Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I

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1 Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I Dorothee D. HAROSKE and Leszek SKRZYPCZAK Mathematical Institute Friedrich-Schiller-University Jena D Jena Germany haroske@minet.uni-jena.de Faculty of Mathematics & Computer Science Adam Mickiewicz University Ul. Umultowska Poznań Poland lskrzyp@amu.edu.pl Received: July 0, 2007 Accepted: August 30, 2007 ABSTRACT We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt A p class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases. Key words: wavelet bases, Muckenhoupt weighted function spaces, compact embeddings, entropy numbers, approximation numbers Mathematics Subject Classification: 46E35, 42C40, 42B35, 4A46, 47B06. Introduction In recent years, some attention has been paid to compactness of embeddings of function spaces of Sobolev type as well as to analytic and geometric quantities describing this compactness, in particular, corresponding approximation and entropy numbers. As an application D. E. Edmunds and H. Triebel [2] proposed a program to investigate the spectral properties of certain pseudo-differential operators based on the The research was partially supported by the program DAAD-MEN D/06/2558. author was also supported by the DFG Heisenberg fellowship HA 2794/- The first Rev. Mat. Complut , no., ISSN:

2 asymptotic behavior 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 H. Triebel, see [9], and were continued and extended by Th. Kühn, H.-G. Leopold, W. Sickel and the second author in the series of papers [23 25]. In all the above papers the authors considered the class of so-called admissible weights. These are smooth weights with no singular points. One can take wx = + x 2 α/2, α R, x R n, as a prominent example. In this paper we follow a different approach and consider weights from the Muckenhoupt class A. In contrast to admissible weights the A weights may have local singularities, which can influence properties of the embeddings of function spaces. Now the weight wx = x α, α> n, may serve as a classical example. Weighted Besov and Triebel-Lizorkin spaces with Muckenhoupt weights are well known concepts, see [3 6, 5, 3, 32] and, more recently, [, 2, 8]. But the compactness of Sobolev embeddings of such spaces were not yet studied in detail. The present paper fills in this gap. First we give a necessary and sufficient condition on the parameters and weights of the Besov spaces which guarantees compactness of the corresponding embeddings. Then we determine the exact asymptotic behavior of entropy and approximation numbers of the embeddings of the spaces with weights that have purely polynomial growth, both near some singular point and at infinity. In both cases we use the technique of discretization, i.e., we reduce the problem to the corresponding problem for some suitable sequence spaces. This can be done in terms of wavelet bases. Then one obtains estimates of the following type: if the weight is of type x α if x, wx x β with α> n, β > 0, if x >, and A s p,q stands for either Besov spaces Bp,q s or Triebel-Lizorkin spaces Fp,q s with s 2 s,0<, <, 0<q,q 2, then id : A s R n,w A s2 R n is compact if, and only if, β > n p and α δ>max, n p, where δ = s s 2 n + n and p = max, 0, as usual. For the entropy numbers of this embedding we can prove that for k N, and e k id : A s R n,w A s2 R n k β np + p p 2 e k id : A s R n,w A s2 R n k s s 2 n if if β <δ, β >δ. 2008: vol. 2, num., pags

3 There are parallel results for the limiting case β = δ and for approximation numbers. It is remarkable that apart from the criterion for compactness of the embedding the parameter α, connected with the local singularity has no further influence on the degree of compactness measured in terms of entropy or approximation numbers, respectively. The study of entropy and approximation numbers of embeddings in the context of more general weights is postponed; likewise applications are out of the scope of the present paper. The paper is organized as follows. In section we recall basic facts about Muckenhoupt weights and weighted spaces needed later on. We also prove the wavelet characterization of Besov spaces via compactly supported wavelets. Section 2 is devoted to the continuity and compactness of the embeddings. For weights of purely polynomial growth we find simpler conditions. In the last two sections we determine exact asymptotic behavior of the entropy and approximation numbers for purely polynomial weights.. 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}, byc the complex plane, by R n euclidean n-space, n N, and by Z n the set of all lattice points in R n having integer components. The positive part of a real function f is given by f + x = maxfx, 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,c 2 > 0 such that c 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 n-dimensional Lebesgue measure in the sequel. If not otherwise indicated, log is always taken with respect to base 2... Muckenhoupt weights We briefly recall some fundamentals on Muckenhoupt classes A p. Definition.. Let w be a positive, locally integrable function on R n, and <p<. Then w belongs to the Muckenhoupt class A p, if there exists a constant 0 <A< such that for all balls B the following inequality holds: /p /p wx dx wx p /p dx A, B B B B : vol. 2, num., pags

4 where p is the dual exponent to p given by /p +/p = and B stands for the Lebesgue measure of the ball B. The limiting cases p = and p = can be incorporated as follows. By a weight w we shall always mean a locally integrable function w L loc R n, positive a.e. in the sequel. Let M stand for the Hardy-Littlewood maximal operator given by Mfx = sup fy dy, x R n, Bx, r Bx,r B where B is the collection of all open balls Bx,r Bx, r = y R n : y x <r}, r > 0. Definition.2. A weight w belongs to the Muckenhoupt class A if there exists a constant 0 <A< such that the inequality Mwx Awx holds for almost all x R n. The Muckenhoupt class A is given by A = p> A p. Since the pioneering work of Muckenhoupt [27 29], these classes of weight functions have been studied in great detail, we refer, in particular, to the monographs [6; 36, chap. 5; 37; 38, chap. 9], for a complete account on the theory of Muckenhoupt weights. We use the abbreviation wω = wx dx, Ω where Ω R n is some bounded, measurable set. For convenience, we recall a few basic properties only; in particular, the class A p is stable with respect to translation, dilation and multiplication by a positive scalar. Moreover, it is known: Lemma.3. Let <p<. i If w A p, then we have w p /p A p, where /p +/p =. ii w A p possesses the doubling property, i.e., there exists a constant c>0 such that wb 2 cwb holds for arbitrary balls B = Bx, r and B 2 = Bx, 2r with x R n, r>0. iii Let <. Then we have A p A p : vol. 2, num., pags

5 iv If w A p, then there exists some number r<psuch that w A r. Note that the somehow surprising property iv is closely connected with the so-called reverse Hölder inequality, a fundamental feature of A p weights, see [36, chap. 5, Prop. 3, Cor.]. In our case this fact will re-emerge in the number r w := infr :w A r }, w A, 2 that plays an essential role later on. Obviously, r w <, and w A rw if, and only if, r w = in view of iv. In the sequel we shall use decompositions of A p weights into A weights several times, for that reason we collect a few related facts, see [3, Lemma 2.3; 36, chap. 5,.4,.9, 5.3, 6.], and also [38, chap. 9, Thm. 2., secs. 4, 5]. Lemma.4. i Let, <, w A p, w 2 A p2, and θ [0, ]. Let w p θ = w w θ 2, p = θ + θ. Then w A p. ii The minimum, maximum, and the sum of finitely many A weights yields again an A weight. iii Let w and w 2 be A weights, and p<. Then w = w w p 2 A p. Conversely, suppose that w A p, then there exist v,v 2 A such that w = v v p 2. iv A positive, locally integrable function w on R n belongs to A p, p<, if, and only if, /p c fy dy f p xwx dx 3 B B wb B holds for all nonnegative f and all balls B. Of course, Lemma.3 i can be understood as a special case of Lemma.4 i. Moreover, let E B and f = χ E, then 3 implies that E B c whenever w A p, p<. /p we, E B, 4 wb : vol. 2, num., pags

6 Examples.5. i One of the most prominent examples of a Muckenhoupt weight w A p, p<, is given by wx = x ϱ, with wx = x ϱ n <ϱ<np, if <p<, A p if, and only if, n <ϱ 0, if p =. Thus r w =+ ϱ+ n and w A r w if ϱ 0, whereas w/ A rw for ϱ>0. ii Let vx = x α log β 2 + x or vx = x α log β 2 + x. Then, also in view of Lemma.4 one verifies that β R, if n<α<0, v A if β 0, if α =0, whereas the counterpart for <p< reads as v A p if n<α<np, β R, see also [3, Lemma 2.3]. Similarly as above, r v =+ α+ n. iii Finally, wx = x n α A p if, and only if, <α<p, where x = x,...,x n R n and p <. This is a special case of [3, Lemma 2.3], see also [8, Prop. 2.8]. We return to these examples and combinations of them in the sequel..2. Function spaces of type Bp,q s Rn,w and Fp,q s Rn,w with w A Let w A be a Muckenhoupt weight, and 0 <p<. Then the weighted Lebesgue space L p R n,w contains all measurable functions such that /p f L p R n,w = fx p wx dx R n is finite. Note that for p = one obtains the classical unweighted Lebesgue space, L R n,w=l R n, w A. 5 We thus restrict ourselves to p< in what follows. The Schwartz space SR n and its dual S R n of all complex-valued tempered distributions have their usual meaning here. Let ϕ 0 = ϕ SR n be such that supp ϕ y R n : y < 2} and ϕx = if x, 2008: vol. 2, num., pags

7 and for each j N let ϕ j x =ϕ2 j x ϕ2 j+ x. Then ϕ j } j=0 forms a smooth dyadic resolution of unity. Given any f S R n, we denote by Ff or ˆf, and F f or f, its Fourier transform and its inverse Fourier transform, respectively. Let f S R n, then the compact support of ϕ j ˆf implies by the Paley-Wiener-Schwartz theorem that ϕ j ˆf is an entire analytic function on R n. Definition.6. Let 0 <q, 0 <p<, s R, and ϕ j } j a smooth dyadic resolution of unity. Assume w A. i The weighted Besov space Bp,qR s n,w is the set of all distributions f S R n such that /q f Bp,qR s n,w = 2 jsq F ϕ j Ff L p R n,w q 6 j=0 is finite. In the limiting case q = the usual modification is required. ii The weighted Triebel-Lizorkin space F s p,qr n,w is the set of all distributions f S R n such that /q f Fp,qR s n,w = 2 jsq F ϕ j Ff q L p R n,w j=0 is finite. In the limiting case q = the usual modification is required. Remark.7. The spaces Bp,qR s n,w and Fp,qR s n,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, and SR n Bp,qR s n,w S R n, similarly for the F -case, where the first embedding is dense if q< ; see [3]. Moreover, for w 0 A we re-obtain the usual unweighted Besov and Triebel-Lizorkin spaces; we refer, in particular, to the series of monographs by Triebel [39 42], for a comprehensive treatment of the unweighted spaces. The above spaces with weights of type w A have been studied systematically by Bui et al. in [3 6]. It turned out that many of the results from the unweighted situation have weighted counterparts: e.g., we have Fp,2R 0 n,w=h p R n,w, 0 <p<, where the latter are Hardy spaces, see [3, Thm..4], and, in particular, h p R n,w=l p R n,w=fp,2r 0 n,w, <p<, w A p, see [37, chap. VI, Thm. ]. Concerning classical Sobolev spaces Wp k R n,w built upon L p R n,win the usual way it holds W k p R n,w=f k p,2r n,w, k N 0, <p<, w A p, see [3, Thm. 2.8]. Further results, concerning, for instance, embeddings, real interpolation, extrapolation, lift operators, duality assertions can be found in [3, 4, 6, 32] : vol. 2, num., pags

8 Rychkov extended in [33] the above class of weights in order to incorporate locally regular weights, creating in that way the class A loc p. Recent works are due to Roudenko [5, 3, 32] and Bownik [, 2]. We partly rely on our approach [8]. We collect some more or less immediate embedding results for weighted spaces of the above type that will be used later. For that purpose we adopt the nowadays usual custom to write A s p,q instead of Bp,q s or Fp,q, s respectively, when both scales of spaces are meant simultaneously in some context. Proposition.8. Let 0 <q, 0 <p<, s R, and w A. i Let <s s 0 < and 0 <q 0 q. Then ii We have A s0 p,qr n,w A s p,qr n,w and A s p,q 0 R n,w A s p,q R n,w. B s p,minp,q Rn,w F s p,qr n,w B s p,maxp,q Rn,w. iii Assume that there are numbers c>0, d>0, such that, for all balls, wbx, r cr d, 0 <r, x R n. 7 Let 0 <p 0 < <, <s <s 0 <, with Then s 0 d p 0 = s d. 8 and B s0 p 0,qR n,w B s,qr n,w, 9 F s0 p 0, R n,w F s,qr n,w. 0 iv Let w satisfy 7 and let 0 <p 0 <p< <, <s <s<s 0 < satisfy s 0 d p 0 = s d p = s d. Then B s0 p 0,pR n,w F s p,qr n,w B s,pr n,w. Proof. Parts i iii coincide with [3, Thm. 2.6] where, in particular, assumption 7 is denoted by w M d. As for iv we use the partial result F s0 p 0, R n,w B s,p 0 R n,w : vol. 2, num., pags

9 from [3, Thm. 2.6] together with 0 and interpolation results for weighted F -spaces in [3, Thm. 3.5]. Obviously, the right-hand side of is a consequence of 2 together with i for A = F. As for the left-hand side, choose, for given s 0, p 0, and s, suitable numbers σ, σ 2, and 0 <θ< such that for appropriate r, r 2, σ >s 0 >σ 2 >s, s 0 = θσ + θσ 2 and σ i d p 0 = s d r i, i =, 2. According to 0, Fp σi 0, R n,w Fr s i,qr n,w, i =, 2. On the other hand, the interpolation results [3, Thm. 3.5] read in our case as F σ p 0, R n,w,fp σ2 0, R n,w θ,p = Bs0 p 0,pR n,w and F s r,qr n,w,fr s 2,qR n,w θ,p = F p,qr s n,w, which concludes the proof of. Remark.9. The above embeddings i and ii are natural extensions from the unweighted case w, see [39, Prop /2], whereas iii and iv with d = n have their unweighted counterparts in [35, Thm. 3.2.; 39, Thm. 2.7.]. Remark.0. We shall need an extension of the unweighted Besov spaces Bp,qR s n to so-called weak-bp,q s spaces. By this we mean the modification of 6 with w when L p R n is replaced by the Lorentz space L p, R n, /q f weak-bp,qr s n = 2 jsq F ϕ j Ff L p, R n q. 3 j=0 Note that f L p, R n sup t p f t, t>0 where f is the non-increasing rearrangement of f, as usual, f t =inf s 0: x R n : fx >s} t }, t 0. We recall the definition of atoms. Let for m Z n and ν N 0, Q ν,m denote the n-dimensional cube with sides parallel to the axes of coordinates, centered at 2 ν m and with side length 2 ν. For 0 <p<, ν N 0, and m Z n we denote by χ ν,m p the p-normalized characteristic function of the cube Q ν,m, hence χ p ν,m L p R n =. χ p ν,mx =2 νn p χν,m x = 2 νn p for x Q ν,m, 0 for x/ Q ν,m, : vol. 2, num., pags

10 Definition.. Let K N 0 and b>. i The complex-valued function a C K R n is said to be an K -atom if supp a bq 0,m for some m Z n, and D α ax for α K, x R n. ii Let s R, 0 < p, and L + N 0. The complex-valued function a C K R n is said to be an s, p K,L -atom if for some ν N 0, supp a bq ν,m for some m Z n, D α ax 2 νs n p + α ν for α K, x R n, R n x β ax dx = 0 for β L. We shall denote an atom ax supported in some Q ν,m by a ν,m in the sequel. For 0 <p<, 0<q, w A, we introduce suitable sequence spaces b pq w by b p,q w = λ = λ ν,m } ν,m : λ ν,m C, /q } λ b p,q w = λ ν,m χ p ν,m L p R n,w q <. 4 ν=0 m Z n Then the atomic decomposition result used below reads as follows. Proposition.2. Let 0 <p<, 0 <q, s R, and w A be a weight with r w given by 2. LetK, L + N 0 with [ ] rw K +[s] + and L max, n p s. + Then f S R n belongs to B s pqr n,w if, and only if, it can be written as a series f = ν=0 m Z n λ νm a ν,m x, converging in S R n, 5 where a ν,m x are K -atoms ν =0or s, p K,L -atoms ν N and λ b pq w. Furthermore inf λ b pq w is an equivalent quasi-norm in B s pqr n,w, where the infimum ranges over all admissible representations 5. The above result coincides with [8, Thm. 3.0], see also [, Theorem 5.0]. 2008: vol. 2, num., pags

11 .3. Wavelet characterizations of Besov spaces with A weights Nowadays there is a variety of excellent textbooks on wavelet theory and hence we may assume that the reader is familiar with basic assertions. For general background material on wavelets we refer, in particular, to [0, 2, 26, 43]. Let φ be an orthogonal scaling function on R with compact support and of sufficiently high regularity. Let ψ be an associated wavelet. Then the tensor-product approach yields a scaling function φ and associated wavelets ψ,...,ψ 2 n, all defined now on R n. We suppose φ C N R and supp φ [ N 2,N 2 ] for certain natural numbers N and N 2. This implies φ, ψ i C N R n and supp φ, supp ψ i [ N 3,N 3 ] n, i =,...,2 n. 6 We shall use the standard abbreviations φ ν,m x =2 νn/2 φ2 ν x m and ψ i,ν,m x =2 νn/2 ψ i 2 ν x m. Apart from function spaces with weights we introduce sequence spaces with weights. Let σ R. We extend 4 by b σ p,qw := λ = λ ν,m } ν,m : λ ν,m C, λ b σ p,qw = 2 νσq λ ν,m χ p ν,m L /q } pr n,w q < ν=0 m Z n and l p w := λ = λ m } m : λ m C, λ l p w = λ m χ p 0,m L pr n,w }. < m Z n If σ = 0 we write b p,q w instead of b σ p,qw; moreover, if w we write b σ p,q instead of b σ p,qw. For smooth weights and compactly supported wavelets it makes sense to consider the Fourier-wavelet coefficients of tempered distributions f S R n with respect to such an orthonormal basis. Theorem.3. Let 0 <p,q and let s R. Let φ be a scaling function and let ψ i, i =,...,2 n, be the corresponding wavelets satisfying 6. We assume : vol. 2, num., pags

12 that s <N. Then a distribution f S R n belongs to Bp,qR s n,w, if, and only if, f Bp,qR s n,w = f,φ 0,m } m Z n l p w + f,ψ i,ν,m } ν N0,m Z n b σ p,qw <, 7 2 n i= where σ = s + n 2 n p. Furthermore, f Bs p,qr n,w may be used as an equivalent quasi-norm in B s p,qr n,w. Proof. The idea of the proof is standard, see [3, Theorem 0.2]. First we assume that 7 holds. There exists a positive constant c such that functions a i,ν,m x =c 2 νs+ n 2 n p ψ i,ν,m are s, p-atoms and a 0,m x =c φ 0,m are -atoms. The distribution f can be represented in the following way: f = m Z n f,φ 0,m φ 0,m + = m Z n c f,φ 0,m a 0,m + 2 n i=0 2 n i= ν N 0 m Z n f,ψ i,ν,m ψ i,ν,m c2 νs+ n ν N 0 m Z n see also Remark.4 below. So Proposition.2 implies f Bp,qR s n,w C f,φ 0,m χ p ν,m L pr n,w m Z n 2 n + 2 n p f,ψ i,ν,m χ p ν,m L pr n,w i=0 ν=0 m Z n 2 νs+ n n 2 n p f,ψ i,ν,m a i,ν,m, 8 q /q C f,φ 0,m } lp w 2 m Z + f,ψ n i,ν,m } b σ ν N 0,m Z n p,q w. Now let f Bp,qR s n,w. We can define an equivalent quasi-norm in the Besov spaces with A weights using the inhomogeneous ϕ-transform, see [, 4]. Let Φ SR n and Φ 0 SR n satisfy supp Φ [ π, π] \0}, supp Φ 0 [ π, π], and sup j N Φ2 j ξ, Φ 0 ξ } > 0 for all ξ R n. Given a pair Φ 0, Φ SR n satisfying the above conditions one can find functions Ψ 0, Ψ SR n satisfying the same conditions and such that Φ 0 ξ Ψ 0 ξ + Φ2 j ξ Ψ2 j ξ = j= i=0 2008: vol. 2, num., pags

13 for all ξ R n, see [4]. Let Φ j,m x = 2 nj/2 Φ2 j x m, j N, and m Z n. Moreover, we put Φ 0,m x =Φ 0 x m, m Z n. In a similar way we define Ψ j,m. The inhomogeneous ϕ-transform S Φ is a map taking each f S R n into the sequence S Φ f =S Φ f j,m defined by S Φ f j,m = f,φ j,m, j N 0, and m Z n. The inhomogeneous inverse ϕ-transform T Ψ is a map taking each sequence λ = λ j,m } j N0,m Z n to T Ψ λ = λ j,m Ψ j,m, j=0 m Z n with convergence in S R n. We have the following representation formula, f = T Ψ S Φ f 9 for any f S R n. The operator S Φ is a bounded operator from Bp,qR s n,w into b σ p,qw and T Ψ is a bounded operator from b σ p,qw ontobp,qr s n,w. Moreover, T Ψ S Φ is the identity operator on Bp,qR s n,w. Thus f,φ j,m } j,m b σ p,qw C f Bp,qR s n,w. 20 Applying the formula 9 to the function ψ i,ν,m we get f,ψ i,ν,m = j=0 k Z n f,ψ j,k ψ i,ν,m, Φ j,k. A similar formula holds for f,φ 0,m. The numbers λ ν,m,j,k = ψ i,ν,m, Φ j,k form the almost diagonal matrix in the sense of Frazier and Jawerth, see [4, Lemmas 3.6, 3.8 and Remarks in 2]. Thus the almost diagonal operator related to the matrix is bounded in b σ p,qw, see [, 5.4]. Since the coefficients f,ψ j,k satisfy also the inequality 20 we get f B s p,qr n,w C f,ψj,m } j,m b σ p,q w C f B s p,q R n,w. This finishes the proof. Remark.4. The representability 8 of f S R n may not be clear immediately. However, using Hölder s inequality and the already mentioned reverse Hölder inequality of A weights [36, chap. 5, Prop. 3, Cor.] one can reduce the argument to the corresponding one for admissible weights, say, of type x α, α R, since B s p,qr n,w can be squeezed in between spaces of type B s p i,qr n, x αi, i =, 2 for suitably chosen α i, p i, i =, 2. But then [20] implies the representability 8 at the expense of some higher smoothness and cancellation needed for the atomic decomposition argument according to Proposition.2, see also [42] : vol. 2, num., pags

14 2. Continuity and compactness of embeddings We start with a general result on weighted embeddings and discuss its consequences in different settings afterwards. Proposition 2.. Let w and w 2 be two A weights and let <s 2 s <, 0 <,, 0 <q,q 2. We put p := and + q :=. q 2 q + i There is a continuous embedding Bp s,q R n,w Bp s2 2,q 2 R n,w 2 if, and only if, 2 νs s 2 } w 2 Q ν,m /p2 w Q ν,m /p } m l p ν l q. 2 ii The embedding B s R n,w B s2 R n,w 2 is compact if, and only if, 2 holds and, in addition, lim ν 2 νs s2 w 2 Q ν,m /p2 w Q ν,m /p } m l p =0 if q =, and lim w 2Q ν,m /p2 w Q ν,m /p = for all ν N 0 if p =. 22 m Proof. It follows from the last theorem that the mapping T : f f,φ 0,m } m Z n, f,ψ i,ν,m } ν N0,m Z n,i=,...,2 n is an isomorphism of B s p,qr n,wontol p w 2 n i= b σ p,qw, σ = s + n 2 n p. It can be easily seen that the last sequence space is isomorphic to b σ p,qw. Consequently we have the following commutative diagrams, B s R n,w T b σ w b σ w T B s R n,w Id B s2 R n,w 2 S id w 2 b σ2 and id b σ2 w 2 S B s2 Id R n,w 2, where T and S are the corresponding isomorphisms and σ i = s i + n 2 n p i, i =, 2. On the other hand, one can easily verify that the expression q/p /q 2 νσq λ ν,m νn wq ν,m ν=0 m Z n 2008: vol. 2, num., pags

15 is an equivalent norm in b σ p,qw. But wq ν,m > 0 for any ν and m. Sowecan reduce the investigation of the embeddings of two weighted sequence spaces to the study of embeddings of a weighted space into an unweighted one, using the following commutative diagrams, b σ w A b σ w /w 2 b σ w /w 2 A b σ w Id b σ2 w 2 A id b σ2 and id b σ2 A Id w 2, b σ2 where b σ2 denotes an unweighted space, i.e., with weight w. But the necessary and sufficient conditions for the boundedness and compactness of the embeddings b σ w /w 2 b σ2 are known, see [24, Theorem ]. Taking w ν,m = 2 νn wq ν,m /p in the last mentioned theorem we get the result. Remark 2.2. In view of 5 it is clear that we obtain unweighted Besov spaces if = =. Then by, w Q ν,m =w 2 Q ν,m =2 νn for all ν N 0 and m Z n, such that 2 leads to p =, i.e.,, and δ := s n s 2 + n > 0, 23 with the extension to δ =0ifq q 2, i.e., q =. Moreover, by 22, the embedding is never compact as is well-known in this case. Furthermore, i generalizes Proposition.8 iii, in particular 9, since taking w = w 2 = w satisfying 7, we obtain that wq ν,m c2 νd for all m Z n which immediately leads to p = in 2, i.e.,. Moreover, 8 then yields that 2 νs s2+d p = for all ν N 0. Hence q =, that is, q q 2. Thus Proposition 2. i implies 9. Examples 2.3. We collect some elementary examples and explicate the proposition in their context. i Let w α x = x α, x R n, n <α<. 24 It is well known that w α A if α 0 and w α A r provided that α<nr. If α>0, then the embedding B s R n,w α B s2 R n α is continuous if, and only if, > n p and δ α if q = or δ> α if q <. α The embedding is compact if, and only if, > n p and δ> α.if n <α<0, then the embedding is not continuous : vol. 2, num., pags

16 ii We consider 24 with n <α<0; here one can deal with embeddings Bp s,q R n Bp s2 2,q 2 R n,w α. α The embedding is continuous if, and only if, < n p and δ α if q = or δ> α if q α <. The embedding is compact if, and only if, < n p and δ> α. iii If w α,n x = x n α, x =x,...,x n R n, α > 0, then the embedding Bp s,q R n,w α,n Bp s2 2,q 2 R n is continuous if, and only if, p = and δ α if q = or δ> α if q <. The embedding is never compact. Further examples are studied in detail in the next sections. As already mentioned, we may restrict ourselves to the situation when only the source space is weighted, and the target space unweighted. However, for comparison with the unweighted case we shall study two types of embeddings; firstly and essentially we concentrate on Bp s,q R n,w Bp s2 2,q 2 R n, 25 where w A. We shall assume in the sequel that < for convenience, since otherwise we have Bp s,q R n,w=bp s,q R n, recall 5, and we arrive at the unweighted situation in 25 which is well-known already. Therefore we stick here to the general assumptions <s 2 s <, 0 < <, 0 <, 0 <q,q Secondly, we shall occasionally formulate some results in the double-weighted situation, in particular, corresponding to the setting B s R n,w B s2 R n,w 27 with <s 2 s <, 0 <, <, 0 <q,q Example 2.4. Obviously all Examples 2.3 have their immediate counterparts for embeddings of type 27, e.g., B s R n,w α B s2 R n,w α, n <α<, 29 is continuous if, and only if, p = that is, and δ α 0if q =, orδ>α p 0ifq <. The embedding 29 is compact if, and only if, δ>α > 0. If n <α<0, then the embedding is never compact. 2008: vol. 2, num., pags

17 Remark 2.5. The continuity assertion of the above example is well-known in the unweighted case α = 0. However, there is no compactness in unweighted situations possible, unlike in Weights of purely polynomial growth We consider weights of polynomial growth both near zero and infinity of the form x α if x, w α,β x = x β with α> n, β > n. 30 if x >, Obviously this refines the approach 24, i.e., for α = β> nwe arrive at Example 2.3 i, w α,α = w α. Note that r wα,β in this case. =+ maxα,β,0 n Proposition 2.6. Let w α, w β be given by 24, respectively, and w α,β by 30. i Let w α A r and w β A r, r<. Then w α,β A r. ii Let the parameters be given by 26. The embedding Bp s,q R n,w α,β Bp s2 2,q 2 R n is continuous if, and only if, either β 0 if p =, β or > n p if p 3 <, and one of the following conditions is satisfied: α δ max, 0 if q =, p =, α δ max, n p α δ>max, n p if q =, p <, otherwise. n p α, iii The embedding A s R n,w α,β A s2 R n is compact if, and only if, β > n n p and δ>max p, α Proof. According to Lemma.4, the minimum and maximum of two A weights is also an A weight. Moreover, w r A r if w A. Using the above facts one can prove part i : vol. 2, num., pags

18 Parts ii and iii for Besov spaces follow easily from Proposition 2. since for any α> n we have w α Q ν,0 2 νn+α and w α Q ν,m 2 νn+α m α if m 0. Part iii for Triebel-Lizorkin spaces follows from the Besov case in combination with Proposition.8 iii. Remark 2.7. When p <, the restriction β > n p in 3 cannot be weakened; however, the limiting case β = n p can be included in the framework of weak-besov spaces, recall their definition in 3. This will be contained as a special case in Proposition 2.0 below. Obviously, Proposition.8 and part ii of Proposition 2.6 imply continuity assertion in the F -case, too, where some care is needed whenever the q-parameters are involved. However, we are mainly interested in compact embeddings in the sequel and omit further discussion. We turn to the double-weighted situation now. Proposition 2.8. Let w α,β and w α2,β 2 be given by 30. i Let the parameters be given by 28. The embedding Bp s,q R n,w α,β Bp s2 2,q 2 R n,w α2,β 2 is continuous if, and only if, β either β 2 0 if p =, or β β 2 > n p if p <, and one of the following conditions is satisfied: α δ max δ max δ>max α 2 α α, 0 α 2, n p α 2, n p if q =, p =, if q =, p <, otherwise. n p α α 2, 32 ii The embedding A s R n,w α,β A s2 R n,w α2,β 2 is compact if, and only if, β β 2 > n α p and δ>max α 2, n p. Proof. The argument is parallel to that one given for Proposition 2.6 above. 2008: vol. 2, num., pags

19 Remark 2.9. If we take α 2 = β 2 = 0 we get Proposition 2.6. In case of α = α 2 = α and β = β 2 = β we obtain an embedding of type 27, in that way extending Example 2.4 where we considered the special case α = β. Then the embedding Bp s,q R n,w α,β Bp s2 2,q 2 R n,w α,β is continuous if, and only if,, β 0, and one of the following conditions is satisfied: δ maxα, 0 δ>maxα, 0 if q =, if q <. The embedding Bp s,q R n,w α,β Bp s2 2,q 2 R n,w α,β is compact if, and only if, β>0, <, and δ>maxα, 0. As already mentioned, we deal with the limiting case β = n p separately; in view of 3 it is excluded in the context of target spaces B s p,qr n apart from p =. But in the context of weak-besov spaces we obtain the following extension. Proposition 2.0. Let <s 2 s <, 0 < <, 0 <q,q 2, α> n, β>0, and assume α β + s s 2. Let be given by = + β n. Then B s R n,w α,β weak-b s2 R n 33 if either s 2 <s α β +, 0 <q 2, or s 2 = s α β +, q q 2. Proof. Ste. By elementary embeddings monotonicity of the spaces it is sufficient to prove 33 for the case s 2 = s α β+, and q = q 2 = q only, that is, using notation 30, we have to show that B s,qr n,w α,β weak-b s2,qr n, = + β, s 2 = s α β +, 34 n with s R, 0< <, 0<q, α> n, β> : vol. 2, num., pags

20 Ste. We first assume α β such that s 2 = s = s and 34 reads as B s,qr n,w α,β weak-b s,qr n, The argument is based on the observation that = + β, α β. n L p R n,w L p2, R n 35 if w /p L r, R n, r =. Recall that real interpolation together with Hölder s inequality gives L p R n L r, R n L p2, R n in the sense that for all h L p R n, g L r, R n, then hg L p2, R n with hg L p2, R n c h L p R n g L r, R n. 36 Now let f L p R n,w. Then h = fw /p L p R n. Assume for the moment that g = w /p L r, R n with r = np β. Then 36 implies 35. f L p2, R n c f L p2, R n c fw /p L p R n w /p L r, R n c f L p R n,w. It remains to show that w /p α,β Then, for α 0, and so g L r, R n sup 0<t< gx =w /p α,β x = L r, R n for n <α β, β>0. Let x α if x, g t x β if x >. t α n if t, t β n if t>, t r g t + sup t r g t sup t β α n t 0<t< + sup t β β n t for 0 α β. In case of n <α<0 we get at least g t ct r, t>0, leading to g = w /p α,β L r, R n again. Step 3. Assume α>β>0 such that we have s 2 = s α β <s in 34. In view of Proposition 2.8 i we have B s,qr n,w α,β B s2,qr n,w β,β, δ = s s 2 = α β > 0, 2008: vol. 2, num., pags

21 see 32. Moreover, Ste yields B s2,qr n,w β,β weak-b s2,qr n, = + β, n and this concludes the proof General weights Now we deal with general weights w A r, r<, and will check what Proposition 2. means in this context. Essentially, we concentrate on two types of embeddings: either the target space is unweighted, id w : B s R n,w B s2 R n, 37 with 26, or both source space and target space share the same weight, id ww : B s R n,w B s2 R n,w, 38 with 28, recall also Proposition.8. We begin with a short preparation in order to apply Proposition 2. for such situations. Hence we have to consider expressions of type w 2 Q ν,m /p2 w Q ν,m /p only, that is, 2 νn/p2 wq ν,m /p in case of 37, and wq ν,m in case of 38. Let w A r, r<. Then, by 4, wq ν,m c2 νnr wq 0,l for all Q ν,m Q 0,l, ν N 0, m,l Z n. 39 So, for any κ < 0, and thus wq ν,m κ c 2 νnrκ wq 0,l κ, Q ν,m Q 0,l, wq ν,m κ } m l c2 νrnκ inf wq 0,l κ, κ < 0, ν N0, 40 l with w A r, r<. Moreover, for arbitrary γ>0, 39 leads to lim m wq ν,m γ = for all ν N 0 if, and only if, lim l wq 0,l =. 4 Summarizing the above considerations, we find that for embeddings of type 37 or 38 the conditions inf wq 0,l c>0, 42 l and lim wq 0,l =, 43 l : vol. 2, num., pags

22 are essential when, i.e., p =. Ifp <, then careful calculation leads to wq ν,m κ } m l p c 2 νrnκ+ν n p wq 0,l κp /p = c 2 νrnκ+ν n p wq 0,l κ } l l p for κ < 0, ν N 0, as the counterpart of 40. This corresponds to 37 with κ = / < 0. Therefore the adequate replacement of 42 for p < reads as wq 0,l /p} l l p <. 44 Note that there cannot be a continuous embedding of type 38 if >, w A, and at least 42 is assumed to hold, since 39 implies then for some r, r<, that wq ν,m c 2 νnr for all ν N 0, m Z n, such that wq ν,m } p 2 l p = m m l /p wq ν,m diverges for any ν N 0. So we are left to consider the case as far as 38 is concerned. We begin with the case 37. Corollary 2.. Let the parameters be given by 26 with. Let w A with r w given by 2. i Let δ> n r w. 45 Then id w in 37 is continuous if, and only if, 42 is satisfied. The embedding id w in 37 is compact if, and only if, 43 holds. ii Let δ<0, then Bp s,q R n,w is not embedded in Bp s2 2,q 2 R n. iii Let δ =0. When q <, then Bp s,q R n,w is not embedded in Bp s2 2,q 2 R n. When q = and w A rw, that is, r w =, then id w in 37 is continuous if, and only if, 42 is satisfied. iv If 45 is not satisfied, then for every r > r w there exists an A r weight v satisfying 42 such that the space Bp s,q R n,v is not embedded in Bp s2 2,q 2 R n. Proof. Ste. If 42 does not hold, then obviously there is no embedding independent of δ in view of 2 and our above discussion. Similarly, if 43 does not hold, then the embedding cannot be compact in view of 22 and 4 with γ =/. 2008: vol. 2, num., pags

23 Ste. Let 45 be satisfied and r>r w such that δ> n r > n r w, 46 then w A r. Thus 40 with κ = / < 0 leads for 2 with w w, w 2 to the estimate 2 νs s2 w 2 Q ν,m /p2 w Q ν,m /p} l m c 2 νs s2+ n p nr 2 p inf wq 0,l / = c 2 νδ n p r inf wq 0,l /. 47 l So the continuity and compactness of the embedding follow from Proposition 2. in view of 42 and 46. Together with Ste this concludes the proof of i. As far as iii with q = and w A rw = A is concerned, we can adapt the above argument by taking r = r w = such that 47 and 2 with q = give the result. Step 3. Concerning ii and iii it remains to show that for δ<0orδ = 0 and q < there is no embedding 37 even if 42 is satisfied. Since wq 0,m 2 nν min wq ν,l, l:q ν,l Q 0,m one obtains wq 0,m /p } m l 2 nν/p wq ν,l /p } l l. Thus, 2 νs s2 ν n wqν,l /p} } l l l q ν 2 νδ wq 0,m /p} } l m l q ν = wq 0,m /p} l m 2 νδ } lq ν = if δ<0orδ 0 and q <. l Step 4. If 45 is not satisfied, then for any r>r w there is some 0 <αsuch that δ< α < n r. Hence v = w α A r and the embedding B s R n,w α B s2 R n does not hold, see Example 2.3 i. Remark 2.2. It is obvious that we obtain by i iii a complete characterization with respect to δ in case of w A only, i.e., when r w =. Otherwise there remains the gap 0 <δ n r w : vol. 2, num., pags

24 apart from the complementing assertion iv, of course. However, it is not surprising that general features of w like r w and 42 are not appropriately adapted for the interplay with the parameters 26 as required in Proposition 2.. Reviewing, for instance, Proposition 2.6 and its proof one realizes that more information of the weight is used than reflected by r w and 42 only. For example, Corollary 2. covers only the cases β 0 and δ > maxα,β in Proposition 2.6 with, thus neglecting the admitted situations when maxα,0 δ maxα,β. But this requires further information on the weight, as already mentioned. Corollary 2.3. Let the parameters be given by 26 with >. Let w A and r w be given by 2. i Let δ> n p + n r w. The embedding id w in 37 is compact if, and only if, 44 holds. ii Let δ< n p, then B s R n,w is not embedded in Bp s2 2,q 2 R n. iii Let δ = n p. When q <, then B s R n,w is not embedded in Bp s2 2,q 2 R n. When q = and w A rw, that is, r w =, then id w in 37 is continuous if, and only if, 44 is satisfied. n p iv If <δ< n r w, then for every r>r w there exists an A r weight v satisfying 44 such that the space Bp s,q R n,v is not embedded in Bp s2 2,q 2 R n. Proof. This is a consequence of our above considerations and parallel arguments as presented for the case p =. As for iv, our assumption on r implies that we can always find some number α with n r > α >δ> n p, such that v = w α A r serves as an example in view of Example 2.3 i. We turn to the double-weighted situation now and restrict ourselves to the case w = w 2, i.e., when both spaces are weighted in the same way. Corollary 2.4. Let the parameters be given by 28 with <. Let w A and r w be given by 2. i Let The embedding n δ>r w n. 48 id ww : B s R n,w B s2 R n,w : vol. 2, num., pags

25 is continuous if, and only if, 42 is satisfied. The embedding 49 is compact if, and only if, 43 is satisfied. ii Let δ<0, then Bp s,q R n,w is not embedded in Bp s2 2,q 2 R n,w. iii Let δ =0. When q <, then Bp s,q R n,w is not embedded in Bp s2 2,q 2 R n,w. When q = and w A rw, that is, r w =, then id ww in 49 is continuous if, and only if, 42 is satisfied. iv If the condition 48 does not hold, then for every r>r w there exists an A r weight v satisfying 42 such that the space Bp s,q R n,v is not embedded in Bp s2 2,q 2 R n,v. Proof. The proof of parts i iii is completely parallel to the proof of Corollary 2., where we apply 40 for κ = < 0, and 4 with γ =. If 48 does not hold, then there is some 0 <αsuch that δ<α n < r n. Then v = w α A r and the embedding Bp s,q R n,w α Bp s2 2,q 2 R n,w α does not hold, see Example 2.4. Remark 2.5. Note that the compactness in i is in some sense surprising as it is different from the unweighted situation w where one cannot have a compact embedding as is well-known. Of course, there is no contradiction as 43 is not satisfied in this case. Moreover, Corollary 2.4 refines Proposition.8 iii in some sense: Assume that 42 is satisfied; then since for an arbitrary ball Bx, ϱ with radius 0 <ϱ< there is some m Z n such that Bx, ϱ Q 0,m apart from a universal constant and 4 implies for w A r that wbx, ϱ cϱ nr wq 0,m, we obtain 7 with d = rn. The limiting case δ =r n n in 48 coincides with 8 for d = nr such that 9 covers the continuity of the embedding 49. We end this section with a somehow astonishing result dealing with the situation =. It turns out that there is no direct influence of the weights on the continuity or compactness of the embedding 49. Corollary 2.6. Let the parameters be given by 28 with =. Let w A. Then the embedding 49 is continuous if, and only if, s s 2 > 0 if q <, s s 2 0 if q =. The embedding 49 is never compact. Proof. This is an immediate consequence of 2 and : vol. 2, num., pags

26 3. Entropy numbers of compact embeddings Let X, Y be two quasi-banach spaces and let T : X Y be a bounded linear operator. The k-th dyadic entropy number of T, k N, is defined as e k T =infε >0:T B X can be covered by 2 k balls of radius ε in Y }, where B X denotes the closed unit ball in X. Due to the well known fact that T : X Y is compact if, and only if, lim e kt : X Y =0, k the entropy numbers can be viewed as a quantification of the notion of compactness. On the other hand, the k-th approximation number of T is defined as a k T =inf T L : rank L<k}. If a k T 0 for k, then T is compact. So the asymptotic behavior of approximation numbers also gives us the quantitative analysis of compactness of the operator. Further properties like multiplicativity and additivity, as well as applications of entropy and approximation numbers can be found in [9,, 2, 30]. We consider the weights w A such that x α if x, wx w α,β x = x β with α> n, β > if x >, It follows from Theorem.3 that the investigation of the asymptotic behavior of entropy numbers of the embedding Bp s,q R n,w Bp s2 2,q 2 R n 5 can be reduced to the estimation of the asymptotic behavior of entropy numbers of embeddings of corresponding sequence spaces b s w b s2, see the proof of Proposition 2.. First we regard the part near zero. To make the notation more transparent we introduce the following spaces, l q 2 jθ l p w = λ = λ j,m } j,m : λ j,m C, j,m N 0, l q 2 jθ l γ2nj p w = l q 2 jθ lγ2 j p w = λ l q 2 jθ l p w = 2 jθq j=0 m=0 } s j,l } j,l l q 2 jθ l p w : s j,l =0ifl>γ2 nj, } s j,l } j,l l q 2 jθ l p w : s j,l =0ifl γ2 nj, q } λ j,m p p q wj, m <, 52 with the usual modification in 52 when p = and/or q =, γ N. We put w ξ j, l =l ξ if l 0 and w ξ j, 0=,j N : vol. 2, num., pags

27 Lemma 3.. Let 0 < <, 0 <, 0 <q,q 2, ξ R, θ>0, and γ N. Assume ξ > θ n + p. Then there are positive constants c and C such that for all k N the estimates ck θ n + ξ + hold. e k id : l q 2 jθ l γ2jn w ξ l q2 l γ2nj Ck θ n + ξ p + p p 2 Proof. Ste. Preparations. For ξ = 0 unweighted case the result is known, see [4, Thm. 8.2]. So we assume that ξ 0. Let Λ := λ = λ j,l } j,l : λ j,l C, j N 0, 0 l γ2 nj }, 53 and B = l q 2 jθ l γ2nj w ξ, B 2 = l q2 l γ2nj. 54 Let P j :Λ Λ be the canonical projection onto j-level, i.e., for λ = λ j,l } we put P j λ l := λ k,l if k = j, 0 otherwise, l N To shorten the notation we put /p =/ /. Elementary properties of the entropy numbers yield and e k Id : l γ2 nj e k Pj : B B 2 2 jθ e k Id : l γ2nj w ξ l γ2nj, 56 w ξ l γ2nj = ek Dσ : l γ2nj l γ2nj, 57 where D σ is the diagonal operator defined by the sequence σ l = l ξ/p σ 0 =. if l>0 and Ste. The estimate from above. We use the notation of operator ideals, see [9,30] for details. Here we recall only what we need for the proof. For a given bounded linear operator T LX, Y, where X and Y are Banach spaces, and a positive real number r we put L r, T e := sup k /r e k T. 58 k N The last expression is an operator quasi-norm. Using 56 and 57 we find L r, P e j : B B 2 c 2 jθ L e r, D σ : l γ2nj l γ2nj : vol. 2, num., pags

28 Subste.. We fix j in this substep and put N j = γ2 nj. Let σl denote the non-increasing rearrangement of σ l, l =0,...,N j.thus σl σ l if ξ 0, = N j l ξ/p if ξ<0. It should be clear that a k Dσ : l Nj and a k Dσ : l Nj =0ifk>Nj +. For any β>0 there exists C β > 0 such that sup l β e l Dσ : l Nj l k σ k if k N j +, 60 Cβ sup l k l β a l Dσ : l Nj, 6 see [8, Thm. ; 9, p. 96] and its extension to quasi-banach spaces in [2, Thm..3.3]. Let ξ>0. Taking β = ξ we get from 60 and 6 that k ξ e k Dσ : l Nj c sup Now 62 and the multiplicativity of entropy numbers imply l k l ξ a l D σ C ξ,p. 62 Consequently, k r e2k Dσ : l Nj Ck r ξ e k id : l N j. L r, e Dσ : l Nj cl e s, id : l N j 63 if s = r ξ > 0. If 0 <, then Schütt s characterization of the asymptotic behavior of the entropy numbers e k id : l N p l N, see [34], implies L e s, id : l N p l N N s if c s p > 0, log N /s if s p Under the assumption r > ξ + p we conclude from 63 and 64 that L r, e Dσ : l Nj C 2 jn r ξ p p, 65 where the constant C depends on γ, but is independent of j. If 0 < < <, then L s, e id : l N p l N cn s p, : vol. 2, num., pags

29 and 63 and 66 imply for r > max ξ + p, 0 that L r, e Dσ : l Nj C 2 jn r ξ p p. 67 Let ξ<0. Then sup l β a l Dσ : l Nj sup l β N j l ξ/p l k l k Consequently, k β e k Dσ : l Nj Now using 68 we obtain k r e2k Dσ : l Nj mink, c β,ξ,p N j β N j mink, c β,ξ,p N j ξ/p k β N C j k ξ/ if k<cβ,ξ,p N j, β,ξ,p N β ξ/p j if k c β,ξ,p N j. Cβ,ξ,p N β ξ/p j. 68 β ξ CN j k r β e k id : l N j. 69 Let us choose first r such that r > max0, p, and now β > 0 such that s = r β>max0, p. Then 68 and 64 or 66, respectively, imply L r, e β ξ Dσ : l Nj CN j N j + r β p C2 jn r ξ p p, 70 where the constant C depends on γ, but is independent of j. Subste.2. Now, for given M N 0, let P := M P j and Q := j=0 j j j=m+ P j. 7 The expression L r, T e is a quasi-norm therefore there exists a number 0 <ϱ such that ϱ L r, e T j L e r, T j ϱ, 72 see König [22,.c.5]. Hence, 59 and yield L e r, P : B B 2 ϱ M j=0 L e r, P j : B B 2 ϱ c M j=0 2 jnϱ r ξ + p θ n c 2 2 nmϱ r ξ + p θ n : vol. 2, num., pags

30 with a constant c 2 independent of M, if r ξ + p + θ n. Hence We proceed similarly to 73 and obtain L e r, Q : B B 2 ϱ c e 2 nm P : B B 2 c 3 2 nm θ n + ξ + p. 74 j=m+ 2 jnϱ r ξ + p θ n c 2 2 nmϱ r ξ + p θ n if max 0, p, ξ + < p r < ξ + p + θ n. This is always possible in view of our assumptions. Hence e 2 nm Q : B B 2 c 3 2 nm θ n + ξ + p. 75 Summarizing we get from 74 and 75 replacing nm by M, e 2 M+id : B B 2 e 2 M P : B B 2 +e 2 M Q : B B 2 c 2 M θ n + ξ p + p. Now by monotonicity of the entropy numbers the estimate from above follows. Step 3. To estimate the entropy numbers from below we regard the diagonal operator D σ between finite-dimensional sequence spaces l Nj and l Nj. Let an operator Q j : l Nj w ξ B be given by Q j λ u,l := λ l if u = j, 0 otherwise, u N 0, l N 0. It should be clear that Q j 2 jθ. The identity operator Id : l Nj w ξ factorized by Id = P j id Q j, can be where the operator P j is regarded as an operator acting between B 2 and l Nj. Since P j =, we get e k Id : l N j w ξ Qj P j e k id : B B 2 2 jθ e k id : B B : vol. 2, num., pags

31 First we consider the case ξ>0. By the result of Gordon, König, and Schütt, see [7] or [22, p. 3], we have su k/2l σ 0 σ σ l /l e k+ Dσ : l Nj l N Let Stirling s formula yields C ξ,p 6 su k/2l σ 0 σ σ l /l. l N = sup e ξ/p λξ/p λ>0 2. λ/2 e ξ 2 6 2π 3 ξ/ k ξ/ e k+ Dσ : l γ2nj l γ2nj 6Cξ,p k ξ/p 77 and the constant C ξ,p is independent of j and γ = N j 2 nj. Now using 57 and 77 we get ck ξ/p e k Dσ : l Nj e k Dσ : l Nj p id 2 : l N j C2 jn/p e k Id : l N j w ξ. 78 Thus 76 and 78 imply for k =2 nj, When ξ<0, note that for σ l = σ l e 2k Id : l N j c2 jn θ n + ξ + p e 2 nj id : B B 2. we have Id = D σ D σ. So by 57 and 77 we get ek D σ : l Nj ek Dσ : l Nj Ck ξ/p e k Id : l N j w ξ. We take k =2 nj N.Schütt s lower estimates of e k Id : l j and 76 imply c2 nj p e2 nj+ Id : l N j,0<p, c 2 nj ξ e 2 nj Id : l N j w ξ c 2 2 nj ξ + θ e 2 nj id : B B 2. N If 0 < one should use the estimate e k Id : l j and 76. This finishes the proof. 2 k 2N j N p j Before we present our result concerning the asymptotic behavior of entropy numbers for the compact embedding 5 with weights of type 50, we recall the following result of [23, Thm. ; 24, Cors. 4.5, 4.6]. Let the weight w β x, β>0, be given by w β x = + x 2 β/2, x R n : vol. 2, num., pags

32 Proposition 3.2. Let the parameters satisfy 26 with δ>0, recall 23. Assume β>0. Then the embedding l q 2 jδ l p w β l q2 l p2 is compact if, and only if, min δ, βp > n p. In that case one has for δ β, e k l q 2 jδ l p w β l q2 l p2 k minδ,β/ n p +, k N. In case of δ = β, one has for τ = s s2 n + q 2 q > 0 that e k l q 2 jδ l p w β l q2 l p2 whereas τ<0leads to e k l q 2 jδ l p w β l q2 l p2 k s s 2 n + log k τ, k N, k s s 2 n, k N. Remark 3.3. Note that the compactness assertion coincides with Proposition 2.6 iii with α = 0. Embeddings of functions spaces with weights of type 79 have been studied by many authors for several years, in particular the limiting case δ = β attracted a lot of attention; we do not want to report on this history here. There are two-sided estimates for the case δ = β/, τ = 0, in [23]. Recall that A s p,q stands for either Bp,q s or Fp,q s if no distinction is needed. Theorem 3.4. Let the parameters satisfy 26 and let the weight w A be of type 50 with β > n n, α > n, and δ>max p p, α. 80 i If β <δ, then e k A s R n,w A s2 R n k β np + p p 2. ii If β >δ, then e k A s R n,w A s2 R n k s s 2 n. 2008: vol. 2, num., pags

33 iii If β = δ and τ = s s2 n + q 2 q > 0, then e k B s R n,w B s2 R n k s s 2 n + log k τ. iv If β = δ and τ<0, then e k B s R n,w B s2 R n k s s 2 n. Remark 3.5. It is surprising that α, independently of its value within the given bounds, does not influence the asymptotic behavior of entropy numbers. Proof. First we consider the Besov spaces. It follows from Theorem.3 that e k B s R n,w Bp s2 2,q 2 R n e k b σ w b σ2, where σ i = s i + n 2 n p i, see the proof of Proposition 2.. So we can deal with sequence spaces. It should be clear that it is sufficient to regard the weights w = w α,β. Moreover, e k b σ w α,β b σ2 ek b δ p,q w α,β b p2,q 2 e k lq 2 jδ l p w α,β l q2 l p2, where δ = σ σ 2 = s n s 2 + n, see 23, and if l =0, w α,β j, l = 2 j l α/n if 2 j l<, 2 j l β/n if 2 j l. We divide the identity operator into two parts where Id : l q 2 jδ l p w α,β l q2 l p2 Id=Id +Id 2, Id : l q 2 jδ l 2jn w α,β l q2 l p2 and Id 2 : l q 2 jδ l2 jn w α,β l q2 l p2. Now it follows from Lemma 3. with θ = δ α and ξ = α n that e k Id k δ n + p p 2 = k s s 2 n. The estimate of e k Id 2 follows from the estimates for the weights w β x = + x 2 β/2, see Proposition 3.2. The corresponding estimates for Triebel-Lizorkin spaces follow by Proposition.8 and the properties of entropy numbers : vol. 2, num., pags

34 We turn to the related double-weighted situation. above we get for entropy numbers: Using the same method as Theorem 3.6. Let the parameters satisfy 28 and let the weights w α,β,w α2,β 2 A be of type 50 with α. β β 2 > n p and δ>max α 2, n p i If β β2 <δ, then e k A s R n,w α,β A s2 R n,w α2,β 2 k n n+β p n+β 2 p 2. ii If β β2 >δ, then e k A s R n,w α,β A s2 R n,w α2,β 2 k s s 2 n. iii If β β2 = δ and τ = s s2 n + q 2 q > 0, then e k B s R n,w α,β B s2 R n,w α2,β 2 k s s 2 n + log k τ. iv If β β2 = δ and τ<0, then e k B s R n,w α,β B s2 R n,w α2,β 2 k s s 2 n. Remark 3.7. The proof of the above theorem is the same as the proof of Theorem 3.4. In particular we use Lemma 3. and Proposition 3.2. There is nothing substantially new in this approach, but it has some interesting consequences concerning embeddings of type 49. In particular, if we take α 2 = β 2 = 0, then Theorem 3.6 coincides with Theorem 3.4. If α = α 2 = α and β = β 2 = β and β>0, <, δ > maxα, 0, then e k B s R n,w α,β Bp s2 2,q 2 R n,w α,β where τ = s s2 n + q 2 q as above. k n+β n k s s 2 k s s 2 k s s 2 p, β <δ, n, β >δ, n + log k τ, β =δ, τ > 0, n, β =δ, τ < 0, 2008: vol. 2, num., pags