Sharp Sobolev type embeddings on the entire Euclidean space

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.


Introduction
An embedding theorem of Sobolev type amounts to a statement asserting that a certain degree of integrability of the (weak) derivatives of a function entails extra integrability of the function itself. A basic formulation concerns the space of weakly differentiable functions W 1,p (R n ), endowed with the norm u W 1,p (R n ) = u L p (R n ) + ∇u L p (R n ) . Membership of u in W 1,p (R n ) guarantees that u ∈ L np n−p (R n ) if 1 ≤ p < n, or u ∈ L ∞ (R n ), if p > n. Since np n−p > p, this is locally a stronger property than the a priori assumption that u ∈ L p (R n ), but it is weaker, and hence does not add any further information, near infinity. The same phenomenon occurs in a higher-order version of this result for W m,p (R n ), the Sobolev space of those m-times weakly differentiable functions u such that the norm is finite. Here, ∇ k u denotes the vector of all derivatives of u of order k, and, in particular, ∇ 1 u stands for ∇u and ∇ 0 u for u. Indeed, one has that if either m ≥ n, or 1 ≤ m < n and p > n m . Embedding (1.2) is still valid if R n is replaced with an open subset Ω with finite measure and regular boundary. The intersection with L p (Ω) is irrelevant in the target spaces of the resulting embedding, and L np n−mp (Ω), or L ∞ (Ω) are optimal among all Lebesgue target spaces. By contrast, no optimal Lebesgue target space exists in (1.2).
The existence of optimal target spaces in the Sobolev embedding for W m,p (R n ) can be restored if the class of admissible targets is enlarged, for instance, to all Orlicz spaces. This class allows to describe different degrees of integrability -not necessarily of power type -locally and near infinity. The use of Orlicz spaces naturally emerges in the borderline missing case in (1.2), corresponding to the exponents 1 < p = n m , and enables one to cover the full range of exponent m and p. The resulting embedding takes the form where L B (R n ) is the Orlicz space associated with a Young function B obeying Here, ≈ denotes equivalence in the sense of Young functions -see Section 2. Actually, embedding (1.3) is equivalent to (1.2) if either 1 ≤ p < n m , or m ≥ n, or 1 ≤ m < n and p > n m . The case when 1 < p = n m is a consequence of a result of [31,34,38], which, for m = 1, is also contained in [37]. The space L B (R n ) is the optimal (i.e. smallest) target in (1.3) among all Orlicz spaces. However, embedding (1.3) can still be improved when either 1 ≤ p < n m , or 1 < p = n m , provided that the class of admissible target spaces is further broadened. For instance, if 1 ≤ p < n m , then where L np n−mp ,p (R n ) is a Lorentz space. The intersection space in (1.4) is the best possible among all rearrangement-invariant spaces. A parallel result can be shown to hold in the limiting situation corresponding to 1 < p = n m , and involves spaces of Lorentz-Zygmund type. This follows as a special case of the results of the present paper, whose purpose is to address the problem of optimal embeddings, in the whole of R n , for Sobolev spaces built upon arbitrary rearrangement-invariant spaces. Precisely, given any rearrangement-invariant space X(R n ), we find the smallest rearrangement-invariant space Y (R n ) which renders the embedding (1.5) W m X(R n ) → Y (R n ) true. Here, W m X(R n ) denotes the m-th order Sobolev type space built upon X(R n ), and equipped with the norm defined as in (1.1), with L p (R n ) replaced with X(R n ). A rearrangement-invariant space is, in a sense, a space of measurable functions endowed with a norm depending only on the measure of the level sets of the functions. A precise definition will be recalled in the next section.
Questions of this kind have been investigated in the literature in the case when R n is replaced with a domain of finite measure. The optimal target problem has been solved in full generality for these domains in [18,23]. Apart from the examples mentioned above, few special instances are known in the entire R n . In this connection, let us mention that first-order sharp Orlicz-Sobolev embeddings in R n are established in [12], and that the paper [39] contains the borderline case m = 1, p = n of (1.4).
A key result in our approach is what we call a reduction principle, asserting the equivalence between a Sobolev embedding of the form (1.5), and a couple of one-dimensional inequalities involving the representation norms of X(R n ) and Y (R n ) on (0, ∞). This is the content of Theorem 3.3.
The optimal rearrangement-invariant target space Y (R n ) for W m X(R n ) in (1.5) is exhibited in Theorem 3.1. The conclusion shows that the phenomenon recalled above for the standard Sobolev spaces W m,p (R n ) carries over to any space W m X(R n ): no higher integrability of a function near infinity follows from membership of its derivatives in X(R n ), whatever rearrangement-invariant space is X(R n ). Loosely speaking, the norm in the optimal target space behaves locally like the optimal target norm for embeddings of the space W m X(R n ) with R n replaced by a bounded subset, and like the norm of X(R n ) near infinity. We stress that, since the norm in X(R n ) is not necessarily of integral type, a precise definition of the norm of the optimal target space is not straightforward, and the proofs of Theorems 3.1 and 3.3 call for new ingredients compared to the finite-measure framework.
Theorems 3.1 and 3.3 can be applied to derive optimal embeddings for customary and unconventional spaces of Sobolev type. In particular, we are able to identify the optimal Orlicz target and the optimal rearrangement-invariant target in Orlicz-Sobolev embeddings (Theorems 3.5 and 3.9), and the optimal rearrangement-invariant target in Lorentz-Sobolev embeddings (Theorem 3.11).

Background
Let E ⊂ R n be a Lebesgue measurable set. We denote by M(E) the set of all Lebesgue measurable functions u : E → [−∞, ∞]. Here, vertical bars | · | stand for Lebesgue measure. We also define We also define u * * : The operation of rearrangement is neither linear, nor sublinear. However, We say that a functional · X(0,L) : M + (0, L) → [0, ∞] is a function norm if, for every f , g and {f k } in M + (0, L), and every λ ≥ 0, the following properties hold: ( (P5) for every set F ⊂ (0, L) of finite measure there exists a positive constant C F such that F f (s) ds ≤ C F f X(0,L) for every f ∈ M + (0, L). Here, and in what follows, χ F denotes the characteristic function of a set F. If, in addition, (P6) f X(0,L) = g X(0,L) whenever f * = g * , we say that · X(0,L) is a rearrangement-invariant function norm.

The generalized Hölder inequality
holds for every f ∈ M + (0, L) and g ∈ M + (0, L). Given any t > 0, the dilation operator E λ is defined at f ∈ M(0, L) by The inequality holds for every rearrangement-invariant function norm · X(0,L) , and for every f ∈ M + (0, L).
A "localized" notion of a rearrangement-invariant function norm will be needed for our purposes. The localized rearrangement-invariant function norm of · X(0,∞) is denoted by · X (0,1) and defined as follows. Given a function f ∈ M + (0, 1), we call f its continuation to (0, ∞) by 0 outside (0, 1), namely Then, we define the functional · X(0,1) by for f ∈ M + (0, 1). One can verify that · X(0,1) is actually a rearrangement-invariant function norm. Given a measurable set E ⊂ R n and a rearrangement-invariant function norm · X(0,|E|) , the space X(E) is defined as the collection of all functions u ∈ M(E) such that the quantity is finite. The functional · X(E) defines a norm on X(E), and the latter is a Banach space endowed with this norm, called a rearrangement-invariant space. Moreover, X(E) ⊂ M 0 (E) for any rearrangementinvariant space X(E).
The rearrangement-invariant space X ′ (E) built upon the function norm · X ′ (0,|E|) is called the associate space of X(E).
Given any rearrangement-invariant spaces X(E) and Y (E), one has that with the same embedding norms. We refer the reader to [4] for a comprehensive treatment of rearrangement-invariant spaces. In remaining part of this section, we recall the definition of some rearrangement-invariant function norms and spaces that, besides the Lebesgue spaces, will be called into play in our discussion. Let us begin with the Orlicz spaces, whose definition makes use of the notion of Young function.
for f ∈ M + (0, L). In particular, given a measurable set E ⊂ R n , one has that A Young function A is said to dominate another Young function B near infinity [resp. near zero] [resp. globally] if there exist positive constants c > 0 and t 0 > 0 such that If |E| < ∞, then Given 1 ≤ p, q ≤ ∞, we define the Lorentz functional · L p,q (0,L) by . This functional is a rearrangement-invariant function norm provided that 1 ≤ q ≤ p, namely when q ≥ 1 and the weight function s 1 p − 1 q is non-increasing. In general, it is known to be equivalent, up to multiplicative constants, to a rearrangement-invariant function norm if (and only if) either p = q = 1, or 1 < p < ∞ and 1 ≤ q ≤ ∞, or p = q = ∞. The rearrangement-invariant space on a measurable set E ⊂ R n , built upon the latter rearrangement-invariant function norm, is the standard Lorentz space L p,q (E). Thus, L p,q (E) consists of all functions u ∈ M(E) such that the functional u L p,q (E) , defined as is finite. The notion of Lorentz functional can be generalized on replacing the function s 1 p − 1 q by a more general weight w(s) ∈ M + (0, L) in (2.6). The resulting functional is classically denoted by · Λ q (w)(0,L) , and reads f Λ q (w)(0,L) = f * (t)w(t) L q (0,L) for f ∈ M + (0, L). A characterization of those exponents q and weights w for which the latter functional is equivalent to a rearrangement-invariant function norm is known. In particular, when q ∈ (1, ∞), this is the case if and only if there exists a positive constant C such that This follows from a result of [33,Theorem 3.1]. In any of these cases, we shall denote by Λ q (w)(E) the corresponding rearrangement-invariant space on a measurable set E ⊂ R n . A further extension of the notion of the Lorentz functional · Λ q (w)(0,L) is obtained when the role of the function norm · L q (0,L) is played by a more general Orlicz function norm · L A (0,L) . The resulting functional will be denoted by · Λ A (w)(0,L) , and defined as for f ∈ M + (0, L). A complete characterization of those Young functions A and weights w for which the functional · Λ A (w)(0,L) is equivalent to a rearrangement-invariant function norm seems not to be available in the literature. However, this functional is certainly equivalent to a rearrangement-invariant function norm whenever the weight w is equivalent, up to multiplicative constants, to a non-increasing function. This is the only case that will be needed in our applications. For such a choice of weights w, we shall denote by Λ A (w)(E) the corresponding rearrangement-invariant space on a measurable set E ⊂ R n . A comprehensive treatment of Lorentz type functionals can be found in [11] and [30].

Main results
Our first main result exhibits the optimal rearrangement-invariant space into which any assigned arbitrary-order Sobolev space is embedded. Given an open set Ω in R n and any rearrangement- The space W m X(Ω) is a Banach space endowed with the norm given by for u ∈ Ω. In (3.1), and in what follows, the rearrangement-invariant norm of a vector is understood as the norm of its length. Given a rearrangement-invariant space X(R n ), we say that Y (R n ) is the optimal rearrangementinvariant target space in the Sobolev embedding if it is the smallest one that renders (3.2) true, namely, if for every rearrangement-invariant space The construction of the optimal space Y (R n ) in (3.2) requires a few steps. A first ingredient is the function norm · Z(0,1) obeying where the localized rearrangement-invariant function norm · X(0,1) is defined as in (2.3), and · X ′ (0,1) stands for its associated function norm. The function norm · Z(0,1) determines the optimal rearrangement-invariant space in the Sobolev embedding for the space W m X(Ω) in any regular open set in R n (with |Ω| = 1), see [18,23]. Next, we extend · Z(0,1) to a function norm · X m (0,∞) on (0, ∞) by setting The optimal rearrangement-invariant target space Y (R n ) in embedding (3.2) is the space X m opt (R n ) defined as This follows via Proposition 4.1, Section 4. Hence, if |supp u| < ∞, then u ∈ X m opt (R n ) if and only if u ∈ X m (R n ). In this sense the optimal space X m opt (R n ) is determined by X m (R n ) locally, and by X(R n ) near infinity.
Theorem 3.1 (Optimal Sobolev embedding). Let n ≥ 2, m ∈ N, and let X(R n ) be a rearrangementinvariant space. Then Theorem 3.1 has the following consequence.
Our second main result is a reduction theorem for Sobolev embeddings on the entire R n .
if and only if there exists a constant C such that (3.10) for every non-increasing function f : (0, ∞) → [0, ∞). In the remaining part of this section we present applications of our general results to two important families of Sobolev type spaces: the Orlicz-Sobolev and the Lorentz-Sobolev spaces. In the light of Corollary 3.2, we shall mainly restrict our attention to the case when m < n.
Let us begin with embeddings for Orlicz-Sobolev spaces built upon a Young function A. The notation W m,A (R n ) will also be adopted instead of W m L A (R n ). This kind of spaces is of use in applications to the theory of partial differential equations, whose nonlinearity is not necessarily of power type, which also arise in mathematical models for diverse physical phenomena -see e.g. [1,3,5,6,7,8,9,22,24,25,27,35,36,40]. We first focus on embeddings of W m L A (R n ) into Orlicz spaces. We show that an optimal target space in this class always exists, and we exhibit it explicitly. Define the function H n m : for τ ≥ 0, and the Young function A n m as denotes the (generalized) left-continuous inverse of H n m (see [13,15]). Theorem 3.5 (Optimal Orlicz target in Orlicz-Sobolev embeddings). Let n ≥ 2, m ∈ N, and let A be a Young function. Assume that m < n, and let A n m be the Young function defined by (3.13). Let A opt be a Young function such that and L Aopt (R n ) is the optimal Orlicz target space in (3.14).
Remark 3.6. In equations (3.12) and (3.13), we may assume, without loss of generality, that so that H is well defined. Indeed, since only the behavior of A n m , and hence of H, near infinity is relevant in view of our result, A can be replaced in (3.12) and (3.13) (if necessary) by a Young function equivalent near infinity, which renders (3.15) true. Any such a replacement results into an equivalent function A n m near infinity. Remark 3.7. Notice that, if then H −1 (t) = ∞ for large t, and equation (3.13) has accordingly to be interpreted as A n m (t) = ∞ for large t.
Remark 3.8. Condition (3.16) can be shown to agree with (3.7) when X(R n ) = L A (R n ), and embedding (3.14) recovers the conclusion of Corollary 3.2 in this case. Indeed, by Remark 3.7, under assumption (3.7) one has that Owing to Lemma 5.2 below, the resulting Orlicz target space Hence, the target space L Aopt (R n ) is in fact also optimal among all rearrangement-invariant spaces in (3.14).
We next determine the optimal target space for embeddings of W m,A (R n ) among all rearrangementinvariant spaces. Such a space turns out to be a Lorentz-Orlicz space defined as follows. Let A be a Young function. In the light of Remark 3.8, we may restrict our attention to the case when m < n, and Define the Young function A by and a is the function appearing in (2.5). Notice that condition (3.15) is equivalent to requiring that Thus, by a reason analogous to that explained in Remark 3.6, there is no loss of generality in assuming (3.19).
Theorem 3.9 (Optimal rearrangement-invariant target in Orlicz-Sobolev embeddings). Let n ≥ 2 and m ∈ N, and let A be a Young function. Assume that m < n, and condition (3.17) is fulfilled, and let A be the Young function given by (3.18). Let E be a Young function such that and let v : (0, ∞) → (0, ∞) be the function defined as Then and the Orlicz-Lorentz space Λ E (v)(R n ) is the optimal rearrangement-invariant target space in (3.21).
Example 3.10. We focus here on Orlicz-Sobolev spaces built upon a special family of Orlicz spaces, called Zygmund spaces and denoted by L p (log L) α (R n ), where either p = 1 and α ≥ 0, or p > 1 and α ∈ R. They are associated with a Young function equivalent to t p (1 + log + t) α , where log + (t) = max{log t, 0}. Of course, L p (log L) 0 (R n ) = L p (R n ). Owing to Theorem 3.5 and [15, Example 3.4], one has that where Moreover, L G (R n ) is optimal among all Orlicz spaces. Furthermore, by Theorem 3.9 and [15, Example 3.10], where v is given by (3.20), and Moreover, the Orlicz-Lorentz space Λ E (v) is optimal among all rearrangement-invariant spaces. The optimal rearrangement-invariant target space when either p > n m , or p = n m and α > n m − 1 agrees with the Orlicz space L G (R n ) with G defined as above, and hence with (L ∞ ∩ L p (log L) α )(R n ).
We conclude this section with an optimal embedding theorem for Lorentz-Sobolev spaces in R n , which relies upon Theorem 3.1.
(ii) Let m < n and 1 < q ≤ ∞. Then (iii) Let either m > n, or m ≤ n and p > n m , or m ≤ n, p = n m and q = 1. Then and Moreover, in each case, the target space is optimal among all rearrangement-invariant spaces.

Proofs of the main results
Let · X(0,∞) be a rearrangement-invariant function norm, and let · X(0,1) be its localized rearrangement-invariant function norm on (0, 1) given by (2.3). The rearrangement-invariant function norm · Z(0,1) defined as in (3.3) is the optimal one for which the Hardy type inequality holds with some positive constant C independent of nonnegative functions f ∈ X(0, 1), see [23, Theorem A]. The following proposition tells us that such a norm is always at least as strong as that of X(0, 1).
Proposition 4.1. Let n ≥ 2 and let m ∈ N. Assume that · X(0,∞) is a rearrangement-invariant function norm, and let · Z(0,1) be rearrangement-invariant function norm defined as in (3.3). Then Proof. Embedding (4.2) is equivalent to By the very definition of Z(0, 1), embedding (4.3) is in turn equivalent to the inequality for some constant C and for every f ∈ M + (0, 1). It is easily verified that the operator is sublinear, and bounded in L 1 (0, 1) and in L ∞ (0, 1). An interpolation theorem by Calderón [4, Theorem 2.12, Chapter 3] then tells us that it is bounded in any rearrangement-invariant space. Hence, inequality (4.4) follows.
The next auxiliary result will be critical in the proof of the sharpness of our embeddings. Throughout, we shall use the relation [ ] between two expressions to denote that the former bounds [is bounded by] the latter, up to a positive constant. The notation ≃ is adopted to denote that the relations and hold simultaneously. Notice the different meanings of the relation ≃ and the relation ≈ of equivalence between Young functions introduced in Section 2.
Proof. Let f be as in the statement. Assume, in addition, that f is constant on (0, 2L), and denote by f 0 ∈ R the constant value of f on (0, 2L). If s ≥ 2L, then . Inequalities (4.5) and (4.7) imply that .
Consider functions f of the form For this choice of f , one has that .
On the other hand, .
A key step towards the general embedding theorem of Theorem 3.1 is a first-order embedding, for somewhat more general non-homogeneous Sobolev type spaces defined as follows. Given two rearrangement-invariant function norms · X(0,∞) and · Y (0,∞) , we define the space W 1 (X, Y )(R n ) as W 1 (X, Y )(R n ) = {u ∈ X(R n ) : u is weakly differentiable in R n , and |∇u| ∈ Y (R n )} endowed with the norm Theorem 4.3 (Non-homogeneous first-order Sobolev embeddings). Let · X(0,∞) and · Y (0,∞) be rearrangement-invariant function norms. Let · Y 1 (0,∞) be defined as in (3.4), with m = 1 and X(R n ) replaced by Y (R n ). Then Proof. We begin by showing that there exists a constant C such that for every function u such that |suppu| ≤ 1 and |∇u| ∈ Y (R n ). Indeed, a general form of the Pólya-Szegö principle on the decrease of gradient norms under spherically symmetric symmetrization ensures that, for any such function u, the decreasing rearrangement u * is locally absolutely continuous in (0, ∞), and where ω n denotes the Lebesgue measure of the unit ball in R n -see e.g. [16,Lemma 4.1]. Moreover, there exists a constant C such that where the first equality holds since u * vanishes in (1, ∞), the inequality is a consequence of the Hardy type inequality (4.1), the second equality holds since u * (1) = 0, and the last inequality by the very definition of the norm in Y 1 (R n ). Inequality (4.13) is a consequence of (4.14) and (4.15). Now, let u ∈ W 1 (X, Y )(R n ) such that (4.16) u W 1 (X,Y )(R n ) ≤ 1. Then, where ϕ X is the fundamental function of · X(0,∞) defined as in (2.2). Let ϕ −1 X denote its generalized left-continuous inverse. One has that lim t→0 + ϕ −1 X (t) = 0, and ϕ −1 X (ϕ X (t)) ≥ t for t > 0. Therefore, by (4.17), Choose t 0 such that ϕ −1 X (1/t 0 ) ≤ 1, whence |{|u| > t 0 }| ≤ 1.
We are now in a position the accomplish the proof of Theorem 3.1.
Proof of Theorem 3.1. We shall prove by induction on m that for some constant C, and for every u ∈ W m X(R n ). This inequality, combined with the trivial embedding W m X(R n ) → X(R n ), yields embedding (3.6). If m = 1, then inequality (4.20) is a straightforward consequence of Theorem 4.3 applied in the special case when Y (R n ) = X(R n ). Now, assume that (4.20) is fulfilled for some m ∈ N. Let u ∈ W m+1 X(R n ). By induction assumption applied to the function u x i for i = 1, . . . , n, for some constants C and C ′ . Consequently, for some constant C. From Theorem 4.3 with Y (R n ) = X m (R n ), and inequality (4.21), we obtain for some constants C and C ′ . From [18, Corollary 9.6] one can deduce that for every u ∈ W m+1 X(R n ). Inequality (4.20), with m replaced by m + 1, follows from (4.22) and (4.23). It remains to prove the optimality of the space (X m ∩ X)(R n ). Assume that S(R n ) is another rearrangement-invariant space such that Then there exists a constant C such that for every u ∈ W m X(R n ). We have to show that or, equivalently, that for some constant C, and every f ∈ M + (0, ∞). Inequality (4.25) will follow if we show that for some constant C, and for every f ∈ M + (0, ∞). Let B be the ball in R n , centered at 0, such that |B| = 1. We claim that inequality (4.24) implies that for some constant C, and for every v ∈ W m X(B). Here, X(B) and S(B) denote the rearrangementinvariant spaces built upon the function norms · X (0,1) and · S(0,1) defined as in (2.3). To verify this claim, one can make use of the fact that there exists a bounded extension operator T : for some constant C and for every v ∈ W m X(B). Coupling (4.24) with (4.29) we deduce that for some constants C and C ′ , and for every v ∈ W m X(B). Hence, inequality (4.28) follows. Inequality (4.28) in turn implies that for some positive constant C, and every g ∈ M + (0, 1). This implication can be found in the proof of [23, Theorem A]. For completeness, we provide a proof hereafter, that also fixes some details in that of [23]. Let us preliminarily note that we can restrict our attention to the case when m < n. Indeed, if m ≥ n, inequality (4.30) holds with · S(0,1) = · L ∞ (0,1) , and hence for every rearrangement-invariant norm · S(0,∞) . Given any bounded function f ∈ M + (0, 1), define Set M = ω − 1 n n . We need to derive a pointwise estimate for |D m u|. As a preliminary step, consider any function v : where g : (0, M ) → [0, ∞) is an m-times weakly differentiable function. One can show that every ℓ-th order derivative of v, with 1 ≤ ℓ ≤ m, is a linear combination of terms of the form where α 1 , . . . , α i ∈ {1, . . . , n}, and Here, g (j) denotes the j-th order derivative of g. As a consequence, Here, we have made use of (4.37), on X(0, 1) for every j as above. The boundedness of these operators follows from Calderón interpolation theorem, owing to their boundedness in L 1 (0, 1) and L ∞ (0, 1). Inequality (4.30) follows from (4.38). Inequality (4.30) implies, via the optimality of the norm · Z(0,1) in (4.1), that for every g ∈ M + (0, 1). Given any function f ∈ M + (0, ∞), we can apply the latter inequality to f * χ (0, 1) , and obtain f * χ (0,1) S(0,1) ≤ C f * χ (0,1) Z(0,1) .
The last proof of this section concerns Theorem 3.3.
5. Proofs of Theorems 3.5, 3.9 and 3.11 The proofs of our results about Orlicz-Sobolev embeddings require a couple of preliminary lemmas.
Proof. Define the rearrangement-invariant function norm · X(0,∞) as as a set equality. Assume first that f ∈ X(0, ∞). We have that where the second inequality holds since H is equivalent to F near infinity. By Lemma 5.1, ∞) , where the second inequality holds since F dominates G near infinity, and the third one since F and H agree near infinity. Since f ∈ L H (0, ∞), then f ∈ L G (1, ∞) by Lemma 5.1. Thus, the right-hand side of (5.5) is finite, whence f ∈ X(0, ∞).
The main ingredients for a proof of Theorem 3.9 are now at our disposal.
Proof of Theorem 3.5. Since A opt is equivalent to A n m near infinity, by [13, Inequality (2.7)] inequality (5.9) holds with B = A opt . Moreover, since A opt is equivalent to A near zero, condition (5.10) is satisfied with B = A opt . Proposition 5.3 thus tells us that embedding (3.14) holds. On the other hand, if B is any Young function such that W m,A (R n ) → L B (R n ), then, by Proposition 5.3, conditions (5.9) and (5.10) are fulfilled. The former ensures that A n m dominates B near infinity -see [15, Proof of Theorem 3.1]. The latter tells us that A dominates B near 0. Altogether, A opt dominates B globally, whence L Aopt (R n ) → L B (R n ).