm · L p,q and we have that · L p,q = p −1/q · L p,q . Hence, so we recover the classical result.

Remark 6.5
The space L p(·),q is not rearrangement-invariant for arbitrary p(·) ∈ P and 0 < q ≤ ∞ satisfying the assumptions of Corollary 6.3. Similarly, as in the case of variable Lebesgue spaces (see Remark 5.9), this result can be seen as the extension of our result connecting the growth envelope function E X G and fundamental function ϕ X in rearrangement-invariant spaces, see Remark 2.3, to more general spaces.

Additional Index
Now let us consider the additional index.
Proof The upper estimate can be reached again by the help of Corollary 6.8, the embedding L p(·),q (Ω) → L p − ,q (Ω) and Proposition 2.4. For the lower estimate, suppose that q < ∞ (if q = ∞, then the proof is similar). We will show that for all 1 < v < q, the inequality does not hold for all functions f from L p(·),q (Ω). Since 1/v > 1/q, there exists α > 0, such that p − /q < α < p − /v. Let us consider the same sequence of functions s k as in the proof of Theorem 5.19. First, we will show that sup k> j 0 s k L p(·),q < ∞, where j 0 satisfies that B 2 − j 0 (x 0 ) ⊂ Ω. We can suppose that μ B 2 − j 0 (x 0 ) < 1. For all b j < u < b j+1 ( j = j 0 , . . . , k − 1), by (5.6), and therefore That is, v ≥ q, and therefore u L p(·),q (Ω) G = q, which proves the theorem.

Applications
Using the growth envelopes of the function spaces L− → p , L− → p ,q , L p(·) , and L p(·),q , some Hardy-type inequalities and limiting embeddings can be obtained.
In particular, if κ is an arbitrary non-negative function on (0, ε], then (7.1) holds if, and only if, κ is bounded.
Concerning the mixed Lorentz spaces, we get similar results. if v = ∞. In particular, if κ is an arbitrary non-negative function on (0, ε], then (7.2) holds if, and only if, κ is bounded.
For the variable Lebesgue and Lorentz spaces, we need to assume further conditions on the exponent function. if v = ∞. In particular, if κ is an arbitrary non-negative function on (0, ε], then (7.4) holds if, and only if, κ is bounded.
For the variable Lebesgue spaces, we have the following corollaries.
Proof To verify the claims, let us use Corollaries 5.18 and 6.10. The proof is analogous to the proof of Corollary 7.5.
For the embedding L p(·) → L r (or L p(·),q → L r ,s ), it is necessary that r ≤ p − . If μ(Ω) < ∞ and r ≤ p − , then the smallest space from the spaces {L r (Ω) : r ≤ p − } is L p − (Ω) = L p − , p − (Ω). Similarly, for fixed 0 < s ≤ ∞, the smallest space from {L r ,s (Ω) : r ≤ p − } is L p − ,s (Ω). Therefore, if μ(Ω) < ∞ and we look for the smallest classical Lebesgue or classical Lorentz space, which contains L p(·) (Ω) (or L p(·),q (Ω), respectively), then the first index must be p − . What about the second index? In this case we make use of the additional index which yields the following. Proof The proof is analogous to the proof of Corollary 7.6. For 1., Corollary 5.21 is used and Corollary 6.12 is applied for 2.
In conclusion, the smallest classical Lorentz space, containing the variable Lebesgue space L p(·) (Ω), is the space L p − , p − (Ω) and the smallest classical Lorentz space, which contains the variable Lorentz space L p(·),q (Ω), is L p − ,q (Ω). By Corollaries 7.5, 7.6, 7.7 and 7.8 we have the following necessary conditions for the following embeddings.