On fractional Orlicz–Sobolev spaces

Some recent results on the theory of fractional Orlicz–Sobolev spaces are surveyed. They concern Sobolev type embeddings for these spaces with an optimal Orlicz target, related Hardy type inequalities, and criteria for compact embeddings. The limits of these spaces when the smoothness parameter s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,1)$$\end{document} tends to either of the endpoints of its range are also discussed. This note is based on recent papers of ours, where additional material and proofs can be found.


Introduction
One of the available notions of Sobolev spaces of fractional order calls into play the Gagliardo-Slobodeckij seminorm. Given an open set ⊂ R n , with n ∈ N, and numbers s ∈ (0, 1) and p ∈ [1, ∞), this seminorm will be denoted by | · | s, p, , and is defined as is finite. Standard properties of the spaces W s, p ( ) are classical. The last two decades have witnessed an increasing number of investigations on these spaces because of their use in the analysis of nonlocal elliptic and parabolic equations, whose study has received an enormous impulse in the same period -see e.g. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][24][25][26][27]31,34,35,[38][39][40][43][44][45][46][47][48][49][50][51][52][53][54]56,57,59,60,64]. The aim of this note is to survey a few recent results, contained in [1][2][3][4], on some aspects of fractional Orlicz-Sobolev spaces. They constitute an extension of the spaces W s, p ( ), in that the role of the power function t p is performed by a more general finite-valued Young function A(t), namely a convex function from [0, ∞) into [0, ∞), vanishing at 0. The fractional Orlicz-Sobolev space, of order s ∈ (0, 1), associated with a Young function A, will be denoted by W s,A ( ), and is built upon the Luxemburg type seminorm | · | s,A, given by ≤ C|u| s, p,R n (1.5) for every measurable function u : R n → R decaying to 0 near infinity. A companion result holds if R n is replaced by any bounded open set with a sufficiently regular boundary ∂ , for any function u ∈ W s, p ( ), provided that the seminorm |u| s, p,R n is replaced by the norm u W s, p ( ) . Sharp extensions of these Sobolev type inequalities and ensuing embeddings to the spaces W s,A ( ) are presented in Sect. 3. For instance, the optimal Orlicz target space L B (R n ) in the inequality for some constant C and every measurable function u : R n → R decaying to 0 near infinity, is exhibited. Compact embeddings are also characterized. Here, we shall limit ourselves to consider target spaces of Orlicz type. However, inequalities and embeddings involving even stronger rearrangement-invariant norms are available. For these results we refer to [1,4], where proofs of the material collected in this paper can also be found. Let us add that in those papers optimal embeddings for higher-order fractional spaces W s,A ( ) associated with any s ∈ (0, n)\N are established as well.
A second issue that will be addressed concerns the limits as s → 1 − and s → 0 + of the space W s,A (R n ). It is well known that setting s = 1 in the definition of the space W s, p (R n ) does not recover the first-order Sobolev space W 1, p (R n ). Moreover, the Lebesgue space L p (R n ) cannot be obtained on choosing s = 0 in the definition of W s,A (R n ). Still, the seminorm ∇u L p (R n ) and the norm u L p (R n ) of a function are reproduced as limits as s → 1 − and s → 0 + , respectively, of the seminorm |u| s, p,R n , provided that the latter is suitably normalized by a multiplicative factor depending on s, p and n.
Specifically, a version in the whole of R n of a result by Bourgain-Brezis-Mironescu [9,10] tells us that, if p ∈ [1, ∞), then Here, S n−1 denotes the (n − 1)-dimensional unit sphere in R n , H n−1 denotes the (n − 1)-dimensional Hausdorff measure, e is any point on S n−1 , and the dot " · " stands for scalar product in R n . Conversely, if p ∈ (1, ∞), u ∈ L p (R n ) and the limit (or even the liminf) on the left-hand side of (1.6) is finite, then u ∈ W 1, p (R n ). The case when p = 1 is excluded from the latter result, but has a counterpart with W 1,1 (R n ) replaced by BV (R n ), the space of functions of bounded variation in R n . A slight variant of these facts is proved in [62]. In the precise form appearing above, they follow as special cases of results of [3]. A version of Eq. (1.6) with R n replaced by a bounded regular domain can be found in [32].
The limit as s → 0 + is the subject of a theorem by Maz'ya-Shaposhnikova [50], which ensures that for each p ∈ [1, ∞), and for every function u decaying to 0 near infinity and making the double integral finite for some s ∈ (0, 1). Here, ω n denotes the Lebesgue measure of the unit ball in R n . Equation (1.8) has to be interpreted in the sense that u ∈ L p (R n ) if and only if the limit on the left-hand side if finite, and that, in the latter case, the equality holds. Section 4 is devoted to counterparts, established in [3] and [2], of these results in the Orlicz framework. Namely, it deals with the limits (1.10) Interestingly, the conclusions about these limits share some features with those in (1.6) and (1.8), but also present some diversities. In particular, as shown by counterexamples, certain results can only hold under the additional 2 -condition on A, or are affected by some restrictions in the general case.

Function spaces
A Young function A is said to satisfy the 2 -condition -briefly A ∈ 2 -globally if there exists a positive constant C such that for t ≥ 0. If A is finite-valued and there exists t 0 > 0 such that inequality (2.4) holds for t ≥ t 0 , then we say that A satisfies the 2 -condition near infinity. Let be a measurable subset of R n , with n ≥ 1, having Lebesgue measure | |.
for every u ∈ L A ( ), where C is the same constant as in (2.1). If | | < ∞, and A dominates B near infinity, then inequality (2.5) continues to hold for some constant C = C(A, B, t 0 , | |).
The alternative notation A(L)( ) will also be employed, in the place of L A ( ), to denote the Orlicz space associated with a Young function equivalent to A.
If A is finite-valued, then the space E A ( ) agrees with the closure in L A ( ) of the space of bounded functions with bounded support in . Trivially, This inclusion holds as equality if and only if either | | < ∞ and A ∈ 2 near infinity, or | | = ∞ and A ∈ 2 globally. Assume now that is an open subset of R n . We denote by V 1,A ( ) the homogeneous Orlicz-Sobolev space given by Here, ∇u denotes the gradient of u. The notation W 1,A ( ) is adopted for the classical Orlicz-Sobolev space defined by The space W 1,A ( ) is a Banach space equipped with the norm The space of functions of bounded variation on is denoted by BV ( ). It consists of all functions in L 1 ( ) whose distributional gradient is a vector-valued Radon measure Du with finite total variation Du ( ) on . The space BV ( ) is a Banach space, endowed with the norm defined as Given a function u ∈ BV ( ), we denote by ∇u the absolutely continuous part of Du with respect to the Lebesgue measure, and by D s u its singular part. One has that where D s u ( ) stands for the total variation of the measure D s u over .
More generally, assume that A is a Young function with a linear growth near infinity, in the sense that Then a functional J A, associated with A can be defined on BV ( ) as The functional J A, agrees on BV ( ) with the relaxed functional of One has that the functional J A, is lower semicontinuous in BV ( ) with respect to convergence in L 1 loc ( ). Moreover, for every function u ∈ BV ( ), there exists a sequence {u m } ⊂ C 1 ( ) such that The homogeneous fractional Orlicz-Sobolev space V s,A ( ) is defined as where | · | s,A, is the seminorm given by (1.3). The subspace of those functions in V s,A ( ) that decay near infinity is denoted by If | | < ∞ and s ∈ (0, 1), we also define the space and is a Banach space equipped with the norm given by (1.4).
We conclude by mentioning a fractional-order Pólya-Szegő principle, which implies the decrease of the fractional Orlicz-Sobolev seminorm under symmetric rearrangement of functions u. Recall that the symmetric rearrangement u of a function u ∈ M d (R n ) is defined as the radially decreasing function about 0 which is equidistributed with u.
In the case when A is a power, inequality (2.9) can be traced back to [5,6]. The result for Young functions A satisfying the 2 -condition and functions u ∈ W s,A (R n ) is proved in [33]. The general version stated in Theorem 2.1 can be found in [1]. An earlier related contribution, dealing with functions of one-variable, is [42].

Sobolev type inequalities
Our first theorem provides us with the optimal -i.e. smallest possible -Orlicz target space in the Sobolev embedding for the space V s,A d (R n ). Such an optimal space is built upon the Young function A n s associated with A, n and s as follows. Let s ∈ (0, 1) and let A be a Young function such that Then, A n s is given by

4)
and there exists a constant C = C(n, s) such that is the optimal target space in inequality (3.5) among all Orlicz spaces. Remark 1 Assumption (3.2) on the Young function A cannot be dispensed with in Theorem 3.1. Actually, one can show that it is necessary for an embedding of the space V s,A d (R n ) to hold into any rearrangement-invariant space. Assumption (3.1) amounts to requiring that A has a subcritical growth with respect to the smoothness parameter s. It generalizes the condition p < n s required for the classical inequality (1.5).

Remark 2
The fractional Orlicz-Sobolev inequality (3.5) precisely matches the integer-order inequality established in [29] (see also [28] for an alternative form). Indeed, setting s = 1 in formula (3.3) for the function A n s recovers the Young function which defines the optimal Orlicz target space in the Orlicz-Sobolev inequality for W 1,A (R n ).
We now present an application of Theorem 3.1 to a family of Young functions whose behaviour near zero and near infinity is of power-logarithmic type. Although quite simple, these model Young functions enable us to recover results available in the literature and to exhibit genuinely new inequalities.

Example 1 Consider a Young function A such that
where either p 0 > 1 and α 0 ∈ R, or p 0 = 1 and α 0 ≤ 0, and either p > 1 and α ∈ R, or p = 1 and α ≥ 0. Here, equivalence is meant in the sense of Young functions. Let s ∈ (0, 1). The function A satisfies assumption (3.1) if either 1 ≤ p < n s and α is as above, or p = n s and α ≤ n s − 1, and satisfies assumption (3.2) if either 1 ≤ p 0 < n s and α 0 is as above, or p 0 = n s and α 0 > n s − 1.    ∈ (0, 1). Assume that A is a Young function satisfying conditions (3.1) and (3.2) and let A be the Young function given by (3.11). Then there exists a constant C = C(n, s) such that lim s→1 − C(n, s) < ∞ and for every function u ∈ M d (R n ).
Let us mention that a Hardy-type inequality for one-dimensional fractional Orlicz-Sobolev spaces has recently been established in [58].
A version of Theorem 3.1 holds even if R n is replaced by an open set ⊂ R n , provided that the latter enjoys suitable regularity properties. For instance, it suffices to assume that is a bounded Lipschitz domain. This is the subject of the next result, which, like the other results of this section dealing with subsets of R n , is established in [4]. the target spaces being optimal among all Orlicz spaces. Embedding (3.15) recovers or extends to the fractional case various results available in the literature. The case corresponding to (3.10) is classical. Integer-order Sobolev embeddings parallel to (3.15) are special instances of the general results of [30], which, in their turn, include various borderline cases established in [36,41,55,61,63]. In fact, the paper [36], and some sequel contributions by the same authors, also deal with fractional embeddings, but for spaces defined in terms of potentials instead of difference quotients.

Theorem 3.4 (Extension operator for fractional Orlicz-Sobolev spaces) Let be a bounded Lipschitz domain in R n . Assume that s ∈ (0, 1) and let A be a Young function. Then there exist a linear extension operator E : W s,A ( ) → W s,A (R n ) and a constant C = C( ) such that
Moreover, there exists a constant C = C (s, ) such that The Poincaré type inequality stated in the next proposition, of independent interest, has a role in the Proof of Theorem 3.4.

Proposition 1 (Fractional Orlicz-Poincaré inequality) Let be a bounded open set
in R n . Assume that s ∈ (0, 1) and that A is a Young function. If u ∈ V s,A ( ), then u ∈ L A ( ). Moreover, there exists a constant C = C(s, ) such that u − u L A ( ) ≤ C|u| s,A, for every function u ∈ V s,A ( ). Furthermore, The last result of this section is a criterion for the compactness of a fractional Orlicz-Sobolev embedding into an Orlicz space. A necessary and sufficient condition amounts to requiring that the Young function that defines the latter space grows essentially more slowly near infinity (in the sense of (2.2)) than the Young function that defines the optimal Orlicz target for a merely continuous embedding given by Theorem 3.1. This is the content of the following theorem.
The assertion that embedding (3.16) is compact means that every bounded sequence in V s,A d (R n ) has a subsequence whose restriction to E converges in L B (E) for every bounded measurable set E in R n . Let us notice that the equivalence of properties (i) and (ii) is not explicitly mentioned in [4]. Its proof follows, modulo minor variants, along the same lines as that of the equivalence of (i) and (iii).

Example 4 Assume that is a bounded
if p = n s and α < n s − 1 lim t→∞ if p = n s and α = n s − 1.
If A and B are as above, an analogous result holds for the embedding V s,A d (R n ) → L B loc (R n ).

Limits as s → 0 + and s → 1 −
Here we are concerned with the question of existence and values of the limits (1.9) and (1.10). Let us begin by addressing the problem of the limit as s → 0 + . A result from [2] tells us that, if the Young function A satisfies the 2 -condition, and the function u ∈ V s,A d (R n ) for some s ∈ (0, 1), then the limit in (1.10) does exist, and equals the integral of a Young function of |u| over R n . Interestingly, such a Young function is not just a constant multiple of A in general. It is instead the Young function A given by the formula Observe that the Young functions A and A are equivalent, since for t ≥ 0, owing to the monotonicity of A(t) and A(t)/t. In particular, if A(t) = t p for some p ≥ 1, then

Theorem 4.1 Let A be a Young function satisfying the
Plainly, owing to Eq. (4.1), Theorem 4.1 reproduces the Maz'ya-Shaposhnikova result (1.8) when A(t) = t p for some p ≥ 1.
Let us mention that, under the additional ∇ 2 -condition on A, an earlier partial result in this connection had been established in [23], where bounds for the lim inf s→0 + and lim sup s→0 + of the expression under the limit in (4.2) are given.
We emphasize that the 2 -condition imposed on A in Theorem 4.1 is not just a technicality. The next result shows that its conclusion can indeed fail if the 2condition is removed.

Theorem 4.2
There exist Young functions A, which do not satisfy the 2 -condition, and corresponding functions u : R n → R such that u ∈ V s,A d (R n ) for every s ∈ (0, 1) and We refer to the paper [2] for Proofs of Theorems 4.1 and 4.2. Let us just mention here that the Young functions A and the functions u announced in the statement of Theorem 4.2, that demonstrate the possible failure of Eq. (4.2), can be chosen with the following properties: and for large |x|, . . x n ), for suitably related constants γ ≥ 1 and λ ∈ (1, 2). We finally focus on the limit as s → 1 − . As recalled in Sect. 1, in the case of standard Sobolev spaces associated with the exponent p, the result takes a different form depending on whether p = 1 or p ∈ (1, ∞). More precisely, whereas Eq. (1.6) holds under the assumption that u ∈ W 1, p (R n ) for every p ∈ [1, ∞), the fact that u ∈ L p (R n ) and the limit in (1.6) is finite ensure that u ∈ W 1, p (R n ) if p ∈ (1, ∞), but just that u ∈ BV (R n ) if p = 1. Also, under the latter assumption, Eq. (1.6) continues to hold with R n |∇u|dx replaced by Du (R n ) on the right-hand side.
A parallel phenomenon occurs in the ambient of Orlicz-Sobolev spaces. Similarly to the limit (1.10), the Young function to be applied to the modulus of the gradient to describe the limit (1.9) is not a mere multiple of A. The relevant function will be denoted by A • : [0, ∞) → [0, ∞), and is defined as where e is any fixed vector in S n−1 . One can show that A • is a Young function equivalent to A. Specifically, there exist constants c 1 = c 1 (n) and c 2 = c 2 (n) such that Observe that, if A(t) = t p , then dx, is finite for sufficiently large λ. However, under the 2 -condition on A, one has that W 1,A (R n ) = W 1,A (R n ), and hence Eq. (4.4) holds for every λ > 0, including λ = 1.
In the framework of Orlicz spaces associated with a Young function A, an analogue of the distinction between p = 1 and p ∈ (1, ∞) for powers is properly formulated in terms of the limit at infinity and/or at 0 of the (non-decreasing) function A(t)/t. In particular, a converse to Theorem 4.3 holds under the superlinear growth condition on A near infinity 5) and the sublinear decay condition at 0 Plainly, if A(t) = t p , either of conditions (4.5) and (4.6) is equivalent to requiring that p > 1. (4.5) and (4.6).

Theorem 4.4 Let A be a finite-valued Young function. Assume that A fulfills conditions
for some λ > 0, then u ∈ W 1,A (R n ).
In the case when A has a linear growth near infinity or near 0, Theorems 4.3 and 4.4, respectively, have counterparts in the framework of functions of bounded variation. Assume that A is a Young function for which equation (4.5) fails, and hence condition (2.6) holds. Since the function A • given by (4.3) is equivalent to A, Eq. (2.6) also holds if A is replaced by A • . Let a ∞ • be the number defined as in (2.8), with A replaced by A • , namely The following result tells us that, under (2.6), if u ∈ BV (R n ) then the limit in (4.4) equals the functional J A • ,R n (u) defined as in (2.7), namely the relaxed functional of R n A • (|∇u|) dx with respect to convergence in L 1 loc (R n ).

Conflict of interest
The authors declare that they have no conflict of interest.
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