On grand Sobolev spaces and pointwise description of Banach function spaces

We study weighted grand Sobolev spaces, defined on arbitrary open sets (of finite or infinite measure). We give a new characterization of grand Sobolev spaces in terms of pointwise inequality. The same description is valid as well for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded.


Introduction
Let Ω ⊂ R n be an open connected set having finite or infinite Lebesgue measure, which we will denote as |Ω|. The Lebesgue space L q (Ω), 1 ≤ q < ∞, is defined to be the space of all measurable functions f on Ω such that and the Sobolev space W 1,q (Ω) is defined to be the space of all f ∈ L q (Ω) such that the weak gradient 1 ∇f belongs to L q (Ω) with a finite norm f W 1,q (Ω) := f L q (Ω) + ∇f L q (Ω) .
If w is a weight, i.e. a measurable, positive, and finite almost everywhere (a.e.) function, the weighted Lebesgue space L q (Ω, w) and weighted Sobolev space Key words and phrases. Banach function space, grand Sobolev space, maximal function, pointwise description. 1 Recall that ∇f = (g1, g2, . . . , gn) : Ω → R n is a weak gradient of f : Ω → R if f , ∇f ∈ L 1 loc (Ω) and the identitŷ holds for all test functions ϕ ∈ C ∞ 0 (Ω), k = 1, . . . , n. 1 respectively. Sobolev functions are known to have a pointwise characterization, which is a classic nowadays.
Theorem A1. Let 1 < q < ∞. A function f belongs to W 1,q (R n ) if and only if there exists a non negative g ∈ L q (R n ) such that the inequality |f (x) − f (y)| ≤ |x − y|(g(x) + g(y)) (1.1) holds for all x, y outside of some set Σ ⊂ R n of measure zero.
The necessity part of Theorem A1 has been firstly obtained in [3], while the sufficiency has been proved independently in [25] and [49]. Afterwards, this description turned out to be useful in various aspects of analysis such as Sobolev spaces with higher-order derivatives [4], Hardy-Sobolev spaces [38], Sobolev spaces on Carnot groups [49], the theory of Mappings with Bounded Distortion on Carnot Groups [50], and many others. Moreover, since the formula (1.1) does not involve the notion of derivative, it turned out to be a starting point to define a counterpart of Sobolev spaces on metric structures and to investigate its properties [25], for comprehensive coverage of the theory of Sobolev spaces on the metric measure spaces the reader is referred to [28] and references therein. Another natural direction of developing the topic is to consider Sobolev spaces based on other functional classes than those of Lebesgue functions. Thus, in [47] a pointwise description in terms of the Young function was obtained for Orlicz-Sobolev mappings.
In this paper we ask if the characterization (1.1) can be obtained for grand Sobolev spaces, spaces that are slightly larger then Sobolev ones. A grand Lebesgue space L n) (Ω), where Ω is a bounded domain in R n , was first introduced in [30], to study the question of the integrability of the Jacobian of an orientationpreserving mapping belonging to the Sobolev space W 1,n (Ω). Grand Lebesgue spaces have been thoroughly studied in the 1D-case when Ω = I = (0, 1), some recent results can be found in [1,6,13,15,32,33] (see Section 2 for definitions and properties). The grand version of Sobolev spaces W 1,q) (Ω) was defined and studied in [44] (see Section 3). For further discussion of grand Lebesgue and Sobolev spaces we refer the reader to [5,7,11,14,16,18-21, 23,24,31,36,37,39,45] and [8, §7.2]. Note, that the situation when |Ω| = ∞ is not easy, neither for L q) (Ω) nor for W 1,q) (Ω). The space L q) (Ω) over the unbounded domain Ω was introduced and developed in [43,48].
In the present paper, we define the weighted grand Sobolev space W 1,q) a (Ω, w), when Ω may be unbounded. The main result of the paper is a pointwise estimate in the spirit of (1.1) stated in Theorem 4.5 in Section 4. Moreover, a generalization in the case of Banach function spaces, provided that the Hardy-Littlewood maximal operator is bounded, is given in Theorem 5.3, and a number of new pointwise characterizations for various spaces is also available, see Corollaries 5.4-5.8.

Grand Lebesgue spaces
Let us recall a definition of grand Lebesgue spaces, given in [30].
Let Ω ⊂ R n be a bounded domain for n ≥ 2. For 1 < q < ∞, the grand Lebesgue space L q) (Ω) consists of all measurable functions g, g ∈ 1<p<q L p (Ω), such that (2.1) In some cases it is more convenient to consider a multiplier 1 |Ω| in front of the integral in (2.1).
The spaces L q) (Ω) are rearrangement invariant Banach function spaces and the following continuous embeddings hold: For any given 1 < q < ∞, the inclusion L q (Ω) ⊂ L q) (Ω) is strict and, moreover, the spaces L q) (Ω) are not reflexive [13].
To define and deal with grand Lebesgue space L q) (Ω) in case of unbounded domains Ω ⊂ R n , i.e. |Ω| ≤ ∞, one needs to introduce a class of weights.
where Q ⊂ R n is a cube with edges parallel to the coordinate axes and denotes integral average of f over Q, i.e.
The class A q (R n ), 1 < q < ∞, possesses the following properties which are consequences of the Hölder inequality and the reverse Hölder inequality, see for instance [9,Theorem IV].
(v) If w ∈ A q (R n ), then there exists α > 1 such that w α ∈ A q (R n ). Now we are ready to define a generalized version of grand Lebesgue spaces, following [43,48].
Definition 2.4. Let 1 < q < ∞ and w, a be weights such that wa ε ∈ L 1 loc (Ω), for all ε ∈ (0, q − 1). The generalized grand Lebesgue space L q) a (Ω, w) consists of all measurable functions g defined on Ω such that a (Ω, w) is called a grandisator in [43,48]. It is introduced for the close control of the behavior of g ∈ L q) a (Ω, w) at infinity.
To work with weights in the context of grand spaces the following lemma (see [48,Lemma 5]) appears to be useful.
It has been proved in [48,Lemma 3] that the following chain of embeddings holds if and only if a ∈ L q (Ω, w). Some of the properties possessed by the space L q) a (Ω, w) are proved below: Proof. Properties (i)-(iii) can be easily checked and we leave them as an exercise for the reader.
To prove (v) let us consider a compact K ⊂ Ω and f ∈ L q) a (Ω, w). By Lemma A3 we know that there exists 0 < ε 0 < δ such that wa ε 0 ∈ A q−ε 0 (Ω). Then by the Hölder inequality, we obtain (2.5) Remark 2.7. In view of Proposition 2.6, the space L q) a (Ω, w) is "very close" to become a Banach function space (refer to Definition 5.1 for the notion of Banach function space).

Grand Sobolev spaces
We begin with the following definition.
In [44] the grand Sobolev space was defined by the means of the norm However, the norms (3.1) and (3.2) are equivalent.
Proposition 3.2. The following estimates hold Proof. We use Hölder's inequality for sums and obtain Also, we have and the assertion follows. Now, we define the generalized grand Sobolev space on Ω ⊂ R n , |Ω| ≤ ∞, as follows. Definition 3.3. Let 1 < q < ∞ and w, a be weight functions on Ω such that a (Ω, w) and Proposition 3.2, it can be shown that Remark 3.5. If the weight w ∈ A q (Ω) then the weighted Sobolev space W 1,q (Ω, w) is a Banach space and functions f ∈ W 1,q (Ω, w) belong also W 1,1 loc (Ω), see [34, Proposition 2.1]. Similar properties are valid for grand Sobolev spaces W 1,q) a (Ω, w). .

4.
Maximal operator and the pointwise estimate for generalized grand Sobolev functions Definition 4.1. Let g ∈ L 1 loc (R n ) and 0 < t ≤ ∞. We define the Hardy-Littlewood maximal operator or simply the maximal operator by where B(x, r) is a ball with radius r > 0 and center x ∈ R n . In the case t = ∞, we write M g(x) instead of M ∞ g(x).
In the celebrated paper [40] it was proved that the maximal operator M : L q (R n , w) → L q (R n , w) is bounded for 1 < q < ∞ if and only if w ∈ A q (R n ). Later, the class A q (R n ) turned out to be very useful since it also characterizes the L q -boundedness of several other operators such as the Hilbert transform [29]. In fact, the same class characterizes also the boundedness of the maximal operator on grand Lebesgue spaces. In particular, the following theorem was proved in [15].
where supremum is taken over all intervals J ⊂ I.
The following theorem, proved in [48], gives the sufficient condition for the maximal operator M to be bounded in the space L q) a (R n , w).
Theorem A5. Let 1 < q < ∞ and δ > 0 be such that a δ ∈ A q (R n ) for some a ∈ L q (R n , w). If w ∈ A q (R n ), then the maximal operator is bounded on L q) for all g ∈ L q) a (R n , w), where K q is a positive constant.

Remark 4.2.
It is unknown if the condition w ∈ A q (R n ) is also necessary for the boundedness of M on L q) a (R n , w).
As a consequence of Theorem A5 we obtain the boundedness of the maximal operator on grand Lebesgue spaces in the case of a bounded domain Ω ⊂ R n . Proposition 4.3. Let 1 < q < ∞, Ω ⊂ R n , be a bounded domain and t ∈ (0, ∞) be a fixed number. Then the inequality holds for all g ∈ L q) (Ω) with some constant K q independent of g.
Proof. We take a weight a ≡ 1 in R n and a weight w : R n → (0, ∞) such that w ∈ A q (R n ) ∩ L 1 (R n ) and w ≡ 1 in some cube Q ⊃ Ω big enough such that B(x, t) ⊂ Q for any x ∈ Ω. Then we have wa ε = w ∈ L 1 loc (Ω) for all ε ∈ (0, q − 1).
The following estimate was obtained in [22,Lemma 7.16]. For convenience of the reader we provide here the statement and a new proof of it.
In view of the estimatê we can apply standard approximation arguments and to justify inequality (4.3) for any f ∈ W 1,1 loc (R n ).
Theorem 4.5. Let 1 < q < ∞ and δ > 0 be such that a δ ∈ A q (R n ) for some a ∈ L q (R n , w), and w ∈ A q (R n ). Then f ∈ W 1,q) a (R n , w) if and only if f ∈ L q) a (R n , w) and there exists a non negative function g ∈ L q) a (R n , w) such that the inequality |f (x) − f (y)| ≤ |x − y|(g(x) + g(y)) (4.4) holds for x, y ∈ R n a.e.
Proof. Proposition 3.6 ensures that (4.3) holds for f ∈ W 1,q) a (R n , w). Now, for x, y ∈ R n , consider a ball B of the least radius with center ξ = 1 2 (x + y) such that x, y ∈ B and |x − ξ| = |y − ξ|. Applying (4.3), we obtain for all x, y ∈ B \ S B .
Let f ∈ W 1,q) (R n , w). Then by Proposition 4.4, the estimate It is obvious that 2B ⊂ B 1 = B(x, t) for t = 3|x − ξ|. Thus, we get the estimate, which first appeared in [27], where ω n is the volume of the unit ball. Similarly, I 2 can be estimated. Using these estimates, (4.5) gives a (R n , w). Hence, by (4.5) we obtain (4.4).
Conversely, by conditions of the theorem the inequality (4.4) holds with some functions f ∈ L q) a (R n , w) and g ∈ L q) a (R n , w) for almost all x, y ∈ R n \ S where S is some set of zero measure. By Proposition 3.6 we can assume that f , g ∈ L 1 loc (R n ). For any k ∈ N we define a set Then for all points x, y ∈ A k we have Therefore f is a Lipschitz function on the set A k in the conventional sense.
Applying the Kirszbraun extension theorem [12, 3.1, Theorem 1], we obtain an extensionf k : R n → R of f : A k → R to a Lipschitz function on R n with the same Lipschitz constant. In particular, for all points x, y ∈ R n \ A k we have Take an arbitrary vector e i of the standard basis in R n , i = 1, 2, ..., n. We will write points x ∈ R n as x = (x, x i ) =x + x i e i wherex ∈ R n−1 and x i ∈ R. For anyx ∈ R n−1 the restriction R ∋ x i →f k (x + x i e i ) is a Lipschitz function with respect to x i ∈ R. Hence it is absolutely continuous on every line parallel to i-coordinate axis and therefore it has the partial derivative df k dx i (x + x i e i ) for almost all x i ∈ R. Thus, by Tonelli's theorem (see, for example, [46,Theorem 13.8]) the partial derivative df k dx i (x) exists for almost all x ∈ R n , i = 1, 2, ..., n.

Again by Tonelli's theorem the intersection
{x + x i e i : x i ∈ R} ∩ A k is measurable for almost allx ∈ R n−1 . Takex ∈ R n−1 such that (1) the intersection {x + x i e i : x i ∈ R} ∩ A k is measurable and has a positive Lebesgue measure; (2) the restriction of g to the line {x + x i e i : x i ∈ R} belongs to the class L 1 loc . Now let t ∈ R be a value such that Properties (1)-(2) are fulfilled for almost allx ∈ R n−1 such that the intersection is {x + x i e i : x i ∈ R} ∩ A k is not empty. For fixedx ∈ R n−1 properties (3)-(6) hold for almost all t such thatx + te i ∈ A k . Therefore, by Tonelli's theorem, properties (1)-(6) are satisfied for almost all x ∈ A k .
Our immediate goal is to evaluate a derivative: The second line is a consequence of (4.6) and the Rademacher differentiability theorem. Let x =x + te i ∈ A k be a point meeting all the above-mentioned properties (1)- (6). In view of (4.4) for the function we have an estimate of the difference relation provided x + τ e i ∈ A k : The last row of the relations (4.8) shows that the estimate for the derivative h ′ (0) depends on the behavior of the difference g(x + τ e i ) − g(x). Since x is the Lebesgue point of the restriction g : We fix an arbitrary number ε > 0. Using the Chebyshev inequality, we deduce as δ → 0. Therefore, we obtain as δ → 0.
As soon as the point x is a density point with respect to the intersection {x + x i e i : x i ∈ R} ∩ A k , we come to (4.10) We introduce the notation By virtue of (4.9), the point 0 is a density point with respect to the set T .
Similarly to the previous one, due to (4.10), the point 0 is also a density point with respect to the set From the definition of a density point, we conclude that 0 is the density point of the intersection T ∩ P . From here, (4.9) and (4.10) we derive the relations for all points τ ∈ [−δ, δ] ∩ (T ∩ P ). Passing to the limit in (4.11) as τ → 0, τ ∈ [−δ, δ] ∩ (T ∩ P ), we get Since here ε > 0 is an arbitrary positive number, the inequality (4.7) is proven.
As long as A k ⊂ A k+1 , k = 1, 2, . . ., and R n \ ∞ k=1 A k = 0, for almost all x ∈ R n there exist limits lim k→∞f k (x) = f (x), and, for i = 1, 2, . . . , n, we define Fixing an arbitrary point x 0 ∈ A k and taking into account the inequality k ≤ g(x) at x ∈ R n \ A k we have the following estimates: (4.12) Moreover |w i (x)| ≤ 2g(x) for almost all x ∈ R n by (4.7) and the inequality Take an arbitrary test function ϕ ∈ C ∞ 0 (Ω). By the conditions of Theorem 4.5, the function f is integrable on supp ϕ. Sincef k has the first generalized derivatives, we havê On the compact set {x : ϕ(x) = 0} the sequencesf k (x) and ∂f k ∂x i (x) have the majorants max Therefore, by the Lebesgue dominated convergence theorem we obtain for all ϕ ∈ C ∞ 0 (Ω). Consequently, w i (x) is the generalized derivative ∂f ∂x i (x), i = 1, . . . , n, of the function f and, by virtue of (4.7) and Proposition 2.6(ii), we have the estimate ∇f L q) a (R n ,w) ≤ 2 √ n g L q) a (R n ,w) . Therefore, f ∈ L q) a (R n , w). Thus, it is proven that f ∈ W 1,q) a (R n , w).
Remark 4.6. The proof of Theorem 4.5 is based on arguments of the paper [49] where the similar assertion for Sobolev functions, defined on Carnot groups, was given. See also [25] for an independent proof for Sobolev functions defined on Euclidean spaces.
The following result is the version of Theorem 4.5 for bounded domains.
Theorem 4.7. Let 1 < q < ∞ and Ω ⊂ R n be a bounded domain. A function f belongs to W 1,q) (Ω) if and only if f ∈ L q) (Ω) and there exist a non negative function g ∈ L q) (Ω) and a set S ⊂ Ω of measure zero such that the inequality holds for all points x, y ∈ Ω \ S with B(x, 3|x − y|) ⊂ Ω.
In view of Proposition 4.3, the case Ω ⊂ R n , where |Ω| < ∞ can be proved with similar arguments making evident changes. Remark 4.8. If f ∈ W 1,q) (Ω) then |f | ∈ W 1,q) (Ω) because by Theorem 4.7, there exist a non negative function g ∈ L q) (Ω) and a set S ⊂ Ω of measure zero such that holds for all points x, y ∈ Ω \ S with B(x, 3|x − y|) ⊂ Ω.

Pointwise estimates for Banach function spaces
Note, that the proof of Theorem 4.5 is based on the following facts: a (R n , w) satisfies the lattice property, i.e. if |f | ≤ |g| a.e. then f ≤ g .
Hence, this proof can be applied almost verbatim for proving a similar result for function spaces, which meet conditions (⋆)-(⋆ ⋆ ⋆). It includes many and various spaces, for example, grand Lesbegue, Musielak-Orlicz, Lorentz and Marcinkiewicz spaces, as well as Lebesgue spaces with variable exponents. In particular, it includes the general concept, that covers many different spaces at once, namely, the theory of Banach function spaces. We refer the reader to [2] and [42] for details. (i) f = 0 if and only if f = 0 a.e., αf = |α| f and f +g ≤ f + g ; (ii) if |f | ≤ |g| a.e. then f ≤ g ; (iii) if 0 ≤ f n ր f then f n ր f ; (iv) for every measurable E ⊂ R, µ(E) < ∞: χ E < ∞; (v) for every measurable E ⊂ R, there exists a constant C E > 0 (independent of f ), such that´E |f | dµ ≤ C E f .
The space X(R) = {f ∈ M : f < ∞} with norm · is called a Banach function space.
Remark 5.2. The property (⋆) is a local version of (v) in the sense that it considers only compact sets E ⊂ R.
Let Ω ⊂ R n be an open set and X(Ω) be a Banach function space w.r.t. the Lebesgue measure. The Sobolev space W X(Ω) denotes the space of weakly differentiable mappings f with f , ∇f ∈ X(Ω). This space is equipped with a norm f W X(Ω) := f X(Ω) + ∇f X(Ω) .
It is easy to see that conditions (⋆) and (⋆ ⋆ ⋆) are fulfilled for any Banach function space X. Hence, we can formulate the theorem in a general way as follows.
Theorem 5.3. Let Ω ⊂ R n be an open set and X(Ω) be a Banach function space such that the Hardy-Littlewood maximal operator is bounded in X(Ω). Then a function f belongs to W X(Ω) if and only if f ∈ X(Ω) and there exist a non negative function g ∈ X(Ω) such that the inequality holds for almost all x, y ∈ Ω with B(x, 3|x − y|) ⊂ Ω.
Let us consider several specific examples.
Corollary 5. 4. A function f belong to Lorentz-Sobolev space W L p,q (R n ) with p > 1 and q ≥ 1 if and only if f ∈ L p,q (R n ) and there exist a non negative function g ∈ L p,q (R n ) such that the inequality (5.1) is fulfilled for all x, y ∈ R n \S where S is a set of measure zero.
Properties of Lorentz spaces can be found in [2,Chapter 4.4]. For the research on the maximal operator in Orlicz spaces the reader is referred to [35] and [41,Theorem 3.3].
Remark 5.6. In [47, Theorem 1.2] the characterization in spirit of [26] is given for the Orlicz-Sobolev space W L A (R n ) with the Young function A and a complimentary A * satisfying the ∆ 2 -condition. The right hand side of (5.1) is expressed in terms of the Young function: |f (x) − f (y)| ≤ C|x − y|(A −1 (M σ|x−y| A(g)(x)) + A −1 (M σ|x−y| A(g)(y))), for some C > 0 and σ ≥ 1. Here A −1 is a generalized inverse of A.
The theory of Lebesgue and Sobolev spaces with variable exponent and, in particular, conditions about boundedness of maximal operator can be found in [10, Section 4.3]).
Corollary 5.8. Let X be a rearrangement invariant Banach function space with the upper Boyd index β X < 1. A function f belong to W X(Ω) if and only if f ∈ X(Ω) and there exist a non negative function g ∈ X(Ω) such that the inequality (5.1) holds for all x, y ∈ Ω \ S with B(x, 3|x − y|) ⊂ Ω, where S ⊂ Ω is a set of measure zero.
Remark 5.9. If X is a rearrangement invariant Banach function space then the maximal operator is bounded if and only if the upper Boyd index β X < 1, see [2, Chapter 3, Definition 5.12 and Theorem 5.17]. The formulas for calculating the Boyd indices of classical function spaces may be found in, for example, [17].