Compactness of higher-order Sobolev embeddings

We study higher-order compact Sobolev embeddings on a domain $\Omega \subseteq \mathbb R^n$ endowed with a probability measure $\nu$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(\Omega,\nu)$ and $Y(\Omega,\nu)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(\Omega,\nu)$ into $Y(\Omega,\nu)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(\Omega,\nu)$. We then apply this result to the characterization of higher-order compact Sobolev embeddings on concrete measure spaces, including John domains, Maz'ya classes of Euclidean domains and product probability spaces, whose standard example is the Gauss space.


Introduction
Embeddings of Sobolev spaces into other function spaces play a very important role in modern functional analysis. Although Sobolev spaces on the Euclidean space R n and on bounded Euclidean domains having a Lipschitz boundary are discussed most frequently, it turns out that Sobolev spaces on various other domains, possibly endowed with more general measures than just with the Lebesgue one, are of interest as well. For instance, the class of John domains (see Section 3 for a definition), which is strictly larger than the class of domains having a Lipschitz boundary, appears in connection with the study of holomorphic dynamical systems and quasiconformal mappings. It was shown that Sobolev inequalities on John domains have the same form as in the standard case of Lipschitz domains, see [4,12,15,9]. Furthermore, among quite a wide class of Euclidean domains, John domains are exactly those for which the Sobolev inequality holds in this form [7]. Another important example is the Gauss space, that is, R n endowed with the Gauss measure γ n defined by dγ n (x) = (2π) − n 2 e −|x| 2 2 dx.
In contrast to the Euclidean setting, Sobolev inequalities on the Gauss space are dimension-free, which yields the possibility to extend them also to infinite dimensions. This is of use in the study of quantum fields, since this study can often be reduced to Sobolev inequalities in infinitely many variables. One possible way how to prove Sobolev embeddings is to derive them from isoperimetric inequalities for the underlying domains. This connection between Sobolev embeddings and isoperimetric inequalities was first found by Maz'ya in [18] and [19]. His discovery then led to an extensive research on this topic, which resulted in a number of important contributions that are considered classical these days (see, e.g., those by Moser [21], Talenti [25], Aubin [1], and Brézis and Lieb [6]), and which has continued until now.
Let us note that, until a very recent time, almost all available results on the interplay between Sobolev embeddings and isoperimetric inequalities involved only first-order embeddings. In our recent paper with Andrea Cianchi and Luboš Pick [9] we have developed a method based on deriving higher-order Sobolev embeddings via subsequent iteration of first-order ones, which enables us to derive also higher-order Sobolev embeddings from isoperimetric inequalities. Furthermore, and more significantly, for customary underlying domains (e.g., for John domains and for the Gauss space, which we have already briefly mentioned) the results obtained by this method are sharp in the context of the class of rearrangement-invariant spaces.
In the present paper we show that not only continuous higher-order Sobolev embeddings but also the compact ones, can be derived from isoperimetric properties of the underlying domains, and that the results obtained in this way are sharp in many customary situations.
Let us now describe the subject of the paper more precisely. We shall study compact Sobolev embeddings on a domain Ω in R n endowed with a probability measure ν which is absolutely continuous with respect to the Lebesgue measure. We also require that the density of ν fulfils some technical assumptions, see Section 3 for more details. For any ν-measurable set E ⊆ Ω we denote by P ν (E, Ω) its perimeter in Ω with respect to ν (a precise definition can be found in Section 3 again). The isoperimetric properties of (Ω, ν) are described by the so-called isoperimetric function of (Ω, ν), denoted I Ω,ν . It is the largest function on [0, 1] with values in [0, ∞] which is nondecreasing on [0, 1 2 ], nonincreasing on [ 1 2 , 1] and for which the isoperimetric inequality P ν (E, Ω) ≥ I Ω,ν (ν(E)) holds for every ν-measurable E ⊆ Ω.
The question of finding the exact form of I Ω,ν is very difficult and has been solved only in few special cases, such as the Euclidean ball [20] and the Gauss space [5]. The asymptotic behaviour of I Ω,ν at 0, in which we are interested, can be however evaluated more easily, and is therefore known for quite a wide class of domains, including Euclidean John domains (see [12] combined with [20, Corollary 5.2.3, p. 297]) or product probability spaces [2], which extend the Gauss space.
Given m ∈ N and a rearrangement-invariant space X(Ω, ν), we will consider the m-th order Sobolev space V m X(Ω, ν) consisting of all mtimes weakly differentiable functions on Ω whose m-th order weak derivatives belong to the space X(Ω, ν). A precise definition of the notion rearrangement-invariant space can be found in Section 2, we just briefly recall that a rearrangement-invariant space is, roughly speaking, a Banach space consisting of ν-measurable functions on Ω in which the norm of a function depends only on the measure of its level sets. A basic example of rearrangement-invariant spaces are Lebesgue spaces; besides them, the class of rearrangement-invariant spaces includes many further families of function spaces, such as Orlicz spaces, Lorentz spaces, etc.
In [9] we have shown that a continuous embedding of the Sobolev space V m X(Ω, ν) into a rearrangement-invariant space Y (Ω, ν) is implied by a certain one-dimensional inequality depending on the representation norms · X(0,1) and · Y (0,1) of X(Ω, ν) and Y (Ω, ν), respectively, on m and on the asymptotic behaviour of I Ω,ν at 0, described in terms of a nondecreasing function I giving a lower bound for the isoperimetric function at 0. We remark that this inequality can be understood as boundedness of a certain integral operator from the representation space X(0, 1) into Y (0, 1). The above mentioned operator will be de- We note that while H m I is (possibly) a kernel operator, K m I is just a weighted Hardy-type operator, which is far easier to work with. We also recall that important customary examples are available for the cases when (1.2) is valid as well as for the cases when (1.2) fails.
The main aim of the present paper is to prove that compactness of the operator H m I from X(0, 1) into Y (0, 1) implies the compact embedding of V m X(Ω, ν) into Y (Ω, ν) (Theorem 5.1). We will also show (Theorem 5.3) that if (1.2) is fulfilled, then the same result holds with H m I replaced by K m I . The proof of Theorem 5.1 strongly depends on the use of almost-compact embeddings, called also absolutely continuous embeddings in some literature. They have been studied, e.g., in [11] and [24]. It is well known that such embeddings have a great significance for deriving compact Sobolev embeddings.
In many customary situations, the sufficient condition in terms of the operator H m I turns out to be also necessary for compactness of the corresponding Sobolev embedding. We demonstrate this fact on the cases of Euclidean John domains, product probability spaces, and Maz'ya classes of domains. The latter classes consist of those bounded Euclidean domains whose isoperimetric function is bounded from below by a multiple of some fixed power function. Unlike the case of John domains and product probability spaces, in which the necessity holds for each individual domain, for Maz'ya classes the sharpness is fulfilled in a wider sense: there is one domain in each class for which the necessity holds.
The structure of the paper is as follows. In the next section we introduce rearrangement-invariant spaces and their almost-compact embeddings. Section 3 contains a description of the measure spaces that will come into play, of their isoperimetric properties and of Sobolev spaces built upon rearrangement-invariant spaces over these measure spaces.
Section 5 contains the main results of the paper that have been already described above.
In Section 6 we apply the results of Section 5 to the characterization of compact Sobolev embeddings on John domains (Theorem 6.1), on Maz'ya classes of domains (Theorem 6.2) and on product probability spaces (Theorem 6.4). The final Section 7 then provides examples of compact Sobolev embeddings for concrete pairs of rearrangementinvariant spaces over the measure spaces discussed in Section 6.

Rearrangement-invariant spaces
In this section we recall some basic facts from the theory of rearrangement-invariant spaces. Our standard general reference is [3].
Let (R, µ) be a nonatomic measure space satisfying µ(R) = 1. In fact, R will always be a domain in R n for some n ∈ N. If the measure µ is omitted, we assume that it is the n-dimensional Lebesgue measure on R. We denote by M(R, µ) the collection of all µ-measurable functions on R having its values in [−∞, ∞]. We also set M + (R, µ) = {f ∈ M(R, µ) : f ≥ 0 on R}.
If two functions f, g ∈ M(R, µ) fulfil µ f = µ g (or, equivalently, f * µ = g * µ ), we say that f and g are equimeasurable and write f ∼ µ g.
We now summarize some basic properties of rearrangement-invariant spaces. We first note that each function f ∈ X(R, µ) is finite µ-a.e. on R. Furthermore, the Fatou lemma [3, Chapter 1, Lemma 1.5(iii)] yields that whenever (f k ) ∞ k=1 is a sequence in X(R, µ) converging to some function f µ-a.e. and fulfilling that lim inf k→∞ f k X(R,µ) < ∞, then f ∈ X(R, µ) and Moreover, if (f k ) ∞ k=1 is a sequence which converges to some function f in the norm of the space X(R, µ), then (f k ) ∞ k=1 converges to f in measure. In particular, there is a subsequence of (f k ) ∞ k=1 which converges to f µ-a.e. on R.
Given a rearrangement-invariant norm · X(0,1) , we shall consider the functional · X (0,1) : Then · X (0,1) is also a rearrangement-invariant norm, called the associate norm of · X(0,1) . The corresponding rearrangement-invariant space X (R, µ) is called the associate space of X(R, µ). It is not hard to observe that If · X(0,1) and · Y (0,1) are rearrangement-invariant norms, then the continuous embedding X(R, µ) → Y (R, µ) holds if and only if X(R, µ) ⊆ Y (R, µ), see [3, Chapter 1, Theorem 1.8]. We shall write X(R, µ) = Y (R, µ) if the set of functions belonging to X(R, µ) coincides with the set of functions belonging to Y (R, µ). In this case, the norms · X(R,µ) and · Y (R,µ) are equivalent, in the sense that there are positive constants C 1 , C 2 such that Furthermore, according to [3, Chapter 1, Proposition 2.10], the embed- Suppose that · X(0,1) is a rearrangement-invariant norm. Then the fundamental function ϕ X of · X(0,1) is defined by Owing to [3, Corollary 5.3, Chapter 2], ϕ X is quasiconcave, in the sense that ϕ X is nondecreasing on (0, 1] and ϕ X (t) t is nonincreasing on (0, 1]. We say that a function f ∈ X(R, µ) has an absolutely continuous An easy observation yields that this can be equivalently reformulated by lim a→0+ The collection of all functions having an absolutely continuous norm in X(R, µ) is denoted by X a (R, µ). Further, we say that a subset S of X(R, µ) is of uniformly absolutely Suppose that · X(0,1) and · Y (0,1) are rearrangement-invariant norms. We say that X(R, µ) is almost-compactly embedded into Y (R, µ) and write X(R, µ) * Observe that X(R, µ) * → Y (R, µ) holds if and only if the unit ball of X(R, µ) is of uniformly absolutely continuous norm in Y (R, µ). We shall make use of two characterizations of X(R, µ) * and lim [11,Section 4,Property 5]. Let us now give some examples of rearrangement-invariant norms. A basic example are the Lebesgue norms · L p (0,1) , p ∈ [1, ∞], defined for all f ∈ M(0, 1) by The corresponding rearrangement-invariant spaces L p (R, µ) are then called the Lebesgue spaces. Recall that for each rearrangement-invariant space X(R, µ) the embeddings hold. We denote by C X the constant from the latter embedding, that is, we have and C X is the least real number for which (2.2) is satisfied. It is a well-known fact that a rearrangement-invariant space X(R, µ) is different from L ∞ (R, µ) if and only if lim s→0+ ϕ X (s) = 0. Furthermore, owing to [24,Theorems 5.2 and 5.3], L ∞ (R, µ) * → X(R, µ) is characterized by X(R, µ) = L ∞ (R, µ), and X(R, µ) One can consider also more general functionals · L p,q (0,1) and · L p,q;α (0,1) which were studied, e.g., in [10] and [22]. They are given for any f ∈ M(0, 1) by respectively. Here, we assume that p ∈ [1, ∞], q ∈ [1, ∞], α ∈ R, and use the convention that 1/∞ = 0. Note that · L p (0,1) = · L p,p (0,1) and · L p,q (0,1) = · L p,q;0 (0,1) for every such p and q. However, it turns out that under these assumptions on p, q, and α, · L p,q (0,1) and · L p,q;α (0,1) do not have to be rearrangement-invariant norms. To ensure that · L p,q;α (0,1) is equivalent to a rearrangement-invariant norm, we need to assume that one of the following conditions is satisfied: In this case, · L p,q (0,1) is called a Lorentz norm, · L p,q;α (0,1) is called a Lorentz-Zygmund norm and the corresponding rearrangement-invariant spaces L p,q (R, µ) and L p,q;α (R, µ) are called Lorentz spaces and Lorentz-Zygmund spaces, respectively. Furthermore, if · L p 1 ,q 1 ;α 1 (0,1) and · L p 2 ,q 2 ;α 2 (0,1) are equivalent to rearrangement-invariant norms then holds if and only if p 1 > p 2 , or p 1 = p 2 and one of the following conditions is satisfied: (2.7)

Sobolev spaces
Let n ∈ N and let Ω be a domain in R n endowed with a measure ν satisfying ν(Ω) = 1. We assume that ν is absolutely continuous with respect to the n-dimensional Lebesgue measure λ n , and we denote by ω the density of ν with respect to λ n (that is, whenever E ⊆ Ω is ν-measurable, we have ν(E) = E ω(x) dx). The function ω is supposed to be Borel measurable and fulfilling that for a.e. x ∈ Ω there is an open ball B x centered in x such that B x ⊆ Ω and ess inf Bx ω > 0.
Notice that a subset of Ω (or a function defined on Ω) is ν-measurable if and only if it is Lebesgue measurable. We shall write measurable instead of Lebesgue measurable in what follows.
Let us now give a few examples of compatible triplets.
Suppose that n ∈ N, n ≥ 2. We recall that a bounded domain Ω ⊆ R n is called a John domain if there exist a constant c ∈ (0, 1) and a point x 0 ∈ Ω such that for every x ∈ Ω there are l > 0 and a rectifiable curve : [0, l] → Ω, parametrized by arclength, such that (0) = x, (l) = x 0 , and In what follows, we shall consider (with no loss of generality) only John domains whose Lebesgue measure is equal to 1. It is known that each John domain satisfies where n = n n−1 . Therefore, if we denote I(t) = t 1 n , t ∈ (0, 1], then (Ω, λ n , I) is a compatible triplet. 1]. We denote by J α the Maz'ya class of all bounded Euclidean domains Ω ⊆ R n with λ n (Ω) = 1 fulfilling that there is a positive constant C, possibly depending on Ω, such that Set I α (t) = t α , t ∈ (0, 1]. Then (Ω, λ n , I α ) is another example of a compatible triplet.
As a final example we mention product probability spaces, namely, R n with the product probability measure defined as follows.
The main example of product probability measures we have just defined is the n-dimensional Gauss measure which can be obtained by setting More generally, measures associated with for some β ∈ [1,2] are also examples of product probability measures. They are called the Boltzmann measures. For each β ∈ [1, 2], such n-dimensional measure is denoted by γ n,β . We of course have γ n,2 = γ n .
We shall now define Sobolev spaces built upon rearrangement-invariant spaces over (Ω, ν). The measure space (Ω, ν) is required to satisfy all the above mentioned properties and, moreover, the inequality (3.7) I Ω,ν (t) ≥ Ct, t ∈ [0, 1/2], has to be fulfilled for some positive constant C independent of t. Notice that condition (3.7) is satisfied whenever there is a function I for which (Ω, ν, I) is a compatible triplet. Let m ∈ N and let u be an m-times weakly differentiable function on Ω. Given k ∈ {1, 2, . . . , m}, we denote by ∇ k u the vector of all k-th order weak derivatives of u. Moreover, we set ∇ 0 u = u. Then the m-th order Sobolev space built upon a rearrangement-invariant space X(Ω, ν) is the set V m X(Ω, ν) = {u : u is an m-times weakly differentiable function on Ω such that |∇ m u| ∈ X(Ω, ν)}.
We now state a theorem and a proposition which were proved in [9] and which will be used in what follows.
The set W m X(Ω, ν) equipped with the norm is easily seen to be a normed linear space. We always have the continuous embedding W m X(Ω, ν) → V m X(Ω, ν). The reverse embedding is not true in general, however, we have the following: . Suppose that (Ω, ν) is as in the first paragraph of the present section and, moreover, that Let m ∈ N and let · X(0,1) be a rearrangement-invariant norm. Then In particular, if (Ω, ν, I) is a compatible triplet such that then V m X(Ω, ν) = W m X(Ω, ν) for every m ∈ N and for every rearrangement-invariant norm · X(0,1) . Indeed, property (C3) of compatible triplets yields that there is c ∈ (0, 2) for which The result now follows from Proposition 3.3.

Compact operators
In this section we give several characterizations of compactness of certain one-dimensional operator on rearrangement-invariant spaces. These characterizations play a central role in the proofs of our main results in the following Section 5. Moreover, the results of this section will be used to characterize compactness of this operator on concrete classes of rearrangement-invariant spaces (see Section 7).
Let J : (0, 1] → (0, ∞) be a measurable function satisfying We set 1] J(t), a ∈ (0, 1), and observe that for every a ∈ (0, 1), We shall consider the operator H J defined by and the operator R J defined by Furthermore, given j ∈ N, we define the operators H j J and R j j by see [9,Remarks 8.2]. For technical reasons, we also set H 0 J = R 0 J = Id. We remark that the operators H j J and R j J are associate in the sense that for every f ∈ M + (0, 1) and g ∈ M + (0, 1) we have We also observe that whenever j ∈ N and f ∈ M(0, 1) then H j J f is nonincreasing on (0, 1). Finally, given a ∈ (0, 1], the equality which follows from the change of variables formula, will be of use. Given two rearrangement-invariant norms · X(0,1) and · Y (0,1) , we shall write H j J : X(0, 1) → Y (0, 1) in order to denote that the operator H j J is bounded from X(0, 1) into Y (0, 1). Our goal is to find necessary and sufficient conditions for compactness of H j J from X(0, 1) into Y (0, 1), denoted by (4.8) H j J : X(0, 1) →→ Y (0, 1). The first result in this connection is the following: Theorem 4.1. Let J : (0, 1] → (0, ∞) be a measurable function satisfying (4.1) and let j ∈ N. Suppose that · X(0,1) and · Y (0,1) are rearrangement-invariant norms. Consider the following two conditions:  1 we obtain two more necessary and sufficient conditions for (4.8). Their equivalence to (4.8) holds in a slightly less general setting, but the strength of these characterizations rests on the possibility to reformulate them as almostcompact embeddings between certain rearrangement-invariant spaces. This connection between compactness of H j J and almost-compactness of an embedding becomes a key tool for the proof of our main result, Theorem 5.1.
We shall now introduce a family of rearrangement-invariant spaces whose almost-compact embeddings are suitable for characterization of (4.8).
The following theorem characterizes (4.8) by means of the space X r j,J (0, 1).
We shall now define another family of rearrangement-invariant spaces whose almost-compact embeddings will be used for characterization of (4.8).
Let · Y (0,1) be a rearrangement-invariant norm fulfilling Then · Y d j,J (0,1) is a rearrangement-invariant norm and the corresponding rearrangement-invariant space Y d j,J (0, 1) is the optimal domain for Y (0, 1) with respect to the operator H j J , in the sense of the following: Proposition 4.5. Let J : (0, 1] → (0, ∞) be a measurable function satisfying (4.1) and let j ∈ N.
The last result of this section provides a necessary and sufficient condition for compactness of the operator H j J given in terms of the optimal domain space. Theorem 4.6. Let J : (0, 1] → (0, ∞) be a measurable function satisfying (4.1) and let j ∈ N. Suppose that · X(0,1) is a rearrangementinvariant norm such that X(0, 1) = L 1 (0, 1) and · Y (0,1) is a rearrangement-invariant norm fulfilling (4.16). Then the following conditions are equivalent: (a) Using the definition of the associate norm and (4.6) we deduce that condition (ii) of Theorem 4.6 is equivalent to equality is of uniformly absolutely continuous norm in X (0, 1).
(b) It is not hard to verify that conditions (ii) and (iii) of Theorem 4.6 are never fulfilled with X(0, 1) = L 1 (0, 1). However, this is not the case of condition (i) since we have already observed in Remark 4.4 that (i) is satisfied with J ≡ 1, X(0, 1) = L 1 (0, 1), and Y (0, 1) = L ∞ (0, 1) whenever j > 1. Furthermore, in contrast to Theorem 4.2, which holds in the exceptional case Y (0, 1) = L ∞ (0, 1) for quite a wide class of functions J, there are only very few functions J for which Theorem 4.6 is fulfilled with X(0, 1) = L 1 (0, 1). We shall now characterize all nondecreasing functions having this property (recall that the case when the function J is nondecreasing is the most significant from the point of view of applications to compact Sobolev embeddings).
Fix j ∈ N. In order to decide whether there is a rearrangementinvariant space Y (0, 1) for which (4.19) holds, it is enough to study whether which is characterized in the last section of the present paper (see The- (no matter what j is). Combining this with (4.1) we deduce that the only case when (4.21) is not fulfilled for some j ∈ N (and hence Theorem 4.6 is fulfilled with X(0, 1) = L 1 (0, 1)) is the one when there is a set M ⊆ (0, 1) such that 0 ∈ M and J(s) ≈ s on M .
Furthermore, having only the information that (4.22) holds, we cannot decide whether (4.14) is satisfied with X(0, 1) = L 1 (0, 1) or not. An example supporting this was already presented in Remark 4.4.
Remark 4.9. The classical result due to Luxemburg and Zaanen [17] relates compactness of a kernel integral operator to its absolute continuity, and to absolute continuity of the associate operator. Let us describe this result in some more detail, and then compare it to our Theorems 4.2 and 4.6.
Let X(0, 1) and Y (0, 1) be rearrangement-invariant spaces, let T be a kernel integral operator, and let T be the operator associate to T (in a similar sense in which our operator R j J is associate to H j J , see [17] for a precise definition). Assume that T f ∈ Y a (0, 1) for every f ∈ X(0, 1), and T g ∈ X a (0, 1) for every g ∈ Y (0, 1). In [17] it is proved (even in a more general setting) that we have the equivalence of the following three conditions: (a) T : If we set T = H j J , then its associate operator T is the operator R j J . We have observed in Remarks 4.3 and 4.7 that in this case condition (b) is exactly condition (ii) of Theorem 4.2 and condition (c) is identical to condition (ii) of Theorem 4.6. The main difference between our result and the one proved in [17] is the following: when proving that (a) implies (b) or (c), we do not need to assume that either the operator H j J , or R j J , has its range in the set of functions of absolutely continuous norm, since this fact already follows from (a) (under the indispensable assumption that Y (0, 1) = L ∞ (0, 1) or X(0, 1) = L 1 (0, 1), respectively). It can be easily observed that such a claim fails when T is an arbitrary kernel integral operator.
Therefore, given k ∈ N, we can find a nonnegative measurable function f k on (0, 1) such that f k X(0,1) ≤ 1 and Since the sequence ( converges to some function f in the norm of the space Y (0, 1). Moreover, the subsequence can be found in such a way that (H j J ( Since the function is nondecreasing on (0, 1), we obtain (ii), as required. Now, suppose that (ii) holds and (4.11) is satisfied. If Conversely, assume that Then, in particular, and therefore also .
Passing to limit when a tends to 0, we obtain (iii), as required.
Proof of Theorem 4.2: (i) ⇒ (ii) According to Lemma 4.10, condition (i) implies , thanks to Lemma 4.10. Passing to limit when b tends to 0, we get as required.
(ii) ⇒ (i) The proof is completely analogous to that of [23, Theorem 3.1, implication (ii) ⇒ (i)], even with simplifications following from the fact that we consider rearrangement-invariant spaces over a finite interval.
(ii) ⇔ (iii) Using the definition of the associate norm and the equality (4.6), we get Note that the second last equality holds because χ (0,a) H j J g is nonincreasing on (0, 1) for every a ∈ (0, 1) and g ∈ X(0, 1). Thus, we have proved that (ii) holds if and only if Y (0, 1) * → (X r j,J ) (0, 1). Since the latter condition is equivalent to (iii), the proof is complete.
Suppose that condition (4.11) is satisfied. Then the implication (ii) ⇒ (i) follows from the proof of Theorem 4.2. So, assume that Y (0, 1) = L ∞ (0, 1), 1 0 dr J(r) < ∞ and (ii) holds. We observe that to prove (i), it is enough to show that for every a ∈ (0, 1), the operator so H j J will be a norm limit of compact operators, and thus itself a compact operator.
Fix a ∈ (0, 1). For every f ∈ X(0, 1) we can consider the function H j J,a f to be defined by (4.5) (with f replaced by χ (a,1) f ) on the entire [0, 1]. Then H j J,a f is continuous on [0, 1], and it follows from (4.25) (with b = a) and from the fact that Then, using the result of the previous paragraph with j replaced by j − 1, we get The last expression goes to 0 when t 2 − t 1 tends to 0 thanks to the absolute continuity of the Lebesgue integral. This proves that the image by H j J,a of the unit ball of X(0, 1) is equicontinuous. Let j = 1 and X(0, 1) = L 1 (0, 1). Then we deduce that which goes to 0 when t 2 −t 1 tends to 0 thanks to the almost-compact embedding X(0, 1) * → L 1 (0, 1). This proves the equicontinuity in this case. Arzela-Ascoli theorem now yields that H j J,a maps the unit ball of X(0, 1) into a relatively compact set in C([0, 1]). Since the space C([0, 1]) is continuously embedded into L ∞ (0, 1), the operator H j J,a is compact from X(0, 1) into L ∞ (0, 1), as required.
Finally, suppose that X(0, 1) = L 1 (0, 1), Y (0, 1) = L ∞ (0, 1), and j = 1. We have To complete the proof, we will show that if X(0, 1) = L 1 (0, 1), Y (0, 1) = L ∞ (0, 1), and j = 1 then condition (i) is not satisfied. Indeed, since 1 J > 0 on (0, 1), there is ε > 0 and a set M ⊆ (0, 1) of measure 1 2 Therefore, the sequence (f n ) ∞ n=1 is bounded in X(0, 1). Let m, n ∈ N, m < n. Since both H J f m and H J f n are continuous on (0, 1), we have Consequently, there is no subsequence Proof of Proposition 4.5: Suppose that · Y (0,1) is a rearrangement-invariant norm fulfilling (4.16). We start by showing that · Y d j,J (0,1) is a rearrangement-invariant norm. Properties (P5), (P6) as well as the first two properties in (P1) trivially hold. Since the functional · L 1 (0,1) satisfies the axioms of rearrangement-invariant norms, we only have to verify that the functional · Z(0,1) defined by fulfils the triangle inequality and properties (P2)-(P4). However, (P2) and (P3) can be proved exactly in the same way as it is done in [8, proof of Lemma 4.2]. Furthermore, using the fact that each nonnegative function equimeasurable to 1 is equal to 1 a.e., and applying the equality (4.7) with a = 1, we get which proves (P4). Thus, it only remains to verify the triangle inequality. Suppose that u, v are nonnegative simple functions on (0, 1). We will show that Assume that a nonnegative function h on (0, 1) satisfies h ∼ u + v. Then it is not hard to observe that there exist nonnegative simple functions h u and h v on (0, 1) such that Passing to supremum over all h we get (4.28).
Let f, g ∈ M + (0, 1). Then there are two sequences of nonnegative simple functions (u n ) ∞ n=1 and (v n ) ∞ n=1 such that u n ↑ f and v n ↑ g. Then also u n + v n ↑ f + g. Thus, using the property (P3) for · Z(0,1) (which has already been verified) and the inequality (4.28), we obtain as required.

Main results
In the present section we state and prove the main results of this paper. They concern derivation of m-th compact Sobolev embeddings from compactness of the one-dimensional operator H m I defined in the previous section. Here, I stands for a function which is related to the underlying measure space (Ω, ν) by the fact the (Ω, ν, I) is a compatible triplet (recall that the notion of a compatible triplet was introduced in Definition 3.1).
Theorem 5.1. Assume that (Ω, ν, I) is a compatible triplet. Let m ∈ N and let · X(0,1) and · Y (0,1) be rearrangement-invariant norms. Then Let us remark that further characterization of (5.1) and (5.2) can be obtained by applying Theorems 4.2 and 4.6 with J = I and j = m. This claim can be proved by methods of [9,Section 9]. Namely, we first observe that, for every a ∈ (0, 1] and f ∈ M(0, 1), where the functional · X d (0,1) is defined by Then it suffices to show that for every a ∈ (0, 1], We note that the only nontrivial inequality in (5.5) is the second one, which was proved in [9, Theorem 9.5] in the case when a = 1. Equalities (5.6) and (5.7) were proved for a = 1 in [9, Corollary 9.8]. All the proofs can be easily extended also to general a ∈ (0, 1]. Suppose that I : (0, 1] → (0, ∞) is a nondecreasing function satisfying (3.1) and let m ∈ N. Set (5.8) J(t) = (I(t)) m t m−1 , t ∈ (0, 1]. We observe that J is measurable on (0, 1] and fulfils (4.1). We can therefore consider operators K m I and S m I defined by K m I = H J and S m I = R J , respectively. Then Then (5.13) is satisfied for all pairs of rearrangement-invariant norms · X(0,1) and · Y (0,1) .
Analogously to the general case, which we dealt with in Theorem 5.1, one can obtain further characterization of (5.11) and (5.12) by applying  ∞) is a nondecreasing function satisfying (3.1) and m ∈ N is such that (5.14) is fulfilled, then it will follow from the proof of Theorem 5.3 that (5.12) is satisfied for all pairs of rearrangement-invariant norms · X(0,1) and · Y (0,1) . However, Theorem 4.1 applied with J as in (5.8) and j = 1 yields that (5.11) is not satisfied if X(0, 1) = L 1 (0, 1) and Y (0, 1) = L ∞ (0, 1). Therefore, in contrast to the part (a), in the part (b) we do not have the equivalence of (5.11) and (5.12). Moreover, compactness of the operator K m I seems not to be appropriate to characterize compact Sobolev embeddings in this case, and condition (5.12) turns out to be a suitable substitute for (5.11).
(ii) Notice that to prove the equivalence of (5.11) and (5.12) in the part (a), we do not need to assume that I satisfies (5.9).
The remaining part of this section is devoted to proofs of Theorems 5.1 and 5.3. We start with an auxiliary result which shows that, in our setting, the unit ball of each Sobolev space is compact in measure.
Proof: For a.e. x ∈ Ω we can find an open ball B x centered in x such that B x ⊆ Ω and ess inf Bx ω > 0. Denote by N the set of points in Ω for which such a ball does not exist. Then ν(N ) = 0 and we have , it is also bounded in V 1 L 1 (Ω, ν). Hence, for every j ∈ N and k ∈ N we have . Denote u 0 k = u k , k ∈ N. By induction, for every j ∈ N we will construct a subsequence (u j k ) ∞ k=1 of the sequence (u j−1 k ) ∞ k=1 converging a.e. on B xj . Suppose that, for some j ∈ N, we have already found the sequence (u is bounded in V 1 L 1 (B xj ) and the compact embedding V 1 L 1 (B xj ) → → L 1 (B xj ) holds, we can find a subsequence (u j k ) ∞ k=1 of (u j−1 k ) ∞ k=1 converging in L 1 (B xj ). Passing, if necessary, to another subsequence, (u j k ) ∞ k=1 can be found in such a way that it converges a.e. on B xj . Now, the diagonal sequence (u k k ) ∞ k=1 converges a.e. (or, what is the same, ν-a.e.) on ∞ j=1 B xj = Ω \ N . Since ν(N ) = 0, (u k k ) ∞ k=1 converges ν-a.e. on Ω, as required. Furthermore, it is a well known fact that each sequence converging ν-a.e. is convergent in measure.
We also need the following Proof: We first observe that condition (4.16) is fulfilled with j = 1, J = I, and · Y (0,1) = · L ∞ (0,1) . Indeed, we have  Since f * is a nonnegative function equimeasurable to f , The proof is complete.
Conversely, assume that Y (0, 1) = L ∞ (0, 1) and 1 0 1/I(s) ds < ∞ (recall that the assumption (5.1) is still in progress). We start with the case when m = 1. The proof of Lemma 5.6 then yields that condition (4.16) is fulfilled with J = I and j = 1. Furthermore, since the operator H I is not compact from L 1 (0, 1) into L ∞ (0, 1) (see the last part of the proof of Theorem 4.1), we have X(0, 1) = L 1 (0, 1). Thus, due to Theorem 4.6,  Let (u k ) ∞ k=1 be a bounded sequence in V 1 X(Ω, ν). Then it is bounded also in W 1 X(Ω, ν). Without loss of generality we may assume that Due to Lemma 5.5, there is a subsequence (v k ) ∞ k=1 of the sequence (u k ) ∞ k=1 which converges in measure to some function v. Our aim is to show that (v k ) ∞ k=1 is a Cauchy sequence in L ∞ (Ω, ν). Then, thanks to the completeness of L ∞ (Ω, ν), (v k ) ∞ k=1 will converge to v in the norm of the space L ∞ (Ω, ν). This will prove that V 1 X(Ω, ν) is compactly embedded into L ∞ (Ω, ν).
Fix ε > 0 and observe that for all k, l ∈ N we have Since v k , v l and the constant function ε/2 are weakly differentiable on Ω, |v k − v | − ε/2 is weakly differentiable on Ω as well and a.e. on Ω. Furthermore, max{|v k − v | − ε/2, 0} is weakly differentiable on Ω and a.e. on Ω. Thus, a.e. on Ω (and therefore also ν-a.e. on Ω, since ν is absolutely continuous with respect to the n-dimensional Lebesgue measure).
Thanks to (5.17), there is δ > 0 such that Since (v k ) ∞ k=1 converges in measure to v, we can find k 0 ∈ N such that for every k ≥ k 0 We observe that for all k, ≥ k 0 , Consequently, by (5.24) and (5.25), Hence, (v k ) ∞ k=1 is a Cauchy sequence in L ∞ (Ω, ν), as required. Finally, assume that m > 1. According to Lemma 5.6, for every g ∈ M(0, 1) we have Let (u k ) ∞ k=1 be a bounded sequence in V m X(Ω, ν). Then (u k ) ∞ k=1 is bounded in L 1 (Ω, ν), so ( Ω u k dν) ∞ k=1 is a bounded sequence of real numbers and we can find a subsequence (u 0 k ) ∞ k=1 of (u k ) ∞ k=1 such that the sequence ( Ω u 0 k dν) ∞ k=1 is convergent. For i = 1, 2, . . . , n, consider the sequence (D i u 0 k ) ∞ k=1 consisting of weak derivatives with respect to the i-th variable of elements of the sequence (u 0 k ) ∞ k=1 . Owing to the boundedness of (u k ) ∞ k=1 in V m X(Ω, ν), all these sequences are bounded in V m−1 X(Ω, ν). Now, the compact embedding (5.28) yields that we can inductively find sequences . Since a subsequence of a convergent sequence is still convergent, we have, in particular, that (D j u n k ) ∞ k=1 is a Cauchy sequence in (L ∞ ) d 1,I (Ω, ν) for every j ∈ {1, 2, . . . , n}.
Consider the function J defined by (5.8). We claim that, given t ∈ (0, 1), Indeed, we trivially have Conversely, because I is nondecreasing on (0, 1], for every s ∈ (0, t) Passing to supremum over all s ∈ (0, t), we obtain Suppose that (5.32) is not satisfied (i.e., part (a) is in progress). Since K m I = H J , an application of Theorem 4.1 yields that (5.11) is equivalent to (5.12). Furthermore, according to the first part of the proof, each of (5.11) and (5.12) implies (5.13).

Compactness of Sobolev embeddings on concrete measure spaces
In this section we characterize compact Sobolev embeddings on Euclidean John domains, on Maz'ya classes of Euclidean domains and, finally, on product probability spaces, whose standard example is the Gauss space. Recall that definitions and basic properties of the above mentioned measure spaces can be found in Section 3.
We note that the equivalence of (6.1) and (i) in Theorem 6.1 is already known in the special case when Ω is a domain having a Lipschitz boundary and m < n, see [13].
Let us now focus on Maz'ya classes of domains in R n , n ≥ 2. When dealing with m-th order Sobolev embeddings on a domain from the Maz'ya class J α (for some m ∈ N and α ∈ [ 1 n , 1]), we shall use the operator T m α given by 1), and by Theorem 6.2. Let n ∈ N, n ≥ 2, let m ∈ N and let α ∈ [ 1 n , 1]. Suppose that · X(0,1) and · Y (0,1) are rearrangement-invariant norms. If m(1 − α) ≤ 1 (notice that this is true for every m ∈ N provided that α = 1) then the fact that is equivalent to each of the following conditions: is satisfied for all pairs of rearrangement-invariant norms · X(0,1) and · Y (0,1) except of those for which X(0, 1) = L 1 (0, 1) and Y (0, 1) = L ∞ (0, 1). Furthermore, if m(1 − α) > 1 then condition (6.2) is fulfilled independently of the choice of · X(0,1) and · Y (0,1) . Remarks 6.3. (a) It will follow from the proof of Theorem 6.1 that its statement is true for all domains Ω belonging to the Maz'ya class J 1 n (this class contains, in particular, all John domains).
(b) Let m, n ∈ N, n ≥ 2, let α ∈ [ 1 n , 1] and let Ω be a domain in R n belonging to the Maz'ya class J α . Suppose that · X(0,1) and · Y (0,1) are rearrangement-invariant norms. Consider the following two assertions: If α = 1 n then conditions (i) and (ii) are equivalent (this follows from Theorem 6.1 combined with the part (a) of this remark). However, such an equivalence is no longer true when α > 1 n . This can be easily observed since each Maz'ya class J α contains, in particular, all John domains. Compactness of Sobolev embeddings on John domains is characterized by compactness of the operator Q m n , which does not coincide with compactness of T m α . On the other hand, given an arbitrary α ∈ ( 1 n , 1], there is one domain Ω ∈ J α for which the equivalence of (i) and (ii) holds for all rearrangement-invariant norms · X(0,1) and · Y (0,1) (an example of such a domain can be found in Proposition 6.6).
(c) The operators Q m n and T m α can be described via the operators "H" and "K" defined in Sections 4 and 5, respectively, in the following way: Hence, Theorems 4.2 and 4.6 applied to an appropriate operator "H" provide further characterization of compactness of Q m n and T m α .
We finally focus on product probability spaces (R n , µ Φ,n ), where n ∈ N and the measure µ Φ,n is defined by (3.3) if n = 1 and by (3.4) if n > 1. Given m ∈ N, we characterize compact Sobolev embeddings on (R n , µ Φ,n ) in terms of compactness of the operator H m LΦ , with L Φ as in (3.5). The operator H m LΦ therefore has the form f ∈ M(0, 1), t ∈ (0, 1).
We also prove that compactness of the operator H m LΦ coincides with compactness of the somewhat simpler operator P m Φ , defined by We note that the operator P m Φ was introduced in [9] where it was shown that boundedness of H m LΦ is equivalent to boundedness of P m Φ . Furthermore, we show that anologues of Theorems 4.1, 4.2, and 4.6 hold for the operator P m Φ , although it does not have the form H j J for some j ∈ N and some function J. In order to do this, we define two families of rearrangement-invariant norms, playing the role of optimal range and optimal domain norms with respect to the operator P m Φ . Let · X(0,1) be a rearrangement-invariant norm. Given m ∈ N, consider the rearrangement-invariant norm · Xm(0,1) whose associate norm fulfils for every f ∈ M(0, 1). Then the functional · X r m,Φ (0,1) , given for every f ∈ M(0, 1) by , is a rearrangement-invariant norm and we have X r m,Φ (0, 1) = X r m,LΦ (0, 1), up to equivalent norms, see [9, Theorem 7.3 and its proof].
Further, let · Y (0,1) be a rearrangement-invariant norm fulfilling The fact that the functional · Y d m,Φ (0,1) is actually a rearrangementinvariant norm can be proved in the same way as it is done for the functional · Y d j,J (0,1) in the proof of Proposition 4.5. Theorem 6.4. Let n, m ∈ N and let Φ be as in Section 3. Suppose that · X(0,1) and · Y (0,1) are rearrangement-invariant norms. Then the following conditions are equivalent: Furthermore, if X(0, 1) = L 1 (0, 1) and (6.3) is satisfied, then (i)-(vi) are equivalent to each of the following conditions: Observe that Theorem 6.4 yields that, in contrast to the Euclidean setting, compact Sobolev embeddings on (R n , µ Φ,n ) do not depend on the dimension n, in the sense that we have the equivalence of the following two assertions.
Let us now prove the results we have stated. The proofs are based on the results of the previous section, and on the following: Proposition 6.5. Assume that (Ω, ν) is as in Section 3. Let m ∈ N and let · X(0,1) and · Y (0,1) be rearrangement-invariant norms satisfying Let α ∈ (0, 1]. Denote Suppose that L is an operator defined on X α + , with values in V m X(Ω, ν), fulfilling that for some positive constant C and for all f ∈ X α + . Set Hf = (Lf ) * ν , f ∈ X α + . Assume that for some real valued function K satisfying that K(·, t) is nonnegative and measurable on (t, 1) for every t ∈ (0, 1). Then Proof: We first observe that whenever k is a positive integer satisfying 1/k ≤ α and f ∈ X(0, 1) ∩ M + (0, 1), then χ (0,1/k) f ∈ X α + and the functions L(χ (0,1/k) f ) and H(χ (0,1/k) f ) are thus well defined. Since condition (6.4) implies V m X(Ω, ν) → Y (Ω, ν), for every k ∈ N satisfying where C is the constant from (6.5) and C is the constant from the embedding V m X(Ω, ν) → Y (Ω, ν). Consequently, for every k as above we can find a function f k ∈ M + (0, 1) such that f k X(0,1) ≤ 1 and Since the sequence ( converges to some function g in the norm of the space Y (Ω, ν). Then, in particular, L(χ (0,1/k ) f k ) → g in measure.
Proof of Theorem 6.1: Consider the function I(t) = t 1 n , t ∈ (0, 1]. It was observed in Section 3 that (Ω, λ n , I) is a compatible triplet. Furthermore, the function I satisfies (5.9). We have that lim t→0+ t m−1 (I(t)) m = lim t→0+ t m n −1 = 0 holds if and only if m > n. In such a case, the proof follows directly from Theorem 5.3.
Suppose that m ≤ n. Since Q m n = K m I , Theorem 5.3 gives the equivalence of (i) and (ii) and also shows that each of the conditions (i) and (ii) implies (6.1). Hence, it only remains to prove that (6.1) implies (ii).
Assume that m = n and X(0, 1) = L 1 (0, 1). Then there is nothing to prove, since condition (ii) (or, equivalently, (i)) always holds. Indeed, we have and L ∞ (0, 1) → Y (0, 1). Therefore, Q m m : L 1 (0, 1) → Y (0, 1). The conclusion now follows from Remark 4.8 and from the fact that Q m m = H 1 . Suppose that (6.1) is satisfied and m < n, or m = n and X(0, 1) = L 1 (0, 1). Let B R be an open ball of radius R > 0 such that B R ⊆ Ω. Without loss of generality we may assume that B R is centered at 0 and that κ n R n ≤ 1, where κ n denotes the Lebesgue measure of the unit ball in R n . Let f be a function belonging to the set X κnR n + defined in Proposition 6.5. Then we set It is not hard to observe that Lf is an m-times weakly differentiable function on Ω. By subsequent applications of the Fubini theorem, we obtain Denote Hf = (Lf ) * λn . Then n (s − t) m−1 ds, t ∈ (0, 1).
One can show similarly as in [14, proof of .
If m < n then it follows from [14, proof of Theorem A] that for every i ∈ {1, 2, . . . , m − 1}, In the remaining case when m = n and X(0, 1) = L 1 (0, 1) we obtain by the Fubini theorem that Furthermore, by (6.11), Altogether, we obtain up to multiplicative constants independent of f ∈ X κnR n + . The operator L therefore satisfies (6.5). Proposition 6.5 now gives that (6.12) lim Since the constant function 1 fulfils (5.9), the equivalence (5.30) implies that for all a ∈ (0, κ n R n ) and for all f ∈ X(0, 1), up to multiplicative constants depending on m. The assertion (ii) follows by combined using of (6.12) and (6.13). Finally, in order to obtain a characterization of (6.1) in the case when m = n, it suffices to describe when (i) holds with m = n. We have already shown that if X(0, 1) = L 1 (0, 1) then (i) is satisfied. Furthermore, it follows from (6.10), from the embedding X(0, 1) → L 1 (0, 1), from where κ n−1 denotes the Lebesgue measure of the unit ball in R n−1 .
Let Ω α be the domain in R n given by Then λ n (Ω α ) = 1 and I Ωα (s) ≈ s α , s ∈ [0, 1 2 ]. Proof of Theorem 6.2: Set I α (t) = t α , t ∈ (0, 1]. As observed in Section 3, (Ω, λ n , I α ) is a compatible triplet for each domain Ω ∈ J α . Suppose that α ∈ [ 1 n , 1). Then (5.9) is fulfilled with I = I α and we have m(1 − α) ≤ 1 then Theorem 5.3 combined with the fact that T m α = K m Iα yields that (i) is equivalent to (ii) and that each of the conditions (i) and (ii) implies (6.2). Further, if α = 1 then the equivalence of (i) and (ii) and the implication (i) (or (ii)) implies (6.2) follow from Theorem 5.1 and from the fact that T m 1 = H m I1 . Thus, it suffices to prove that, in all cases, (6.2) implies (ii).
Let f be any function in X(0, 1) ∩ M + (0, 1) (or, what is the same, let f be an arbitrary function belonging to the set X 1 + defined in Proposition 6.5). For x = (x 1 , . . . , x n ) ∈ Ω α , we set Then Lf is an m-times weakly differentiable function on Ω α and, owing to (6.14), we have Furthermore, it follows from [9, proof of Theorem 6.4] that L satisfies (6.5). Hence, Proposition 6.5 implies that (6.15) lim If α = 1 then (6.15) is exactly condition (ii) which we wanted to verify. If α ∈ [ 1 n , 1) then the equivalence (5.30) (with I = I α ) yields that for every f ∈ X(0, 1) and every a ∈ (0, 1) we have up to multiplicative constants depending on m and α. Condition (ii) now follows by combined using of (6.15) and (6.16).
Proof of Theorem 6.4: We have observed in Section 3 that (R n , µ Φ,n , L Φ ) is a compatible triplet. Theorem 5.1 therefore yields that (ii) implies (i).
Conversely, suppose that (i) is fulfilled. Let f be an arbitrary function belonging to the set X 1/2 + defined in Proposition 6.5. For every x = (x 1 , . . . , x n ) ∈ R n we set Then Lf is an m-times weakly differentiable function on R n . Denote Hf = (Lf ) * µΦ,n . Then, thanks to the equality µ Φ,n ({(x 1 , x 2 , . . . , x n ) ∈ R n : we have It follows from [9, proof of Theorem 7.1] that L satisfies (6.5). Thus, Proposition 6.5 implies that (6.17) lim Since I Φ (s) ≈ L Φ (s), s ∈ (0, 1 2 ], we deduce that Hf ≈ H m LΦ f for all f ∈ X 1/2 + , up to multiplicative constants independent of f . Hence, condition (6.17) is equivalent to (6.18) lim which is equivalent to (ii) according to Theorem 5.1. We have thus proved the equivalence of (i) and (ii). Due to Remark 5.2, (ii) is equivalent to (6.19) lim Further, according to [9,Proposition 11.2], H m LΦ g ≈ P m Φ g is fulfilled for all nonnegative nonincreasing functions g, up to multiplicative constants depending on m. Therefore, (6.19) holds if and only if (6.20) lim Since for every g ∈ M(0, 1), [9,Proposition 11.2], we have that for every a ∈ (0, 1), Using this chain of inequalities and the equivalence of (6.18) and (6.20), we obtain that (iv) is equivalent to (6.18), and therefore also to (ii).
Owing to Theorem 4.2 and to the fact that By (6.21), condition (6.22) implies (v). Trivially, (v) implies (iv), and, thanks to the result of the previous paragraph, (iv) implies (ii). Consequently, (v) is equivalent to (ii). Condition (vi) is equivalent to (ii) owing to Theorem 4.2 and to the fact that X r m,Φ (0, 1) = X r m,LΦ (0, 1). The implication (iii) ⇒ (iv) can be proved in the same way as the implication (i) ⇒ (ii) in Lemma 4.10. We have already proved that (iv) implies (v). Let us now show that (v) implies (iii).
Finally, assume that X(0, 1) = L 1 (0, 1) and condition (6.3) is satisfied. According to Theorem 4.6, (ii) is equivalent to (6.23) lim Using (6.21) we deduce that (6.23) implies (vii). Trivially, (vii) implies (iv). Since we have already shown that (iv) is equivalent to (ii), we arrive at the equivalence of (vii) and (ii). We now claim that the space Y d m,Φ (0, 1) is the largest rearrangement-invariant space from which the operator P m Φ is bounded into Y (0, 1). This fact can be proved in the same way as it is done in the proof of Proposition 4.5 for the rearrangement-invariant space Y d j,J (0, 1) and the operator H j J . Since boundedness of P m Φ coincides with boundedness of H m LΦ (see [9,Proposition 11.3]), we obtain that Y d m,Φ (0, 1) is the optimal domain for Y (0, 1) with respect to the operator H m LΦ , and therefore, by Proposition 4.5, Y d m,Φ (0, 1) = Y d m,LΦ (0, 1). Consequently, Theorem 4.6 yields that (viii) is equivalent to (ii). The proof is complete.

Examples
In the present section we characterize compact Sobolev embeddings on domains from Maz'ya classes, and on product probability spaces, for some of the customary rearrangement-invariant norms. The case of John domains is not discussed explicitly, however, results for John domains can be derived from corresponding results for Maz'ya classes of domains, by applying the equivalence of the following two conditions: We recall that this equivalence follows from Theorems 6.1 and 6.2.
In the first part of this section we shall study when the compact Sobolev embedding holds, provided that (Ω, ν) is either a Euclidean domain belonging to the Maz'ya class J α for some α ∈ [ 1 n , 1], or a product probability space, and one of the rearrangement-invariant spaces X(Ω, ν) and Y (Ω, ν) is equal to L 1 (Ω, ν) or L ∞ (Ω, ν) (the largest and the smallest rearrangement-invariant space, respectively). To obtain a tool for dealing with this problem, we characterize for a given nondecreasing function I the validity of condition in each of the four cases when one of the spaces X(0, 1) and Y (0, 1) coincides with L 1 (0, 1) or L ∞ (0, 1). We start with the two "L 1 -cases". It turns out that in this situation the operator H m I can be replaced by the simpler operator K m I , without assuming any restrictons on I (compare to Theorem 5.3). The following theorem characterizes m-th order compact Sobolev embeddings on domains from the Maz'ya class J α in the two "L 1 -cases". It can be obtained by combined using of Theorems 6.2 and 7.1 (with I(s) = s α ). Let us note that here, and also in all further results on Maz'ya classes of domains later in this section, we assume that m(1−α) < 1. This can be done with no loss of generality, since the case when m(1 − α) ≥ 1 was sufficiently described in Theorem 6.2.
(a) The condition is satisfied for every Ω ∈ J α if and only if This is never fulfilled for α = 1. (b) Suppose that X(0, 1) = L 1 (0, 1). Then the condition is satisfied for every Ω ∈ J α . Furthermore, if X(0, 1) = L 1 (0, 1) then (7.6) is fulfilled for every Ω ∈ J α if and only if α ∈ [ 1 n , 1). An analogous result for product probability spaces is provided in the next theorem. Theorem 7.3. Let n, m ∈ N, and let Φ be as in Section 3. Suppose that · X(0,1) is a rearrangement-invariant norm and denote by ϕ X the fundamental function of · X(0,1) .
(a) The condition is satisfied if and only if (b) Suppose that X(0, 1) = L 1 (0, 1). Then the condition is satisfied. Furthermore, if X(0, 1) = L 1 (0, 1) then (7.9) is fulfilled if and only if Note that Theorem 7.3 follows from Theorem 6.4, Theorem 4.1 (applied with J = L Φ and j = m), and Theorem 7.1 (applied with I = L Φ ). We also need to use the equivalence which was proved in [9, Lemma 11.1(iii)], and the following chain: Remark 7.4. It follows from the convexity of Φ and from the fact that Φ(0) = 0 that the function s → s Φ(s) is nonincreasing on (0, ∞). Hence, lim s→∞ s Φ(s) exists. In particular, if (7.10) is not fulfilled then lim s→∞ s Φ(s) ∈ (0, ∞). Combining this with the monotonicity of s Φ(s) we obtain that in this situation, Φ(s) ≈ s on (a, ∞) for every a ∈ (0, ∞), up to multiplicative constants possibly depending on Φ and a.
Let us now focus on the two "L ∞ -cases". Similarly as in the "L 1cases", we start with a one-dimensional result concerning the validity of condition (7.2), now with X(0, 1) or Y (0, 1) equal to L ∞ (0, 1). In contrast to Theorem 7.1, in this situation we cannot equivalently replace the operator H m I by K m I (a counterexample follows from Remarks 5.4(iv)). We will change the notation from I to J and from m to j, and we will not assume any monotonicity of J. Then, by setting J(s) = I(s) m s m−1 , s ∈ (0, 1], and j = 1, our result applies also to the characterization of condition (7.2) with H m I replaced by K m I (notice that if the function I satisfies (5.9) then condition (7.2) is not affected by replacing H m I by K m I ). Theorem 7.5. Let j ∈ N and let J : (0, 1] → (0, ∞) be a measurable function satisfying (4.1). Suppose that · X(0,1) is a rearrangement-invariant norm.
(a) If is satisfied for all j ∈ N and for all rearrangement-invariant norms · X(0,1) . In the case that This is never fulfilled in the case that The previous theorem combined with Theorem 6.2 easily leads to the following result on m-th order compact Sobolev embeddings on domains from the Maz'ya class J α in the "L ∞ -cases". We note that Theorem 7.5 has to be applied with j = 1 and J(s) = s 1−m(1−α) , s ∈ (0, 1], if α ∈ [ 1 n , 1), and with j = m and J(s) = s, s ∈ (0, 1], if α = 1. Theorem 7.6. Let n ∈ N, n ≥ 2, and let m ∈ N. Suppose that · X(0,1) is a rearrangement-invariant norm.
An analogous result for the product probability space (R n , µ Φ,n ) can be derived from Theorems 6.4 and 7.5 (with J = L Φ and j = m), by making use of the equivalence Theorem 7.7. Let n, m ∈ N, and let Φ be as in Section 3. Suppose that · X(0,1) is a rearrangement-invariant norm.
(a) The condition is satisfied if and only if is never fulfilled.
We shall now study the compact Sobolev embedding (7.1), provided that (Ω, ν) is either a Euclidean Maz'ya domain, or a product probability space, and both X(Ω, ν) and Y (Ω, ν) are Lebesgue spaces. We shall consider also the more general situation when both X(Ω, ν) and Y (Ω, ν) are Lorentz spaces (in the case when (Ω, ν) is the Maz'ya domain), or Lorentz-Zygmund spaces (in the case when (Ω, ν) is the Boltzmann space, a particular example of product probability spaces). We note that Lorentz spaces in the former case and Lorentz-Zygmund spaces in the latter case naturally arise as optimal targets of Lebesgue spaces in the Sobolev embeddings on the corresponding domains, see [9, Theorems 6.9 and 7.12].
The result for Maz'ya classes of domains takes the following form.
(i) The compact embedding holds for every Ω ∈ J α . (ii) The compact embedding holds for every Ω ∈ J α . (iii) One of the following conditions is satisfied: We now focus on compact Sobolev embeddings in context of Lebesgue spaces over product probability spaces. Interestingly, we can often speak about optimal compact embeddings in this connection. Theorem 7.9. Let n, m ∈ N, let Φ be as in Section 3, and let p, q ∈ [1, ∞].
(i) Suppose that lim s→∞ s Φ(s) = 0. Then holds if and only if q ≤ p and q < ∞. In particular, if p < ∞ then L p (R n , µ Φ,n ) is the optimal (i.e., the smallest) Lebesgue space into which V m L p (R n , µ Φ,n ) is compactly embedded. Notice that, according to Remark 7.4, parts (i) and (ii) of Theorem 7.9 indeed cover all cases of the function Φ.
(i) Suppose that p 1 < ∞. Then (7.27) V m L p1,q1;α1 (R n , γ n,β ) → → L p2,q2;α2 (R n , γ n,β ) holds if and only if p 1 > p 2 , or p 1 = p 2 and one of the following conditions is satisfied: (ii) Suppose that p 1 = ∞. Then (7.27) holds if and only if p 2 < ∞, or We finish the paper by proving those results of this section which have not been verified yet. We need the following auxiliary lemma. Lemma 7.11. Suppose that m ∈ N and I : (0, 1] → (0, ∞) is a nondecreasing function fulfilling (3.1). Then for every f ∈ M(0, 1) and a ∈ (0, 1), is satisfied. Since both H m I and K m I have the form H j J for a suitable choice of j and J, we obtain that (i) and (ii) are satisfied as well (and, in particular, that they are equivalent).