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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 Rd never admit a compact, let alone nuclear embedding. But replacing Rd by a bounded Lipschitz domain Ω ⊂ Rd, 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 s questions. More precisely, for Besov spaces on bounded Lipschitz domains, Bp,q(Ω), it is well known that B Bs1 ֒ Bs2 idΩ : p1,q1 (Ω) → p2,q2 (Ω) is nuclear if, and only if, 1 1 s − s > d − d max − , 0 , 1 2 p p 2 1 where 1 ≤ pi, qi ≤ ∞, si ∈ 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 B and entropy numbers of idΩ . Now we study its nuclearity and find almost complete char- acterisations 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 The- orem 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 p1 = 1, p2 = ∞, and s1 − s2 > d, while in the case b(Ω) < ∞ the conditions look more similar to the above ones for bounded domains; e.g., for B such quasi-bounded domains idΩ is nuclear, if 1 1 b(Ω) s − s − d − > , 1 2 p p t(p ,p ) 1 2 1 2 1 where 1 ≤ pi, qi ≤ ∞, i = 1, 2, s1 > s2, and the new quantity t(p1,p2) is defined by = t(p1,p2) 1 − max( 1 − 1 , 0). For bounded Lipschitz domains Ω one has b(Ω) = d and then the above- p1 p2 mentioned 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 ℓp. In our situation, however, Mj it turned out that we need more general vector-valued sequence spaces of type ℓq(βj ℓp ), cf. Definition 1.1 below. So we were led to the study of the embedding

ℓ β ℓMj ֒ ℓ ℓMj idβ : q1 ( j p1 ) → q2 ( p2 )

2 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 ≤ pi, qi ≤ ∞, i =1, 2,

1 t −1 (p1,p2) idβ is nuclear if, and only if, βj Mj ∈ ℓt(q1,q2). N j∈ 0 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 Mj ≡ 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. Fi- nally this also leads to a corresponding result for Triebel-Lizorkin spaces on quasi-bounded domains.

1 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 N0 the set N∪{0}, and by Zd the set of all lattice points in Rd having integer components. R Let a+ = max(a, 0), a ∈ . For two positive real sequences (ak)k∈N0 and (bk)k∈N0 we mean by ak ∼ bk that there exist constants c1,c2 > 0 such that c1ak ≤ bk ≤ c2ak for all k ∈ N0; 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.

1.1 Sequence spaces We begin with the deﬁnition of weighted vector-valued sequence spaces. We consider a gen- ∞ ∞ eral weight sequence (βj )j=0 of positive real numbers and and a sequence (Mj )j=0 of positive integers that are dimensions of ﬁnite-dimensional vector spaces.

Deﬁnition 1.1. Let 0 < p,q ≤ ∞, (βj )j∈N0 be a weight sequence, that is βj > 0, and (Mj )j∈N0 be a sequence of natural numbers. Then

Mj C ℓq βj ℓp = x = (xj,k)j∈N0,k=1,...,Mj : xj,k ∈ ,

n ∞ Mj q 1 p q Mj q p x|ℓq βjℓp = βj |xj,k| < ∞ , j=0 k=1 X X o with the usual modiﬁcations if p = ∞ and/or q = ∞.

Mj Remark 1.2. The spaces ℓq(βj ℓp ) are quasi-Banach spaces (Banach spaces if p, q ≥ 1). If Mj Mj ≡ 1, then the space ℓq(βj ℓp ) coincides with the usual weighted space ℓq (βj ), that is, for βj ≡ 1, nothing else than ℓq. 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

3 the sequel: for 0

Mj Mj idβ : ℓq1 βj ℓp1 → ℓq2 ℓp2 . (1.2) It was proved in [13, 14] that the embedding is compact if, and only if,

1 p∗ −1 ∗ βj Mj ∈ ℓq , (1.3) N j∈ 0 ∗ where for q = ∞ the space ℓ∞ has to be replaced by c0. In Section 2 below we study its nuclearity.

1.2 Function spaces and compact Sobolev embeddings

d We now consider some function spaces. Let 0 < p ≤ ∞. Then the Lebesgue space Lp(R ) contains all measurable functions such that

1/p d p kf |Lp(R )k = |f(x)| dx , 0 Schwartz space S(Rd) and its dual S′(Rd) of all complex-valued tempered distribu- d tions have their usual meaning here. Let ϕ0 = ϕ ∈ S(R ) be such that supp ϕ ⊂ y ∈ Rd : |y| < 2 and ϕ(x)=1 if |x|≤ 1 , (1.5) N −j −j+1 ∞ and for each j ∈ let ϕj (x) = ϕ(2 x) − ϕ(2 x). Then (ϕj )j=0 forms a smooth dyadic resolution of unity. Given any f ∈ S′(Rd), we denote by Ff and F −1f its Fourier transform and its inverse Fourier transform, respectively. R Deﬁnition 1.4. Let 0 < p,q ≤ ∞, s ∈ and (ϕj )j a smooth dyadic resolution of unity. s Rd ′ Rd (i) The Besov space Bp,q( ) is the set of all distributions f ∈ S ( ) such that

s d js −1 d f |B (R ) = 2 F (ϕj Ff)|Lp(R ) |ℓq (1.6) p,q j∈N0

is ﬁnite. s Rd (ii) Assume 0

is finite. s Rd s Rd Remark 1.5. The spaces Bp,q( ) and Fp,q( ) are independent of the particular choice of the smooth dyadic resolution of unity (ϕj )j appearing in their definitions. They are quasi-Banach Rd s Rd ′ Rd - spaces (Banach spaces for p, q ≥ 1), and S( ) ֒→ Bp,q( ) ֒→ S ( ), similarly for the F case, where the first embedding is dense if p,q < ∞; we refer, in particular, to the series of monographs by Triebel [26–29] for a comprehensive treatment of these spaces. k Rd Rd Concerning (classical) Sobolev spaces Wp ( ) built upon Lp( ) in the usual way, it holds k Rd k Rd N Wp ( )= Fp,2( ), k ∈ 0, 1

4 s s s Convention. We adopt the nowadays usual custom to write Ap,q instead of Bp,q or Fp,q, respectively, when both scales of spaces are meant simultaneously in some context (but al- ways 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. If 0 < q ≤ ∞, 0 < q0 ≤ q

s1 d s2 d Rd A R ֒ A R id : p1,q1 ( ) → p2,q2 ( ) 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 Rd such that Ω 6= Rd. Such a set will be called an arbitrary domain. ′ s We denote the collection of all distributions on Ω by D (Ω). As usual the spaces Bp,q(Ω) and s Fp,q(Ω) are deﬁned on Ω by restriction, i.e., using our above convention,

s ′ s Rd Ap,q(Ω) = {f ∈ D (Ω) : ∃ g ∈ Ap,q( ) with g|Ω = f}, equipped with the quotient norm, as usual,

s s Rd s Rd kf|Ap,q(Ω)k = inf kg|Ap,q( )k : g ∈ Ap,q( ) with g|Ω = f .

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].

d Proposition 1.7. Let Ω ⊂ R be a bounded Lipschitz domain and si ∈ R, 0 d − . (1.11) 1 2 p p 1 2 + 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. js jd Remark 1.8. Note that for βj = 2 and Mj ∼ 2 , with d ∈ N, the above sequence space Mj js 2jd ¯s s ℓq(βj ℓp )= ℓq(2 ℓp ) is isomorphic to the subspace Bp,q(Ω) of the Besov space Bp,q(Ω) deﬁned on a bounded Lipschitz domain Ω in Rd. 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.

2 Nuclear embeddings in general vector-valued sequence spaces

Our ﬁrst main goal in this paper is to study nuclear embeddings between sequence spaces of Mj the type ℓq(βj ℓp ) introduced above. So we ﬁrst recall some fundamentals of the concept and important results we rely on in the sequel.

5 2.1 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. [19–21] and also [22] for some history. There exist extensions of the concept to r-nuclear operators, 0 trace class S1(H1,H2), consisting of those T with singular numbers (sn(T ))n ∈ ℓ1. In the next proposition we recall well-known properties of nuclear operators needed in the sequel, cf. [12, Chapter 1]. Proposition 2.1. (i) If X is an n-dimensional Banach space, n ∈ N, then ν(id : X → X)= n.

n (ii) For any Banach space X and any bounded linear operator T : ℓ∞ → X we have n ν(T )= kTeik. i=1 X

(iii) If T ∈ L(X, Y ) is a nuclear operator and S ∈ L(X0,X) and R ∈ L(Y, Y0), then RTS is a nuclear operator and

ν(RTS) ≤kRkν(T )kSk (ideal property).

Already in the early years there was a strong interest to ﬁnd natural examples of nuclear operators beyond diagonal operators in ℓp spaces, for which a complete answer was obtained in [25]. Let τ = (τj )j∈N be a scalar sequence and denote by Dτ the corresponding diagonal operator, Dτ : x = (xj )j 7→ (τj xj )j , acting between ℓp spaces. Let us introduce the following notation: for r1, r2 ∈ [1, ∞], let t(r1, r2) be given by

1 1, if 1 ≤ r ≤ r ≤ ∞, = 2 1 (2.1) t 1 1 (r1, r2) 1 − + , if 1 ≤ r1 ≤ r2 ≤ ∞. ( r1 r2

Hence 1 ≤ t(r1, r2) ≤ ∞, and 1 1 1 1 1 1 =1 − − ≥ = − , t(r , r ) r r r∗ r r 1 2 1 2 + 2 1 + ∗ with t(r1, r2)= r if, and only if, {r1, r2} = {1, ∞}. We heavily rely in our arguments below on the following remarkable result by Tong [25].

Proposition 2.2 ( [25, Thms. 4.3, 4.4]). Let 1 ≤ r1, r2 ≤ ∞ and Dτ be the above diagonal operator. t (i) Then Dτ is nuclear if, and only if, τ = (τj )j ∈ ℓt(r1,r2), with ℓt(r1,r2) = c0 if (r1, r2) = ∞. Moreover,

ν(Dτ : ℓr1 → ℓr2 )= kτ|ℓt(r1,r2)k. N n n n n n (ii) Let n ∈ and Dτ : ℓr1 → ℓr2 be the corresponding diagonal operator Dτ : x = (xj )j=1 7→ n (τj xj )j=1. Then ν Dn ℓn ℓn τ n ℓn . (2.2) ( τ : r1 → r2 )= ( j )j=1| t(r1,r2)

Example 2.3. In the special case of τ ≡ 1, i.e., Dτ = id, (i) is not applicable and (ii) reads as

n n n if 1 ≤ r2 ≤ r1 ≤ ∞, ν(id : ℓ → ℓ )= 1 1 r1 r2 1− r + r (n 1 2 if 1 ≤ r1 ≤ r2 ≤ ∞.

n n In particular, ν(id : ℓ1 → ℓ∞)=1.

Remark 2.4. We refer also to [19] for the case r1 =1, r2 = ∞.

6 2.2 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 ≤ pi, qi ≤ ∞, i = 1, 2. Then idβ given by (1.2) is nuclear if, and only if, the compactness condition (1.3) is replaced by

1 t −1 (p1,p2) βj Mj ∈ ℓt(q1,q2), (2.3) N j∈ 0 where for t(q1, q2)= ∞ the space ℓ∞ has to be replaced by c0. 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 n ∈ N0, 1 ≤ p, q ≤ ∞, we put

n Mj C ℓq βj ℓp = x = (xj,k)0≤j≤n,k=1,...,Mj : xj,k ∈ , (2.4)

n n Mj q 1 p q Mj q p x|ℓq βj ℓp = βj |xj,k| , j=0 k=1 X X o 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 βj ≡ 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. j−1 N N Let M = (Mj)j∈ 0 be a sequence of natural numbers and let n ∈ 0. Denote by αj = l=0 Ml, j ∈ N0, with M−1 := 0, i.e., α0 = 0. Let Ij = {αj +1,...,αj+1}, that is, I0 = {1,...,M0}, and P N = αn+1. We put

∞ q 1 M M p p q l (l j )= x = (x ) N : x ∈ C, x|l (l j ) = |x | < ∞ , (2.5) q p  k k∈ k q p k  j=0 k∈I  X Xj  n q 1  p q  ln(lMj )= x = (x ) : x ∈ C, x|ln(lMj ) = |x |p , (2.6) q p  k 1≤k≤N k q p k  j=0 k∈I  X Xj 

 l lMj  appropriately modiﬁed for p = ∞ and/or q = ∞. Then q( p ) is a vector space isometrically M j ln lMj isomorphic to ℓq(ℓp ) and q ( p ) is a ﬁnite-dimensional vector space isometrically isomorphic n Mj ln lMj to ℓq (ℓp ), with dim q ( p )= N. N Lemma 2.5. Let n ∈ 0, 1 ≤ pi, qi ≤ ∞, i = 1, 2, and M = (Mj )j∈N0 and N as above. Let ln lMj ln lMj λ = (λk)k=1,...,N be complex numbers and Dλ : q1 ( p1 ) → q2 ( p2 ) be the diagonal operator given N N by Dλ : (xk)k=1 7→ (λkxk)k=1. Then

n Mj n Mj n Mj N D l l l l λ l ∗ l ∗ , λ λ , k λ : q1 ( p1 ) → q2 ( p2 )k = k | q ( p )k = ( k)k=1 where p∗ and q∗ are given by (1.1).

n Mj n Mj n Mj Proof. D D l l l l D λ l ∗ l ∗ The upper estimate of k λk = k λ : q1 ( p1 ) → q2 ( p2 )k, that is, k λk≤k | q ( p )k, follows easily from H¨older’s inequality if p2

7 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 CN given by the matrix A, then D(T ) will denote the operator given by the matrix D(A).

n M N N l l j Lemma 2.6. Let n ∈ 0, 1 ≤ pi, qi ≤ ∞, i = 1, 2, and M = (Mj )j∈ 0 as above. If T : q1 ( p1 ) → ln lMj q2 ( p2 ) is a linear operator, then kD(T )k≤kT k. Proof. Let A be the matrix of T . Let ω ∈ C and let U(ω) be the diagonal matrix with entries 1,ω,ω2,...,ωN−1 down its diagonal. If |ω| = 1, then the matrix U(ω) generates an operator in CN ln lMj 2πi/N that is an isometry in the norm of q∗ ( p∗ ) for all p, q. We take ω = e . Using the identity N−1 j j=0 ω =0 and elementary calculations one concludes, cf. [2], that

P N−1 1 D(A)= U(ω)j AU(¯ω)j . (2.7) N j=0 X The operators U(ω)j and U(¯ω)j are isometries therefore the last formula implies kD(T )k = kD(A)k≤kAk = kT k. N Lemma 2.7. Let 1 ≤ pi, qi ≤ ∞, i = 1, 2, n ∈ 0 and M = (Mj )j∈N0 as above. Assume that ln lMj ln lMj D : q1 ( p1 ) → q2 ( p2 ) is a diagonal linear operator deﬁned by the diagonal matrix A with entries λ1,...,λN . Then n Mj N ν(D)= kλ|lt (l )k, λ = (λk) , (q1 ,q2) t(p1 ,p2) k=1 where t(p1,p2) and t(q1, q2) are given by (2.1).

ln lMj ln lMj Proof. Step 1. Both spaces q1 ( p1 ) and q2 ( p2 ) are finite-dimensional, therefore it is sufficient to consider the finite representations of any nuclear operator T , i.e.,

k T x′ y , x′ ln lMj ′ y ln lMj , = ℓ ⊗ ℓ ℓ ∈ ( q1 ( p1 )) and ℓ ∈ q2 ( p2 ) (2.8) Xℓ=1 cf. [12, p.19]. ′ ln lMj , ln lMj First we show that the space N q1 ( p1 ) q2 ( p2 ) , that is, the dual space to N ln (lMj ), ln (lMj ) , is isometrically isomorphic to L ln (lMj ), ln (lMj ) . This can be proved via q1 p1 q2 p2 q2 p2 q1 p1 trace-duality. For any operator S ∈ L ln (lMj ), ln (lMj ) the composition operator ST is again a q2 p2 q1 p1 nuclear operator which belongs to N ln (lMj ), ln (lMj ) , and q1 p1 q1 p1

k ST x′ Sy , x′ ln lMj ′ Sy ln lMj . = ℓ ⊗ ℓ ℓ ∈ ( q1 ( p1 )) and ℓ ∈ q1 ( p1 ) Xℓ=1 Then k ′ ϕS (T ) := tr(ST )= xℓ(Syℓ) Xℓ=1 ln lMj ln lMj deﬁnes a linear functional on N q1 ( p1 ), q2 ( 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

′ ϕ ln lMj , ln lMj S ln lMj , ln lMj . k S|N q1 ( p1 ) q2 ( p2 ) k≤k |L q2 ( p2 ) q1 ( p1 ) k The reverse inequality can be proved by applying the functional ϕS to rank-one operators T = x′ ⊗ y with kx′k = kyk =1. Thus the mapping

′ ln lMj ln lMj ln lMj ln lMj Φ: L q2 ( p2 ), q1 ( p1 ) → N q1 ( p1 ), q2 ( p2 ) , S 7→ ϕS, 8 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

′ ln lMj ln lMj ln lMj ln lMj kϕSk = kϕS|N q1 ( p1 ), q2 ( p2 ) k = kS|L q2 ( p2 ), q1 ( p1 ) k = kSk, such that, in particular,

′ ν D ϕ D ϕ ln lMj , ln lMj , ϕ ( ) = sup{| ( )| : ∈ N q1 ( p1 ) q2 ( p2 ) k k≤ 1} SD S ln lMj , ln lMj , S . = sup{| tr( )| : ∈ L q2 ( p2 ) q1 ( p1 ) k k≤ 1}

Step 2. S ln lMj , ln lMj S S Let ∈ L q2 ( p2 ) q1 ( p1 ) with k k ≤ 1. We denote its diagonal part by D( ): n Mj n Mj N l (l ) → l (l ), represented by the entries b = (bℓ) . Lemma 2.5 shows that the subspace q2 p2 q1 p1 ℓ=1 n Mj n Mj n Mj l l , l l l ∗ l ∗ of diagonal operators in L q2 ( p2 ) q1 ( p1 ) is isometrically isomorphic to q˜ ( p˜ ), and

n Mj n Mj n Mj S l l , l l b l ∗ l ∗ , kD( )|L q2 ( p2 ) q1 ( p1 ) k = k | q˜ ( p˜ )k where 1 1 1 1 1 1 := − and := − . p˜∗ p p q˜∗ q q 1 2 + 1 2 + Note that 1 1 1 1 1 1 . ∗ =1 − = ′ and ∗ =1 − = ′ p˜ t(p1,p2) t(p1,p2) q˜ t(q1, q2) t(q1, q2) Consequently,

n Mj n Mj n Mj kD(S)|L l (l ), l (l ) k = kb|lt ′ (l ′ )k. (2.9) q2 p2 q1 p1 (q1,q2) t(p1,p2)

Since tr(SD) = tr(D(S)D) and kD (S)k≤kSk by Lemma 2.6, we get for any such operator S with kSk≤ 1 by H¨older’s inequality that

Thus the ﬁrst step implies that

ln lMj ln lMj ν(D) = sup{| tr(SD)| : kS|L( q2 ( p2 )), q1 ( p1 )k≤ 1 } n Mj ≤ kλ|lt (l )k. (2.10) (q1,q2) t(p1,p2)

S S ln lMj , ln lMj Conversely, we may restrict ourselves to diagonal operators = D( ) in L q2 ( p2 ) q1 ( p1 ) and beneﬁt from the sharpness of the H¨older inequality to obtain the estimate from below, n Mj ν(D) ≥kλ|lt (l )k. (q1 ,q2) t(p1 ,p2)

This concludes the argument.

M Let p , q , i , , M M N as above, and let D l l j Proposition 2.8. 1 ≤ i i ≤ ∞ = 1 2 = ( j )j∈ 0 λ : q1 ( p1 ) → M l l j be a diagonal linear operator deﬁned by the sequence λ λ N. Then the operator D q2 ( p2 ) = ( j )j∈ λ Mj is nuclear if, and only if, λ ∈ lt (l ) and λ ∈ c if t(q , q )= ∞. Moreover, (q1 ,q2) t(p1 ,p2) 0 1 2

Mj ν(D )= kλ|lt (l )k. λ (q1,q2) t(p1,p2)

9 l lMj Proof. Step 1. First we show that the nuclearity of Dλ implies that λ ∈ t(q1 ,q2)( p1,p2 ) with the additional requirement that λ ∈ c0 if t(q1, q2)= ∞. In particular, we shall obtain that

Mj ν(D ) ≥kλ|lt (l )k. (2.11) λ (q1,q2) t(p1,p2)

n N P l lMj ln lMj x , x ,... Let for ∈ 0, n : q2 ( p2 ) → q2 ( p2 ) denote a projection given by ( 1 2 ) 7→ ln lMj l lMj (x1,...,xN(n)) and let Sn : q1 ( p1 ) → q1 ( p1 ) denote an embedding given by (x1,...,xN(n)) 7→ l lMj l lMj (x1, x2,...,xN(n), 0, 0,...). Both operators have norm 1. If Dλ : q1 ( p1 ) → q2 ( p2 ) is a nuclear l lMj l lMj operator, then by Proposition 2.1(iii) PnDλSn : q1 ( p1 ) → q2 ( p2 ) is nuclear and

ν(PnDλSn) ≤kPnkν(Dλ)kSnk = ν(Dλ). (2.12)

Hence Lemma 2.7 implies

Mj Mj kλ|lt(q ,q )(lt )k = lim k(λ1, λ2,...,λN(n))|lt(q ,q )(lt )k 1 2 (p1,p2) n→∞ 1 2 (p1 ,p2)

= lim ν(PnDλSn) ≤ ν(Dλ). n→∞

If t(q1, q2) = ∞, we need to show, in addition, that λ ∈ c0. However, if λ 6∈ c0, then one can easily prove that the operator Dλ is not compact. So it is also not nuclear.

Mj Step 2. It remains to show the converse direction, i.e., the sufficiency of λ ∈ lt (l ) (q1 ,q2) t(p1 ,p2) for the nuclearity of Dλ, and the inequality converse to (2.11). Suppose that we have a diagonal Mj operator Dλ defined by the sequence λ ∈ lt (l ), with λ ∈ c0 if t(q1, q2) = ∞. Then the (q1,q2) t(p1,p2) Mj sequence (λ1, λ2,...,λ , 0, 0,... ) is a Cauchy sequence in lt (l ). Thus Lemma N(n) n (q1,q2) t(p1,p2) (n) 2.7 implies that the sequence of diagonal operators Dλ that are defined by the sequences λ , λ ,...,λ , , ,... l lMj , l lMj ( 1 2 N(n) 0 0 ) n, is a Cauchy sequence in N ( q1 ( p1 ) q2 ( p2 ). The space of nuclear (n) operators is a Banach space, so the sequence (Dλ )n is convergent to a nuclear operator D in the nuclear norm. On the other hand one can easily see that e (n) lim D (x1, x2,...) = (λ1x1, λ2x2,...)= Dλx. n→∞ λ

(n) So the sequence (Dλ )n converges to D = Dλ in the sense of pointwise convergence. The (n) convergence in the sense of the nuclear norm is stronger, so (Dλ )n converges to Dλ also in l lMj l lMj e N q1 ( p1 ), q2 ( p2 ) and

(n) ν(Dλ) = lim ν(D ) n→∞ λ Mj = lim k(λ1, λ2,...,λN(n), 0, 0,... )|lt(q ,q )(lt )k n→∞ 1 2 (p1,p2) Mj = kλ|lt (l )k. (q1,q2) t(p1,p2)

Now we are ready to present our main outcome in this section. It generalises Tong’s result [25] as recalled in Proposition 2.2 to the setting of generalised vector-valued sequence spaces.

Theorem 2.9. Let 1 ≤ pi, qi ≤ ∞, i =1, 2, (βj )j∈N0 be an arbitrary weight sequence and (Mj )j∈N0 be a sequence of natural numbers. Then the embedding

Mj Mj idβ : ℓq1 βj ℓp1 → ℓq2 ℓp2 (2.13) is nuclear if, and only if,

1 t −1 (p1,p2) βj Mj ∈ ℓt(q1,q2), (2.14) N j∈ 0

10 where for t(q1, q2)= ∞ the space ℓ∞ has to be replaced by c0. In that case,

1 t −1 (p1,p2) ν(idβ)= βj Mj |ℓt(q1,q2) . j∈N 0

M l lMj j Proof. We apply Proposition 2.8. The space q( p ) is isometrically isomorphic to ℓq(ℓp ). So l lMj l lMj −1 the embedding idβ corresponds to a diagonal operator Dλ from q1 ( p1 ) to q2 ( p2 ) with λl = βj if l ∈ Ij and this gives

1 Mj −1 t(p ,p ) ν(id )= ν(D )= kλ|lt (l )k = β M 1 2 |ℓt . β λ (q1 ,q2) t(p1 ,p2) j j (q1,q2) j∈N 0

−1 Remark 2.10. In case of Mj ≡ 1, βj = τj , Theorem 2.9 generalises Tong’s result [25], cf. Proposition 2.2(i), in a natural way.

Remark 2.11. Note that in the extremal cases {p1,p2} = {1, ∞} and {q1, q2} = {1, ∞}, that is, ∗ ∗ whenever t(p1,p2) = p and t(q1, q2) = q , then compactness and nuclearity of the embedding idβ coincide, recall condition (1.3). Moreover, when p1 = 1 and p2 = ∞, then t(p1,p2) = ∞ −1 N N and condition (2.14) is reduced to (βj )j∈ 0 ∈ ℓt(q1,q2), independent of (Mj)j∈ 0 , where for t(q1, q2)= ∞ the space ℓ∞ has to be replaced by c0. 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 factori- sation. We sketch it for the case q1 ≤ p1 ≤ p2 ≤ q2. Note that in this situation 1 1 1 1 1 1 =1 − + and =1 − + . t(p1,p2) p1 p2 t(q1, q2) q1 q2 We decompose

M M −1 j j N N Dβ : ℓq1 (ℓp1 ) → ℓq2 ℓp2 ,Dβ : (xj,m)j∈ 0,m=1,...,Mj 7→ (βj xj,m)j∈ 0 ,m=1,...,Mj into Mj Dβ Mj ℓq1 (ℓp1 ) −−−−−−−→ ℓq2 (ℓp2 )

D1 D2

Mj D0 x Mj ℓq1 (ℓq1 ) −−−−−−→ ℓq2(ℓq2 ) y  with

D ℓ ℓMj ℓ ℓMj , 1 : q1 ( p1 ) → q1 ( q1 ) 1 1 p − q N 1 1 D1 : (xj,m)j∈ 0,m=1,...,Mj 7→ Mj xj,m , N j∈ 0,m=1,...,Mj

Mj Mj D2 : ℓq2 (ℓq2 ) → ℓq2 (ℓp2 ), 1 1 q − p N 2 2 D2 : (xj,m)j∈ 0,m=1,...,Mj 7→ Mj xj,m , N j∈ 0,m=1,...,Mj

Mj Mj D0 : ℓq1 (ℓq1 ) → ℓq2 (ℓq2 ), 1 − 1 −1 t(p ,p ) t(q ,q ) N 1 2 1 2 D0 : (xj,m)j∈ 0,m=1,...,Mj 7→ βj Mj xj,m , N j∈ 0,m=1,...,Mj 1 1 1 1 1 1 such that Dβ = D2 ◦ D0 ◦ D1, using also − = − + − + in this case. Thus t(p1,p2) t(q1,q2) p1 q1 q2 p2 Proposition 2.1(iii) leads to ν(Dβ) ≤kD1k kD2k ν(D0). (2.15)

11 We estimate kD1k. By our assumption p1 ≥ q1 H¨older’s inequality leads for x = (xj,m)j,m ∈ Mj ℓq1 (ℓp1 ) to

Mj ∞ 1 1 1 ( − )q1 q Mj p1 q1 q1 1 Mj kD1x|ℓq1 (ℓq1 )k = Mj |xj,m| ≤kx|ℓq1 (ℓp1 )k, j=0 m=1 X X hence kD1k≤ 1. Likewise, since p2 ≤ q2, by another application of H¨older’s inequality,

Mj q ∞ 1 1 2 1 ( − )q2 p q Mj q2 p2 p2 2 2 Mj kD2x|ℓq2 (ℓp2 )k = Mj |xj,m| ≤kx|ℓq2 (ℓq2 )k j=0 m=1 X X

Mj for any x = (xj,m)j,m ∈ ℓq2 (ℓq2 ). Thus kD2k≤ 1 and (2.15) implies ν(Dβ) ≤ ν(D0). We would like Mj to use Proposition 2.2(ii) and have to identify ℓqr (ℓqr ) therefore with ℓqr , r = 1, 2. This can be seen by a bijection like N (xj,m)j∈N0,m=1,...,Mj ↔ (yk)k∈N, yk = yk(j,m), j ∈ 0, m =1,...,Mj , j−1

k(j, m)= Ml + m, Xl=0 j−1 N recall M−1 =0, i.e., k(0,m)= m. Using our previous notation αj = l=0 Ml, j ∈ 0, with α0 =0, we get k(j, m)= αj +m, αj+1 −αj = Mj . For the rewritten sequence (xj,m)j∈N ,m=1,...M = (yk)k∈N, P 0 j k = k(j, m), let D0 denote the corresponding diagonal operator, acting now as

N e D0 : ℓq1 → ℓq2 , D0 : (yk)k∈ 7→ (γkyk)k∈N , such that D0x = D0y if x =e (xj,m)j,m = (yk)ek = y in the above identiﬁcation. More precisely, 1 1 e t − t −1 (p1,p2) (q1,q2) using the notation γj = βj Mj for the moment, then γj xj,m = γkyk when k = k(j, m). Consequently,e by Prop. 2.2(ii), e Mj Mj ν(Dβ) ≤ ν(D0 : ℓq1 (ℓq1 ) → ℓq2 (ℓq2 )) = ν(D0 : ℓq1 → ℓq2 )

∞ 1 ∞ αj+1 1 t(q ,q ) t t(q ,q ) t(q1,q2) 1 2 (q1,q2) 1 2 = |γk| = e γk(j,m) j=0 m=α +1 Xk=1 X Xj ∞ 1 1 et t(q ,q ) t e (q1,q2) 1 2 (q1,q2) = γj Mj = k(γj Mj )j |ℓt(q1,q2)k j=0 X 1 t −1 (p1,p2) = (βj Mj )j |ℓt(q1,q2)

1 t −1 (p1,p2) if q2 < ∞. If q2 = ∞ and q1 =1, then this has to be replaced by (βj Mj )j ∈ c0.

3 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.

3.1 Embeddings of function spaces on bounded domains In Proposition 1.7 we have already recalled the criterion for the compactness of the embedding

As1 As2 . idΩ : p1,q1 (Ω) → p2,q2 (Ω) Recently Triebel proved in [31] the following counterpart for its nuclearity.

12 d Proposition 3.1 ( [4, 11, 31]). Let Ω ⊂ R be a bounded Lipschitz domain, 1 ≤ pi, qi ≤ ∞ (with pi < ∞ in the F -case), si ∈ R. Then the embedding idΩ given by (1.10) is nuclear if, and only if,

1 1 s − s > d − d − . (3.1) 1 2 p p 2 1 + 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 qi, 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 was proved by Pietsch in [18] such that only the limiting case s −s = d−d( 1 − 1 ) was Ω 1 2 p2 p1 + 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 pi, qi ∈ {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 characterisa- tions of idΩ in (1.11) and (3.1) as follows, involving the number t(p1,p2) deﬁned in (2.1). Let 1 ≤ pi, qi ≤ ∞, si ∈ R and d d δ = s1 − − s2 + . p1 p2 Then d s1 s2 idΩ : A (Ω) → A (Ω) is compact ⇐⇒ δ > and p1,q1 p2,q2 p∗ d As1 As2 δ > . idΩ : p1,q1 (Ω) → p2,q2 (Ω) is nuclear ⇐⇒ t(p1,p2)

∗ Hence apart from the extremal cases {p1,p2} = {1, ∞} (when t(p1,p2)= p ) nuclearity is indeed stronger than compactness, i.e.,

As1 As2 idΩ : p1,q1 (Ω) → p2,q2 (Ω) is compact, but not nuclear, if, and only if,

d d <δ . ∗ ≤ p t(p1,p2) We observed similar phenomena in the weighted setting in [11], recall also our discussion in Remark 2.11 for the sequence space situation. Remark . Bs1,α1 Bs2,α2 3.3 In [4] the authors dealt with the nuclearity of the embedding p1,q1 (Ω) → p2,q2 (Ω) where the indices αi represent some additional logarithmic smoothness. They obtained a characterisation for almost all possible settings of the parameters. Finally, in [5] some further d limiting endpoint situations of nuclear embeddings like id : Bp,q(Ω) → Lp(log L)a(Ω) are studied. For some weighted results see also [17] and our recent contribution [11].

3.2 Embeddings of function spaces on quasi-bounded domains Now we study so called quasi-bounded domains and need to introduce them ﬁrst, together with their wavelet characterisation. An unbounded domain Ω in Rd is called quasi-bounded if

lim dist(x, ∂Ω) = 0 . x∈Ω,|x|→∞

An unbounded domain is not quasi-bounded if, and only if, it contains inﬁnitely many pairwise disjoint congruent balls, cf. [1], page 173.

13 Before we can formulate the properties of embeddings of function spaces defined on quasi- s bounded domains, we first need to extend the notion of Ap,q(Ω) as given in Section 1.2. Here ¯s ¯s we follow the ideas of Triebel in [30] and define spaces Fp,q(Ω) and Bp,q(Ω). We put 1 1 σ = d − 1 and σ = d − 1 , 0 < p,q ≤ ∞. p p p,q min(p, q) + + Definition 3.4. Let Ω be an arbitrary domain in Rd with Ω 6= Rd and let

with p< ∞ for the F -spaces. (i) Let

s ′ s Rd Ap,q(Ω) = f ∈ D (Ω) : ∃ g ∈ Ap,q( ) with g|Ω = f and supp g ⊂ Ω , n o equipped withe the quotient norm,

s s Rd kf|Ap,q(Ω)k = inf kg|Ap,q( )k,

s where the inﬁmum is taken overe all g ∈ Ap,q(Ω) with f = g|Ω . (ii) We deﬁne s Fp,q(Ω) if 0 σp,q , ¯s 0 Fp,q(Ω) = Fp,q(Ω) if 1

and  s Bp,q(Ω) if 0

σp , ¯s 0 Bp,q(Ω) = Bp,q(Ω) if 1

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. d −j Let Qj,m denote the dyadic cube in R with side-length 2 , j ∈ N0, given by

−j d −j d Qj,m = [0, 2 ) +2 m, m ∈ Z , j ∈ N0 .

d d Let Ω ⊂ R be a non-empty open set with Ω 6= R . For j ∈ N0 we denote by

k N bj (Ω) = sup k ∈ : Qj,mℓ ⊂ Ω, ℓ=1 [ −j Qj,mℓ being dyadic cubes of side-length 2 .

−j If there is no dyadic cube of size 2 contained in Ω we put bj (Ω) = 0. The following properties of the sequence bj(Ω) are obvious: j∈N0 (i) There exists a constant j0 = j0(Ω) ∈ N0 such that for any j ≥ j0 we have

d(j−j0) 0 < 2 bj0 (Ω) ≤ bj(Ω). (3.2)

(ii) If |Ω| < ∞, then −jd bj (Ω)2 ≤ |Ω| . (3.3)

14 −js It follows from (3.2) that if 0 b. We put j→∞ j→∞

−jt b(Ω) = sup t ∈ R+ : lim sup bj(Ω)2 = ∞ . (3.4) j→∞ Remark 3.5. For any non-empty open set Ω ⊂ Rd we have d ≤ b(Ω) ≤ ∞. If Ω is unbounded and not quasi-bounded, then b(Ω) = ∞. But there are also quasi-bounded domains such that b(Ω) = ∞, cf. [15]. Moreover it follows from (3.2) and (3.3) that if the measure |Ω| is finite, then b(Ω) = d. To illustrate this definition we give simple examples of quasi-bounded domains. 2 Example 3.6. Let α> 0 and β > 0 . We consider the open sets ωα,ω1,β ⊂ R defined as follows 2 −α ωα = {(x, y) ∈ R : |y| < x ,x> 1} , 2 −1 −β ω1,β = {(x, y) ∈ R : |y| < x (log x) ,x>e}. One can easily calculate that

−1 2j(α +1) if 0 <α< 1 , 2j bj (ωα) ∼ j2 if α =1 , 22j if α> 1 , and  22j j1−β if β < 1 , 2j bj(ω1,β) ∼ 2 log j if β =1 , 22j if β > 1 .

In consequence  α−1 +1 if 0 <α< 1 , b(ωα) = (2 if α ≥ 1 .

−jb(ωα) Moreover, the limit limj→∞ bj(ωα)2 is a positive ﬁnite number if α 6=1. But if α =1, then the limit equals inﬁnity. −jb(ω ,β ) In a similar way b(ω1,β)=2 for all β and limj→∞ bj (ω1,β)2 1 = ∞ if 0 <β ≤ 1. Remark 3.7. (i) One can construct a quasi-bounded domain in Rd with prescribed sequence bj (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 6= Rd and any r> 0 we put Ωr = {x ∈ Ω : dist(x, ∂Ω) > r}. (3.5)

If the domain Ω is quasi-bounded, then |Ωr| < ∞ for any r> 0 and log |Ω | b(Ω) = d + lim sup 2 r . (3.6) log r r→0 2

¯s ¯s To give the wavelet characterisation of the spaces Bp,q(Ω) and Fp,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. Definition 3.8. (i) A closed set Γ ⊂ Rd is said to be porous if there exists a number 0 <η< 1 such that one finds for any ball B(x, r) ⊂ Rd centered at x and of radius r with 0

15 (ii) A closed set Γ ⊂ Rd is said to be uniformly porous if it is porous and there is a locally ﬁnite positive Radon measure µ on Rd such that Γ = supp µ and

µ(B(γ, r)) ∼ h(r) , with γ ∈ Γ, 0

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).

Remark 3.9. The closed set Γ is called an α-set if there exists a locally ﬁnite positive Radon measure µ on Rd such that Γ = supp µ and

µ(B(γ, r)) ∼ rα , with γ ∈ Γ, 0

Naturally 0 ≤ α ≤ d. Any α-set with α < d is uniformly porous. Deﬁnition 3.10. Let Ω be an open set in Rd such that Ω 6= Rd and Γ= ∂Ω. (i) The domain Ω is said to be E-thick if one can ﬁnd for any interior cube Qi ⊂ Ω with

i −j i −j ℓ(Q ) ∼ 2 , and dist(Q , Γ) ∼ 2 , j ≥ j0 ∈ N,

a complementing exterior cube Qe ⊂ Rd \ Ω with

e −j e e i −j ℓ(Q ) ∼ 2 , and dist(Q , Γ) ∼ dist(Q ,Q ) ∼ 2 , j ≥ j0 ∈ N .

Qi and Qe denote cubes in Rd 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) ⊂ Rd centred at γ ∈ Γ and of radius r with 0

(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 quasi- bounded but also uniformly E-porous. ¯s If the domain is uniformly E-porous, then one can characterise Ap,q(Ω) spaces in terms of the wavelet expansion of the distributions. Now we give the wavelet characterisation of the ¯s ¯s N N N N spaces Fp,q(Ω) and Bp,q(Ω). Let Nj ∈ , j ∈ 0 and = ∪ {∞}.

jσ Nj C ℓq 2 ℓp := λ = (λj,k)j∈N0,k=1,··· ,Nj : λj,k ∈ ,

n Nj 1/p ∞ jσ Nj jσ p λ ℓq 2 ℓp = 2 |λj,k | ℓq < ∞ j=0 k=1 X o

Nj (usual modifications if p = ∞ and/or q = ∞). If Nj = ∞, then ℓp = ℓp. Theorem 3.12. Let Ω be a uniformly E-porous domain in Rd, Ω 6= Rd. Let s ∈ R, 0 < p,q ≤ ∞, ¯s N and Bp,q(Ω) be defined as in Definition 3.4. Let u ∈ 0. Then there exists an orthonormal basis

j N j u N Φr : j ∈ 0; r =1,...,Nj , Φr ∈ C (Ω), j ∈ 0, r =1,...,Nj , 16 ′ ¯s in L2(Ω), such that if u> max(s, σp − s), then f ∈ D (Ω) is an element of Bp,q(Ω) if, and only if, it can be represented as

∞ Nj d j −jd/2 j j(s− p ) Nj f = λr2 Φr, λ ∈ ℓq 2 ℓp , (3.7) j=0 r=1 X X unconditional convergence being in D′(Ω). ¯s Furthermore, if f ∈ Bp,q(Ω), then the representation (3.7) is unique with λ = λ(f)

j j jd/2 j λr = λr(f)=2 (f, Φr), where (·, ·) is a dual pairing and

d ¯s j(s− p ) Nj I : Bp,q(Ω) ∋ f 7→ λ(f) ∈ ℓq 2 ℓp

j ¯s is an isomorphism. If in addition max (p, q) < ∞ , then Φr is an unconditional basis in Bp,q(Ω). Remark 3.13. The above theorem was proved by Triebel, cf. [30, Theorem 3.23]. He used so called u-wavelet systems, cf. [30, Chapter 2] for the construction of this wavelet system. The sketch of the construction can be found in [15]. If we assume that the domain Ω is only E- ¯s thick, then the theorem holds for Bp,q(Ω) with s 6= 0 [30, Theorem 3.13]. Similar results were ¯s obtained for Fp,q(Ω) spaces.

There is a strict relation between the numbers Nj used in the last theorem and the sequence bj (Ω). Namely it was proved in [15] that

d 2 bj−2(Ω) ≤ Nj ≤ bj (Ω) . (3.8)

Using the above wavelet characterisation we obtained the necessary and sufﬁcient conditions for compactness of embeddings of the function spaces deﬁned on a uniformly E-porous quasi-bounded domain. We refer to [15] for the proof.

d Proposition 3.14. Let Ω be a uniformly E-porous quasi-bounded domain in R and let s1 >s2, ¯s 0 < pi, qi ≤ ∞ (with pi < ∞ in case of A = F ), i = 1, 2, and the spaces Ap,q(Ω) be given by Deﬁnition 3.4.

(i) If b(Ω) = ∞ , then the embedding

¯s1 ¯s2 (Ap1,q1 (Ω) ֒→ Ap2,q2 (Ω) (3.9

is compact if, and only if, p1 ≤ p2 and d d s1 − − s2 + > 0 . (3.10) p1 p2

(ii) Let b(Ω) < ∞. The embedding ¯s1 ¯s2 (Ap1,q1 (Ω) ֒→ Ap2,q2 (Ω) (3.11 is compact if 1 1 b(Ω) s s d > . 1 − 2 − − ∗ (3.12) p1 p2 p ∗ 1 1 If the embedding (3.11) is compact and p = ∞, then s1 − s2 − d( − ) > 0. p1 p2 ∗ 1 1 b(Ω) If the embedding (3.11) is compact and p < ∞, then s1 − s2 − d( − ) ≥ ∗ . p1 p2 p Remark 3.15. We collect some results about the sharpness of the above statement. (i) In case of b(Ω) < ∞ one can prove that (3.12) is a sufﬁcient and necessary condition for ∗ −jb(Ω) compactness of the embeddings, except of the case p < ∞ and lim supj→∞ bj(Ω)2 =0.

17 (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 Rd with ﬁnite Lebesgue measure, then the embedding (3.11) is compact if, and only if,

d d s1 − s2 − − > 0 . p1 p2 + 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 bj(Ω) are finite. But in contrast to the jd domain with a finite measure, the numbers bj (Ω) are not asymptotically equivalent to 2 . (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].

d Theorem 3.16. Let Ω be a uniformly E-porous quasi-bounded domain in R and let s1 > s2. Assume 1 ≤ pi, qi ≤ ∞, i =1, 2. (i) If b(Ω) = ∞ , then the embedding

B ¯s1 ¯s2 (idΩ : Bp1,q1 (Ω) ֒→ Bp2,q2 (Ω) (3.13

is nuclear if, and only if, p1 =1, p2 = ∞ and s1 − s2 > d. B (ii) Let b(Ω) < ∞. The embedding idΩ given by (3.13) is nuclear if 1 1 b(Ω) s1 − s2 − d − > . (3.14) p1 p2 t(p1,p2) Conversely, if the embedding (3.13) is nuclear and t(p1,p2)= ∞, that is, p1 =1 and p2 = ∞, 1 1 then s1 − s2 − d( − ) > 0. p1 p2 1 1 b(Ω) If the embedding (3.13) is nuclear and t(p1,p2) < ∞, then s1 − s2 − d( − ) ≥ . p1 p2 t(p1,p2) Proof. Step 1. At ﬁrst we show that

B idΩ is nuclear if, and only if, (3.15) 1 t −jδ (p1,p2) t (2 bj(Ω) )j∈N0 ∈ ℓt(q1,q2) if (q1, q2) < ∞, 1 t −jδ (p1,p2) t ((2 bj(Ω) )j∈N0 ∈ c0 if (q1, q2)= ∞.

d d Recall δ = (s1 − ) − (s2 − ). To verify this, we consider the following diagram: p1 p2

d I1 j(s1− ) Nj D1 Nj ¯s1 p1 Bp1,q1 (Ω) −−−−−−→ ℓq1 (2 ℓp1 ) −−−−−→ ℓq1 (ℓp1 ) B idΩ Dλ − −  I 1 d D 1  s 2 j(s2− ) Nj 2 Nj B¯ 2 (Ω) ←−−−−−−− ℓ (2 p2 ℓq ) ←−−−−−−− ℓ (ℓp ) p2,q2 y q2 2 qy2 2

18 −1 with I1 and I2 denote the wavelet-isomorphism from Theorem 3.12 and for i =1, 2 we deﬁne

d j(si − ) pi Nj Nj Di : ℓqi (2 ℓpi ) → ℓqi (ℓqi ), d j(si− ) pi Di : (xj,m)j∈N0,m=1,...,Nj 7→ 2 xj,m , j∈N0,m=1,...,Nj d j(si− ) −1 Nj pi Nj Di : ℓqi (ℓpi ) → ℓqi (2 ℓpi ), d −1 −j(si− ) N pi Di : (xj,m)j∈ 0,m=1,...,Nj 7→ 2 xj,m , j∈N0,m=1,...,Nj Nj Nj Dλ : ℓq1 (ℓp1 ) → ℓq2 (ℓp2 ), d d j((s2− )−(s1− )) p2 p1 Dλ : (xj,m)j∈N0,m=1,...,Nj 7→ 2 xj,m , j∈N0,m=1,...,Mj B −1 −1 B −1 −1 such that idΩ = I2 ◦ D2 ◦ Dλ ◦ D1 ◦ I1 and, vice versa, Dλ = D2 ◦ I2 ◦ idΩ ◦I1 ◦ D1 . B B Then Proposition 2.1(iii) applied to idΩ and Dλ, respectively, implies that the embedding idΩ is nuclear if, and only if, the operator Dλ is nuclear, which – in view of Proposition 2.8 – is −jδ the case, if, and only if, the sequence λ with λj,m = 2 for j ∈ N0, m = 1,...,Nj, belongs to Nj lt (l ), replaced by c if t(q , q )= ∞. But by deﬁnition of the spaces, (q1,q2) t(p1,p2) 0 1 2

1 t Nj −jδ (p1,p2) kλ|lt(q ,q )(lt )k = 2 N |ℓt(q ,q ) , (3.16) 1 2 (p1 ,p2) j j∈N0 1 2

1 1 B t(p ,p ) t(p ,p ) −jδ 1 2 N −jδ 1 2 N so idΩ is nuclear, if, and only if, (2 Nj )j∈ 0 ∈ ℓt(q1,q2), replaced by (2 Nj )j∈ 0 ∈ c0 if t(q1, q2)= ∞. d Because of 2 bj−2(Ω) ≤ Nj ≤ bj (Ω) - see (3.8) - we can replace Nj by bj (Ω) in the deﬁnition of λ, and ﬁnally arrive at (3.15). t B In particular, if p1 = 1, p2 = ∞, that is, (p1,p2)= ∞, then δ = s1 − s2 − d, so idΩ is nuclear if, and only if, s1 − s2 > d, which completes the proof in case of (i) and (ii) in this setting. It remains to deal with the remaining cases for t(p1,p2) < ∞ in dependence on b(Ω).

b(Ω) Step 2. Next we assume that b(Ω) < ∞ and t(p1,p2) < ∞. Let δ > and choose s>b(Ω) t(p1,p2) such that δ > s > b(Ω) . Then it follows from the deﬁnition of b(Ω), compare (3.4), that t(p1,p2) t(p1,p2) −js js limj→∞ bj(Ω)2 = 0. This implies that there exists a constant c such that bj (Ω) ≤ c2 . Thus 1 1 s −jδ t −jδ t δ > implies that (2 b (Ω) (p1,p2) ) N ∈ ℓt , with (2 b (Ω) (p1,p2) ) ∈ c in case of t(p1,p2) j j∈ 0 (q1,q2) j 0 t(q1, q2)= ∞. In view of Step 1, in particular (3.15), this concludes the proof of the sufﬁciency B of (3.14) for the nuclearity of idΩ in case (ii). 1 B t −jδ (p1,p2) R Now let idΩ be nuclear. Then (3.15) implies that 2 bj (Ω) → 0, which for any s ∈ is equivalent to saying that s 1 −j(δ− t(p ,p ) ) −js t(p ,p ) 2 1 2 2 bj (Ω) 1 2 −→ 0 . s −js −j(δ− t p ,p ) Assume now s< b(Ω), then lim sup bj (Ω)2 = ∞ and therefore 2 ( 1 2) → 0. Thus j→∞ δ > s for any s

Step 3. Finally we deal with the case b(Ω) = ∞ and know by Step 1, that p1 = 1, p2 = ∞, B s1 −s2 > d is sufficient for the nuclearity of idΩ . So it remains to show that the condition p1 =1 B and p2 = ∞ is also necessary for the nuclearity of idΩ . Note that the necessity of the condition 1 1 δ = s1 − s2 − d( − ) > 0 follows from Proposition 3.14 since otherwise the embedding is not p1 p2 compact. If b(Ω) = ∞, then by definition (3.4), for any s > 0 there is an increasing sequence jk and jks a positive constant c > 0 such that c2 ≤ bjk (Ω). This implies that the condition (3.15) is fulfilled only if δ > 0 and t(p1,p2)= ∞, that is, p1 =1 and p2 = ∞.

−jb(Ω) Remark 3.17. If 0 < lim supj→∞ bj(Ω)2 ≤ ∞ we can replace in Step 2 of the proof s with s

19 Remark 3.18. The condition (3.14) can be rewritten as follows: 1 1 (i) If 1 ≤ p2 ≤ p1 ≤ ∞, then it reads as s1 − s2 >b(Ω) − d( − ) . p2 p1 (ii) If 1 ≤ p ≤ p ≤ ∞, then it reads as s − s >b(Ω) − (b(Ω) − d)( 1 − 1 ). 1 2 1 2 p1 p2 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(p1,p2). 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(p1,p2), too, which in the end means t(p1,p2) = ∞, hence only p1 = 1, p2 = ∞. 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 B- and F -spaces appeared. This is now different to some extent.

d Corollary 3.20. Let Ω be a uniformly E-porous quasi-bounded domain in R and let s1 > s2, 1 ≤ pi < ∞, 1 ≤ qi ≤ ∞, i =1, 2. Then the embedding

F ¯s1 ¯s2 (idΩ : Fp1,q1 (Ω) ֒→ Fp2,q2 (Ω) (3.17 is nuclear if b(Ω) < ∞, and (3.14) is satisﬁed. Conversely, if idF is nuclear, then b(Ω) < ∞, and s − s − d( 1 − 1 ) ≥ b(Ω) . Ω 1 2 p1 p2 t(p1,p2) F In particular, if b(Ω) = ∞, then idΩ is never nuclear.

Proof. Note first, that in case of pi < ∞, i =1, 2, that is, when all spaces involved are properly F B defined, then idΩ given by (3.17) is nuclear if, and only if, idΩ given by (3.13) is nuclear (subject to appropriately adapted fine indices q1, q2). This is due to the embedding (1.9) (adapted to spaces on domains) and the independence of all the conditions in Theorem 3.16 on the fine F indices qi, i =1, 2. Hence, if b(Ω) = ∞ and idΩ given by (3.17) was nuclear, then in view of (1.9), (idB : B¯s1 (Ω) ֒→ B¯s2 (Ω Ω p1,min(p1,q1) p2,max(p2,q2)

is nuclear, which by Theorem 3.16(i) implies, in particular, p2 = ∞, which is not admitted. Assume now b(Ω) < ∞, then by an argument similar to the above one, we obtain that (3.14) F 1 1 b(Ω) is sufﬁcient for the nuclearity of id , and s1 − s2 − d( − ) ≥ necessary. Recall that Ω p1 p2 t(p1,p2) p2 < ∞ implies t(p1,p2) < ∞.

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Dorothee D. Haroske Institute of Mathematics Friedrich Schiller University Jena 07737 Jena Germany [email protected]

Hans-Gerd Leopold Institute of Mathematics Friedrich Schiller University Jena 07737 Jena Germany [email protected]

Leszek Skrzypczak Faculty of Mathematics & Computer Science Adam Mickiewicz University ul. Uniwersytetu Pozna ´nskiego 4 61-614 Pozna ´n Poland [email protected]