Entropy numbers of embeddings of function spaces with...

JOURNAL OF c© 2011, Scientific Horizon

FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 9, Number 2 (2011), 129–178

Entropy numbers of embeddings of function spaces

with Muckenhoupt weights, III.

Some limiting cases

Dorothee D. Haroske and Leszek Skrzypczak

(Communicated by Hans Triebel)

2000 Mathematics Subject Classification. 46E35.

Keywords and phrases. Besov spaces, Triebel-Lizorkin spaces,Muckenhoupt weights, compact embeddings, entropy numbers.

Abstract. We study compact embeddings for weighted spaces of Besovand Triebel-Lizorkin type where the weight belongs to some MuckenhouptAp class. This extends our previous results [25] to more general weightsof logarithmically disturbed polynomial growth, both near some singularpoint and at infinity. We obtain sharp asymptotic estimates for the entropynumbers of this embedding. Essential tools are a discretisation in termsof wavelet bases, as well as a refined study of associated embeddings insequence spaces and interpolation arguments in endpoint situations.

1. Introduction

In recent years, some attention has been paid to compactness ofembeddings of function spaces of Besov and Sobolev type as well asto analytic and geometric quantities describing this compactness,in particular, corresponding approximation and entropy numbers.In [13] Edmunds and Triebel proposed a program to investigate the

130 Entropy numbers of embeddings of function spaces

spectral properties of certain pseudo-differential operators based onthe asymptotic behaviour of entropy and approximation numbers,together with Carl’s inequality and the Birman-Schwinger principle.Similar questions in the context of weighted function spaces of thistype were studied by the first named author and Triebel, cf. [19],and were continued and extended by Kuhn, Leopold, Sickel andthe second author in the series of papers [31–33]. All these paperswere devoted to the class of so-called ‘admissible’ weights: Theseare smooth weights with no singular points, with w(x) = (1+|x|2)α/2 ,α ∈ R , x ∈ Rn , as a prominent example.

We started in [25] a different approach and considered weightsfrom the Muckenhoupt class A∞ which – unlike ‘admissible’weights – may have local singularities, that can influenceembedding properties of such function spaces. Weighted Besovand Triebel-Lizorkin spaces with Muckenhoupt weights are wellknown concepts, cf. [4–7], [15,39,40] and, more recently, [2,3], [24].But (the compactness of) their embeddings were not yet studied indetail.

In [25] we dealt with general transformation methods fromfunction to appropriate sequence spaces provided by a waveletdecomposition; we essentially concentrated on the example weight

(1.1) wα,β(x) ∼{|x|α if |x| ≤ 1 ,

|x|β if |x| > 1 ,with α > −n, β > 0,

of purely polynomial growth both near the origin and for |x| → ∞ .In the general setting for w ∈ A∞ we obtained sharp criteria for thecontinuity or compactness of embeddings of type

id : As1p1,q1(Rn, w) ↪→ As2p2,q2(R

n),

where s2 ≤ s1 , 0 < p1, p2 < ∞ , 0 < q1, q2 ≤ ∞ , and Asp,q stands foreither Besov spaces Bsp,q or Triebel-Lizorkin spaces F sp,q . Moreover,for w = wα,β given by (1.1) we determined the exact asymptoticbehaviour of corresponding entropy and approximation numbers,e.g.,

ek(id : As1p1,q1(R

n, wα,β) ↪→ As2p2,q2(Rn)) ∼ k−min( β

np1+ 1p1

− 1p2,s1−s2n ), k ∈ N,

if βnp1

+ 1p1

− 1p2

�= s1−s2n . (There are further results in [25] also dealing

with ‘double-weighted’ situations and corresponding approximationnumbers.) It turned out that the ‘local’ singularity (represented

D. Haroske, L. Skrzypczak 131

by the parameter α here) has no further influence on the ‘degree’of compactness (measured in terms of entropy or approximationnumbers, respectively). We investigated this phenomenon in [26]in detail, introducing the new concept of the set of singularitiesSsing(w) which – roughly speaking – collects all those x0 ∈ Rn wherethe weight tends to infinity or vanishes. We could prove in [26] thatthe entropy and approximation numbers (for spaces on boundeddomains) do not take care of this singular behaviour of the weightas long as Ssing(w) is bounded and ‘small’ enough.

In both above contributions [25, 26] we were directly led toquestions of limiting embeddings which could not be solved in therespective context. This is the reason for us to return to this subjectin further detail, before we come to applications in the above sense(but out of the scope of the present paper).

Instead of the weight wα,β given by (1.1) we now consider itsrefinement

(1.2) w(α,β)(x) =

{|x|α1(1− log |x|)α2 , if |x| ≤ 1 ,

|x|β1(1 + log |x|)β2 , if |x| > 1 ,

where

(1.3) α = (α1, α2), α1 > −n, α2 ∈ R, β = (β1, β2), β1 > −n, β2 ∈ R.

Our first main result in Proposition 3.9 is the completecharacterisation of the continuity and compactness of

(1.4) id : Bs1p1,q1(Rn, w(α,β)) ↪→ Bs2p2,q2(R

n).

It is compact if, and only if,⎧⎨⎩either β1

p1> n

p∗ , β2 ∈ R,

or β1

p1= n

p∗ ,β2

p1> 1

p∗ ,(1.5)

and ⎧⎨⎩either δ > max(α1

p1, np∗

), α2 ∈ R,

or δ = α1

p1> n

p∗ ,α2

p1> 1

q∗ ,(1.6)

where 1p∗ = max

(1p2

− 1p1, 0)

, 1q∗ = max

(1q2

− 1q1, 0)

, and δ = s1 −np1

− s2 + np2

. Since w(α,β) with α2 = β2 = 0 coincides with wα1,β1


in the sense of (1.1) and studied in [25], it is clear that the limitingcases β1

p1= n

p∗ in (1.5) and δ = α1

p1in (1.6) were excluded so far.

It is especially interesting (and new in this context) to observe theinterplay of the fine indices q1, q2 with the local fine parameter α2

of the weight w(α,β) in (1.6). Such phenomena for Besov spaceswere already known from similar considerations. We essentiallyconcentrate on one-weighted situations here, but refer to double-weighted counterparts occasionally.

Some new idea is connected with Besov or Triebel-Lizorkinspaces built upon Lorentz-Zygmund spaces Lp,r(logL)a(R

n), thusextending the usual approach based on Lp(R

n) = Lp,p(logL)0(Rn).

These spaces appeared in special settings in [12] already and –to some extent – in [8, 13] (for bounded domains) and [18, 21] (onRn ). Another more abstract approach can be found in [10]. In the

present paper these spaces Asq(Lp,r(logL)a)(Rn) appear naturally

as appropriate endpoint spaces for our weight w(α,β) . We shallonly sketch some ideas now and apply particular results, butwe have the strong feeling that these spaces deserve a moresystematic and detailed study, both from the abstract interpolationand extrapolation point of view, and for applications.

Our second main result can be found in Proposition 4.4 wherewe study entropy numbers of the embeddings of certain sequencespaces,

idβ1,β2 : �q1(2jδ�p1(w

β1,β2))↪→ �q2 (�p2)(1.7)

with(1.8)wβ1,β2(j,m) =

(1 + 2−j |m|)β1

(1 + log(1 + 2−j |m|))β2

, j ∈ N0,m ∈ Zn,

for β1 ≥ 0 , β2 ∈ R . This obviously corresponds to embeddingsof weighted function spaces not regarding local singularities. Weachieve an almost complete description of the asymptotic behaviourof the entropy numbers ek(id

β1,β2) for k → ∞ , which is partiallybased on earlier results in [30–33] and [29], but also coversnew cases. These estimates together with local observations arecombined to characterise

ek(id : As1p1,q1(R

n, w(α,β)) ↪→ As2p2,q2(Rn))

for k ∈ N ; this is presented in Theorem 4.7. In particular, we donot only extend the weight class considered in [25] to logarithmic


perturbations, but can now deal with the limiting situations β1

p1= n

p∗

in (1.5) and δ = α1

p1in (1.6) for the first time.

The paper is organised as follows. In Section 2 we recall basicfacts about Muckenhoupt weight classes and weighted functionspaces needed later on. Section 3 is devoted to the continuityand compactness of the embeddings, in particular, when dealingwith w = w(α,β) given by (1.2) with (1.3). The concluding Section 4contains our results on the asymptotic behaviour of the entropynumbers for weighted embeddings.

2. Weighted function spaces

First of all we need to fix some notation. By N we denote the setof natural numbers, by N0 the set N ∪ {0} , and by Zn the set of alllattice points in Rn having integer components.

The positive part of a real function f is given by f+(x) =

max(f(x), 0). For two positive real sequences {ak}k∈N and {bk}k∈N

we mean by ak ∼ bk that there exist constants c1, c2 > 0 such thatc1 ak ≤ bk ≤ c2 ak for all k ∈ N ; similarly for positive functions.

Given two (quasi-) Banach spaces X and Y , we write X ↪→ Y ifX ⊂ 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 base2 .

2.1 Muckenhoupt weights. We briefly recall some fundamentalson Muckenhoupt classes Ap . By a weight w we shall always mean alocally integrable function w ∈ Lloc

1 (Rn), positive a.e. in the sequel.

Definition 2.1. Let w be a weight function on Rn and 1 < p <∞ .Then w belongs to the Muckenhoupt class Ap , if there existsa constant 0 < A < ∞ such that for all balls B the followinginequality holds

(2.1)(

1

|B|∫B

w(x) dx

)1/p(1

|B|∫B

w(x)−p′/p dx

)1/p′

≤ A,

where p′ is the dual exponent to p given by 1/p′ + 1/p = 1 and |B|stands for the Lebesgue measure of the ball B .


The limiting cases p = 1 and p = ∞ can be incorporated asfollows. Let M stand for the Hardy-Littlewood maximal operatorgiven by

(2.2) Mf(x) = supB(x,r)∈B

1

|B(x, r)|∫B(x,r)

|f(y)| dy, x ∈ Rn,

where B is the collection of all open balls

B(x, r) ={y ∈ R

n : |y − x| < r}, r > 0.

Definition 2.2. A weight w belongs to the Muckenhoupt classA1 if there exists a constant 0 < A <∞ such that the inequality

Mw(x) ≤ Aw(x)

holds for almost all x ∈ Rn . The Muckenhoupt class A∞ is given by

(2.3) A∞ =⋃p>1

Ap.

Since the pioneering work of Muckenhoupt [35–37], these classesof weight functions have been studied in great detail, we refer, inparticular, to the monographs [16], [44], [45, Ch. IX], and [43, Ch. V]for a complete account on the theory of Muckenhoupt weights.Let us only mention the important feature of decomposition of Apweights into A1 weights based on the facts that for two A1 weightsw1 , w2 , and 1 ≤ p <∞ , then w = w1w

1−p2 ∈ Ap . Conversely, suppose

that w ∈ Ap , then there exist v1 ∈ A1 , v2 ∈ A1 such that w = v1v1−p2 .

Moreover, it is known that the minimum, maximum, and the sumof finitely many A1 weights yields again an A1 weight. We refer tothe above-mentioned literature for proofs and further details. Asusual, we use the abbreviation

(2.4) w(Ω) =

∫Ω

w(x) dx,

where Ω ⊂ Rn is some bounded, measurable set.

Example 2.3. We restrict ourselves to one typical example onlysince the study of further weight functions is postponed to the nextsections. Let

(2.5) w(x) = |x|� log−κ(2 + |x|) or w(x) = |x|� logκ (2 + |x|−1).


Then,

w ∈ A1 if

{κ ∈ R, if − n < � < 0,

κ ≥ 0, if � = 0,(2.6)

whereas the counterpart for 1 < p <∞ reads as

w ∈ Ap if − n < � < n(p− 1), κ ∈ R,(2.7)

see also [14, Lemma 2.3]. In particular,

(2.8) w�(x) = |x|� ∈ Ap if

{−n < � ≤ 0, if p = 1,

−n < � < n(p− 1), if 1 < p <∞.

2.2 Function spaces of type Bsp,q(Rn, w) and F sp,q(R

n, w) withw ∈ A∞ . Let w ∈ A∞ be a Muckenhoupt weight, and 0 <

p < ∞ . Then the weighted Lebesgue space Lp(Rn, w) contains all

measurable functions such that

(2.9) ‖f |Lp(Rn, w)‖ =

(∫Rn

|f(x)|pw(x) dx)1/p

is finite. Note that for p = ∞ one obtains the classical (unweighted)Lebesgue space,

(2.10) L∞(Rn, w) = L∞(Rn), w ∈ A∞.

Thus we mainly restrict ourselves to p <∞ in what follows.

The Schwartz space S(Rn) and its dual S′(Rn) of all complex-valued tempered distributions have their usual meaning here. Letϕ0 = ϕ ∈ S(Rn) be such that

(2.11) suppϕ ⊂ {y ∈ Rn : |y| < 2} and ϕ(x) = 1 if |x| ≤ 1 ,

and for each j ∈ N let ϕj(x) = ϕ(2−jx) − ϕ(2−j+1x). Then {ϕj}∞j=0

forms a smooth dyadic resolution of unity. Given any f ∈ S′(Rn),we denote by Ff and F−1f its Fourier transform and its inverseFourier transform, respectively. Let f ∈ S ′(Rn), then the compactsupport of ϕjFf implies by the Paley-Wiener-Schwartz theoremthat F−1(ϕjFf) is an entire analytic function on Rn .


Definition 2.4. Let 0 < q ≤ ∞ , 0 < p < ∞ , s ∈ R and {ϕj}j asmooth dyadic resolution of unity. Assume w ∈ A∞ .

(i) The weighted Besov space Bsp,q(Rn, w) is the set of all

distributions f ∈ S′(Rn) such that∥∥f |Bsp,q(Rn, w)∥∥ =

∥∥∥{2js∥∥F−1(ϕjFf)|Lp(Rn, w)∥∥}

j∈N0|�q∥∥∥(2.12)

is finite.(ii) The weighted Triebel - Lizorkin space F sp,q(R

n, w) is the set ofall distributions f ∈ S′(Rn) such that∥∥f |F sp,q(Rn, w)

∥∥ =∥∥∥∥∥{2js|F−1(ϕjFf)(·)|

}j∈N0

|�q∥∥ |Lp(Rn, w)

∥∥∥(2.13)

is finite.

Remark 2.5. The spaces Bsp,q(Rn, w) and F sp,q(R

n, w) areindependent of the particular choice of the smooth dyadicresolution of unity {ϕj}j appearing in their definitions. They arequasi-Banach spaces (Banach spaces for p, q ≥ 1 ), and S(Rn) ↪→Bsp,q(R

n, w) ↪→ S ′(Rn), similarly for the F -case, where the firstembedding is dense if q <∞ ; cf. [4]. Moreover, for w0 ≡ 1 ∈ A∞ weobtain the usual (unweighted) Besov and Triebel-Lizorkin spaces;we refer, in particular, to the series of monographs by TRIEBEL

[46–49] for a comprehensive treatment of the unweighted spaces.The above spaces with weights of type w ∈ A∞ have been studiedsystematically by BUI first in [4,5], with subsequent papers [6,7]. Itturned out that many of the results from the unweighted situationhave weighted counterparts: e.g., we have F 0

p,2(Rn, w) = hp(R

n, w),0 < p < ∞ , where the latter are Hardy spaces, see [4, Thm. 1.4],and, in particular, hp(Rn, w) = Lp(R

n, w) = F 0p,2(R

n, w), 1 < p < ∞ ,w ∈ Ap , see [44, Ch. VI, Thm. 1]. Concerning (classical) Sobolevspaces Wk

p (Rn, w) built upon Lp(R

n, w) in the usual way, it holds

(2.14) W kp (R

n, w) = F kp,2(Rn, w), k ∈ N0, 1 < p <∞, w ∈ Ap,

cf. [4, Thm. 2.8]. Further results can be found in [4, 5, 16, 40].In [41] the above class of weights was extended to the class Aloc

p .Recent works are due to ROUDENKO [15, 39, 40] and BOWNIK [2, 3].We partly rely on our approaches [24–26].

We shall need an extension of the (unweighted) spaces Asp,q(Rn)

to some logarithmic spaces Asq(Lp,r(logL)a)(Rn) built upon Lorentz-

Zygmund spaces.


By this we mean the modification of (2.12) and (2.13) (withw ≡ 1 ) when Lp(R

n) is replaced by the Lorentz-Zygmund spaceLp,r(logL)a(R

n), 0 < p <∞ , 0 < r ≤ ∞ , a ∈ R , given by

(2.15) ‖f |Lp,r(logL)a(Rn)‖ ∼(∫ ∞

0

[t1/p(1 + | log t|)af∗(t)

]r dt

t

)1/r

,

(appropriately modified if r = ∞ ), where f∗ is the non-increasingrearrangement of f , as usual,

f∗(t) = inf {s ≥ 0 : |{x ∈ Rn : |f(x)| > s}| ≤ t} , t ≥ 0.

Plainly, Lp,r(logL)0(Rn) = Lp,r(Rn) with the special case Lp,p(logL)0(Rn) =

Lp(Rn).

Definition 2.6. Let 0 < q ≤ ∞ , 0 < p < ∞ , s ∈ R , 0 < r ≤ ∞ ,a ∈ R , and {ϕj}j a smooth dyadic resolution of unity.

(i) The logarithmic Besov space Bsq(Lp,r(logL)a)(Rn) is the set of

all distributions f ∈ S′(Rn) such that

∥∥f |Bsq(Lp,r(logL)a)(Rn)∥∥ =

∥∥∥{2js∥∥F−1(ϕjFf)|Lp,r(logL)a(Rn)∥∥}

j∈N0|�q∥∥∥(2.16)

is finite.(ii) The logarithmic Triebel-Lizorkin space Fsq (Lp,r(logL)a)(R

n) isthe set of all distributions f ∈ S ′(Rn) such that

∥∥f |F sq (Lp,r(logL)a)(Rn)∥∥ =

∥∥∥{2js∥∥F−1(ϕjFf)(·)|�q∥∥}

j∈N0|Lp,r(logL)a(Rn)

∥∥∥(2.17)

is finite.

Remark 2.7. The above spaces appear in their F -version in caseof p = r already in [12, Def. 4.3], denoted there by F sp,q(logF )a(R

n).In case of p = r and q = 2 these are the logarithmic Sobolevspaces considered also in [8,13] (for bounded domains) and [18,21](on Rn ). Their B -counterparts were discussed in some personalcommunication with H. Triebel; see also the general abstract result[10]. However, we shall reserve the notation Asp,q(logA)a(R

n) forthe spaces obtained by extrapolation and (logarithmic) interpolationaccording to [10]. Since their coincidence (even on boundeddomains and with additional assumptions) is not yet proved in fullgenerality we stick to the clumsy notation introduced above.


Finally, we briefly describe the wavelet characterisations of Besovspaces with A∞ weights proved in [25]. Let for m ∈ Zn and ν ∈ N0 ,Qν,m denote the n-dimensional cube with sides parallel to the axesof coordinates, centered at 2−νm and with side length 2−ν . Apartfrom function spaces with weights we introduce sequence spaceswith weights. For 0 < p < ∞ , 0 < q ≤ ∞ , σ ∈ R , and w ∈ A∞ , weintroduce suitable sequence spaces

bσp,q(w) :=

{λ = {λν,m}ν,m : λν,m ∈ C ,

‖λ |bσp,q(w)‖ ∼∥∥∥{2νσ ( ∑

m∈Zn

|λν,m|p 2νn w(Qν,m)) 1p}ν∈N0

|�q∥∥∥ <∞

}

and

�p(w) :=

{λ = {λm} : λm ∈ C, ‖λ |�p(w)‖ ∼

( ∑m∈Zn

|λm|p 2νn w(Q0,m)) 1p

<∞}.

If w ≡ 1 we write bσp,q instead of bσp,q(w).

Let φ ∈ CN1(R) be an orthogonal scaling function on R withsupp φ ⊂ [−N2, N2] for certain natural numbers N1 and N2 , andψ an associated wavelet. Then the tensor-product ansatz yields ascaling function φ and associated wavelets ψ1, . . . , ψ2n−1 , all definednow on Rn . This implies(2.18)φ, ψi ∈ CN1(Rn) and suppφ, suppψi ⊂ [−N3, N3]

n , i = 1, . . . , 2n−1 .

Using the standard abbreviations φν,m(x) = 2νn/2 φ(2νx − m) andψi,ν,m(x) = 2νn/2 ψi(2

νx − m) we proved in [25, Thm. 1.13] thefollowing wavelet decomposition result.

Theorem 2.8. Let 0 < p, q ≤ ∞ and let s ∈ R . Let φ be a scalingfunction and let ψi , i = 1, . . . , 2n − 1 , be the corresponding waveletssatisfying (2.18). We assume that |s| < N1 . Then a distributionf ∈ S ′(Rn) belongs to Bsp,q(R

n, w) , if, and only if,

‖ f |Bsp,q(Rn, w)‖� =∥∥∥ {〈f, φ0,m〉}m∈Zn

|�p(w)∥∥∥

+

2n−1∑i=1

∥∥∥ {〈f, ψi,ν,m〉}ν∈N0,m∈Zn|bσp,q(w)

∥∥∥


is finite, where σ = s+ n2 − n

p . Furthermore, ‖ f |Bsp,q(Rn, w)‖� may beused as an equivalent (quasi-) norm in Bsp,q(R

n, w) .

3. Continuity and compactness of embeddings

We collect some embedding results for weighted spaces of theabove type that will be used later. For that purpose we adoptthe nowadays usual custom to write Asp,q instead of Bsp,q or F sp,q ,respectively, when both scales of spaces are meant simultaneouslyin some context (but always with the understanding of the samechoice within one and the same embedding, if not otherwise statedexplicitly).

Proposition 3.1. Let 0 < q ≤ ∞ , 0 < p <∞ , s ∈ R and w ∈ A∞ .

(i) Let −∞ < s1 ≤ s0 <∞ and 0 < q0 ≤ q1 ≤ ∞ , then

(3.1) As0p,q(Rn, w) ↪→ As1p,q(R

n, w), and Asp,q0(Rn, w) ↪→ Asp,q1(R

n, w).

(ii) We have

Bsp,min(p,q)(Rn, w) ↪→ F sp,q(R

n, w) ↪→ Bsp,max(p,q)(Rn, w).(3.2)

(iii) Assume that there are numbers c > 0 , d > 0 such that for allballs,

w (B(x, r)) ≥ crd, 0 < r ≤ 1, x ∈ Rn.(3.3)

Let 0 < p0 < p < p1 <∞ , −∞ < s1 < s < s0 <∞ satisfy

s0 − d

p0= s− d

p= s1 − d

p1.(3.4)

Then

Bs0p0,q(Rn, w) ↪→ Bs1p1,q(R

n, w),(3.5)

F s0p0,∞(Rn, w) ↪→ F s1p1,q(Rn, w),(3.6)

and

Bs0p0,p(Rn, w) ↪→ F sp,q(R

n, w) ↪→ Bs1p1,p(Rn, w).(3.7)

Remark 3.2. These embeddings are natural extensions from theunweighted case w ≡ 1 , see [46, Prop. 2.3.2/2, Thm. 2.7.1]


and [42, Thm. 3.2.1]. The above result essentially coincides with [4,Thm. 2.6] and can be found in [25, Prop. 1.8]. We shall benefit fromthis result inasmuch as we can essentially concentrate on Besovspaces which are usually easier to handle, see also Theorem 2.8.This does not apply to limiting cases as will be pointed out later.

We next recall a general result on weighted embeddings and applyit to our model case afterwards.

Proposition 3.3. Let w1 and w2 be two A∞ weights and let−∞ < s2 ≤ s1 <∞ , 0 < p1, p2 ≤ ∞ , 0 < q1, q2 ≤ ∞ . We put

(3.8)1

p∗:=

(1

p2− 1

p1

)+

and1

q∗:=

(1

q2− 1

q1

)+

.

(i) There is a continuous embedding Bs1p1,q1(Rn, w1) ↪→ Bs2p2,q2(R

n, w2)

if, and only if,(3.9){

2−ν(s1−s2) ‖ {(w2(Qν,m))1/p2 (w1(Qν,m))−1/p1}m∈Zn |�p∗‖}ν∈N0

∈ �q∗ .

(ii) The embedding Bs1p1,q1(Rn, w1) ↪→ Bs2p2,q2(R

n, w2) is compact if,and only if, (3.9) holds and, in addition,

(3.10)limν→∞ 2−ν(s1−s2) ‖ {(w2(Qν,m))1/p2(w1(Qν,m))−1/p1}m∈Zn |�p∗‖ = 0 if q∗ = ∞ ,

and(3.11)

lim|m|→∞

(w2(Qν,m))1/p2 (w1(Qν,m))−1/p1 = 0 for all ν ∈ N0 if p∗ = ∞ .

Remark 3.4. The result is proved in [25, Prop. 2.1] based on thewavelet decomposition Theorem 2.8, the commutative diagram

Bs1p1,q1(Rn, w1)

T−−−⇀↽−−−T−1

bσ1p1,q1(w1)

Id⏐⏐� ⏐⏐� id

Bs2p2,q2(Rn, w2)

S↼−−−−⇁S−1

bσ2p2,q2(w2)

with appropriate isomorphisms S and T and the general result [32,Theorem 1]. Similarly, with an appropriate isomorphism A onecan reduce the investigation of the embeddings of two weightedsequence spaces to the study of embeddings of a weighted space


into an unweighted one, using

bσ1p1,q1(w1)

A−−−⇀↽−−−A−1

bσ1p1,q1(w1/w2)

Id⏐⏐� ⏐⏐� id

bσ2p2,q2(w2)

A−1

↼−−−−−−⇁A

bσ2p2,q2

Remark 3.5. In view of (2.10) it is clear that we obtainunweighted Besov spaces if p1 = p2 = ∞ . Then by (2.4), w1(Qν,m) =

w2(Qν,m) = 2−νn for all ν ∈ N0 and m ∈ Zn , such that (3.9) leads top∗ = ∞ , i.e., p1 ≤ p2 , and

(3.12) δ := s1 − n

p1− s2 +

n

p2> 0,

with the extension to δ = 0 if q1 ≤ q2 , i.e., q∗ = ∞ . Moreover, by(3.11), the embedding is never compact (as is well-known in thiscase).

Thus we may restrict ourselves to the situation when only thesource space is weighted, and the target space unweighted,

(3.13) Bs1p1,q1(Rn, w) ↪→ Bs2p2,q2(R

n),

where w ∈ A∞ . Moreover, we shall assume in the sequel that p1 <∞ for convenience, as otherwise we have Bs1p1,q1(R

n, w) = Bs1p1,q1(Rn),

recall (2.10), and we arrive at the unweighted situation in (3.13)which is well-known already. Therefore we stick to the generalassumptions(3.14)

−∞ < s2 ≤ s1 <∞, 0 < p1 <∞, 0 < p2 ≤ ∞, 0 < q1, q2 ≤ ∞.

For later use we recall the corresponding result for a weight of type

wα1,β1(x) =

{|x|α1 , if |x| ≤ 1 ,

|x|β1 , if |x| > 1 ,(3.15)

with α1 > −n , β1 > −n , where we already adapted the notationappropriately for later comparison with our model weight function.Obviously this modifies Example 2.3. In [25, Prop. 2.6] we provedthe following.

Proposition 3.6. Let wα1 , wβ1 be given by (2.8), respectively, andwα1,β1 by (3.15).


(i) Let wα1 ∈ Ar and wβ1 ∈ Ar , 1 ≤ r <∞ , then wα1,β1 ∈ Ar .(ii) Let the parameters be given by (3.14). The embedding

Bs1p1,q1(Rn, wα1,β1) ↪→ Bs2p2,q2(R

n) is continuous if, and only if,

(3.16)

{either β1 ≥ 0 if p∗ = ∞,

or β1

p1> n

p∗ if p∗ <∞,

and one of the following conditions is satisfied,

δ ≥ max

(α1

p1, 0

)if q∗ = ∞, p∗ = ∞,(3.17)

δ ≥ max

(α1

p1,n

p∗

)if q∗ = ∞, p∗ <∞,

n

p∗�= α1

p1,(3.18)

δ > max

(α1

p1,n

p∗

)otherwise.(3.19)

(iii) The embedding As1p1,q1(Rn, wα1,β1) ↪→ As2p2,q2(R

n) is compact if, andonly if,

β1p1

>n

p∗and δ > max

(n

p∗,α1

p1

).(3.20)

Remark 3.7. In [25, Prop. 2.8] we have also dealt with thedouble-weighted situation Bs1p1,q1(R

n, wα1,β1) ↪→ Bs2p2,q2(Rn, wγ1,κ1).

Note that the limiting cases related to the weight, that is, β1

p1= n

p∗

and δ = α1

p1, are excluded in (3.20).

Now we concentrate on our model weight, the following refinedversion of (3.15) in the spirit of Example 2.3,

(3.21) w(α,β)(x) :=

{|x|α1 (1− log |x|)α2 , if |x| ≤ 1 ,

|x|β1(1 + log |x|)β2 , if |x| > 1 ,

where

(3.22) α = (α1, α2), α1 > −n, α2 ∈ R, β = (β1, β2), β1 > −n, β2 ∈ R.

We begin with some auxiliary lemma.

Lemma 3.8. Let γ ∈ R , κ ∈ R , ν ∈ N .


(i) If γ > 0 , then

(3.23)ν∑k=1

2kγkκ ∼ 2νγνκ

(with equivalence constants independent of ν) .(ii) If γ < 0 , then

(3.24)ν∑k=1

2kγkκ ∼ 1

(with equivalence constants independent of ν) .(iii) If γ = 0 , then

(3.25)ν∑k=1

kκ ∼⎧⎨⎩

1 , κ < −1,ν1+κ , κ > −1,log(1 + ν) , κ = −1,

(with equivalence constants independent of ν) .

Proof. The proof is straightforward using only the observationthat for arbitrary ε > 0 there are constants c1 = c1(ε,κ) andc2 = c2(ε,κ) (but independent of k and ν ) such that c−1

1 ≥ 2−kεkκ ≥c12

−νενκ , and c−12 ≤ 2kεkκ ≤ c22

νενκ , k = 1, . . . , ν . �

Proposition 3.9. Let w(α,β) be given by (3.21), (3.22).

(i) Let 1 < r <∞ be such that max(α1, β1) < n(r−1) . Then w(α,β)

belongs to Ar . Moreover, w(α,β) ∈ A1 when

(3.26) max(α1, β1) ≤ 0 and

{α2 ≥ 0 if α1 = 0,

β2 ≤ 0 if β1 = 0.

(ii) Let the parameters be given by (3.14). The embedding

(3.27) idB : Bs1p1,q1(Rn, w(α,β)) ↪→ Bs2p2,q2(R

n)

is continuous if, and only if,

(3.28)

⎧⎪⎪⎨⎪⎪⎩either β1

p1> n

p∗ , β2 ∈ R,

or β1

p1= n

p∗ ,β2

p1> 1

p∗ if p∗ <∞,

β1 = 0, β2 ≥ 0 if p∗ = ∞,


and one of the following conditions is satisfied,

{2−ν(δ−α1

p1)(1 + ν)

−α2p1

}ν∈ �q∗ if

α1

p1>

n

p∗, α2 ∈ R,

(3.29)

{2−ν(δ−

np∗ )}ν∈ �q∗ if

α1

p1<

n

p∗, α2 ∈ R, or

α1

p1=

n

p∗,α2

p1>

1

p∗,

(3.30)

{2−ν(δ−

np∗ )(1 + ν)

1p∗ −α2

p1

}ν∈ �q∗ if

α1

p1=

n

p∗,α2

p1<

1

p∗,

(3.31)

{2−ν(δ−

np∗ ) log

1p∗ (1 + ν)

}ν∈ �q∗ if

α1

p1=

n

p∗,α2

p1=

1

p∗.

(3.32)

(iii) The embedding (3.27) is compact if, and only if,⎧⎨⎩either β1

p1> n

p∗ , β2 ∈ R

or β1

p1= n

p∗ ,β2

p1> 1

p∗ ,(3.33)

and ⎧⎨⎩either δ > max(α1

p1, np∗

), α2 ∈ R,

or δ = α1

p1> n

p∗ ,α2

p1> 1

q∗ .(3.34)

Proof. Step 1. Part (i) can be proved similar to [25, Prop. 2.6],cf. [14, Lemma 2.3]. If w ∈ A1 and a function b is positive a.e. andb, b−1 ∈ L∞(Rn), then bw ∈ A1 . So if a locally integrable, positivea.e. function is equivalent to an A1 weight it is also an A1 weight.Let

w1(x) =

{1, |x| ≤ 1,

(1 + | log |x||)γ1 , |x| > 1,w2(x) =

{1, |x| > 1,

(1 + | log |x||)γ2 , |x| ≤ 1.

Then w1(x) ∼ log(2 + |x|)γ1 , γ1 ∈ R , and w2(x) ∼ log(2 + |x|−1)γ2 ,γ2 ∈ R , so w1 and w2 are A1 weights if γ1 ≤ 0 and γ2 ≥ 0 , seeExample 2.3. In a similar way w3(x) = max (1, |x|γ3(1 + | log |x||)γ4)and w4(x) = min (1, |x|γ5(1 + | log |x||)γ6 ) are A1 weights if γ3, γ5 < 0

and γ4, γ6 ∈ R . Consequently, w(α,β)(x) ∼ wθi (x)w1−θj (x), for suitably


chosen i, j and γ1, . . . , γ6 . Hence w(α,β) ∈ A1 .

Concerning (ii) and (iii) we apply Proposition 3.3(ii), (iii) withw1 = w(α,β) and w2 ≡ 1 ; hence(3.35)

w1(Qν,m) ∼ 2−νn

⎧⎪⎪⎨⎪⎪⎩2−να1(1 + ν)α2 if m = 0,

|2−νm|α1 (1− log |2−νm|)α2 if 1 ≤ |m| < 2ν,

|2−νm|β1 (1 + log |2−νm|)β2 if |m| ≥ 2ν,

and w2(Qν,m) ∼ 2−νn , ν ∈ N0 , m ∈ Zn . This leads to

(3.36)w2(Qν,m)1/p2

w1(Qν,m)1/p1∼ 2

−ν( np2 − np1

)dνm, ν ∈ N0, m ∈ Z

n,

with

(3.37) dνm :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩2ν

α1p1 (1 + ν)−

α2p1 if m = 0,

|2−νm|−α1p1 (1− log |2−νm|)−

α2p1 if 1 ≤ |m| < 2ν,

|2−νm|−β1p1 (1 + log |2−νm|)−

β2p1 if |m| ≥ 2ν.

Then (3.9) can be rewritten as

(3.38){2−νδ ‖{dνm}m |�p∗‖

}ν∈ �q∗ .

Step 2. Let ν ∈ N0 be fixed for the moment; we study ‖{dνm}m |�p∗‖and split it into

Dν1 = ‖{dνm}m=0 |�p∗‖ = dν0 ,(3.39)

Dν2 =

∥∥∥{dνm}m,1≤|m|<2ν |�p∗∥∥∥ ,(3.40)

Dν3 =

∥∥∥{dνm}m,|m|≥2ν |�p∗∥∥∥(3.41)

in obvious notation. Consequently, ‖{dνm}m |�p∗‖ ∼ max (Dν1 , D

ν2 , D

ν3 ) ∼

Dν1 +Dν

2 +Dν3 . Firstly, (3.37) and (3.39) give

(3.42) Dν1 ∼ 2ν

α1p1 (1 + ν)−

α2p1 , p∗ ≤ ∞.

Next we consider Dν3 and obtain for p∗ = ∞ that

Dν3 ∼ sup

|y|≥1

|y|− β1p1 (1 + log |y|)−

β2p1 ≤ c <∞


(independent of ν ) if, and only if,

(3.43) either β1 > 0, β2 ∈ R, or β1 = 0, β2 ≥ 0.

If p∗ <∞ , then

Dν3 ∼

⎛⎝ ∑|m|≥2ν

∣∣2−νm∣∣− β1p1p∗ (

1 + log∣∣2−νm∣∣)− β2

p1p∗

⎞⎠1/p∗

∼⎛⎝ ∞∑l=ν

∑|m|∼2l

∣∣2−νm∣∣−β1p1p∗ (

1 + log∣∣2−νm∣∣)− β2

p1p∗

⎞⎠1/p∗

∼( ∞∑l=ν

2ln2−(l−ν)β1p1 p∗(1 + l− ν)

− β2p1p∗)1/p∗

∼ 2νnp∗

( ∞∑k=1

2−k( β1p1 − n

p∗ )p∗k− β2p1p∗)1/p∗

,

such that Lemma 3.8 implies Dν3 <∞ if, and only if,

(3.44) eitherβ1p1

>n

p∗, β2 ∈ R, or

β1p1

=n

p∗,β2p1

>1

p∗.

Now (3.43) and (3.44) give (3.28) and

(3.45) Dν3 ∼ 2ν

np∗ , p∗ ≤ ∞.

It remains to consider Dν2 . For p∗ = ∞ one immediately verifies

Dν2 ∼ sup

1≤|m|<2ν|2−νm|−α1

p1

(1− log |2−νm|)−α2

p1 ∼ max (Dν1 , D

ν3 ) .(3.46)

If p∗ <∞ , then

Dν2 ∼

⎛⎝ ∑1≤|m|<2ν

∣∣2−νm∣∣−α1p1p∗ (

1− log∣∣2−νm∣∣)−α2

p1p∗

⎞⎠1/p∗

∼⎛⎝ ν∑l=0

∑|m|∼2l

∣∣2−νm∣∣−α1p1p∗ (

1− log∣∣2−νm∣∣)−α2

p1p∗

⎞⎠1/p∗


∼(

ν∑l=0

2ln2(ν−l)α1p1p∗ (1 + ν − l)

−α2p1p∗)1/p∗

∼ 2νnp∗

(ν+1∑k=1

2k(α1p1

− np∗ )p∗k−

α2p1p∗)1/p∗

,

such that Lemma 3.8 implies

Dν2 ∼ 2ν

α1p1 (1 + ν)−

α2p1 if

α1

p1>

n

p∗, α2 ∈ R,

(3.47)

Dν2 ∼ 2ν

np∗ if

α1

p1<

n

p∗, α2 ∈ R, or

α1

p1=

n

p∗,α2

p1>

1

p∗,

(3.48)

Dν2 ∼ 2ν

np∗ (1 + ν)

1p∗ −α2

p1 ifα1

p1=

n

p∗,α2

p1<

1

p∗,

(3.49)

Dν2 ∼ 2ν

np∗ log

1p∗ (1 + ν) if

α1

p1=

n

p∗,α2

p1=

1

p∗.

(3.50)

Summarising, (3.42), (3.45) and (3.46) give

(3.51) ‖{dνm}m |�p∗‖ ∼ max(2ν

α1p1 (1 + ν)−

α2p1 , 2ν

np∗)

for p∗ = ∞ . In view of (3.47), (3.48) we also arrive at (3.51) in caseof p∗ <∞ and

(3.52)α1

p1�= n

p∗, α2 ∈ R, or

α1

p1=

n

p∗,α2

p1>

1

p∗.

Consequently, for p∗ ≤ ∞ and ν ≥ ν0 ,

‖{dνm}m |�p∗‖ ∼ 2να1p1 (1 + ν)−

α2p1 if

α1

p1>

n

p∗, α2 ∈ R,

(3.53)

‖{dνm}m |�p∗‖ ∼ 2νnp∗ if

α1

p1<

n

p∗, α2 ∈ R, or

α1

p1=

n

p∗,α2

p1>

1

p∗,

(3.54)


whereas (3.42), (3.45) and (3.49) lead to

‖{dνm}m |�p∗‖ ∼ 2νnp∗ (1 + ν)

1p∗ −α2

p1 ifα1

p1=

n

p∗,α2

p1<

1

p∗,

(3.55)

and

‖{dνm}m |�p∗‖ ∼ 2νnp∗ log

1p∗ (1 + ν) if

α1

p1=

n

p∗,α2

p1=

1

p∗,

(3.56)

in view of (3.42), (3.45) and (3.50). Now (3.38) together with (3.53)–(3.56) prove (3.29)–(3.32), respectively.

Step 3. It remains to prove (iii). Inspecting Proposition 3.3(iii), inparticular (3.10), gives us the additional condition

(3.57) limν→∞ 2−νδ ‖{dνm}m |�p∗‖ = 0 if q∗ = ∞

in view of (3.36). But (3.53)–(3.56) imply

either δ > max

(α1

p1,n

p∗

), α2 ∈ R, or δ =

α1

p1>

n

p∗, α2 > 0

in that case. Together with the case q∗ < ∞ from (ii) this is just(3.34). Likewise (3.11) requires

(3.58) lim|m|→∞

w1(Qν,m)1/p1

w2(Qν,m)1/p2= ∞ for all ν ∈ N0 if p∗ = ∞,

but in view of (3.36), (3.37) this is nothing else than (3.33) forp∗ = ∞ . �

Remark 3.10. It is sometimes more convenient to simplifyconditions (3.29)–(3.32) in dependence on q∗ < ∞ or q∗ = ∞ ,respectively (and thus parallel to (3.28) in some sense).If q∗ <∞ , then (3.27) if, and only if, (3.28) and(3.59)

either δ > max

(α1

p1,n

p∗

), α2 ∈ R, or δ =

α1

p1>

n

p∗,α2

p1>

1

q∗.


If q∗ = ∞ , then (3.27) if, and only if, (3.28) and(3.60)

either δ > max

(α1

p1,n

p∗

), α2 ∈ R, or δ =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

α1

p1> n

p∗ , α2 ≥ 0,

np∗ >

α1

p1, α2 ∈ R,

np∗ = α1

p1, α2

p1> 1

p∗ ,

np∗ = α1

p1= α2

p1= 0.

In particular, for s1 = s2 = s , r ≤ p , q1 = q2 = q , hence q∗ = ∞ ,δ = n

p∗ = nr − n

p ≥ 0 , we obtain

Bsp,q(Rn, w(α,β)) ↪→ Bsr,q(R

n)(3.61)

if

⎧⎨⎩either α1

np <1r − 1

p , α2 ∈ R,

or α1

np = 1r − 1

p ,α2

p > 1r − 1

p ,and

⎧⎨⎩either β1

np >1r − 1

p , β2 ∈ R,

or β1

np = 1r − 1

p ,β2

p > 1r − 1

p .

(3.62)

There is never a compact embedding of type (3.61).

In the usual ( 1p , s)-diagram, where any (weighted or unweighted)space of type Asp,q is characterised by its parameters s and p

(neglecting q for the moment), we have sketched in the figure belowthe admitted area for the target space Bs2p2,q2(R

n) in order to obtain acontinuous embedding idB for fixed source space Bs1p1,q1(R

n, w(α,β)).For convenience we have chosen the case 0 < α1 < β1 , but themodifications in other cases should be obvious. We shall return tothis figure below.

There is a counterpart of Proposition 3.9 in the double-weightedsituation, that is, in the spirit of Remark 3.7. However, forconvenience we formulate this result only in the special situationwhen s1 = s2 , p1 = p2 , and q1 = q2 .

Corollary 3.11. Let w(α,β) , w(γ,κκκ) be weights of type (3.21) with(3.22), respectively. Let s ∈ R , 0 < p <∞ , 0 < q ≤ ∞ . Then

(3.63) Bsp,q(Rn, w(α,β)) ↪→ Bsp,q(R

n, w(γ,κκκ))

if, and only if,

(3.64) either α1 < γ1, α2 ∈ R, γ2 ∈ R, or α1 = γ1, α2 ≥ γ2,


1p1

+ α1

np1

s1 − np1

− α1

p1

δ = α1

p1

s1 − np1

− β1

p1

δ = β1

p1

1p1

+ β1

np11p1

1p

s1 − np1

ss1

δ = np∗

Bs1p1,q1(Rn, w(α,β))

idB

Bs2p2,q2(Rn)

FIGURE 1. Possible embeddings in case of 0 < α1 < β1

and

(3.65) either β1 > κ1, β2 ∈ R, κ2 ∈ R, or β1 = κ1, β2 ≥ κ2.

Proof. Note that p∗ = q∗ = ∞ , δ = np∗ = 0 . An application of

(3.28), (3.60) to (3.61) with Bsp,q(Rn, w(α−γ,β−κκκ) ↪→ Bsp,q(R

n)) givesthe result. �

We now come to some limiting embedding result adapted to ourweight w(α,β) .

Proposition 3.12. Let 0 < p < ∞ , 0 < q ≤ ∞ , 0 < r ≤ p , u ∈ R ,and w(α,β) given by (3.21) with α1 > −n , β1 > 0 , max(α1, 0) ≤ β1 ,α2, β2 ∈ R . Then

Asp,q(Rn, w(α,β)

)↪→ Asq (Lr,p(logL)u) (R

n)(3.66)

if(3.67)⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

u ∈ R, if 1p + max(α1,0)

np < 1r <

1p + β1

np ,

u ≤ α2

p , if 1p + max(α1,0)

np = 1r and α1 > 0, or α1 = 0, α2 < 0,

u ≤ 0, if 1p + max(α1,0)

np = 1r and α1 < 0, or α1 = 0, α2 ≥ 0,

u ≤ β2

p , if 1p + β1

np = 1r .


Proof. In view of Definition 2.6 it is sufficient to prove

Lp(Rn, w(α,β)

)↪→ Lr,p(logL)u(R

n)(3.68)

whenever (3.67) is satisfied. This result may be well-known, butcan be proved easily by interpolation arguments similar to [22,Lemma 3.33] using real interpolation with function parameteras presented in [34, Example 3, p. 187] together with Holder’sinequality, see also [23, Lemma 3.3]. Thus the classical Holderinequality can be extended to

(3.69) ‖fg|Lr,p(logL)u(Rn)‖ ≤ c ‖f |Lp(Rn)‖ ‖g|Lv,∞(logL)u(Rn)‖,

where 1r = 1

p + 1v , u ∈ R. In order to verify (3.68) under the

assumptions (3.67) it is thus sufficient to show that(w(α,β)

)− 1p ∈ Lv,∞(logL)u(R

n)(3.70)

with 1v = 1

r − 1p and u according to (3.67), since taking this for

granted one may continue (3.69) by

‖f |Lr,p(logL)u(Rn)‖ ≤ c1

∥∥∥f (w(α,β)

) 1p |Lp(Rn)

∥∥∥ ∥∥∥(w(α,β)

)− 1p |Lv,∞(logL)u(R

n)∥∥∥

≤ c2∥∥f |Lp(Rn, w(α,β))

∥∥and obtains (3.68). We verify (3.70). Note that for α1 > 0 or α1 = 0 ,α2 < 0 we obtain

(3.71)[(w(α,β)

)− 1p

]∗(t) ∼

{t−

α1np (1 + | log t|)−α2

p if t ≤ 1 ,

t−β1np (1 + | log t|)− β2

p if t > 1 ,

and the definition (2.15) of Lv,∞(logL)u(Rn) leads to (3.70) if, and

only if,

(3.72) sup0<t<1

t1r− 1

p−α1np (1 + | log t|)u−α2

p + supt>1

t1r− 1

p−β1np (1 + | log t|)u− β2

p

is finite. But this covers the first, second and last line in (3.67). Incase of α1 < 0 or α1 = 0 , α2 ≥ 0 , (3.71) has to be replaced by

(3.73)[(w(α,β)

)− 1p

]∗(t) ∼

{1 if t→ 0 ,

t−β1np (1 + | log t|)− β2

p if t→ ∞ ,


such that the counterpart of (3.72),

(3.74) sup0<t<1

t1r− 1

p (1 + | log t|)u + supt>1

t1r− 1

p−β1np (1 + | log t|)u− β2

p

results in the third line and concludes the proof. �

Remark 3.13. Plainly, if α1 = β1 > 0 in Proposition 3.12, then(3.67) is to be understood such that both the second and last linehold, i.e., we have embedding (3.66) in this case if 1

r = 1p + β1

np =1p +

α1

np >1p and u ≤ min(α2,β2)

p . Moreover, since for r < p we have

Lr,p(logL)u(Rn) ↪→ Lr(R

n) if u >1

r− 1

p> 0,

cf. [1, Thm. 9.3], Proposition 3.12 refines our result (3.61) with(3.62). We complement our above Figure 1 in view of Proposi-tion 3.66.

s1 − np1

− α1

p1

δ = α1

p1

s1 − np1

− β1

p1

δ = β1

p1

s1 − np1

s

1p

δ = np∗

Bs1q1 (Lr,p1(logL)u)(Rn)Bs1p1,q1(R

n, w(α,β)) idB

FIGURE 2. The borderline case s1 = s2

Remark 3.14. In [25] we dealt with weights of type (3.15) whichcan be regarded as special cases of (3.21) with w(α,β) = wα1,β1 ifα2 = β2 = 0 . Using, in addition, our notation (2.16), we proved


in [25, Prop. 2.10] that

Bsp,q(Rn, w(α,β)) ↪→ Bsq (Lr,p(logL)0) (R

n),1

r=

1

p+β1np,

hence a special case of Proposition 3.12. (Note that we formulatedthe result in [25, Prop. 2.10] slightly weaker than our proof wouldallow.)

We end this section with some compactness result for F -spaces.For that purpose we briefly recall some endpoint observation byEDMUNDS and NETRUSOV in their paper [12]: let Ω ⊂ Rn bebounded, say, the open unit cube in Rn (as in [12]), and thespaces F sq (Lp,p(logL)a)(Ω) obtained by restriction from their Rn -counterparts in Definition 2.6(ii), quasi-normed in the usual way.Then [12, Thm. 4.1] implies (in our notation) that for a1, a2 ∈ R witha1 > a2 , 0 < p1, p2 <∞ , 0 < q1, q2 ≤ ∞ , s1, s2 ∈ R with s1 > s2 and

(3.75) δ = s1 − s2 − n

p1+n

p2= 0,

then

(3.76) idΩF,log : F s1q1 (Lp1,p1(logL)a1)(Ω) ↪→ F s2q2 (Lp2,p2(logL)a2)(Ω)

is compact. Of course, (3.76) with a1 = a2 = 0 is the well-known(continuous, but not compact) embedding of F -spaces, see also(3.6) for its weighted generalisation (on Rn ).

Corollary 3.15. Let w(α,β) be given by (3.21), (3.22), and

(3.77) idF : F s1p1,q1(Rn, w(α,β)) ↪→ F s2p2,q2(R

n),

where the parameters are given by (3.14) with p2 <∞ .

(i) Let (3.77) be compact. Then we have (3.33) and

(3.78)

{either δ > max

(α1

p1, np∗

), α2 ∈ R,

or δ = α1

p1> n

p∗ ,α2

p1> 1

p∗ .

(ii) Assume that (3.33) and either

(3.79) δ > max

(α1

p1,n

p∗

)and α2 ∈ R,


or(3.80)

δ =α1

p1>

n

p∗and

{α2

p1> 1

p∗ , if p1 < p2,α2

p1> min

(δn ,

1min(p2,q2)

− 1max(p1,q1)

), if p1 ≥ p2,

are satisfied. Then (3.77) is compact.

Proof. Step 1. In view of the embeddings (3.2) andProposition 3.9(iii) all that is left to verify concerns the limitingsituations when δ = α1

p1> n

p∗ ; in particular, we have to deal withα1 > 0 now. Note that the conditions for α2 in case of p1 ≥ p2 in(3.80) are more restrictive than their counterpart in (3.78), since

min

(δ

n,

1

min(p2, q2)− 1

max(p1, q1)

)≥ 1

p2− 1

p1=

1

p∗.

Step 2. We begin with the necessity part (i) and have to show thatthe compactness of (3.77) leads to α2

p1> 1

p∗ . Since α1 > 0 we canapply Proposition 3.1(iii) with d = n + α1 for w = w(α,β) and d = n

for w ≡ 1 , and conclude from (3.77) the compactness of

Bσ1r1,p1(R

n, w(α,β)) ↪→ Bσ2r2,p2(R

n)

for suitably chosen σ1 > s1 ≥ s2 > σ2 , r1 < p1 , r2 > p2 , with

σ1 − n+ α1

r1= s1 − n+ α1

p1, σ2 − n

r2= s2 − n

p2.(3.81)

Application of Proposition 3.9(iii) with

δ = σ1 − n

r1− σ2 +

n

r2= δ − α1

p1+α1

r1=α1

r1(3.82)

leads to α2

r1> 1

p∗ for arbitrarily chosen r1 < p1 . Hence α2

p1≥ 1

p∗ and itis sufficient to show that the compactness of the embedding (3.77)excludes the case α2

p1= 1

p∗ . Assume first p∗ = ∞ , that is, p1 ≤ p2 .Then the compactness of (3.77) and (3.2) imply the compactness of

Bs1p1,min(p1,q1)(Rn, w(α,β)) ↪→ Bs2p2,max(p2,q2)

(Rn),

such that by Proposition 3.1(iii) the parameters satisfy (3.33), δ =α1

p1> 0 , and thus by (3.34)

α2

p1>

(1

max(p2, q2)− 1

min(p1, q1)

)+

= 0,


excluding α2

p1= 1

p∗ = 0 . Now let p∗ <∞ and α2

p1= 1

p∗ > 0 . In this casewe do not even have a continuous embedding of type (3.77) (not tospeak about compactness). This is parallel to the B -situation whenδ = α1

p1> n

p∗ and α2

p1= 1

q∗ > 0 , see Remark 3.10. To see this we adaptan example from [42, Sect. 5.1] appropriately. Let a function f begiven by

f =

∞∑j=0

aj(ΔNϕ

) (2j(· − xj)

),(3.83)

where aj ∈ C , N ∈ N is sufficiently large, and ϕ ∈ S(Rn) withcompact support near the origin. The points xj are chosen suchthat the support of

(ΔNϕ

) (2j(· − xj)

)and

(ΔNϕ

) (2k(· − xk)

)are

pairwise disjoint for j �= k . According to [42, eq. (5.1.14)],

(3.84)∥∥f |F s2p2,q2(Rn)∥∥ ∼

( ∞∑m=0

2m(s2− np2

)p2 |am|p2)1/p2

.

On the other hand, repeating the same steps as in [42] for weightedspaces, now using the corresponding characterisations by localmeans from [6, 7] (in the version recalled in [24, Thm. 3.5]), weobtain(3.85)∥∥f |F s1p1,q1(Rn, w(α,β))

∥∥ ≤ c( ∞∑m=0

2m(s1− n

p1)p12−mα1(1 +m)α2 |am|p1

)1/p1.

Let fγ be given by (3.83) with am replaced by

(3.86) aγm = 2−m(s2− n

p2)(1 +m)

− 1p2 (1 + log(1 +m))−γ ,

where we may choose γ such that 1p1

< γ < 1p2

; this is possiblesince we are dealing with p∗ < ∞ , i.e., p1 > p2 . Then fγ does notbelong to F s2p2,q2(R

n) as ‖fγ |F s2p2,q2(Rn)‖ diverges for γp2 < 1 . On theother hand, (3.85) reads for δ = α1

p1, α2

p1= 1

p∗ = 1p2

− 1p1

as∥∥fγ |F s1p1,q1(Rn, w(α,β))∥∥

≤ c( ∞∑m=0

2m(δ−α1

p1)p1 (1 +m)

(α2p1

− 1p2

)p1 (1 + log(1 +m))−γp1)1/p1

= c( ∞∑k=1

1

k (1 + log k)γp1

)1/p1


which converges by our choice of γ . Consequently we cannot havean embedding of type (3.77) in this situation.

Step 3. Conversely, we have to prove the compactness of (3.77)under the assumptions of (ii). We may assume again that wedeal with the limiting situations δ = α1

p1> n

p∗ again. Assume

first β1

p1> n

p∗ , β2 ∈ R in (3.33). Since s1 > s2 we may choose

numbers ri and σi , i = 1, 2 , such that p1p∗ < r1

r∗ < min(α2,α1

n ,β1

n ),and s1 > σ1 ≥ σ2 > s2 , satisfying (3.81); then Proposition 3.1(iii) andProposition 3.9(iii) lead to the compactness of (3.77) in view of

F s1p1,q1(Rn, w(α,β)) ↪→ Bσ1

r1,p1(Rn, w(α,β)) ↪→ Bσ2

r2,p2(Rn) ↪→ F s2p2,q2(R

n).

We turn to the case β1

p1= n

p∗ , β2

p1> 1

p∗ , and begin with p1 < p2 , thatis, p∗ = ∞ , β1 = 0 , β2 > 0 , α2 > 0 , δ = α1

p1> 0 , in particular, s1 > s2 .

Again we choose σi and ri , i = 1, 2 , such that s1 > σ1 > σ2 > s2 ,p1 < r1 < r2 < p2 , satisfy (3.81) with (3.82). Then we have thefollowing chain of embeddings,

F s1p1,q1(Rn, w(α,β))

id1−−→ F s1p1,∞(Rn, w(α,β))id2−−→ F σ1

r1,r1(Rn, w(α,β))

id3−−→ F σ2r2,r2(R

n)id4−−→ F s2p2,q2(R

n),

where id1 is continuous by (3.1), and id2 , id4 are continuous dueto (3.6) with w = w(α,β) , d = n+ α1 , and w ≡ 1 , d = n , respectively.Since F σr,r = Bσr,r , id3 is compact in view of Proposition 3.9(iii) with

r∗ = ∞ , β1 = 0 , β2 > 0 , α2 > 0 and δ = α1

r1by (3.82). Thus

(3.77) is compact and it remains to deal with p1 ≥ p2 . Clearly,Proposition 3.9(iii) together with (3.2) establishes the compactnessof (3.77) when

α2

p1>

(1

min(p2, q2)− 1

max(p1, q1)

)≥ 1

p∗.

In order to complete the proof of (3.80) we show that for δ = α1

p1> n

p∗

and α2

p1> δ

n we also obtain compactness of (3.77). We apply somesplitting argument together with (3.76). Let ψ ∈ S(Rn) be compactlysupported near the origin, suppψ ⊂ {x ∈ Rn : |x| < 3} = Ω withψ(x) = 1 for |x| ≤ 2 . By F sp,q(Ω, w) we mean the subspace ofF sp,q(R

n, w),

F sp,q(Ω, w) = {f ∈ F sp,q(Rn, w) : supp f ⊂ Ω},


as usual. We decompose

f = ψf + (1− ψ)f

and consider the maps

Jψ : F s1p1,q1(Rn, w(α,β)) → F s1p1,q1(Ω, w(α,α)), f �→ ψf,

and

Jψ : F s1p1,q1(Rn, w(α,β)) → F s1p1,q1(R

n, w(0,β)), f �→ (1− ψ)f.

The idea is to prove that both, Jψ and Jψ are well-defined and canbe appropriately extended to compact maps into Fs2p2,q2(R

n) suchthat idF = I1 + I2 inherits their compactness finally. We beginwith Jψ . The well-definedness of it is immediate from the atomicdecomposition characterisation of F sp,q(R

n, w), w ∈ A∞ , in [24],since supp(1−ψ)f ∩ B(0, 2) = ∅ such that there is no influence of thelocal part of w(α,β) . Furthermore, since we have already proved thecompactness of idF in case of (3.79), this leads to the compactnessof

I2 : F s1p1,q1(Rn, w(α,β))

Jψ−−→ F s1p1,q1(Rn, w(0,β))

id′F−−→ F s2p2,q2(R

n).

We deal with Jψ . Again the atomic decomposition ensures that∥∥∥ψf |F s1p1,q1(Ω, w(α,β))∥∥∥ ∼

∥∥∥ψf |F s1p1,q1(Ω, w(α,α))∥∥∥

and Jψ is well-defined (in fact, we could have chosen any γ for thetarget space F s1p1,q1(Ω, w(α,γ)), but have taken α = γ for convenience,as will be clear soon). Now we refine this decomposition further andstudy

I1 : F s1p1,q1(Rn, w(α,β))

Jψ−−→ F s1p1,q1(Ω, w(α,α))re−→ F s1p1,q1(Ω, w(α,α))

idΩ−−→ F s2p2,q2(Ω)ext−−→ F s2p2,q2(R

n),

where re and ext are the usual (bounded) restriction and extension(by zero) operators. Thus it is sufficient to show the compactnessof idΩ . By an (obviously) adapted local version of (3.68) (withαi = βi > 0 , i = 1, 2 , cf. Remark 3.13) we know that

Lp1(Ω, w(α,α)) ↪→ Lr,p1(logL)α2/p1(Ω),1

r=

1

p1+

α1

np1,


and, by the definition of Lorentz-Zygmund spaces, see [1, Thm. 9.3],that

Lr,p1(logL)α2/p1(Ω) ↪→ Lr(logL)θ(Ω) for θ <α2

p1− α1

np1,

since r < p1 . Thus, by construction,

F s1p1,q1(Ω, w(α,α)) ↪→ F s1q1 (Lr(logL)θ)(Ω),1

r=

1

p1+α1

np1, θ <

α2

p1− α1

np1.

Now we apply (3.76) with a1 = θ > 0 = a2 and obtain compactnessof

idΩ : F s1p1,q1(Ω, w(α,α)) ↪→ F s1q1 (Lr,r(logL)θ)(Ω)idΩF,log−−−−→ F s2p2,q2(Ω)

since δ = α1

p1implies the required limiting condition s1 − n

r =

s1− np1

− α1

p1= s2− n

p2in (3.75). This finally leads to the compactness

of I1 , hence idF , if we can choose θ appropriately, that is, whenα2

p1> α1

np1= δ

n . �

Remark 3.16. It is obvious that in the above result thesufficiency (ii) of the compactness of idF given by (3.77) and thenecessity (i) only differ by the lower bound of α2 in the limitingsituation when δ = α1

p1> n

p∗ = β1

p1, β2

p1> 1

p∗ , and, in addition,q1 > p1 ≥ p2 or p1 ≥ p2 > q2 . We believe that (i) represents theappropriate criterion for the compactness of (3.77), i.e., that idFis compact if, and only if, conditions (3.33) and (3.78) are satisfied.But this stronger version of (ii) is not yet covered by our above proof.

Remark 3.17. In [25] we studied compact embeddings forweights of type (3.15) and obtained the criterion recalled inProposition 3.6(iii), whereas in [26] we paid attention to generalapproaches for w ∈ A∞ . In [26, Cor. 3.5] we proved that for

δ >n

p∗+n

p1(rw − 1)(3.87)

the embedding idw : As1p1,q1(Rn, w) → As2p2,q2(R

n) is compact if, andonly if,(3.88){w(Q0,m)−1/p1

}m∈Zn

∈ �p∗ with lim|m|→∞

w(Q0,m) = ∞ if p∗ = ∞.

In case of w = w(α,β) the characteristic number rw is given by

rw(α,β)= 1 + max(α1,β1,0)

n . Moreover, in view of (3.35) condition (3.88)


corresponds to (3.33), whereas (3.87), reading in this particularcase as δ > n

p∗ + max(α1,0)p1

only partially covers (3.34). We discussedthis phenomenon in [26] in further detail.

4. Entropy numbers of compact embeddings

We finally study the ‘degree’ of compactness of idB given by (3.27)in detail, using the well-known concept of entropy numbers whichwe shall recall first.

Let X,Y be two quasi-Banach spaces and let T : X → Y be abounded linear operator. The k -th (dyadic) entropy number of T ,k ∈ N , is defined as

ek(T ) = inf{ε > 0 : T (BX) can be covered by 2k−1 balls of radius ε in Y },

where BX denotes the closed unit ball in X . Due to the well knownfact that

T : X → Y is compact if, and only if, limk→∞

ek(T : X → Y ) = 0 ,

the entropy numbers can be viewed as a quantification of the no-tion of compactness. Further properties like multiplicativity andadditivity, as well as applications can be found in [9,11,13,38].

We begin with our main result in [25] for weights of type (3.15)which can be seen as special cases of w(α,β) with α2 = β2 = 0 .

Theorem 4.1. Let the parameters satisfy (3.14) and let the weightwα1,β1 ∈ A∞ be of type (3.15) with

β1p1

>n

p∗ , α1 > −n and δ > max( np∗ ,

α1

p1

).(4.1)

(i) If β1

p1�= δ , then

ek

(As1p1,q1(R

n, wα1,β1) ↪→ As2p2,q2(Rn))

∼ k−min(

β1np1

+ 1p1

− 1p2,s1−s2n )

.

(ii) If β1

p1= δ and τ = s1−s2

n + 1q2

− 1q1

�= 0 , then

ek

(Bs1p1,q1(R

n, wα1,β1) ↪→ Bs2p2,q2(Rn))

∼ k−s1−s2n (1 + log k)τ+ .


Remark 4.2. There is a parallel result for the related double-weighted situation in [25, Thm. 3.6].

We now study compact embeddings of spaces with weights of type(3.21) with (3.22). It follows from Theorem 2.8 that the investigationof the asymptotic behaviour of entropy numbers of the embedding

Bs1p1,q1(Rn, w) ↪→ Bs2p2,q2(R

n)(4.2)

can be reduced to the estimation of the asymptotic behaviour ofentropy numbers of embeddings of corresponding sequence spacesbs1p1,q1(w) ↪→ bs2p2,q2 . To make the notation more transparent weintroduce the following spaces,

�q(2jθ�p(Z

n, w)) =

{λ = {λj,m}j,m : λj,m ∈ C, j ∈ N0, m ∈ Z

n,

‖λ|�q(2jθ�p(Zn, w))‖ =

( ∞∑j=0

2jθq( ∑m∈Zn

|λj,m|pw(j,m)) qp

) 1q

<∞},

(4.3)

�q(2jθ�Mj

p (Zn, w)) =

{{sj,m}j,m ∈ �q(2

jθ�p(Zn, w)) : sj,m = 0 if |m| > Mj

},

(4.4)

�q(2jθ �Mj

p (Zn, w)) =

{{sj,m}j,m ∈ �q(2

jθ�p(Zn, w)) : sj,m = 0 if |m| ≤Mj

},

(4.5)

(with the usual modification in (4.3) when p = ∞ and/or q = ∞ ),Mj ∈ N . In an analogous way we define the space �q(2

jθ�p(N, w))

and its subspaces. Our plan is to adapt arguments used in theproof of Theorem 4.1 in [25] as well as to apply Theorem 4.1 directlybased on Corollary 3.11. For this purpose we recall [25, Lemma 3.1]adapted to our present notation.

Lemma 4.3. Let 0 < p1 < ∞ , 0 < p2 ≤ ∞ , 0 < q1, q2 ≤ ∞ , ξ ∈ R ,θ > 0 and M ∈ N . Let wξ(j,m) = |m|ξ if m �= 0 and wξ(j, 0) = 1 ,j ∈ N0 . Assume ξ

p1> −θ + n

p∗ . Then there are positive constants c

and C such that for all k ∈ N the estimates

c k−( θn+ ξ

np1+ 1p1

− 1p2

) ≤ ek

(id : �q1(2

jθ�M2j

p1 (Zn, wξ)) → �q2(�M2j

p2 (Zn)))

≤ C k−( θn+ ξnp1

+ 1p1

− 1p2

)

hold.


The next tool we need is the following extended version of [32,Cors. 4.18, 4.19] and [33, Thm. 1, Prop. 4]. Let the weight wβ1,β2 ,β1 ≥ 0 , β2 ∈ R , be given by(4.6)wβ1,β2(j,m) =

(1 + 2−j |m|)β1

(1 + log(1 + 2−j |m|))β2

, j ∈ N0,m ∈ Zn,

which corresponds to some weight w(0,β)(x) given by (3.21) withα = 0 = (0, 0), β = (β1, β2), and wβ1,β2(j,m) ∼ w(0,β)(2

−jm), j ∈ N0 ,m ∈ Zn .

Proposition 4.4. Let the parameters satisfy (3.14) with δ > 0 ,recall (3.12).

(i) Assume β1 > 0 , β2 ∈ R . Then the embedding

idβ1,β2 : �q1(2jδ�p1(w

β1,β2))↪→ �q2 (�p2)(4.7)

is compact if, and only if,(4.8)

either min

(δ,β1p1

)>

n

p∗, β2 ∈ R, or δ >

β1p1

=n

p∗,β2p1

>1

p∗.

(ii) In case of min(δ, β1

p1

)> n

p∗ , β2 ∈ R , then for all k ∈ N ,

ek

(idβ1,β2

)∼{k− β1np1

− 1p1

+ 1p2 (1 + log k)

− β2p1 , if n

p∗ <β1

p1< δ,

k−s1−s2n , if n

p∗ < δ < β1

p1,

(4.9)

whereas for np∗ < δ = β1

p1we obtain in case of τ = s1−s2

n + 1q2

−1q1

�= max(0, β2

p1) ,

ek

(idβ1,β2

)∼

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

k−s1−s2n , if β2

p1> max(τ, 0),

k−s1−s2n (1 + log k)

τ−β2p1 , if β2

p1≤ 0 < τ,

or 0 < β2

p1< τ ≤ β2

p1+ 1

q2− 1

q1,

k−s1−s2n (1 + log k)−

β2p1 , if β2 ≤ 0, τ < 0.

(4.10)


(iii) In case of δ > β1

p1= n

p∗ , β2

p1> 1

p∗ , then for p∗ < ∞ we obtainfor all k ∈ N ,

ek

(idβ1,β2

)∼ (1 + log k)

− β2p1

+ 1p∗ ,

(4.11)

whereas for p∗ = ∞ , that is, β1 = 0 , β2 > 0 ,

ek

(id0,β2

)∼⎧⎨⎩k

−β2p1 , if 0 < β2

p1≤ 1

p1− 1

p2,

k− 1p1

+ 1p2 (1 + log k)

− β2p1

+ 1p1

− 1p2 , if β2

p1> 1

p1− 1

p2≥ 0.

(4.12)

Proof. Step 1. Part (i) coincides with [32, Cor. 5.8], and (4.9) iscovered by [32, Cors. 4.18, 4.19]. The equivalence (4.11) was justrecently proved in [29, Thm. 7], and [33, Thm. 1, Prop. 4] gives(4.12). Moreover, the lower estimates in (4.10) were shown in [30,Prop. 2], and the case β2 = 0 coincides with [31, Thm. 2]. Hence itremains to deal with the corresponding upper estimates in (4.10).

Step 2. We always assume np∗ < δ = β1

p1now; therefore

τ =β1np1

+1

p1− 1

p2+

1

q2− 1

q1.(4.13)

Recall that by [31, Thm. 2],

ek(idβ1,0) ∼ k−

s1−s2n , k ∈ N,(4.14)

if β2 = 0 and τ < 0 . Obviously, this implies by monotonicity

ek(idβ1,β2) ≤ c ek(id

β1,0)

and thus the result for β2 ≥ 0 > τ in the first line in (4.10). Asfor the other cases in (4.10) we modify the ideas of the proof of [31,Thm. 2]. So we sketch here the proof and point out appropriatemodifications.

Step 3. We divide idβ1,β2 into two parts idβ1,β2 = idβ1,β2

1 + idβ1,β2

2

where(4.15)(idβ1,β2

1 (λ))j,m

=

{λj,m, if |m| ≤ 2j ,

0, if |m| > 2j ,and idβ1,β2

2 = idβ1,β2 − idβ1,β2

1 .


For the first part we have

ek

(idβ1,β2

1

)∼ ek

(id : �q1

(2jδ�2

j

p1(Zn))→ �q2

(�2j

p2(Zn))) ∼ k−

s1−s2n ,

(4.16)

cf. [48, Thm. 8.2]. On the other hand, similar to [31, Lemma 2]

ek

(idβ1,β2

2

)∼ ek

(id : �q1(�

2j

p1(Zn, wβ1,β2)) → �q2(�

2j

p2(Zn)))

∼ ek

(Dβ1,β2 : �q1(�

2nj

p1 (N)) → �q2(�2nj

p2 (N))),(4.17)

where Dβ1,β2 is a diagonal operator defined by

(Dβ1,β2λ)j,k = k−β1np1 (1 + log(1 + 2−jk1/n))−

β2p1 λj,k, j ∈ N0, k > 2nj .

We consider the projections PN , N ∈ N ,

(PNλ)j,k :=

{λj,k if 0 ≤ j < N and k ∈ N ,0 if j ≥ N ,

and decompose Dβ1,β2 as

Dβ1,β2 = Dβ1,β2PN +Dβ1,β2(Id−PN ) ,

where Id denotes the identity. Applying Holder’s inequality, ifnecessary, yields

(4.18) ‖Dβ1,β2 (Id−PN ) |�q1(�2nj

p1 (N)) → �q2(�2nj

p2 (N))‖ ≤ c1 2−N(

β1p1

− np∗ )

for some constant c1 independent of N .

Step 4. Let β2 < 0 . In order to estimate the entropy numbers ofthe operator Dβ1,β2 PN we consider the diagonal operator

(D(j)β1,β2

λ)k := k− β1np1

(1 + log(1 + 2−jk

1n ))− β2

p1 λk, λ = {λk} ∈ �p1 , j = 0, 1, . . . , N,

(4.19)

and its vector-valued counterparts

Dλ := (D(0)β1,β2

λ1, . . . , D(0)β1,β2

λN ) , λ = (λ1, . . . , λN ) ∈ �Nq1(�p1) ,

(4.20)


Dλ := (D(0)−β1,−β2

D(0)β1,β2

λ1, . . . , D(0)−β1,−β2

D(N−1)β1,β2

λN ) , λ = (λ1, . . . , λN ) ∈ �Nq1(�p1) .

(4.21)

Then D : �Nq1(�p1) → �Nq1(�p1) is a bounded operator with the norm

uniformly bounded and Dβ1,β2 PN = D D PN . Thus

ek(Dβ1,β2PN ) ≤ C ek(D) .(4.22)

It was proved in [28, Thm. 2.2] that if β1

np1> 1

p∗ , then

ek(D(0)β1,β2

: �p1 → �p2) ∼ k−γ(1 + log(1 + k)

)−β2/p1(4.23)

where γ = β1

np1+ 1

p1− 1

p2= s1−s2

n > 0 ; hence in view of (4.13)

τ = γ +1

q2− 1

q1.(4.24)

To simplify notation we write X = �Nq1(�p1) and Y = �Nq2(�p2). LetBX denote the unit ball in X . For arbitrary natural numbersk1, . . . , kN we set

εj := 2 c2 j−1/q1 k−γj (1 + log kj)

− β2p1 and ε :=

⎛⎝ N∑j=1

εq2j

⎞⎠1/q2

.

One can construct an ε-net in Y for D(BX) of cardinality at most2K , where

K := N logN +

N∑j=1

kj ,(4.25)

cf. [31]. Finally we choose kj in an appropriate way. Fix k ∈ N ,(4.26)

N := � k where � is defined by the equation � (β1p1

− n

p∗) = γ.

Then (4.18) becomes

(4.27) ‖Dβ1,β2 (Id−PN ) |�q1(�2nj

p1 (N)) → �q2(�2nj

p2 (N))‖ ≤ c1 2−γk .

Now we distinguish two cases.


Case 1. Let τ < 0 , i.e., γ < 1q1

− 1q2

in view of (4.24). We chooseθ > 1 such that γ θ < 1

q1− 1

q2and set

kj := [j−θ 2k] , j = 1, . . . , N .

Since θ > 1 we obtain K ∼ 2k by (4.25) and ε ≤ c 2−γkk−β2/p1 . Weconclude, for m = c 2k and β2 < 0 that

em(Dβ1,β2) ≤ em(Dβ1,β2PN ) + ‖Dβ1,β2 (Id−PN ) |�q1(�2nj

p1 (N)) → �q2(�2nj

p2 (N))‖≤ c 2−γkk−

β2p1 + c 2−γk

≤ c′ m−γ(1 + logm)−β2p1 .

In view of (4.16) and (4.17) this proves the last line in (4.10), sinceγ = s1−s2

n and β2 < 0 .

Case 2. Let τ > 0 , i.e., γ > 1q1

− 1q2

by (4.24). Select now 0 < θ < 1

such that γ θ > 1q1− 1q2

and set kj as in the first case. Now we obtain

K ∼ 2kk1−θ and ε ≤ c 2−γkk−β2/p1+τ−(1−θ)γ by (4.25). We conclude,for m = c 2kk1−θ and β2 < 0 that


p1 (N)) → �q2(�2nj

p2 (N))‖≤ c 2−γkk−

β2p1

+τ−(1−θ)γ+ c 2−γk

≤ c′ m−γ(1 + logm)τ−β2p1 + c′ m−γ(1 + logm)γ(1−θ)

≤ c′′ m−γ(1 + logm)τ−β2p1 ,

since β2 < 0 and γ(1 − θ) < τ , recall (4.24) and our choice of θ . Inthe same way as above, using (4.16) and (4.17), we obtain part ofthe second line in (4.10), that is, when β2 < 0 < τ .

Step 5. Let β2 > 0 . It remains to consider the case τ > 0 , moreprecisely, we are left to prove(4.28)

ek

(idβ1,β2

)≤ c

{k−

s1−s2n , if β2

p1> τ > 0,


τ−β2p1 , if 0 < β2

p1< τ ≤ β2

p1+ 1

q2− 1

q1.

Using the same diagonal operators (4.19) we get

ek(D(j)β1,β2

: �p1 → �p2) ∼ k−γ(1 + log(1 + 2−jk

1n ))−β2/p1

,(4.29)


γ = β1

np1+ 1p1

− 1p2

= s1−s2n > 0 , cf. [28, Thm. 2.2]. Moreover examining

the proof of the above theorem in [28] one can observe easily thatthe constants in this equivalence are independent of j .

Let Bp denote the unit ball in �p . Let Nj be the εj -net for the setD

(j)β1,β2

(Bp1) in �p2 of cardinality 2kj . If 0 < rj ≤ 1 , then one can find

an εj -net for the set D(j)β1,β2

(rjBp1) in �p2 of the same cardinality 2kj .If

εr :=

⎛⎝ N∑j=1

(rjεj)q2

⎞⎠1/q2

, r = (r1, . . . , rN ),

then one can find the ε-net for

Dβ1,β2

(r1Bp1 × r2Bp1 × . . .× rNBp1

)of cardinality 2k1+...+kN in Y .

In order to construct an ε-net in Y for Dβ1,β2(BX) we proceed asfollows. For any permutation π of the index set {1, 2, . . . , N} we put

Bπ :={λ = (λj)

Nj=1 ∈ BX : ‖λπ(1) |�p1‖ ≥ ‖λπ(2) |�p1‖ ≥ . . . ≥ ‖λπ(N) |�p1‖

}.

Then

(4.30) BX =⋃π

Bπ ,

and λ ∈ Bπ implies ‖λπ(j)|�p1‖ ≤ j−1/q1 . Thus

Bπ ⊂ r1Bp1 × r2Bp1 × . . .× rNBp1 with rj = π−1(j)−1/q1

where π−1 denotes the permutation inverse to π . We put

(4.31) kj = (N − j + 1)−θ2k , with θ > 0, k ∈ N, j = 0, 1, . . . , N.

It follows from the estimates (4.29) that one can assume that

ε1 ≥ ε2 ≥ . . . ≥ εN .(4.32)

So if r(π) = (π−1(1)−1/q1 , . . . , π−1(N)−1/q1), then

εr(π) :=

⎛⎝ N∑j=1

(π−1(j)

−1/q1εj)q2⎞⎠1/q2

≤⎛⎝ N∑j=1

j−q2/q1εq2j

⎞⎠1/q2


≤ c

⎛⎝ N∑j=1

j−q2/q1 k−q2γj

(1 + log(1 + 2−jk

1nj ))− q2β2

p1

⎞⎠1/q2

=: εo .

In the same way as in Step 4 one can construct an εo -net in �Nq2(�p2)

for Dβ1,β2(BX) of cardinality at most 2K , where K is given byformula (4.25).

Let k and N be related by (4.26), then the definition of kj gives

us(1 + log(1 + 2−jk

1nj )) ∼ (N − j + 1), and, consequently,

εo ≤ c 2−kγ

⎛⎝ N∑j=1

j−q2/q1 (N − j + 1)q2θγ−q2β2p1

⎞⎠1/q2

.(4.33)

If we assume θγ − β2

p1≤ 0 , then the last inequality implies

εo ≤ c 2−kγ

⎛⎝ N∑j=1

jq2(θγ−1/q1−β2p1

)

⎞⎠1/q2

.(4.34)

Once more we distinguish two cases.Case 1. Let τ − β2

p1< 0 , i.e., γ < β2

p1+ 1

q1− 1

q2in view of (4.24).

If q2 ≤ q1 , then we choose θ > 1 such that γ θ < β2

p1+ 1

q1− 1

q2< β2

p1.

Since θ > 1 we obtain K ∼ 2k by (4.25) and εo ≤ c 2−γk . Weconclude, for m = c 2k and γ = s1−s2

n ≤ β2

p1that

em(Dβ1,β2) ≤ em(Dβ1,β2PN ) + ‖Dβ1,β2 (Id−PN )|�q1(�2nj

p1 (N)) → �q2(�2nj

p2 (N))‖≤ c 2−γk ≤ c′m−γ .

If q2 > q1 , then we choose q1 = q2 = q2 . Now the estimate followsfrom elementary embeddings of the sequence spaces and the aboveconsiderations. This finishes the proof of the first line in (4.10).

Case 2. It remains to deal with 0 < τ − β2

p1≤ 1

q2− 1

q1. Select now

0 < θ < 1 such that γ θ > β2

p1+ 1

q1− 1

q2. We obtain K ∼ 2kk1−θ

and εo ≤ c 2−γkkθγ+1/q2−1/q1−β2/p1 . Thus for m = c 2kk1−θ andγ = s1−s2

n ≤ β2

p1we conclude that


p1 (N)) → �q2(�2nj

p2 (N))‖≤ c 2−γkkθγ+1/q2−1/q1−β2/p1 + c 2−γk ≤ c′′ m−γ(1 + logm)τ−

β2p1 .


This finally completes the proof of (4.10). �

Remark 4.5. The assumption concerning the upper bounds for τof the form τ < β2

p1+ 1q2− 1q1

in (4.10) is not natural and it is expectednot to be necessary. We conjecture that the estimate holds withoutthis restriction, that is,

ek

(idβ1,β2

)∼ k−

s1−s2n (1 + log k)

(τ+− β2p1

)+

∼

⎧⎪⎪⎨⎪⎪⎩k−

s1−s2n , if β2

p1> max(τ, 0),


τ−β2p1 , if τ > max

(β2

p1, 0),

k−s1−s2n (1 + log k)−

β2p1 , if β2 ≤ 0, τ < 0,

(4.35)

for np∗ < δ = β1

p1and τ �= max(0, β2

p1). But our method of proof gives

us for τ ≥ β2

p1+ 1

q2− 1

q1instead of (4.34) the estimate

εo ≤ c 2−kγNκθ

⎛⎝ N∑j=1

jq2(θγ−1/q1− β2p1

)

⎞⎠1/q2

, where κθ = min

(1

q1, θγ − β2

p1

).

This leads to the estimate

ek

(idβ1,β2

)≤ C k−

s1−s2n (1 + log k)

τ− β2p1

+κ, if

β2p1

≤ β2p1

+1

q2− 1

q1< τ,

where κ = min(

1q1, τ + 1

q1− 1

q2− β2

p1

). For q1 = ∞ this means κ = 0

such that this situation strongly supports our above conjecture:when q1 = ∞ and τ �= max(0, β2

p1), then we have (4.35) in the

limiting case np∗ < δ = β1

p1. This special case may be surprising

at first glance, but (4.33) explains some reason: the disturbingterm j−q2/q1 then disappears in the sum such that we do not haveto balance both entries and the argument is much simplified. Asalready pointed out, we therefore believe that (4.35) describes thedesired asymptotic behaviour correctly and the two-sided estimatefor τ ≥ β2

p1+ 1

q2− 1

q1is only due to our method of proof.

In contrast to this consideration, it is not clear what may happenin the limiting case τ = max(0, β2

p1) in (4.10), i.e., when n

p∗ < δ = β1

p1;

in the majority of cases we have only two-sided estimates here. Butthis is not even solved for the simpler situation β2 = 0 , cf. [31,Thm. 2].


Remark 4.6. We return to the special situation studied in[12] and recalled in front of Corollary 3.15. Under the sameassumptions as there, in particular a1 > a2 and (3.75), now withthe additional restriction that a1 − a2 �= 1

p1− 1

p2, then

ek

(idΩF,log

)∼ k

−min(a1−a2, 1

p1− 1p2

), k ∈ N.(4.36)

Theorem 4.7. Let the parameters satisfy (3.14) and let the weightw(α,β) ∈ A∞ be of type (3.21) with (3.22). We assume that

either min

(δ,β1p1

)>

n

p∗, β2 ∈ R,(4.37)

orβ1p1

=n

p∗,

β2p1

>1

p∗,(4.38)

and

either δ > max

(α1

p1,n

p∗

), α2 ∈ R,(4.39)

or δ =α1

p1>

n

p∗,

α2

p1>s1 − s2n

+δ

n.(4.40)

(i) If np∗ <

β1

p1< δ and β2 ∈ R , then

ek

(As1p1,q1(R

n, w(α,β)) ↪→ As2p2,q2(Rn))

∼ k−β1np1

− 1p1

+ 1p2 (1 + log k)−

β2p1 , k ∈ N.

(ii) If np∗ < δ < β1

p1and β2 ∈ R , then

ek

(As1p1,q1(R

n, w(α,β)) ↪→ As2p2,q2(Rn))

∼ k−s1−s2n , k ∈ N.

(iii) If np∗ < δ = β1

p1, β2 ∈ R , and τ = s1−s2

n + 1q2

− 1q1

�= max(0, β2

p1) ,

then for k ∈ N ,

ek

(Bs1p1,q1(R

n, w(α,β)) ↪→ Bs2p2,q2(Rn))

∼

∼

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩


p1> max(τ, 0),


τ−β2p1 , if β2

p1≤ 0 < τ,

or 0 < β2

p1< τ ≤ β2

p1+ 1

q2− 1

q1,


− β2p1 , if β2 ≤ 0, τ < 0.

(4.41)


(iv) In case of δ > β1

p1= n

p∗ , β2

p1> 1


ek

(As1p1,q1(R

n, w(α,β)) ↪→ As2p2,q2(Rn))

∼ (1 + log k)−β2p1

+ 1p∗ ,


ek

(As1p1,q1(R

n, w(α,β)) ↪→ As2p2,q2(Rn))∼

∼⎧⎨⎩k

−β2p1 , if β2

p1≤ 1

p1− 1

p2,

k−1p1

+ 1p2 (1 + log k)−

β2p1

+ 1p1

− 1p2 , if β2

p1> 1

p1− 1

p2.

Proof. We follow the idea of the proof of Theorem 4.1 and canreduce our consideration to the case

ek

(Bs1p1,q1(R

n, w) ↪→ Bs2p2,q2(Rn))∼ ek

(�q1(2

jδ�p1(w(α,β))) ↪→ �q2(�p2)),

where

w(α,β)(j, l) =

⎧⎪⎨⎪⎩(1 + j)α2 if l = 0,

(2−jl)α1n (1 − log(2−j l))α2 if 2−jl < 1,

(2−jl)β1n (1 + log(2−jl))β2 if 2−jl ≥ 1 .

Clearly, in view of (3.2), we immediately get the F -result in cases(i), (ii), and (iv) since no dependence on the q -parameters appears.

We divide the identity operator Id : �q1(2jδ�p1(w(α,β))) ↪→ �q2(�p2)

into two parts Id = Id1 + Id2, where

Id1 : �q1(2jδ�2

jn

p1 (w(α,β))) ↪→ �q2(�p2) and Id2 : �q1(2jδ �2

jn

p1 (wα,β)) ↪→ �q2(�p2).

In case of (4.39) we can find numbers α1 and α1 such that

α1 > α1 > α1 > −n and max

(n

p∗,α1

p1

)< δ.(4.42)

Thus by Corollary 3.11 with γ = (α1, 0), β = κκκ , we obtain

ek

(�q1(2

jδ�p1(w(α,β))) ↪→ �q2(�p2))∼ ek

(Bs1p1,q1(R

n, w(α,β)) ↪→ Bs2p2,q2(Rn))

≤ c ek

(Bs1p1,q1(R

n, w(γ,β)) ↪→ Bs2p2,q2(Rn))

∼ ek

(�q1(2

jδ�p1(w(γ,β))) ↪→ �q2(�p2)),


where

w(γ,β)(j, l) =

⎧⎪⎨⎪⎩1 if l = 0,

(2−j l)α1n if 2−jl < 1,

(2−j l)β1n (1 + log(2−j l))β2 if 2−jl ≥ 1 .

Now it follows from Lemma 4.3 with θ = δ − α1

p1and ξ = α1

n that

ek(Id1) ≤ c ek(Id1) ∼ k−( δn+ 1p1

− 1p2

) = k−s1−s2n(4.43)

(in obvious notation). On the other hand, interchanging the role ofα and γ = (α1, 0), κκκ = β , in Corollary 3.11 we obtain the inequalityconverse to (4.43),

ek(Id1) ≥ c ek(Id1) ∼ k−s1−s2n ,

i.e., finally,

ek(Id1) ∼ k−s1−s2n , k ∈ N.

In the limiting situation (4.40) the argument for the lower estimateremains valid, whereas it is no longer possible to find α1 with

(4.42) since δ = α1

p1= max

(α1

p1, np∗

)in this situation. Here we

use Remark 4.6 together with the splitting argument presented inStep 3 of the proof of Corollary 3.15. Thus

ek(Id1) ≤ c ek(idΩ : F s1p1,q1(Ω, w(α,α)) ↪→ F s2p2,q2(Ω)

)≤ c′ ek

(idΩF,log : F s1q1 (Lr,r(logL)θ)(Ω) ↪→ F s2p2,q2(Ω)

)≤ c′′ k−min

(θ, 1p2

− 1r

)∼ k−min(θ,

s1−s2n )

≤ C k−s1−s2n ,

if we can choose θ such that 0 < s1−s2n ≤ θ < α2

p1− α1

np1= α2

p1− δ

n .Here we used the approach in Step 3 of the proof of Corollary 3.15together with (4.36) and δ = α1

p1several times.

The estimates of ek(Id2) follow from the estimates for the weightswβ1,β2(x), see Proposition 4.4. �

Remark 4.8. Obviously Theorem 4.7 refines and extendsTheorem 4.1. Moreover, it is obvious that only ‘weak’ weightsinfluence the compactness of the embedding measured in termsof its entropy numbers, more precisely, when either β1

p1< δ or

β1

p1= δ with β2 < 0 . In other words, the weight should not


increase to rapidly for |x| → ∞ . This phenomenon is not onlywell-known from our previous considerations in [25], but also fromparallel observations for so-called ‘admissible’ weights in [31–33]with forerunners in [19,20,27]. In [26] we discussed similar effectsin terms of local singularities of the weight.The most difficult setting is again – as in the above-mentionedrelated papers – the ‘line’ δ = β1

p1where we have not yet obtained

asymptotically sharp estimates in all admitted cases in (iii), not tospeak about the second ‘line’ δ = α1

p1occurring in this context,

at least when 0 < α1 < β1 . In the figure below we have indicatedthe different parameter regions according to Theorem 4.7 in thissituation.

s1 − np1

− α1

p1

δ = α1

p1

s1 − np1

− β1

p1

δ = β1

p1

s1 − np1

s

1p

δ = np∗

Bs2p2,q2(Rn)

idB

(i)

(ii)

(iii)

Bs1p1,q1(Rn, w(α,β))

(iv)

FIGURE 3. The parameter regions according toTheorem 4.7 in case of 0 < α1 < β1

Remark 4.9. We dealt in [26] with more general weights andproved in [26, Thm. 3.7] the independence of the asymptoticbehaviour of related entropy numbers from local singularitiesrepresented in our case by the exponents α = (α1, α2). However, wehave to assume in this general setting that δ is sufficiently large;in our context this leads to δ > n

p∗ + max(0,α1)p1

which thus excludesthe limiting case δ = α1

p1(even for p∗ = ∞ ), recall Remark 3.17.

Studying the unweighted situation on bounded domains, there


are forerunners of (4.36) for Sobolev spaces of logarithmic typein [8,12,13], in particular, one obtains

ek(Hs1p1 (logH)a1(Ω) → Hs2

p2 (logH)a2(Ω)) ∼ k−min(a1−a2, s1−s2n )

(neglecting further assumptions for the moment). Not regardingobvious differences one should thus expect some influence of α2 inthis second limiting case.

Assume that we have α1 = α2 = 0 in Theorem 4.7 such that (4.39)can be rewritten as δ > n

p∗ . Then w(0,β) becomes an admissibleweight in the sense of the papers [19,20,27,31–33] and Theorem 4.7reproduces and extends known results on entropy numbers of suchembeddings.

Corollary 4.10. Let the parameters satisfy (3.14) and let theweight w ∈ A∞ be of type (3.21) with (3.22) and α1 = α2 = 0 . Weassume that

s1 − s2 > n

(1

p1− 1

p2

)+

,(4.44)

and

either min

(δ,β1p1

)>

n

p∗, β2 ∈ R, or

β1p1

=n

p∗,

β2p1

>1

p∗.

(4.45)

(i) If np∗ <

β1

p1< δ and β2 ∈ R , then

ek

(As1p1,q1(R

n, w) ↪→ As2p2,q2(Rn))

∼ k− β1np1

− 1p1

+ 1p2 (1 + log k)

− β2p1 , k ∈ N.

(ii) If np∗ < δ < β1

p1and β2 ∈ R , then

ek

(As1p1,q1(R

n, w) ↪→ As2p2,q2(Rn))

∼ k−s1−s2n , k ∈ N.

(iii) If np∗ < δ = β1

p1, β2 ∈ R , and τ = s1−s2

n + 1q2

− 1q1

�= max(0, β2

p1) ,

then for k ∈ N ,


ek

(Bs1p1,q1(R

n, w) ↪→ Bs2p2,q2(Rn))

∼

∼

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩


p1> max(τ, 0),


τ−β2p1 , if β2

p1≤ 0 < τ,

or 0 < β2

p1< τ ≤ β2

p1+ 1

q2− 1

q1,

k−s1−s2n (1 + log k)−

β2p1 , if β2 ≤ 0, τ < 0.

(4.46)

(iv) In case of δ > β1

p1= n

p∗ , β2

p1> 1


ek

(As1p1,q1(R

n, w) ↪→ As2p2,q2(Rn))

∼ (1 + log k)−β2p1

+ 1p∗ ,


ek

(As1p1,q1(R

n, w) ↪→ As2p2,q2(Rn))∼

∼⎧⎨⎩k

− β2p1 , if β2

p1≤ 1

p1− 1

p2,

k−1p1

+ 1p2 (1 + log k)−

β2p1

+ 1p1

− 1p2 , if β2

p1> 1

p1− 1

p2.

Remark 4.11. Cases (i) and (ii) (for B -spaces) are covered by [32,Thms. 5.9, 5.13], case (iv) with p∗ = ∞ by [33, Cor. 2] and forp∗ <∞ by [29, Thm. 7] with some forerunners in [17,20]. Case (iii)with β2 = 0 can be found in [31, Thm. 2] and some partial resultsfor β2 �= 0 in [30, Cor. 7]. The rest is new. Recall that we have forq1 = ∞ even a complete description in the sense of Remark 4.5. Asmentioned above, we believe that (iii) can be extended in all cases(when τ �= max(0, β2

p1) ) to

ek

(Bs1p1,q1(R

n, w) ↪→ Bs2p2,q2(Rn))∼ k−

s1−s2n (1 + log k)

(τ+− β2p1

)+ ,

but have no proof yet.

Acknowledgement. We are especially indebted to our friend andcolleague Th. Kuhn who quickly corresponded to a question of uswith some new result in [29].


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Mathematical InstituteFriedrich-Schiller-University JenaD-07737 Jena, Germany(E-mail : [email protected])

Faculty of Mathematics & Computer ScienceAdam Mickiewicz University, Ul. Umultowska 8761-614 Poznan, Poland(E-mail : [email protected])

(Received : July 2009)

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