Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $\Omega\subset {\mathbb R}^d$. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have been studied in detail already, also concerning their entropy and $s$-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has been introduce by Grothendieck in 1955 already. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1\leq r\leq\infty$. We can now extend this to the setting of general vector-valued sequence spaces of type $\ell_q(\beta_j \ell_p^{M_j})$ with $1\leq p,q\leq\infty$, $M_j\in {\mathbb N}_0$ and weight sequences with $\beta_j>0$. In particular, we prove a criterion for the embedding $id_\beta : \ell_{q_1}(\beta_j \ell_{p_1}^{M_j}) \hookrightarrow \ell_{q_2}(\ell_{p_2}^{M_j})$ to be nuclear.


Introduction
Grothendieck introduced the concept of nuclearity in [10] more than 60 years ago. It provided the basis for many famous developments in functional analysis afterwards, for instance Enflo used nuclearity in his famous solution [9] of the approximation problem, a long-standing problem of Banach from the Scottish Book. We refer to [19,21], and, in particular, to [22] for further historic details.
Let X, Y be Banach spaces, T ∈ L(X, Y ) a linear and bounded operator. Then T is called nuclear, denoted by T ∈ N (X, Y ), if there exist elements a j ∈ X ′ , the dual space of X, and y j ∈ Y , j ∈ N, such that ∞ j=1 a j X ′ y j Y < ∞ and a nuclear representation T x = ∞ j=1 a j (x)y j for any x ∈ X. Together with the nuclear norm where the infimum is taken over all nuclear representations of T , the space N (X, Y ) becomes a Banach space. It is obvious that nuclear operators are, in particular, compact.
Already in the early years there was a strong interest to study examples of nuclear operators beyond diagonal operators in ℓ p sequence spaces, where a complete answer was obtained in [25]. Concentrating on embedding operators in spaces of Sobolev type, first results can be found, for instance, in [23], [18], [17]. We noticed an increased interest in studies of nuclearity in the last years. Dealing with the Sobolev embedding for spaces on a bounded domain, some of the recent papers we have in mind are [4,5,7,8,31] using quite different techniques however.
There might be several reasons for this. For example, the problem to describe a compact operator outside the Hilbert space setting is a partly open and very important one. It is well known from the remarkable Enflo result [9] that there are compact operators between Banach spaces which cannot be approximated by finite-rank operators. This led to a number of -meanwhile well-established and famous -methods to circumvent this difficulty and find alternative ways to 'measure' the compactness or 'degree' of compactness of an operator. It can be described by the asymptotic behaviour of its approximation or entropy numbers, which are basic tools for many different problems nowadays, e.g. eigenvalue distribution of compact operators in Banach spaces, optimal approximation of Sobolev-type embeddings, but also for numerical questions. In all these problems, the decomposition of a given compact operator into a series is an essential proof technique. It turns out that in many of the recent contributions [4,5,31] studying nuclearity, a key tool in the arguments are new decomposition techniques as well, adapted to the different spaces. Inspired by the nice paper [4] we also used such arguments in our paper [11], and intend to follow this strategy here again.
We have two goals in this paper: we want to inquire into the nature of compact embeddings when the function spaces of Besov and Triebel-Lizorkin type are defined on certain unbounded domains. It is well known, that such function spaces defined on R d never admit a compact, let alone nuclear embedding. But replacing R d by a bounded Lipschitz domain Ω ⊂ R d , then the question of nuclearity has already been solved, cf. [18] (with a forerunner in [23]) for the sufficient conditions, and the recent paper [31] with some forerunner in [18] and partial results in [7,8] for the necessity of the conditions. In [11] we also contributed a little to these questions. More precisely, for Besov spaces on bounded Lipschitz domains, B s p,q (Ω), it is well known that id B Ω : B s1 p1,q1 (Ω) ֒→ B s2 p2,q2 (Ω) is nuclear if, and only if, where 1 ≤ p i , q i ≤ ∞, s i ∈ R, i = 1, 2. So a natural question appears whether for some unbounded domains compactness, or now nuclearity, of the corresponding embedding is possible. This question has already been answered in the affirmative, concerning special, so-called quasi-bounded domains, and its compactness. In [6,15,16] the above-mentioned 'degree' of this compactness has been characterised in terms of the asymptotic behaviour of s-numbers and entropy numbers of id B Ω . Now we study its nuclearity and find almost complete characterisations below, depending on some number b(Ω) which describes the size of such an unbounded domain; we refer for all definitions and details to Section 3. We find, in Theorem 3.16 below, that for domains Ω with b(Ω) = ∞, there is a nuclear embedding if, and only if, we are in the extremal situation p 1 = 1, p 2 = ∞, and s 1 − s 2 > d, while in the case b(Ω) < ∞ the conditions look more similar to the above ones for bounded domains; e.g., for such quasi-bounded domains id B Ω is nuclear, if where 1 ≤ p i , q i ≤ ∞, i = 1, 2, s 1 > s 2 , and the new quantity t(p 1 , p 2 ) is defined by . For bounded Lipschitz domains Ω one has b(Ω) = d and then the abovementioned previous result is recovered. As already indicated, we follow here the general ideas presented in [4] which use decomposition techniques and benefit from Tong's result [25] about nuclear diagonal operators acting in sequence spaces of type ℓ p . In our situation, however, it turned out that we need more general vector-valued sequence spaces of type ℓ q (β j ℓ Mj p ), cf. Definition 1.1 below. So we were led to the study of the embedding in view of its nuclearity. Concerning its compactness this has already been investigated in [13,14], but we did not find the corresponding nuclearity result in the literature and decided to study this question first. We tried to follow Tong's ideas in [25] as much as possible and could finally prove that for 1 ≤ p i , q i ≤ ∞, i = 1, 2, id β is nuclear if, and only if, This is the first main outcome of our paper and, to the best of our knowledge, also the first result in this direction. In case of M j ≡ 1 this coincides with Tong's result [25]. The paper is organised as follows. In Section 1 we recall basic facts about the sequence and function spaces we shall work with, Section 2 is devoted to the question of nuclear embeddings in general vector-valued sequence spaces, with our first main outcome in Theorem 2.9. In Section 3 we return to the setting of function spaces, now appropriately extended to function spaces on certain unbounded, so-called quasi-bounded domains. We are able to prove an almost complete result for the corresponding Besov spaces on quasi-bounded domains in Theorem 3.16, using appropriate wavelet decompositions and our findings in Section 2. Finally this also leads to a corresponding result for Triebel-Lizorkin spaces on quasi-bounded domains.

Sequence and function spaces
We start to define the sequence spaces and function spaces we are interested in. First of all we need to fix some notation. By N we denote the set of natural numbers, by N 0 the set N∪{0}, and by Z d the set of all lattice points in R d having integer components.
Let a + = max(a, 0), a ∈ R. For two positive real sequences (a k ) k∈N0 and (b k ) k∈N0 we mean by a k ∼ b k that there exist constants c 1 , c 2 > 0 such that c 1 a k ≤ b k ≤ c 2 a k for all k ∈ N 0 ; similarly for positive functions.
Given two (quasi-) Banach spaces X and Y , we write X ֒→ Y if X ⊂ Y and the natural embedding of X in Y is continuous.
All unimportant positive constants will be denoted by c, occasionally with subscripts. For convenience, let both dx and | · | stand for the (d-dimensional) Lebesgue measure in the sequel.

Sequence spaces
We begin with the definition of weighted vector-valued sequence spaces. We consider a general weight sequence (β j ) ∞ j=0 of positive real numbers and and a sequence (M j ) ∞ j=0 of positive integers that are dimensions of finite-dimensional vector spaces. Definition 1.1. Let 0 < p, q ≤ ∞, (β j ) j∈N0 be a weight sequence, that is β j > 0, and (M j ) j∈N0 be a sequence of natural numbers. Then with the usual modifications if p = ∞ and/or q = ∞.
We recall necessary and sufficient conditions for the compactness of an embedding of the sequence spaces. Let us introduce the following notation, which will be important for us in the sequel: for 0 < p i , q i ≤ ∞, i = 1, 2, we define (with the understanding that p * = ∞ when p 1 ≤ p 2 , q * = ∞ when q 1 ≤ q 2 ). Recall that c 0 denotes the subspace of ℓ ∞ containing the null sequences.
Remark 1.3. We are especially interested in properties of the embedding It was proved in [13,14] that the embedding is compact if, and only if, where for q * = ∞ the space ℓ ∞ has to be replaced by c 0 . In Section 2 below we study its nuclearity.

Function spaces and compact Sobolev embeddings
We now consider some function spaces. Let 0 < p ≤ ∞. Then the Lebesgue space L p (R d ) contains all measurable functions such that is finite, where for p = ∞ this is the classical Lebesgue space of measurable, essentially bounded functions, L ∞ (R d ).
The Schwartz space S(R d ) and its dual S ′ (R d ) of all complex-valued tempered distributions have their usual meaning here. Let ϕ 0 = ϕ ∈ S(R d ) be such that and for each j ∈ N let ϕ j (x) = ϕ(2 −j x) − ϕ(2 −j+1 x). Then (ϕ j ) ∞ j=0 forms a smooth dyadic resolution of unity. Given any f ∈ S ′ (R d ), we denote by F f and F −1 f its Fourier transform and its inverse Fourier transform, respectively. Definition 1.4. Let 0 < p, q ≤ ∞, s ∈ R and (ϕ j ) j a smooth dyadic resolution of unity.
is finite.
is finite.
Remark 1.5. The spaces B s p,q (R d ) and F s p,q (R d ) are independent of the particular choice of the smooth dyadic resolution of unity (ϕ j ) j appearing in their definitions. They are quasi-Banach spaces (Banach spaces for p, q ≥ 1), and S(R d ) ֒→ B s p,q (R d ) ֒→ S ′ (R d ), similarly for the Fcase, where the first embedding is dense if p, q < ∞; we refer, in particular, to the series of monographs by Triebel [26][27][28][29] for a comprehensive treatment of these spaces. Concerning (classical) Sobolev spaces W k p (R d ) built upon L p (R d ) in the usual way, it holds Convention. We adopt the nowadays usual custom to write A s p,q instead of B s p,q or F s p,q , respectively, when both scales of spaces are meant simultaneously in some context (but always with the understanding of the same choice within one and the same embedding, if not otherwise stated explicitly). Remark 1.6. Occasionally we use the following elementary embeddings.
It is well-known that embeddings of type can never be compact, so in the sequel we turn our attention to embeddings of function spaces on domains which admit compact -and even nuclear -embeddings.

Function spaces on domains
Let Ω be an open set in R d such that Ω = R d . Such a set will be called an arbitrary domain. We denote the collection of all distributions on Ω by D ′ (Ω). As usual the spaces B s p,q (Ω) and F s p,q (Ω) are defined on Ω by restriction, i.e., using our above convention, equipped with the quotient norm, as usual, Consequently we have counterparts of the embeddings mentioned in Remark 1.6.
In the classical setting of bounded Lipschitz domains the compactness of Sobolev embeddings is well known and the following proposition holds, cf. [29,Proposition 4.6].
is compact, if, and only if, In this paper we shall essentially work with a more general class of domains, so-called quasi-bounded domains, that still guarantee the compactness of Sobolev embeddings of the above type. We consider this subject in detail in Section 3 below. Remark 1.8. Note that for β j = 2 js and M j ∼ 2 jd , with d ∈ N, the above sequence space One can prove it using the wavelet decomposition, cf. [30,Sections 4.2.4,4.2.5]. In Section 3 we study the more general setting of quasi-bounded domains which require the more general approach of sequence spaces as introduced above.

Nuclear embeddings in general vector-valued sequence spaces
Our first main goal in this paper is to study nuclear embeddings between sequence spaces of the type ℓ q (β j ℓ Mj p ) introduced above. So we first recall some fundamentals of the concept and important results we rely on in the sequel.

Basic facts concerning nuclearity
Let X, Y be Banach spaces, T ∈ L(X, Y ) a linear and bounded operator. We have already recalled the notion of nuclearity in the Introduction. This concept has been introduced by Grothendieck [10] and was intensively studied afterwards, cf. [19][20][21] and also [22] for some history. There exist extensions of the concept to r-nuclear operators, 0 < r < ∞, where r = 1 refers to the nuclearity. It is well-known that the class of nuclear operators possesses the ideal property. In Hilbert spaces H 1 , H 2 , the nuclear operators N (H 1 , H 2 ) coincide with the trace class S 1 (H 1 , H 2 ), consisting of those T with singular numbers (s n (T )) n ∈ ℓ 1 .
In the next proposition we recall well-known properties of nuclear operators needed in the sequel, cf. [12, Chapter 1].
(ii) For any Banach space X and any bounded linear operator T : ℓ n ∞ → X we have is a nuclear operator and S ∈ L(X 0 , X) and R ∈ L(Y, Y 0 ), then RT S is a nuclear operator and Already in the early years there was a strong interest to find natural examples of nuclear operators beyond diagonal operators in ℓ p spaces, for which a complete answer was obtained in [25]. Let τ = (τ j ) j∈N be a scalar sequence and denote by D τ the corresponding diagonal operator, D τ : x = (x j ) j → (τ j x j ) j , acting between ℓ p spaces. Let us introduce the following notation: for r 1 , r 2 ∈ [1, ∞], let t(r 1 , r 2 ) be given by (2.1) We heavily rely in our arguments below on the following remarkable result by Tong [25].

Nuclearity results for general vector-valued sequence spaces
Our aim now is to prove some 'nuclear' counterpart of the compactness result recalled in Remark 1.3. Roughly speaking, this will read as follows. Assume that 1 ≤ p i , q i ≤ ∞, i = 1, 2. Then id β given by (1.2) is nuclear if, and only if, the compactness condition (1.3) is replaced by where for t(q 1 , q 2 ) = ∞ the space ℓ ∞ has to be replaced by c 0 . But we need some preparation to prove (an even more general version of) this result and postpone it as Theorem 2.9 below, together with some further discussion. In our argument below we rely on the approach in Tong's paper [25] and adapt and extend it appropriately.
We consider the finite-dimensional version of the spaces introduced in Definition 1.1. For appropriately modified for p = ∞ and/or q = ∞. Following the ideas of the proof in [25], we are interested in embeddings of these spaces or, equivalently, in actions of diagonal operators on the spaces with β j ≡ 1. More generally we will work with operators acting on spaces given by matrices. Therefore it will be convenient to rewrite the above definition in the following way. Let M = (M j ) j∈N0 be a sequence of natural numbers and let n ∈ N 0 . Denote by appropriately modified for p = ∞ and/or q = ∞. Then l q (l Mj p ) is a vector space isometrically isomorphic to ℓ q (ℓ Mj p ) and l n q (l Mj p ) is a finite-dimensional vector space isometrically isomorphic to ℓ n q (ℓ Mj p ), with dim l n q (l Mj p ) = N .
where p * and q * are given by (1.1).
Proof. The upper estimate of D λ = D λ : l n q1 (l Mj p1 ) → l n q2 (l Mj p2 ) , that is, D λ ≤ λ|l n q * (l Mj p * ) , follows easily from Hölder's inequality if p 2 < p 1 or q 2 < q 1 , and from the monotonicity of the ℓ r norms otherwise. To prove the opposite inequality one can find sequences that correspond to the equalities in the Hölder inequalities. For details we refer to [3] where a similar, even more general statement is proved. Let A be an N × N complex matrix. Let D(A) denote the diagonal part of A, i.e., a matrix that derives from A by replacing all off-diagonal entries of A by zeros. Analogously, if T is a linear operator on C N given by the matrix A, then D(T ) will denote the operator given by the matrix D(A).
Proof. Let A be the matrix of T . Let ω ∈ C and let U (ω) be the diagonal matrix with entries 1, ω, ω 2 , . . . , ω N −1 down its diagonal. If |ω| = 1, then the matrix U (ω) generates an operator in C N that is an isometry in the norm of l n q * (l Mj p * ) for all p, q. We take ω = e 2πi/N . Using the identity N −1 j=0 ω j = 0 and elementary calculations one concludes, cf. [2], that The operators U (ω) j and U (ω) j are isometries therefore the last formula implies where t(p 1 , p 2 ) and t(q 1 , q 2 ) are given by (2.1). Proof.
Step 1. Both spaces l n q1 (l Mj p1 ) and l n q2 (l Mj p2 ) are finite-dimensional, therefore it is sufficient to consider the finite representations of any nuclear operator T , i.e., x ′ ℓ ∈ (l n q1 (l Mj p1 )) ′ and y ℓ ∈ l n q2 (l Mj p2 ), (2.8) cf. [12, p.19]. First we show that the space N l n q1 (l Mj p1 ), l n q2 (l Mj p2 ) ′ , that is, the dual space to N l n q1 (l Mj p1 ), l n q2 (l Mj p2 ) , is isometrically isomorphic to L l n q2 (l Mj p2 ), l n q1 (l Mj p1 ) . This can be proved via trace-duality. For any operator S ∈ L l n q2 (l Mj p2 ), l n q1 (l Mj p1 ) the composition operator ST is again a nuclear operator which belongs to N l n q1 (l Mj p1 ), l n q1 (l Mj p1 ) , and )) ′ and Sy ℓ ∈ l n q1 (l Mj p1 ). Then defines a linear functional on N l n q1 (l Mj p1 ), l n q2 (l Mj p2 ) . This is well-defined as the trace does not depend on the particular representation of ST and T , respectively. Moreover, due to the ideal property of nuclear operators, recall Proposition 2.1(iii), we have The reverse inequality can be proved by applying the functional ϕ S to rank-one operators T = x ′ ⊗ y with x ′ = y = 1. Thus the mapping is an isometry. It should be clear that both spaces have the same dimension and so the mapping is an isometric isomorphism. This implies that such that, in particular, Step 2. Let S ∈ L l n q2 (l Mj p2 ), l n q1 (l Mj p1 ) with S ≤ 1. We denote its diagonal part by D(S) : l n q2 (l Mj p2 ) → l n q1 (l Mj p1 ), represented by the entries b = (b ℓ ) N ℓ=1 . Lemma 2.5 shows that the subspace of diagonal operators in L l n q2 (l Mj p2 ), l n q1 (l Mj p1 ) is isometrically isomorphic to l ñ q * (l Mj p * ), and Note that Consequently, Since tr(SD) = tr(D(S)D) and D(S) ≤ S by Lemma 2.6, we get for any such operator S with S ≤ 1 by Hölder's inequality that p2) ) .
Thus the first step implies that Conversely, we may restrict ourselves to diagonal operators S = D(S) in L l n q2 (l Mj p2 ), l n q1 (l Mj p1 ) and benefit from the sharpness of the Hölder inequality to obtain the estimate from below, p2) ) .
Proof. We apply Proposition 2.8. The space l q (l Mj p ) is isometrically isomorphic to ℓ q (ℓ Mj p ). So the embedding id β corresponds to a diagonal operator D λ from l q1 (l Mj p1 ) to l q2 (l Mj p2 ) with λ l = β −1 j if l ∈ I j and this gives |ℓ t(q1,q2) .
Remark 2.12. We would like to give an alternative argument inspired by the proof in [4] which works at least in some special cases. The idea is to apply Tong's result [25] and some factorisation. We sketch it for the case q 1 ≤ p 1 ≤ p 2 ≤ q 2 . Note that in this situation We decompose We estimate D 1 . By our assumption p 1 ≥ q 1 Hölder's inequality leads for x = (x j,m ) j,m ∈ ℓ q1 (ℓ Mj p1 ) to hence D 1 ≤ 1. Likewise, since p 2 ≤ q 2 , by another application of Hölder's inequality, for any x = (x j,m ) j,m ∈ ℓ q2 (ℓ Mj q2 ). Thus D 2 ≤ 1 and (2.15) implies ν(D β ) ≤ ν(D 0 ). We would like to use Proposition 2.2(ii) and have to identify ℓ qr (ℓ Mj qr ) therefore with ℓ qr , r = 1, 2. This can be seen by a bijection like recall M −1 = 0, i.e., k(0, m) = m. Using our previous notation α j = j−1 l=0 M l , j ∈ N 0 , with α 0 = 0, we get k(j, m) = α j + m, α j+1 − α j = M j . For the rewritten sequence (x j,m ) j∈N0,m=1,...Mj = (y k ) k∈N , k = k(j, m), let D 0 denote the corresponding diagonal operator, acting now as m ) j,m = (y k ) k = y in the above identification. More precisely, j for the moment, then γ j x j,m = γ k y k when k = k(j, m). Consequently, by Prop. 2.2(ii), if q 2 < ∞. If q 2 = ∞ and q 1 = 1, then this has to be replaced by

Nuclear embeddings of function spaces on domains
Our second main goal in this paper is to study the nuclearity of Sobolev embeddings acting between function spaces on domains. We briefly recall what is known for bounded Lipschitz domains, and concentrate on quasi-bounded domains afterwards.

Embeddings of function spaces on bounded domains
In Proposition 1.7 we have already recalled the criterion for the compactness of the embedding Recently Triebel proved in [31] the following counterpart for its nuclearity.
Remark 3.2. The proposition is stated in [31] for the B-case only, but due to the independence of (3.1) of the fine parameters q i , i = 1, 2, and in view of (the corresponding counterpart of) (1.9) it can be extended immediately to F -spaces. The if-part of the above result is essentially covered by [18] (with a forerunner in [23]). Also part of the necessity of (3.1) for the nuclearity of id Ω was proved by Pietsch in [18] such that only the limiting case ) + was open for many decades. Only recently Edmunds, Gurka and Lang in [7] (with a forerunner in [8]) obtained some answer in the limiting case which was then completely solved in [31]. Note that in [18] some endpoint cases (with p i , q i ∈ {1, ∞}) have already been discussed for embeddings of Sobolev and certain Besov spaces (with p = q) into Lebesgue spaces. In our recent paper [11] we were able to further extend Proposition 3.1 in view of the borderline cases.
For better comparison one can reformulate the compactness and nuclearity characterisations of id Ω in (1.11) and (3.1) as follows, involving the number t(p 1 , p 2 ) defined in (2.1). Let Before we can formulate the properties of embeddings of function spaces defined on quasibounded domains, we first need to extend the notion of A s p,q (Ω) as given in Section 1.2. Here we follow the ideas of Triebel in [30] and define spacesF s p,q (Ω) andB s p,q (Ω). We put

Definition 3.4.
Let Ω be an arbitrary domain in R d with Ω = R d and let with p < ∞ for the F -spaces.
equipped with the quotient norm, where the infimum is taken over all g ∈ A s p,q (Ω) with f = g| Ω .
Next we make use of some quantities describing the quasi-boundedness of the domain. For that reason we introduced in [15] a box packing number b(Ω) of an open set Ω. We recall the definition here.
Let Q j,m denote the dyadic cube in R d with side-length 2 −j , j ∈ N 0 , given by Let Ω ⊂ R d be a non-empty open set with Ω = R d . For j ∈ N 0 we denote by b j (Ω) = sup k ∈ N : If there is no dyadic cube of size 2 −j contained in Ω we put b j (Ω) = 0. The following properties of the sequence b j (Ω) j∈N0 are obvious: (i) There exists a constant j 0 = j 0 (Ω) ∈ N 0 such that for any j ≥ j 0 we have To illustrate this definition we give simple examples of quasi-bounded domains.
Remark 3.7. (i) One can construct a quasi-bounded domain in R d with prescribed sequence b j (Ω), cf. [15]. Some more concrete examples based on this general construction can be found in [16,Section 3].
(ii) Another characterisation of b(Ω) was proved in [16]. For any domain Ω = R d and any r > 0 we put Ω r = {x ∈ Ω : dist(x, ∂Ω) > r}. (3.5) If the domain Ω is quasi-bounded, then |Ω r | < ∞ for any r > 0 and To give the wavelet characterisation of the spacesB s p,q (Ω) andF s p,q (Ω) we need some additional assumptions concerning the underlying domain Ω. Namely we should assume that the domain is E-thick (exterior thick) and E-porous, cf. [30,Chapter 3]. Now we recall the definition starting with porosity. (ii) A closed set Γ ⊂ R d is said to be uniformly porous if it is porous and there is a locally finite positive Radon measure µ on R d such that Γ = supp µ and µ(B(γ, r)) ∼ h(r) , with γ ∈ Γ, 0 < r < 1 , where h : [0, 1] → R is a continuous strictly increasing function with h(0) = 0 and h(1) = 1 (the equivalence constants are independent of γ and r).
Naturally 0 ≤ α ≤ d. Any α-set with α < d is uniformly porous. (i) The domain Ω is said to be E-thick if one can find for any interior cube Q i ⊂ Ω with Q i and Q e denote cubes in R d with sides parallel to the axes of coordinates. Moreover ℓ(Q) denotes the side-length of the cube Q.
(ii) The domain Ω is said to be E-porous if there is a number η with 0 < η < 1 such that one finds for any ball B(γ, r) ⊂ R d centred at γ ∈ Γ and of radius r with 0 < r < 1, a ball B(y, ηr) with B(y, ηr) ⊂ B(γ, r) and B(y, ηr) ∩ Ω = ∅ .

(iii)
The domain Ω is called uniformly E-porous if it is E-porous and Γ is uniformly porous.
Remark 3.11. We collect some observations.
(i) If Ω is E-porous, then Ω is E-thick and |Γ| = 0. On the other hand, if Ω is E-thick and Γ is an α-set, then Ω is uniformly E-porous and d − 1 ≤ α < d.
(ii) There are quasi-bounded domains that are not E-porous or even not E-thick, cf. e.g. [1], page 176 for the example of a quasi-bounded domain with empty exterior.
(iii) The domains given in Example 3.6 and pointed out in Remark 3.7 are not only quasibounded but also uniformly E-porous.
If the domain is uniformly E-porous, then one can characteriseĀ s p,q (Ω) spaces in terms of the wavelet expansion of the distributions. Now we give the wavelet characterisation of the spacesF s p,q (Ω) andB s p,q (Ω). Let N j ∈ N, j ∈ N 0 and N = N ∪ {∞}. (ii) If the domain Ω is not quasi-bounded, then the embedding (3.11) is never compact, cf. [15].
(iii) If Ω is a domain in R d with finite Lebesgue measure, then the embedding (3.11) is compact if, and only if, cf. [15]. Thus for a set of finite Lebesgue measure we get the same conditions for compactness as for bounded smooth domains, recall Proposition 1.7.
(iv) On the other hand, if the domain is not quasi-bounded, then the Sobolev embeddings are never compact. So the most interesting case are the quasi-bounded domains with infinite measure.
If Ω is such a domain, then all numbers b j (Ω) are finite. But in contrast to the domain with a finite measure, the numbers b j (Ω) are not asymptotically equivalent to 2 jd .
(v) One can use the number b(Ω) to describe quantitative properties of corresponding compact embeddings in terms of the asymptotic behaviour of their s-numbers and entropy numbers, cf. [6,15,16]. . (3.14) Conversely, if the embedding (3.13) is nuclear and t(p 1 , p 2 ) = ∞, that is, p 1 = 1 and p 2 = ∞, then s 1 − s 2 − d( 1 p1 − 1 p2 ) > 0. If the embedding (3.13) is nuclear and t(p 1 , p 2 ) < ∞, then To verify this, we consider the following diagram: with I 1 and I −1 2 denote the wavelet-isomorphism from Theorem 3.12 and for i = 1, 2 we define • D −1 1 . Then Proposition 2.1(iii) applied to id B Ω and D λ , respectively, implies that the embedding id B Ω is nuclear if, and only if, the operator D λ is nuclear, which -in view of Proposition 2.8 -is the case, if, and only if, the sequence λ with λ j,m = 2 −jδ for j ∈ N 0 , m = 1, . . . , N j , belongs to l t(q1,q2) (l Nj t(p1,p2) ), replaced by c 0 if t(q 1 , q 2 ) = ∞. But by definition of the spaces, in the definition of λ, and finally arrive at (3.15).
In particular, if p 1 = 1, p 2 = ∞, that is, t(p 1 , p 2 ) = ∞, then δ = s 1 − s 2 − d, so id B Ω is nuclear if, and only if, s 1 − s 2 > d, which completes the proof in case of (i) and (ii) in this setting. It remains to deal with the remaining cases for t(p 1 , p 2 ) < ∞ in dependence on b(Ω).
Step 3. Finally we deal with the case b(Ω) = ∞ and know by Step 1, that p 1 = 1, p 2 = ∞, s 1 − s 2 > d is sufficient for the nuclearity of id B Ω . So it remains to show that the condition p 1 = 1 and p 2 = ∞ is also necessary for the nuclearity of id B Ω . Note that the necessity of the condition δ = s 1 − s 2 − d( 1 p1 − 1 p2 ) > 0 follows from Proposition 3.14 since otherwise the embedding is not compact.
If b(Ω) = ∞, then by definition (3.4), for any s > 0 there is an increasing sequence j k and a positive constant c > 0 such that c2 j k s ≤ b j k (Ω). This implies that the condition (3.15) is fulfilled only if δ > 0 and t(p 1 , p 2 ) = ∞, that is, p 1 = 1 and p 2 = ∞.
Remark 3.17. If 0 < lim sup j→∞ b j (Ω)2 −jb(Ω) ≤ ∞ we can replace in Step 2 of the proof s with s < b(Ω) by b(Ω) itself and obtain that in this case (3.14) is a sufficient and necessary condition for nuclearity of the embeddings id B Ω .
Note that for b(Ω) = d these findings coincide with the condition (3.1) from Proposition 3.1. Moreover, the condition (3.14) corresponds to (3.12), when p * is replaced by t(p 1 , p 2 ).
Remark 3.19. Comparing the quite different behaviour in case (i) and (ii) of Theorem 3.16, one may also interpret it in the sense that, when the quasi-bounded domain becomes 'larger' in the sense that b(Ω) → ∞, then to achieve nuclearity in the sense of (ii) one needs to compensate b(Ω) on the right-hand side of (3.14) by a larger number t(p 1 , p 2 ), too, which in the end means t(p 1 , p 2 ) = ∞, hence only p 1 = 1, p 2 = ∞. This represents the 'smallest' source space and 'largest' target space (with an appropriate interpretation in the context of Sobolev embeddings) which is possible.
We finally deal with the F -case and observe some new phenomenon. Recall that for the compactness result in Proposition 3.14 as well as for the compactness and nuclearity results for spaces on bounded Lipschitz domains, Propositions 1.7 and 3.1, respectively, no difference between Band F -spaces appeared. This is now different to some extent.