Growth envelopes of some variable and mixed function spaces

We study unboundedness properties of functions belonging 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 $p_{i}$ is the smallest, is crucial. More precisely, the growth envelope is given by $E_G(L_{\vec{p}}(\Omega)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},\min\{p_{1}, \ldots, p_{d} \})$ for mixed Lebesgue and $E_G(L_{\vec{p},q}(\Omega)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},q)$ for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces we obtain $E_G(L_{p(\cdot)}(\Omega)) = (t^{-1/p_{-}},p_{-})$, where $p_{-}$ is the essential infimum of $p(\cdot)$, subject to some further assumptions. Similarly, for the variable Lorentz space it holds $E_G(L_{p(\cdot),q}(\Omega)) = (t^{-1/p_{-}},q)$. 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 [37,38] 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 [32,41,31,39] and the embedding W d/p p (Ω) ֒→ L ∞,p (log L) −1 (Ω) was obtained (see [19,7]), where 1 < p < ∞.
The Sobolev embeddings were extended replacing the Sobolev spaces W  [16] 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 [16] and appears first in Triebel's monograph [40]. The concept was studied in detail by Haroske [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 which the inequality holds for all f ∈ X. Here µ G is the Borel measure associated with 1/E X G . The pair E G (X) := E X G , u X G is called the growth envelope of the function space X. In case of classical Lorentz spaces L p,q (R d ) with 0 < p, q ≤ ∞, with q = ∞ if p = ∞, the result reads as E G L p,q (R d ) = t −1/p , q , t > 0.
One generalization of the classical Lebesgue space L p is the mixed Lebesgue space L − → p , where − → p = (p 1 , . . . , p d ) is a vector with positive coordinates. The · − → p -quasi-norm of the function f is defined by where f is defined on Ω, which is the Descartes product of the sets Ω i . These spaces were introduced by Benedek and Panzone and some basic properties of these spaces were proved in [4]. For some 0 < p < ∞, − → p = (p, . . . , p) we get back the classical Lebesgue space L p . Moreover, the mixed Lorentz space L − → p ,q will be defined by the quasi-norm where − → p = (p 1 , . . . , p d ) is a vector and 0 < q < ∞ is a number. Here we use the notation χ A for the characteristic function of a set A. This approach can be seen as a generalization of the classical Lorentz space L p,q . It will turn out, that if the measure of Ω is finite, then for the growth envelopes we have E G (L − → p (Ω)) = t −1/ min{p 1 ,...,p d } , min{p 1 , . . . , p d } , E G (L − → p ,q (Ω)) = t −1/ min{p 1 ,...,p d } , q , see Corollaries 3.7 and 4.5 below. We see that in the case of the mixed Lebesgue and Lorentz spaces, the unboundedness in the worse direction, i.e., in the direction, where p i is the smallest, is crucial.
Moreover, we deal with growth envelopes of variable function spaces. Replacing the constant exponent p in the classical L p -norm by an exponent function p(·), the variable Lebesgue space L p(·) is obtained. The space L p(·) consists of the functions f , whose quasi-norm f p(·) := inf λ > 0 : dx ≤ 1 is finite and Ω ⊂ R d . These spaces were introduced by Kováčik and Rákosník [30] 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. [14] and Cruz-Uribe and Fiorenza [11]. The variable Lebesgue spaces are used for variational integrals with non-standard growth conditions [1,42,43], which are related to modeling of so-called electrorheological fluids [33,34,35]. These spaces are widely used in the theory of harmonic analysis, partial differential equations [8,9,13,15], moreover in fluid dynamics and image processing [2,3,12,17,36], 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 E 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 see Corollary 5.20. For the variable Lorentz spaces when additionally 1 < q ≤ ∞, we obtain in Corollary 6.12, 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 [29], 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 Section 2 we recall the concept of the growth envelopes, collect some of its properties and recall classical examples.
In Sections 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 Section 5. In Section 6 we will prove similar theorems for the variable Lorentz space L p(·),q . Finally, in Section 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 σ-finite 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 Ω → C, its distribution function µ f : [0, ∞) → [0, ∞] is defined as 3 The mixed Lebesgue space Let d ∈ N and (Ω i , A i , µ i ) be measure spaces for i = 1, . . . , d, and − → p := (p 1 , . . . , p d ) with 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).
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] 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 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 −1/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 (6) and (7), if p j < ∞, then α j p j < 1 and therefore By the construction (see (6)), 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 .
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 which finishes the proof.
In conclusion, for the growth envelope function E we obtain the following result.

Now let us study the additional index u
To this, we need the following lemma. The proof can be found in [21].

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 (see [18,Prop. 1.4.9.] and [6]), 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 this by L − → p ,q (Ω). If − → p = (p, . . . , p), where 0 < p < ∞, then for all 0 < q < ∞, by f L (p,...,p) = f p , we see from the definition of · L− → p ,q , that Proof. Let us start with the case q 2 < ∞. Firstly, let us define the constant where we have used that from u ≤ 2 k−1 , it follows that Since q 1 /q 2 ≤ 1 by (12) we obtain If q 2 = ∞, then for all s > 0, and completes the proof. Benedek and Panzone [4]). Thus, for all u > 0, χ {|f |>u} − → r ≤ c χ {|f |>u} − → p and therefore for all 0 < q < ∞, The case q = ∞ can be handled similarly. This means, that if µ(Ω) < ∞ and − → r ≤ − → p , then for all 0 < q ≤ ∞, the embedding L − → p ,q (Ω) ֒→ L − → r ,q (Ω) holds. As a special case, if min{p 1 , . . . , p d } < ∞, we have that (min{p 1 , . . . , p d }, . . . , min{p 1 , . . . , p d }) ≤ − → p and therefore (see (11)), where in case of q = ∞, ∞ 1/∞ := 1.
Proof. Again, suppose that p k = min{p 1 , . . . , p d } and for a fixed t > 0, let s > t. Let us consider the function which proves the proposition.
In terms of the growth envelope function our previous results yield the following. .
which is equivalent with the classical result (4).
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 Using the embedding L − → p ,q (Ω) ֒→ L min {p 1 ,...,p d },q (Ω), Proposition 2.4 and (3), we have that u We put again p l := min{p 1 , . . . , p d } and suppose that q < ∞. We will show, that if v < q, then the inequality ε 0 where 0 < s < 1 and A (l) that is, by (9), This means that f ∈ L − → p ,q (Ω). At the same time, in the proof of Theorem 3.6, we have seen that the left-hand side of (15) is not finite, since γv > 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 {|gs,γ |>t} p l and therefore by the definition of the · Lp l ,∞ quasi-norm and (9), we obtain that that is, f ∈ L − → p ,∞ . Since γ > 0, recall the proof of Theorem 3.6. We have seen, that the integral on the left-hand side of (15) 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 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 where dx denotes the Lebesgue measure. A measurable function f belongs to the variable Lebesgue space L p(·) , if for some λ > 0, ̺ p(·) (f /λ) < ∞. Endowing this space with the quasi-norm 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 [14]) 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(·) (αf ) 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 nonincreasing (moreover, decreasing). Besides that, the · p(·) -quasi-norm of the function f can be estimated by (see [10]) Since ̺ p(·) (χ A ) = µ(A), using (17) and (18) for a characteristic function χ A , we get and the quasi-norm of a characteristic function χ A can be estimated (see (19) and (20)) by The proof of the following theorem for p(·) ∈ P with p − ≥ 1 can be found in [14]. If p − < 1 the proof is similar using inequality (17).
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 [10].
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 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 At and χ As denote their characteristic functions. We may suppose that s ≤ t + 1. Then µ(A s \ A t ) ≤ 1 and by (21), that is, χ As → χ At in the L p(·) -norm. By Lemma 5.4, χ As p(·) → χ At p(·) . Moreover, by Lemma 5.2, χ As p(·) ց χ At p(·) as s ↓ t, that is, inf s>t,At⊂As χ As p(·) = lim s↓t χ As p(·) = χ At p(·) and therefore sup s>t,At⊂As After these preparations we now study 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
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 Proof. Let t > 0. For a fixed number s > t let us choose a set A s ⊂ R d with µ(A s ) = s and consider the functions ϕ s,As := χ As −1 p(·) χ As . First, we see that ϕ s,As p(·) = 1 and using (24), we conclude that ϕ * s,As (t) = χ As −1 p(·) for 0 ≤ t < s. If we consider only the functions ϕ s,As , we get the following lower estimate, We have seen in (23), that this supremum is χ At where the set A t was an arbitrary set with measure t. Thus and the proof is complete.

Upper estimate for
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 |f | ≥ αχ {|f |>α} .
Since ̺ p(·) (χ At ) = µ(A t ) = t < 1, by (21), 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 (21), 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 all (28) By (27) and (28), we obtain for all t < t 0 , Using (26), Theorem 5.2, (21) (with the condition (25)) and (29), 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.
Corollary 5.7 Let p(·) ∈ P with p + < ∞ and suppose that there exists a set 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 (3), we obtain that Using this together 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 We will denote this by r(·) ∈ LH 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(·) ∈ LH 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, log µ (B r (x 0 )) p − (Br (x 0 ))−p + (Br(x 0 )) ≤ C.
Using the previous lemma, we get the following result.
Summing up our previous results we obtain the following.
Then the following assertions are equivalent: Instead of cubes, it is also possible to use balls.
Again, instead of cubes, it is also possible to use balls.
Theorem 5.17 Let p(·) ∈ P with p + < ∞, p(·) ∈ LH 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 (32). Similarly, as in the proof of Theorem 5.6, we use argument by contradiction. By the definition of α and (32), As in the proof of Theorem 5.6, using (34), we have that which is a contradiction. From this, by (33), we have that which proves the theorem.
Using Theorem 5.13 and Theorem 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 point x 0 ∈ Ω, where p(x 0 ) = p − (instead of the condition that p(·) is constant p − on a set), then the equivalence E L p(·) G (t) ∼ t −1/p − is obtained for all 0 < t < ε.

Additional index
We study the index u L p(·) G now. Recall Proposition 2.4.

The variable Lorentz space
Using (10), the space L p(·),q can be defined, where p(·) ∈ P and 0 < q ≤ ∞. The measurable function f : Ω → R belongs to the space L p(·),q , if is finite. Now, as before, we write only L p(·),q (Ω), if the domain is important, for example, if the domain is bounded. It follows from the definition, that for all measurable sets A ⊂ Ω and 0 < q < ∞, (35) and if q = ∞, then χ A L p(·),∞ = χ A p(·) .

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,As := χ As −1 L p(·),q χ As . Then ϕ s,As L p(·),q = 1 and ϕ * s,As (t) = χ As −1 L p(·),q . Using (35) and (23), we get the following lower estimate for 0 < q < ∞ (the case q = ∞ follows analogously): holds for all set A t with measure t. This implies that which proves the proposition.
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 (11), which is not the classical result. But if p(·) = p, then · L p(·),q = · Lp,q , which was only an equivalent norm with the norm · Lp,q and we have that · Lp,q = p −1/q · Lp,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 (21), and therefore where we have used, that µ(B 2 −(j+1) (x 0 )) = 2 −d µ(B 2 −j (x 0 )). Here by Lemma 5.11, By the same way, as in the proof of Theorem 5.19, we have that since α v/p − < 1. 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.
Concerning the mixed Lorentz spaces we get similar results.
Proof. To verify the claims, let us use Corollary 5.18 and Corollary 6.10. The proof is analogous to the proof of Corollary 7.5.