Nuclear embeddings in weighted function spaces

We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results [17,19] where we studied the compactness of corresponding embeddings. The concept of nuclearity goes back to Grothendieck who defined it in [14]. Recently there is a refreshed interest to study such questions [5-8,49]. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong [43] about the nuclearity of diagonal operators acting in $\ell_p$ spaces. In that way we can further contribute to the characterisation of nuclear embeddings on domains obtained in [5,33,34,49].


Introduction
Grothendieck introduced the concept of nuclearity in [14] more than 60 years ago. It paved the way to many famous developments in functional analysis later one, like the theories of nuclear locally convex spaces, operator ideals, eigenvalue distributions, and traces and determinants in Banach spaces. Enflo used nuclearity in his famous solution [10] of the approximation problem, a long-standing problem of Banach from the Scottish Book. We refer to [29,31], and, in particular, to [32] 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 p sequence spaces, where a complete answer was obtained in [43] (with some partial forerunner in [29]). Concentrating on embedding operators in spaces of Sobolev type, first results can be found, for instance, in [28,33,34].
Though the topic was always studied to a certain extent, we realised an increased interest in the last years. Concentrating on the Sobolev embedding for spaces on a bounded domain, some of the recently published papers we have in mind are [5][6][7][8]49] using quite different techniques however.
We observed several directions and 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 [10] 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 papers [5,6,49] studying nuclearity, a key tool in the arguments are new decomposition techniques as well, adapted to the different spaces. So we intend to follow this strategy, too.
Concerning weighted spaces of Besov and Sobolev type, we are in some sense devoted to the program proposed by Edmunds and Triebel [9] to investigate the spectral properties of certain pseudo-differential operators based on the asymptotic behaviour of entropy and approximation numbers, together with Carl's inequality and the Birman-Schwinger principle. Similar questions in the context of weighted function spaces of this type were studied by the first named author and Triebel, cf. [15], and were continued and extended by Kühn, Leopold, Sickel and the second author in the series of papers [21][22][23]. Here the considered weights are always assumed to be 'admissible': These are smooth weights with no singular points, with w(x) = (1 + |x| 2 ) γ/2 , γ ∈ R, x ∈ R d , as a prominent example.
We started in [17] a different approach and considered weights from the Muckenhoupt class A ∞ which -unlike 'admissible' weights -may have local singularities, that can influence embedding properties of such function spaces. Weighted Besov and Triebel-Lizorkin spaces with Muckenhoupt weights are well known concepts, cf. [1-4, 12, 16, 35]. In [17] we dealt with general transformation methods from function to appropriate sequence spaces provided by a wavelet decomposition; we essentially concentrated on the example weight of purely polynomial growth both near the origin and for |x| → ∞. In the general setting for w ∈ A ∞ we obtained sharp criteria for the compactness of embeddings of type id α,β : A s1 p1,q1 (R d , w α,β ) → A s2 p2,q2 (R d ), where s 2 ≤ s 1 , 0 < p 1 , p 2 < ∞, 0 < q 1 , q 2 ≤ ∞, and A s p,q stands for either Besov spaces B s p,q or Triebel-Lizorkin spaces F s p,q . More precisely, we proved in [17] that id α,β is compact if, and only if, In the same paper [17] we determined the exact asymptotic behaviour of corresponding entropy and approximation numbers of id α,β in the compactness case. Now we can refine this characterisation by our new result about the nuclearity of id α,β . One of our main results in the present paper, Theorem 3.12 below, states that id α,β is nuclear if, and only if, where 1 ≤ p 1 < ∞ and 1 ≤ p 2 , q 1 , q 2 ≤ ∞. In [19] we studied the weight Again we obtained the complete characterisation of the compactness of as well as asymptotic results for the corresponding entropy numbers. The intention was not only to generalise the weight function, but also to cover some limiting cases in that way. Our second main result, Theorem 3.22 below, completely answers the question of the nuclearity of id (α,β) , where now even the fine parameters q 1 , q 2 are involved in the criterion. While proving our result we benefit from Tong's observation [43] (and the fine paper [5] which has drawn our attention to it), and the available wavelet decomposition and operator ideal techniques used in our previous papers [17,19] already. Moreover, we used and slightly extended Triebel's result [49] (with forerunners in [33,34]) on the nuclearity of the embedding operator id where Ω ⊂ R d is assumed to be a bounded Lipschitz domain and the spaces A s p,q (Ω) are defined by restriction. In [5] some further limiting cases were studied and we may now add a little more to this limiting question.
Beside embeddings of appropriately weighted spaces and embeddings of spaces on bounded domains, we also consider embeddings of radial spaces which may admit compactness, , for definitions we refer to Section 3.3 below. This has been studied in detail in [38,39]. In particular, we can now gain from the close connection between radial spaces and appropriately weighted spaces established in [38,39]. In that way we are able to prove a criterion of nuclearity of the embedding id R in Theorem 3.27 below.
The paper is organised as follows. In Section 2 we recall basic facts about weight classes and weighted function spaces needed later on. Section 3 is devoted to our main findings about the nuclearity of embeddings: we start with a collection of known results in Section 3.1 which we shall need later; in Section 3.2 we present our new results for the weighted embeddings described above, while in Section 3.3 we turn our attention to radial spaces and nuclearity of embeddings.

Weighted function spaces
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.
The positive part of a real function f is given by f + (x) = max(f (x), 0). For two positive real sequences {a k } k∈N and {b k } k∈N 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; 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.

Weight functions
We shall essentially deal with weight functions of polynomial type. Here we use our preceding results in [17][18][19] which partly rely on general features of Muckenhoupt weights. For that reason we first recall some fundamentals on this special class of weights. By a weight w we shall always mean a locally integrable function w ∈ L loc 1 (R d ), positive a.e. in the sequel. Let M stand for the Hardy-Littlewood maximal operator given by where p is the dual exponent to p given by 1/p + 1/p = 1 and |B| stands for the Lebesgue measure of the ball B.
(ii) Let p = 1. Then w belongs to the Muckenhoupt class A 1 if there exists a constant 0 < A < ∞ such that the inequality holds for almost all x ∈ R d .
Since the pioneering work of Muckenhoupt [25][26][27], these classes of weight functions have been studied in great detail, we refer, in particular, to the monographs [13], [42], [44,Ch. IX], and [41, Ch. V] for a complete account on the theory of Muckenhoupt weights. As usual, we use the abbreviation where Ω ⊂ R d is some bounded, measurable set.
We need some refined study of the singularity behaviour of Muckenhoupt A ∞ weights. Let for m ∈ Z d and j ∈ N 0 , Q j,m denote the d-dimensional cube with sides parallel to the axes of coordinates, centered at 2 −j m and with side length 2 −j . In [18] we introduced the following notion of their set of singularities S sing (w).

Definition 2.3.
For w ∈ A ∞ we define the set of singularities S sing (w) by Recall the following result. Remark 2.5. S sing (w) is a special case of S sing (w 1 , w 2 ) defined in [18] with w 2 ≡ 1, w 1 ≡ w. There we also proved some forerunner of Proposition 2.4. Let us explicitly recall a very useful consequence of the above result, cf. [20,Cor. 2.7]. We call a cube (or ball) Q ⊂ R d regularity cube (or regularity ball) of a given weight w, if the weight is regular there, that is, if there exist positive constants c 1 , c 2 such that for all x ∈ Q it holds c 1 ≤ w(x) ≤ c 2 , i.e., w ∼ 1 on Q.
Hence the above proposition implies that for any w ∈ A ∞ any cube or ball Q ⊂ R d contains a regularity cube or ball Q ⊂ Q.
Remark 2.6. In [15] we studied so-called 'admissible' weights. These are smooth weights with no singular points. One can take as a prominent example. For the precise definition we refer to [15] and the references given therein.

Weighted function spaces of type
Let w ∈ A ∞ be a Muckenhoupt weight, and 0 < p < ∞. Then the weighted Lebesgue space is finite. Note that for p = ∞ one obtains the classical (unweighted) Lebesgue space, Thus we mainly restrict ourselves to p < ∞ in what follows.
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 supp ϕ ⊂ y ∈ R d : |y| < 2 and ϕ(x) = 1 if |x| ≤ 1 , 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 Ff and F −1 f its Fourier transform and its inverse Fourier transform, respectively. Definition 2.7. Let 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and {ϕ j } j a smooth dyadic resolution of unity. Assume w ∈ A ∞ .
is finite.
is finite.
Remark 2.8. The spaces B s p,q (R d , w) and F s p,q (R d , w) 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 , w) → S (R d ), similarly for the F -case, where the first embedding is dense if q < ∞; cf. [3]. Moreover, for w 0 ≡ 1 ∈ A ∞ we obtain the usual (unweighted) Besov and Triebel-Lizorkin spaces; we refer, in particular, to the series of monographs by Triebel [45][46][47][48] for a comprehensive treatment of the unweighted spaces.
The above spaces with weights of type w ∈ A ∞ have been studied systematically by Bui first in [3,4]. It turned out that many of the results from the unweighted situation have weighted counterparts: e.g., we have F 0 where the latter are Hardy spaces, see [3,Thm. 1.4], and, in particular, In [37] the above class of weights was extended to the class A loc p . We partly rely on our approaches [16][17][18].
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 2.9. Occasionally we use the following embeddings which are natural extensions from the unweighted case. Finally, we briefly describe the wavelet characterisations of Besov spaces with A ∞ weights proved in [17]. Let for m ∈ Z d and j ∈ N 0 the cubes Q j,m be as above. Apart from function spaces with weights we introduce sequence spaces with weights: for certain natural numbers N 1 and N 2 , and ψ an associated wavelet. Then the tensor-product ansatz yields a scaling function φ and associated wavelets ψ 1 , . . . , (2.14) Using the standard abbreviations φ j,m (x) = 2 jd/2 φ(2 j x − m) and ψ i,j,m (x) = 2 jd/2 ψ i (2 j x − m) we proved in [17] the following wavelet decomposition result. . Let 0 < p, q ≤ ∞ and let s ∈ R. Let φ be a scaling function and let ψ i , i = 1, . . . , 2 d − 1, be the corresponding wavelets satisfying (2.14). We assume that |s| < N 1 .
Then a distribution f ∈ S (R d ) belongs to B s p,q (R d , w), if, and only if,

Compact embeddings
We collect some compact embedding results for weighted spaces of the above type that will be used later. For that purpose, let us introduce the following notation: for We restrict ourselves to the situation when only the source space is weighted, and the target space unweighted, where w ∈ A ∞ . The weight we now consider is either w α,β given by (2.4), or w (α,β) given by (2.5) (with the special case w log γ as in (2.7)). Moreover, we shall assume in the sequel that , recall (2.9), and we arrive at the unweighted situation in (2.17) which is well-known already.
We first recall the result for Example 2.2(i).
Remark 2.12. Let us briefly point out the main argument in [17][18][19] concerning compactness assertions as we shall follow a similar idea when dealing with nuclearity below. We rely on a reduction of the function space embeddings to corresponding sequence space embeddings based on the wavelet decomposition Theorem 2.10: we make use of the commutative diagram with appropriate isomorphisms S and T . Similarly, with an appropriate isomorphism A it is sufficient to investigate the embedding of a weighted sequence space into an unweighted one, using This will be our starting point below.

Proposition 2.14
is compact if, and only if, Let the spaces B s p,q (Ω) and F s p,q (Ω) be defined by restriction. It is well known that id is compact, if, and only if,

Nuclear embeddings
Our main goal in this paper is to study nuclear embeddings between the weighted spaces introduced above. So we first recall some fundamentals of the concept and important results we rely on in the sequel.

The concept and recent results
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 any nuclear operator can be approximated by finite rank operators, hence nuclear operators are, in particular, compact.
Remark 3.1. This concept has been introduced by Grothendieck [14] and was intensively studied afterwards, cf. [29][30][31] and also [32] for some history. At that time applications were intended to better understand, for instance, nuclear locally convex spaces, operator ideals, eigenvalues of compact operators in Banach spaces. There exist extensions of the concept to r-nuclear operators, 0 < r < ∞, where r = 1 refers to the nuclearity. It is well-known that N (X, Y ) possesses the ideal property. In Hilbert spaces H 1 , H 2 , the nuclear operators We collect some more or less well-known facts needed in the sequel.
(ii) For any Banach space X and any bounded linear operator T : n ∞ → X we have is a nuclear operator and S ∈ L(X 0 , X) and R ∈ L(Y, Y 0 ), then ST R is a nuclear operator and Already in the early years there was a strong interest to find further examples of nuclear operators beyond diagonal operators in p spaces, where a complete answer was obtained in [43]. 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 p spaces. Let us introduce the following notation: for numbers r 1 , r 2 ∈ [1, ∞], let t(r 1 , r 2 ) be given by with t(r 1 , r 2 ) = r * if, and only if, {r 1 , r 2 } = {1, ∞}.
(ii) Let n ∈ N and D n τ : n r1 → n r2 be the corresponding diagonal operator D n τ : Example 3.4. In the special case of τ ≡ 1, i.e., D τ = id, (i) is not applicable and (ii) reads as In particular, ν(id : n 1 → n ∞ ) = 1.
We return to the situation of compact embeddings of spaces on domains, as described in Remark 2.16. Recently Triebel proved in [49] the following counterpart for its nuclearity.
Remark 3.7. The proposition is stated in [49] for the B-case only, but due to the independence of (3.3) of the fine parameters q i , i = 1, 2, and in view of (the corresponding counterpart of) (2.13) it can be extended immediately to F -spaces. The if-part of the above result is essentially covered by [33] (with a forerunner in [34]). Also part of the necessity of (3.3) for the nuclearity of id Ω was proved by Pietsch in [33] 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 [49]. In [5] the authors dealt with the nuclearity of the embedding B s1,α1 p1,q1 (Ω) → B s2,α2 p2,q2 (Ω) where the indices α i represent some additional logarithmic smoothness. They obtained a characterisation for almost all possible settings of the parameters. Note that in [33] some endpoint cases (with p i , q i ∈ {1, ∞}) were already discussed for embeddings of Sobolev and certain Besov spaces (with p = q) into Lebesgue spaces. We are able to further extend Proposition 3.6 in Corollary 3.17 below.
Remark 3.8. In [6] some further limiting endpoint situations of nuclear embeddings like id : B d p,q (Ω) → L p (log L) a (Ω) are studied. For some weighted results see also [28]. Remark 3.9. For later comparison we may reformulate the compactness and nuclearity characterisations of id Ω in (2.25) and (3.3) as follows, involving the number t(p 1 , p 2 ) defined in (3.1). .
We shall observe similar phenomena in the weighted setting later.

Weighted spaces
We begin with some general implication from Proposition 3.6 for Muckenhoupt weights w ∈ A ∞ . Here we benefit from the regularity result Proposition 2.4, in particular, the observation recalled in Remark 2.5.
is nuclear, then .
Proof. Assume that id w is nuclear and Ω is a regularity ball for w which always exists according to Remark 2.5. Consider now the spaces A s1 p1,q1 (Ω) and A s2 p2,q2 (Ω) defined by restriction (and equipped with the equivalent norm induced by the regularity ball), together with the corresponding linear and bounded extension operator, cf. [36]. Then Proposition 3.2(iii) implies the nuclearity of id Ω : A s1 p1,q1 (Ω) → A s2 p2,q2 (Ω) which leads to (3.4) by Proposition 3.6.
Remark 3.11. Later we can slightly extend the above result and incorporate limiting cases p i , q i ∈ {1, ∞}, see Corollary 3.19 below. Note, that the above result is in general a necessary condition for nuclearity only, as the simple example w ≡ 1 ∈ A ∞ shows: in that case the unweighted embedding id : A s1 p1,q1 (R d ) → A s2 p2,q2 (R d ) is known to be never compact (let alone nuclear), no matter what the other parameters s i , p i , q i are.
We return to the weight function w α,β in Example 2.2(i) and give the counterpart of Proposition 2.11. Theorem 3.12. Let α > −d, β > −d, w α,β be given by (2.4). Assume that 1 ≤ p 1 < ∞, 1 ≤ p 2 ≤ ∞, and 1 ≤ q i ≤ ∞, s i ∈ R, i = 1, 2. Then the embedding id α,β : A s1 p1,q1 (R d , w α,β ) → A s2 p2,q2 (R d ) is nuclear if, and only if, Remark 3.13. Note that dealing with the weighted setting we have the same phenomenon in Theorem 3.12 compared with the compactness result Proposition 2.11, as described in Remark 3.9 for the situation of spaces on bounded domains: the stronger nuclearity condition (3.5) is exactly achieved when p * is replaced by t(p 1 , p 2 ).
Proof. First note that in view of (2.13) and the independence of (3.5) from the fine parameters q i , i = 1, 2, together with Proposition 3.2(iii), it is sufficient to consider the case A = B, i.e., the Besov spaces.
Step 1. We first deal with the sufficiency of (3.5) for the nuclearity. We return to Remark 2.12 where we explained our general strategy. Thus, to show the nuclearity of id α,β it is equivalent to proving the nuclearity of which is obviously equivalent to the nuclearity of which in view of σ 1 − σ 2 = δ can be written as Note that (3.7) We define the projection such that (in a slight abuse of notation) we can understand id 1 as We split Now we study the nuclearity of id 1 and id 2 . We further decompose id 1 into where P j is the projection onto 2 jd p1 (|m| α ), hence id j : 2 jd p1 (|m| α ) → p2 is the embedding on level j, and Q j is the embedding of p2 into q2 ( p2 ) with Q j : p2 → q2 ( p2 ) = 1. Thus Proposition 3.2(iii) yields Consequently, (3.8) and (3.9) lead to Next we decompose id j into certain diagonal operators and the natural embedding, where we applied Proposition 3.3, in particular (3.2). Thus It remains to calculate the latter norm. First assume that t(p 1 , p 2 ) < ∞. In this case, (3.12) Thus (3.10), (3.11) and (3.12) result in p2) , α p1 ) as assumed by (3.5). If t(p 1 , p 2 ) = ∞, i.e., if p 1 = 1 and p 2 = ∞, then ν(id j )≤1 if α ≥ 0 and ν(id j )≤2 −jα if α < 0. In consequence, Next we deal with such that (in a slight abuse of notation) we can understand id 2 as Again we decompose where P j is the projection onto p1 (|m| β ), hence is the embedding on level j, and Q j is the embedding of p2 into q2 ( p2 ) with Q j : p2 → q2 ( p2 ) = 1. Proposition 3.2(iii) together with (3.13) yield (3.14) if id j is a nuclear map. So we proceed similar as above, where we applied Proposition 3.3(i). Hence It remains to calculate that norm. If t(p 1 , p 2 ) < ∞, then in view of (the second part of) (3.5) again.
If t(p 1 , p 2 ) = ∞, that is, p 1 = 1 and p 2 = ∞, then id recall Proposition 3.3(i). This requires β > 0 and leads to ν( id This concludes the argument for the sufficiency part. Step 2. Now we show the necessity of (3.5) for the nuclearity of id α,β and begin with the global behaviour of the weight and have to prove that the nuclearity of id α,β implies β p1 > d t(p1,p2) . So assume β p1 ≤ d t(p1,p2) . We return to our above construction. Let k ∈ N and consider the following commutative diagram Similar as above, let where we applied Proposition 3.3, in particular (3.2). Then On the other hand, parallel to (3.12), for arbitrary k ∈ N. But this leads to a contradiction in (3.17) for ν(id 2 ) < ∞ in the considered cases. Thus β p1 > d t(p1,p2) . We are left to deal with the local part of the weight which is related to the second condition in (3.5). Since any nuclear map is compact, the nuclearity of id α,β implies its compactness which by Proposition 2.11 leads to δ > α p1 . It remains to show δ > d t(p1,p2) in all admitted cases of the parameters. If 1 < p i , q i < ∞, i = 1, 2, this is an immediate consequence of Corollary 3.10. If t(p 1 , p 2 ) = ∞, i.e., p 1 = 1 and p 2 = ∞, then t(p 1 , p 2 ) = p * and the statement follows from Proposition 2.11 again. We are left to deal with the limiting cases of p i and q i , i = 1, 2, in case of t(p 1 , p 2 ) < ∞.
First, let us consider the following commutative diagram and . Both operators Q k and Π k are bounded. Moreover Hence, letting k → ∞, we obtain δ ≥ d t(p1,p2) . It remains to exclude the case δ = d t(p1,p2) > 0. The operator id 1 is nuclear, so there exist We choose 0 < ε < 1 and take i o such that The operator id ε is nuclear and ν(id ε ) < ε. Moreover, if λ ∈ X ε , then λ = id 1 (λ) = id ε (λ). We consider the subspaces X k = Π k ( 2 kd p1 ), k ∈ N, cf. (3.20). If k is such that 2 dk > i o , then dim (X ε ∩ X k ) ≥ 2 dk − i o , since codim X ε ≤ i o . Now we can repeat the argument used in the diagram (3.19) for the operator id ε . More precisely, if k ε = dim (X ε ∩ X k ), then we have the following commutative diagram with Π k,ε and Q k,ε defined in a similar way as above, i.e., Π k,ε is the restriction of Π k to kε p1 . Note that Π k,ε is a linear bijection of kε p1 onto X ε ∩ X k , and id 1 = id ε on X k ∩ X ε . Thus On the other hand, in view of (3.2), Together with (3.23) and in view of our assumption δ = d t(p1,p2) we thus arrive at Taking k → ∞ with fixed ε and i o we get the contradiction.
Remark 3.14. We briefly want to discuss the above result and compare it with the compactness criterion as recalled in Proposition 2.11. In view of the parameters (2.18) we now naturally have to assume the Banach case situation, i.e., p i , q i ≥ 1, i = 1, 2, when studying nuclearity. Moreover, as an easy observation shows, it might well happen that for certain parameter settings the compact embedding id α,β can never be nuclear, independent of the target space. This is, for instance, the case when 1 p1 + β p1d < 1, as then (3.5) for β is never satisfied. Moreover, this excludes, in particular, an application of Theorem 3.12 to the situation of Sobolev spaces, id W α,β : W k1 p1 (R d , w α,β ) → W k2 p2 (R d ), 1 < p i < ∞, and k i ∈ N 0 , i = 1, 2, based on (2.12) and Theorem 3.12 with A = F . Here we would need w α,β ∈ A p1 which, by Example 2.2(i), reads as −d < α, β < d(p 1 − 1). But, as just observed, this contradicts (3.5) for β. So it very much depends on the source space, including the weight parameters, whether or not in some compactness case the embedding id α,β is even nuclear. nuclear To illustrate the difference between compactness and nuclearity of id α,β in the area of parameters in the usual ( 1 p , s) diagram above, where any space A s p,q is indicated by its smoothness and integrability (neglecting the fine index q), we have chosen the situation when β dp 1 > 1 > α dp 1 In the sense of Remark 2.13 we can immediately conclude the nuclearity result for embeddings of spaces with admissible weights. . (3.25) In view of (2.20) we observe the phenomenon again that the nuclearity characterisation is distinct from the compactness one by replacing p * by t(p 1 , p 2 ) only. In particular, when t(p 1 , p 2 ) = p * , that is, when p 1 = 1 and p 2 = ∞ (recall that we always assume p 1 < ∞), thus A = F , then nuclearity and compactness conditions coincide. In that case t(p 1 , p 2 ) = p * = ∞ and Theorem 3.12 together with Proposition 2.11 imply the following result.
The following conditions are equivalent We can also extend Proposition 3.6 to limiting cases p 1 , p 2 , q 1 , q 2 equal to 1 or ∞. The generalisation follows easily from Theorem 3.12 for domains with the extension property, in particular for bounded Lipschitz domains. The sufficiency part has already been obtained in [5,Thm. 4.2], we may now complete the argument for the necessity part and thus partly extend [5,Cor. 4.6], i.e., when α 1 = α 2 = 0 in the notation used in [5]. .
Proof. Since the q-parameters play no role it is sufficient to prove the corollary for Besov spaces. The corresponding statement for the F -spaces follows then by elementary embeddings.
For the sufficiency part we benefit from the result [5,Thm. 4.2] (with α 1 = α 2 = 0 in their notation). The necessity can be proved in a way similar to the local part in the Step 2 of the proof of Theorem 3.12. Using the standard wavelet basis argument with Daubechies wavelets we can factorise the embedding q1 (2 jδ 2 jd p1 ) → q2 ( 2 jd p2 ) through the embedding id Ω : B s1 p1,q1 (Ω) → B s2 p2,q2 (Ω). Then we can argue in the same way as in Step 2, (3.19)-(3.24) of the proof of the last theorem.
Remark 3.18. Parallel to Corollary 3.16 we can thus state that for arbitrary q 1 , q 2 ∈ [1, ∞], Remark 3.20. In the sense of Remark 3.11 we can add a further simple argument now, showing that the above criterion is a necessary one for nuclearity only: when p 1 = ∞ and w ∈ A ∞ (arbitrary), then in view of (2.9) the above embedding id w is an unweighted one which is never compact (let alone nuclear). Now we study the counterpart of Theorem 3.12 for the weight function w (α,β) in Example 2.2(ii). For convenience we recall the following well-known fact, which can also be found in [19,Lemma 3.8].
always with equivalence constants independent of j. Now we can give the counterpart of the compactness result Proposition 2.14.
We proceed essentially parallel to the arguments presented in the proof of Theorem 3.12. So again we may restrict ourselves to the study of the corresponding sequence spaces where the counterparts of (3.6) and (3.7) now read as and (3.30) We split id = id 1 + id 2 as above, where only the weight w α,β has to be replaced by w (α,β) . First we consider the non-limiting case δ > max α1 p1 , d t(p1,p2) . By the same arguments as in the proof of Theorem 3.12 we arrive at the following counterpart of (3.10), (3.31) The counterpart of (3.11) is We calculate the norm. First we assume that t(p 1 , p 2 ) < ∞. Thus such that (3.32) and Lemma 3.21 imply . (3.37) We study the different cases to estimate ν(id 1 ) by (3.31). In case of (3.34) we obtain that , δ > α 1 p 1 and α 2 ∈ R.
Moreover P j = 1 and the norm T j is uniformly bounded in j ∈ N 0 . The operators be the corresponding nuclear decompositions. We define the following (double) sequences, c k, = {a k,i b ,n } i∈N,n∈Ij , k, ∈ N 0 , and z k, = {y k,i z ,n } i∈N,n∈Ij , k, ∈ N 0 .
The next statement is a direct consequence of Proposition 2.14, in particular, Remark 2.15, Theorem 3.22 and Remark 3.23.

Radial spaces
So far we considered embeddings within the scale of spaces A s p,q which are compact -and studied the question whether they are even nuclear. In case of spaces on bounded domains Ω or weighted spaces on R d it is well-known that compactness can appear, unlike in case of unweighted spaces on R d . Furthermore, such Sobolev-type embeddings can also be compact in presence of symmetries, i.e., if we restrict our attention to subspaces consisting of distributions that satisfy certain symmetry conditions, in particular, if they are radial. We want to consider this setting now. Here the sufficient and necessary conditions for the nuclearity of the compact embeddings can be easily proved due to the relation between subspaces of radial distributions and appropriately weighted spaces. Indeed the conditions follow from Theorem 3.12. We start with recalling the definition of radial subspaces of Besov and Triebel-Lizorkin spaces.
Let Φ be an isometry of R d . For g ∈ S(R d ) we put g Φ (x) = g(Φx). If f ∈ S (R d ), then f Φ is a tempered distribution defined by where Φ −1 denotes the isometry inverse to Φ.

Definition 3.25.
Let SO(R d ) be the group of rotations around the origin in R d . We say that the tempered distribution f is invariant with respect to SO(R d ) if f Φ = f for any Φ ∈ SO(R d ).
For any possible s, p, q we put