Approximation and entropy numbers in Besov spaces of generalized smoothness

Transcription

1 Journal of Approximation Theory ) wwwelseviercom/locate/at Approximation and entropy numbers in Besov spaces of generalized smoothness Fernando Cobos a,, Thomas Kühn b a Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, E Madrid, Spain b Mathematisches Institut, Faultät für Mathemati und Informati, Universität Leipzig, Johannisgasse 26, D Leipzig, Germany Received 28 August 2007; accepted 22 November 2007 Communicated by Andrei Martinez-Finelshtein Available online 5 January 2008 Dedicated to Professor Paul L Butzer on the occasion of his 80th birthday Abstract We determine the exact asymptotic behaviour of entropy and approximation numbers of the limiting restriction operator J : B s,ψ R d ) B s p,q 2 Ω), defined by J f ) = f Ω HereΩ is a non-empty bounded domain in R d, ψ is an increasing slowly varying function, 0 < p <, 0 < q 1, q 2, s R,andB s,ψ R d ) is the Besov space of generalized smoothness given by the function t s ψt) Our results improve and extend those established by Leopold [Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol 213, Marcel Deer, New Yor, 2000, pp ] 2008 Elsevier Inc All rights reserved MSC: 46E35; 47B06 Keywords: Besov spaces; Generalized smoothness; Compact embeddings; Entropy numbers; Approximation numbers 1 Introduction Besov spaces B s p,q appear in a natural way in approximation theory They play an important role in rational and spline approximation as one can see in the monographs by Butzer and Berens Authors have been supported in part by the Ministerio de Educación y Ciencia of Spain MTM ) Corresponding author addresses: cobos@matucmes F Cobos), uehn@mathuni-leipzigde T Kühn) /$ - see front matter 2008 Elsevier Inc All rights reserved doi:101016/at

2 F Cobos, T Kühn / Journal of Approximation Theory ) [4], Bergh and Löfström [2], Peetre [24], Petrushev and Popov [25], and the references given there During the last 30 years, the complete solution of some natural questions has required the introduction of Besov spaces of generalized smoothness B p,q These spaces are defined similar to B s p,q but replacing the regularity index s by a certain function with given properties This change allows to modify the smoothness properties of the space When t) = t s 1 + log t ) b, s, b R, we write B s,b p,q Note that if b = 0 we recover the usual spaces B s p,q Besov spaces of generalized smoothness have been investigated by many authors in different contexts We refer to the papers by Merucci [23] and by Cobos and Fernandez [8] for their connection with interpolation theory In the paper by Faras and Leopold [12] one can find historical remars and many other references, some of them related to the role that spaces of generalized smoothness play in probability theory and in the theory of stochastic processes Besov spaces B p,q are also of interest in fractal analysis and the related spectral theory At this point, we refer to the monographs by Triebel [28,30] and the references given there Our motivation for dealing with Besov spaces of generalized smoothness comes from the investigation of compactness of limiting embeddings Let Ω R d be a non-empty bounded domain with sufficiently smooth boundary, and let 0 < p 1, p 2, q 1, q 2, < s 2 < s 1 < with s 1 s 2 > d max1/p 1 1/p 2, 0) It is nown that the embedding id B : B s 1 p1,q 1 Ω) B s 2 p2,q 2 Ω) is compact and its entropy numbers satisfy e id B ) s 1 s 2 )/d see [11]; terminology and notation are explained in Section 2) The behaviour of approximation numbers has been determined see [11]) as well In the limit case s 1 d/p 1 = s 2 d/p 2 with s 1 s 2 and 0 < q 1 q 2, the embedding is continuous but not compact To overcome this obstruction, Leopold [22] modified the smoothness of the initial space, replacing B s 1 p1,q 1 Ω) by B s 1,b p 1,q 1 Ω) He showed that the embedding id B : B s 1,b p 1,q 1 Ω) B s 2 p2,q 2 Ω) is compact provided that b > max1/q 1 1/q 2, 0) As for the behaviour of entropy numbers in the special case s 1 = s 2 and p 1 = p 2, he proved that e id B ) log ) b if q 1 q 2 and b > 0 However, if q 1 > q 2 and b > 1/q 2 1/q 1, he only established the estimate log ) b e id B ) log ) b+1/q 2 1/q 1, 2) leaving open the exact behaviour of entropy numbers We shall show here that for any non-empty bounded domain Ω R d the entropy numbers of the restriction operator J : B s,b R d ) B s p,q 2 Ω), defined by J f ) = f Ω, behave asymptotically lie e J) log ) b+1/q 2 1/q 1 In particular, if Ω is sufficiently smooth so that there is an extension operator from B s,b Ω)into B s,b R d ), one has the same result for the embedding id B : B s,b Ω) B s p,q 2 Ω) Consequently, the exact behaviour of the entropy numbers in 2) is given by the upper bound, e id B ) log ) b+1/q 2 1/q 1 In addition we shall prove that the approximation numbers of J behave in the same way as its entropy numbers Furthermore, our techniques of proof also apply to the more general case J : B s,ψ R d ) B s p,q 2 Ω) with ψ being an increasing slowly varying function; we derive sharp results on approximation and entropy numbers in this situation, too 1)

3 58 F Cobos, T Kühn / Journal of Approximation Theory ) We start by establishing a result on equivalence of approximation and entropy numbers for abstract operators between quasi-banach spaces This result is of independent interest Then we apply it to embeddings between certain vector-valued sequence spaces, and finally we derive the results on function spaces by using wavelet bases The paper is organized as follows In Section 2 we recall some general facts on entropy numbers, approximation numbers, and Besov spaces B p,q Equivalence results between entropy and approximation numbers of abstract operators will be given in Section 3 Section 4 contains the results on approximation and entropy numbers of embeddings between sequence spaces and between function spaces 2 Preliminaries Subsequently, given two sequences b ), d ) of non-negative real numbers we write b d if there is a constant c > 0 such that b cd for all N, while b d means b d b For non-negative functions and ψ on 0, ) the notation ψ has a similar meaning; there are positive constants c 1 and c 2 such that c 1 t) ψt) c 2 t)forallt > 0 Let X and Y be quasi-banach spaces and let T LX, Y ) be a bounded linear operator from X into Y For N, theth approximation number a T )oft is defined by a T ) = inf { T R : R LX, Y ) with ran R < } Here ran R is the dimensionofthe range of R The th dyadic) entropy number e T )oft is the infimum of all ε > 0 such that there are y 1,, y q Y with q 2 1 for which q T B X ) y + εb Y ) =1 holds, where B X, B Y are the closed unit balls of X and Y, respectively see [26,6,11]) Clearly, the following monotonicity property holds T =a 1 T ) a 2 T ) 0 and T e 1 T ) e 2 T ) 0 Moreover, entropy and approximation numbers are multiplicative, that is, for all, l N a +l 1 S T ) a S)a l T ), e +l 1 S T ) e S)e l T ) Note that T is compact if and only if lim e T ) = 0 The asymptotic decay of the sequence e T )) can be considered as a measure of the degree of compactness of THowever,thereare compact operators T such that lim a T ) > 0see[11]) Several authors have investigated the relationships between entropy and approximation numbers see [11, Section 133]) In the next section we shall show a new result in this direction Although we do not deal here with eigenvalues, it is worth to recall that there are close connections between these quantities and eigenvalues, which are the basis of many applications Let X be a complex) quasi-banach space and let T LX, X) be a compact operator We denote by λ T )) the sequence of eigenvalues of T, counted according to their multiplicity and ordered by decreasing modulus If T has only a finite number of eigenvalues and M is the sum of their multiplicities, then we put λ T ) = 0forall > M The celebrated Carl Triebel inequality establishes that λ T ) 2e T ), N see [5,7,11]) This inequality and estimate 1) were

4 F Cobos, T Kühn / Journal of Approximation Theory ) the starting points for Edmunds and Triebel in [11] to investigate eigenvalue distributions of degenerate elliptic differential and pseudodifferential operators As for approximation numbers, it was shown by König see [14])thatifX is a Banach space then λ T ) =lim m m a T m ) Next we recall the definitions of the function spaces that we shall need in the sequel We wor on the d-dimensional Euclidean space R d and on bounded domains Ω, that is, bounded open subsets of R d By SR d )ands R d ) we denote the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on R d, and the space of tempered distributions on R d, respectively The symbol F stands for the Fourier transform on S R d ), and F 1 for the inverse Fourier transform Let φ 0 be a C function on R d with φ 0 x) = 1if x 1 and supp φ 0 {x R d : x < 2} For N and x R d, put φ x) = φ 0 2 x) φ x) Then one has a dyadic resolution of unity, =0 φ x) = 1forallx R d By B we denote the class of all continuous functions :0, ) 0, ) with 1) = 1 and such that tu) t) = sup < for every t > 0 u>0 u) For B and 0 < p, q, thebesov space of generalized smoothness B p,qr d ) consists of all f S R d ) having a finite quasi-norm 1/q ) q f Bp,q R d ) = 2 ) F 1 φ F f ) L p =0 with the usual modification if q = Note that this definition also maes sense if is only equivalent to a function in B Moreover, as an immediate consequence of Hölder s inequality, we have the following result: Given 1, 2 B, 0< p < and 0 < q 1, q 2, and defining 1/q = max1/q 2 1/q 1, 0), there is a bounded embedding B 1 R d ) B 2 p,q 2 R d ) if and only if ) 2 2 )/ 1 2 ) q 3) Spaces B p,qr d ) were considered in [23,8,1], among other papers For t) = t s with s R, we recover the usual Besov spaces B s p,q Rd ), see [4,24,27,28,30] A case that will be of special interest for us is when t) t s ψt), where ψ is a slowly varying function and s R Recall that a Lebesgue measurable function ψ :0, ) 0, )issaidtobeslowly varying if ψut) lim = 1 forallu > 0 t ψt) see [3,10]) When t) t s ψt) we write B s,ψ p,qr d ) instead of B p,qr d ) If ψt) = 1 + log t ) b with b R, we simply put B s,b p,qr d ) Spaces B s,b p,qr d ) arise by extrapolation procedures within the scale of usual Besov spaces see [9])

5 60 F Cobos, T Kühn / Journal of Approximation Theory ) A related, but more restrictive concept than slow variation is due to Triebel see [28, Section 222]) He called a function Ψ :0, 1] 0, ) admissible, ifψ is monotone and Ψ2 ) Ψ2 2 ) In this case the function { Ψt) if0 < t 1, ϱ Ψ t) = Ψt 1 ) if1 t < is equivalent to a slowly varying function Spaces B s,ψ p,q R d ), defined as B p,qr d ) with t) = t s ϱ Ψ t), are important in fractal analysis see [28, Sections 22 and 23], [30, Section 195], and the references given there) In our later considerations wavelet representations of Besov spaces will be an essential tool, allowing us to transform our problem in function spaces to the simpler context of sequence spaces Recently this technique has been successfully applied to deal with similar problems for weighted Besov and Triebel Lizorin spaces see eg [13,19 21,17]) Before we briefly describe the wavelet representation of Besov spaces B p,qr d ), we introduce sequence spaces b p,q which are naturally associated to this construction Let N 0 = N {0}, put L 0 = 1andL = 2 d 1if NGiven B and 0 < p, q, the space b p,q consists of all sequences λ = λ lm ), indexed by N 0,1 l L,andm Z d, such that the quasi-norm λ b = ) q/p 1/q 2 )2 d/p) q λ lm p p,q l m is finite, with the usual modification if p = or q = It is well nown that for every r N there exist compactly supported functions ψ 0, ψ l C r R d ), 1 l 2 d 1, satisfying the moment conditions xα ψ l x) dx = 0 for allα N d R d 0 with α r, such that the system {2 d/2 ψ lm : N 0, 1 l L, m Z d} is an orthonormal basis in L 2 R d ), where the functions ψ lm are defined by { ψ0 x m) if = 0, l = 1, m Z d, ψ lm = ψ l 2 1 x m) if N, 1 l 2 d 1, m Z d For the usual spaces B s p,q Rd ), the following result can be found in the paper by Triebel [29],his boo [30], and the references therein The extension to Besov spaces of generalized smoothness B p,qr d ) is due to Almeida [1] Let B, 0< p <, and0< q Then there exists a number r, p) > 0 such that for any system {ψ lm } as above with r > r, p) the following holds: A distribution f S R d ) belongs to B p,qr d ) if and only if it can be represented as f = λ lm ψ lm with λ f ):=λ lm ) b p,q, 4) lm

6 F Cobos, T Kühn / Journal of Approximation Theory ) where the series converges unconditionally in S R d ) and the coefficients are uniquely determined by λ lm = 2 d f, ψ lm Moreover, λ f ) b p,q defines a quasi-norm which is equivalent to f B p,q R d ) Now let Ω R d be a bounded domain As usual the space B p,qω) is defined by restriction of B p,qr d )toω, and equipped with the quasi-norm f B p,q Ω) = inf { g B p,q R d ) : g B p,qr d ), g Ω = f 5) } 6) For any finite or countable) index set I, we denote by p I ) the space of all complex sequences y = y i ) i I with finite quasi-norm { y p = i I y i p) 1/p if 0 < p <, sup i I y i if p = If I = N or I ={1,, M}, we write simply p or M p, respectively Finally, given 0 < p, q, w > 0, and M N, let q w M p ) denote the space of all vector-valued sequences x = x ) N0 with x M p and x q w M p ) = w x p ) q =0 1/q < with the usual modification if q = ) If all w = 1, we simply write q M p ) 3 On equivalence of approximation and entropy numbers In general, the asymptotic behaviour of the approximation numbers of an operator may be quite different from the asymptotic behaviour of its entropy numbers The following examples illustrate this fact Example 31 For the diagonal operator D : 2 2 complex spaces), Dx = 2 n x n )for x = x n ), one has see [6, p 106]) a D) = 2 and e D) 2 Example 32 Consider now the diagonal operator D : 1 2 defined by Dx = log1 + n)) 1/2 x n ) By [26, Theorem 11117],wehave 1/2 n + 1 a D) = sup n=1 log ) n log1 + )) 1/2 and according to the results in [15,16], e D) 1/2 However, under certain assumptions entropy and approximation numbers are equivalent Our next result gives a sufficient condition for this equivalence

7 62 F Cobos, T Kühn / Journal of Approximation Theory ) Theorem 33 Let X, Y be quasi-banach spaces and let T LX, Y ) be a bounded linear operator Assume that b ) is a decreasing sequence of positive real numbers satisfying that lim b = 0 and b b 2 7) Then the one-sided inequalities a T ) b e T ) 8) imply the equivalence a T ) b e T ) 9) Proof A result of Triebel, which extends a previous inequality due to Carl see [11, Section 133]), shows that for any positive increasing function f : N R with f ) f 2), there is a constant C > 0 depending only on f and the quasi-triangle constant of Y such that for all m N max f )e T ) C max f )a T ) 10) 1 m 1 m Applying this inequality with f ) = 1/b, we obtain the desired upper estimate for entropy numbers 1 b m e m T ) max 1 m 1 b e T ) C max 1 m 1 b a T ) C Hence, to complete the proof it suffices to prove the lower estimate for approximation numbers By 7) there exists a constant C b > 1suchthatb C b b 2 for all N Setting α = log 2 C b > 0, this implies that α b C b m α b m whenever 1 m 11) On the other hand, assumption 8) yields that there are positive constants C 1, C 2 such that C 1 a T ) b C 2 e T ) forall N 12) Whence, applying 10) now with f ) = α+1, we get that, with a constant C > 0 depending only on α and Y, b m m α+1 max C 2 1 m α+1 e T ) C max 1 m α+1 a T ) ) = C max max 1 δm α+1 a T ), max δm m α+1 a T ), where 0 < δ < 1 will be chosen later appropriately Due to 11)and12)wehaveforany δm, α+1 a T ) δm α b C 1 δc b C 1 mm α b m Hence, taing δ < C 1 /C 2 CC b ), we see that C max 1 δm α+1 a T ) < b m C 2 m α+1

8 F Cobos, T Kühn / Journal of Approximation Theory ) This yields that b m m α+1 C max C 2 δm m α+1 a T ) Cm α+1 a [δm] T ), where [ ] is the greatest integer function Consequently, a [δm] T ) b m C 2 C b [δm], which gives a m T ) b m This completes the proof Note that the assumption b b 2 is essential in Theorem 33 For instance, the operator in Example 31 satisfies 8) with b = 2, while 9) fails In the next section we shall also need, in addition to Theorem 33, the following recent result of the second named of the present authors [18] Lemma 34 Let 0 < q < p and 1/r = 1/q 1/p Assume that σ m ) r is a decreasing sequence of positive real numbers and let D σ : p q be the diagonal operator defined by D σ x m ) = σ m x m ) Put w m = =m σ r )1/r If w m w 2m then e m D σ ) w m In the Banach case 1 q < p, Pietsch showed that a m D σ ) = w m, even without the doubling condition w m w 2m see [26, Theorem 11114]) However, Lemma 34 does not remain true without that assumption see [18]) 4 Compact embeddings Our aim is to prove asymptotically sharp estimates for approximation and entropy numbers of the restriction operator J : B s,ψ R d ) B s p,q 2 Ω) defined by Jf = f Ω, where Ω R d is a bounded domain, ψ an increasing slowly varying function, 0 < p <, 0 < q 1, q 2, ands R For spaces on R d we have, as a special case of the general result 3), that there is a bounded embedding B s,ψ R d ) B s p,q 2 R d ) if and only if ψ2 ) 1) q where 1/q = max1/q 2 1/q 1, 0) 13) Observe that this condition is also equivalent to the existence of a bounded embedding id : q1 ψ2 ) M p ) q2 M p ), with arbitrary M N Using the wavelet description of Besov spaces, we establish now two factorization diagrams for the operator J, which will reduce the study of J to the investigation of embeddings in certain sequence spaces To this end we introduce the index sets and I ={l, m) :1 l L, supp ψ lm ) Ω } Ĩ ={l, m) :1 l L, supp ψ lm ) Ω}

9 64 F Cobos, T Kühn / Journal of Approximation Theory ) Since the wavelets ψ lm are compactly supported and Ω is bounded with non-empty interior, one easily verifies that card I 2 d card Ĩ 14) Now let us consider the operators S : B s,ψ R d ) q1 ψ2 ) p I )) and T : q2 p I )) B s p,q 2 Ω) defined by and Sf = ) 2 s+ d1 1/p) f, ψ lm N 0,l,m) I T μ lm ) = =0 l,m) I 2 s d1 1/p) μ lm ψ lm Ω It follows from 4), 5) and the definition of the quasi-norm in B s p,q 2 Ω) that the operators S and T are bounded If condition 13) holds, then the identity id b : q1 ψ2 ) p I )) q2 p I )) is bounded and we have the following commutative diagram Consequently, J is bounded as well, and by the multiplicativity of entropy and approximation numbers we have e J) e id b ) and a J) a id b ) Conversely, let the operators T : q1 ψ2 ) p Ĩ )) B s,ψ R d ) and S : B s p,q 2 Ω) q2 p Ĩ )) be defined by T η lm ) = =0 l,m) Ĩ 2 s d1 1/p) η lm ψ lm and S f = 2 s+ d1 1/p) f, ψ lm ) N 0,l,m) Ĩ

10 F Cobos, T Kühn / Journal of Approximation Theory ) Note that S is well defined because supp ψ lm ) Ω for any l, m) Ĩ Similar arguments as above show that T and S are bounded Hence, if J is bounded, we conclude from the commutative diagram that the identity ĩd b is bounded, ie condition 13) holds Moreover we have e ĩd b ) e J) and a ĩd b ) a J) Summarizing our considerations, we have shown that the restriction operator J is bounded if and only if 13) holds, and in this case one has e ĩd b ) e J) e id b ) and a ĩd b ) a J) a id b ) 15) Before we can determine the exact asymptotic behaviour of the entropy and approximation numbers of J, we need some results on diagonal operators between vector-valued sequence spaces Lemma 41 Let M 2 d,0< p 2 p 1, 0< q 2 < q 1, and set 1/p = 1/p 2 1/p 1, 1/q = 1/q 2 1/q 1 Let σ ) q be a decreasing sequence such that σ σ +1 Then the diagonal operator D σ : q1 l M p 1 ) q2 l M p 2 ) satisfies defined by D σ x ) = σ M 1/p x ) a D σ ) b and e D σ ) b where b = 2 d Proof For N N,weset N = N 1 =0 M +1 Let P N : q1 l M p 1 ) q1 l M p 1 ) be the proection onto the first N coordinates, P N x = x 0, x 1,, x N 1, 0, 0, ) for x = x ) N0 Since ran P N < N,wehave a N D σ ) D σ D σ P N The norm can be estimated using Hölder s inequality We get that 1/q a N D σ ) = b 2 Nd 16) σ q =N σ q 1/q

11 66 F Cobos, T Kühn / Journal of Approximation Theory ) From σ σ +1 we infer that b b 2,andM 2 d implies N 2 Nd Hence 16) and the monotonicity of approximation numbers yield that a D σ ) b Finally, applying 10) with f ) = 1/b, we conclude that e D σ ) b Lemma 42 Let M 2 d,0< p <, and 0 < q 1 q 2 Let σ ) be a decreasing sequence with σ σ +1 Then for the diagonal operator D σ : q1 l M p ) q2 l M p ) defined by D σ x ) = σ x ) one has { } a D σ ) e D σ ) b where b = sup σ :2 d Proof Again we have b b 2, and according to Theorem 33 it suffices to prove the inequalities a D σ ) b and e D σ ) b For the approximation numbers we use the same arguments as in the preceding proof, observing that now a D σ ) D σ D σ P N =σ N = b 2 Nd, since σ ) is decreasing As above we conclude that a D σ ) b It remains to show that e D σ ) b Let R : M r p q1 M p ) be the map that associates x Let to x the sequence having all coordinates equal to 0 but the rth coordinate which is σr 1 Q : q2 M p ) M p be the proection onto the rth coordinate Using the commutative diagram we get by multiplicativity of entropy numbers e id) R e D σ ) Q =σ 1 r e D σ ) Tae = M r 2 rd Since e id) = e id : M r p M r p ) 1 see [26,11]), it follows that e 2 rdd σ ) cσ r = cb 2 rd for some constant c > 0andallr N Using b b 2, we conclude that e D σ ) b Lemma 43 Let M 2 d,0< p, 0< q 2 < q 1 and 1/q = 1/q 2 1/q 1 Let σ ) q be a decreasing sequence with σ σ +1 Then for the diagonal operator D σ : q1 l M p ) q2 l M p ) defined by D σ x ) = σ x ) one has a D σ ) e D σ ) b = 2 d σ q 1/q

12 F Cobos, T Kühn / Journal of Approximation Theory ) Proof Lemma 41 gives a D σ ) b ;andσ σ +1 implies b b 2 By Theorem 33 it suffices therefore to show that e D σ ) b To this end choose 0 <w<minp, q 2 ) and put 1/u = 1/w 1/q 2 Let D 1 : M ) q1 M be the operators defined by D 1 x ) = σ q/q 1 p ), D 2 : q2 M p M 1/p x ), D 2 x ) = ) w M w ) σ q/u M 1/p 1/w x ) For the entropy numbers of these operators Lemma 41 yields the estimates e D 1 ) 1 σ 1/q q = b q/q 1 and e D 2 ) 1/u σ q = b q/u 2 d 2 d Let D : M ) w M w ) be the composition D = D 2 D σ D 1 Then ) Dx ) = σ q1/q 1+1/q+1/u) M 1/w x = σ q/w M 1/w x ), and the following diagram commutes Viewing M )and w M w )as and w, respectively, the operator D can be realized as a diagonal operator from into w with diagonal entries δ r = σ q/w M 1/w for 1 M < r =0 M, =0 that means, each entry σ q/w M 1/w appears M times in the sequence δ r ) Our assumption σ σ +1 implies r=m δr w )1/w r=2m δr w )1/w Therefore we can apply Lemma 34 to the operator D, and we get for all N N and N = N 1 =0 M + 1 the estimate e N D) w δr w = σ q = b q 2 Nd r N N Observing that N 2 Nd,thisgivese D) b q/w By the multiplicativity of entropy numbers and using b b 3 we obtain b q/w b q/w 3 e 3 D) e D 1 ) e D σ ) e D 2 ) b q/q 1 e D σ ) b q/u This shows e D σ ) b q1/w 1/q 1 1/u) = b, and the proof is finished

13 68 F Cobos, T Kühn / Journal of Approximation Theory ) Now we determine the exact asymptotic behaviour of entropy and approximation numbers of the identity id : q1 ψ2 ) M p ) q2 M p ), with M 2 d Inviewof14), it is clear that the identities id b and ĩd b used in the first two factorization diagrams of this section are of this form Theorem 44 Let 0 < p <, 0< q 1, q 2, 1/q = max1/q 2 1/q 1, 0), let ψ be an increasing slowly varying function such that ψ2 ) q) q, and let M 2 d Then for the identity id : q1 ψ2 ) M p ) q2 M p ) one has { ψ 1/d ) 1 if q 1 q 2, a id) e id) 1/d ψt) q dt/t ) 1/q if q 1 > q 2 Proof Set σ = ψ2 ) 1 for N 0 It is easy to chec that a id) = a D σ )ande id) = e D σ ), where D σ : q1 M p ) q2 M p ) is the diagonal operator D σ x ) = σ x ) If q 1 q 2 we can apply Lemma 42 Indeed, since ψ is slowly varying, we have in particular ψ2t) 1 ψt) 1, whence σ σ +1 Moreover, using that ψ is increasing, we get { } { } b = sup σ :2 d = sup ψ2 ) 1 :2 d = ψ 1/d ) 1 Therefore, Lemma 42 implies a D σ ) e D σ ) ψ 1/d ) 1 In the case q 1 > q 2 we use Lemma 43 Since ψ is increasing and satisfies ψ2t) ψt), we obtain that b q = 2 d σ q = 2 d ψ2 ) q 2 d 2 +1 Consequently, Lemma 43 yields ) 1/q a D σ ) e D σ ) ψt) q dt/t 1/d The proof is finished 2 ψt) q dt/t 1/d ψt) q dt/t Applying Theorem 44 to the identities id b and ĩd b we see that the approximation and entropy numbers of these operators are equivalent, and combining this with the inequalities 15) we immediately obtain our main result Theorem 45 Let Ω be a bounded domain in R d, let 0 < p <, 0< q 1, q 2, 1/q = max1/q 2 1/q 1, 0), s R and let ψ be an increasing slowly varying function with ψ2 ) 1 ) q Let the operator J : B s,ψ R d ) B s p,q 2 Ω) be given by J f ) = f Ω Then { ψ 1/d ) 1 if q 1 q 2, a J) e J) 1/d ψt) q dt/t ) 1/q if q 1 > q 2 In particular, the operator J is compact if and only if ψ2 ) 1) { c0 if q 1 q 2, q if q 1 > q 2

14 F Cobos, T Kühn / Journal of Approximation Theory ) Clearly, if Ω is sufficiently smooth so that there is an extension operator from B s,ψ Ω) into B s,ψ R d ), then an analogous result to Theorem 45 holds for the embedding id B : B s,ψ Ω) B s p,q 2 Ω) Specializing Theorem 45 for the case ψt) = 1 + log t ) b we can complement the results established by Leopold in [22, Theorem 31], by giving the exact order of decay for entropy numbers when q 1 > q 2 and the exact behaviour of approximation numbers for any 0 < q 1, q 2 Corollary 46 Let Ω, p, q 1, q 2, q, sbeasintheorem45, and let b > 1/q Then the operator J : B s,b R d ) B s p,q 2 Ω) is compact and a J) e J) { log ) b if q 1 q 2, log ) b+1/q if q 1 > q 2 We can also cover the case q 1 > q 2, b = 1/q if we modify the function ψ by inserting a double logarithmic factor Corollary 47 Let Ω, p, q 1, q 2, q, s be as in Theorem 45, where q 2 < q 1 Let ψt) = 1 + log t ) 1/q 1 + log1 + log t ) ) b with b > 1/q Then the operator J : B s,ψ R d ) B s p,q 2 Ω) is compact and a J) e J) log log ) b+1/q Clearly Theorem 45 also applies to slowly varying functions ψ which are not admissible for the definition see the Preliminaries) We give a typical example of such functions Corollary 48 Let Ω and p, q 1, q 2, q be as in Theorem 45, and let ψt) = exp clog1 + t)) α) where c > 0 and 0 < α < 1 Then the operator J : B s,ψ R d ) B s p,q 2 Ω) is compact and a J) e J) ψ 1/d ) 1 log ) 1 α)/q Of course, the list of examples could be extended For other possible candidates we refer to [20,17], where such functions appeared as weights in weighted function spaces We conclude the paper by the following remar: Because we were interested in limiting embeddings, we wored in this paper with slowly varying functions ψ But apart from monotonicity) we used in the proofs only the doubling condition ψt) ψ2t), a property which also holds for functions in the much wider class B That means, implicitly we have shown an analogous version of Theorem 45 for more general restriction operators J : B 1 R d ) B 2 p,q 2 Ω) with arbitrary functions 1, 2 B such that t) := 2 t)/ 1 t) is decreasing and 2 ) ) q For instance, in the non-limiting case t) = t s ψt) with s > 0andψslowly varying, we obtain a J) e J) s/d ψ 1/d ) We leave the details to the reader

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