approximation numbers of sobolev embeddings of radial...

JOURNAL OF c© 2007, Scientific Horizon

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

Volume 5, Number 1 (2007), 27-48

Approximation numbers of Sobolev embeddings of

radial functions on isotropic manifolds

Leszek Skrzypczak and Bernadeta Tomasz

(Communicated by Hans Triebel)

2000 Mathematics Subject Classification. 46E35, 47B06.

Keywords and phrases. Approximation numbers, Sobolev embeddings, radial

functions.

Abstract. We regard the compact Sobolev embeddings between Besov and

Sobolev spaces of radial functions on noncompact symmetric spaces of rank one.

The asymptotic formula for the behaviour of approximation numbers of these

embeddings is described.

1. Introduction

Approximation numbers measure the closeness by which a boundedoperator may be approximated by linear maps of finite range, whereasentropy numbers measure compactness of the operator by means of finitecoverings of an image of the unit ball. Both, approximation and entropynumbers, of compact Sobolev embeddings between function spaces ofSobolev and Besov type on the Euclidean case have been investigated byseveral authors: D. E. Edmunds and H. Triebel cf. [4], [5], D. Haroske andH. Triebel [10], [11] , D. Haroske [9], and A.M. Caetano [1]. Some of theseresults are described in [6], where one can find also the applications to thespectral theory of the pseudo-differential operators.

28 Approximation numbers of Sobolev embeddings

It was noticed in late seventies of the last century by W. A. Strauss[27] and S. Coleman, V. Glazer, and A. Martin [3] that radiality impliescompactness of Sobolev embeddings, cf. also [19] and [13], [12] forcorresponding results on Riemannian manifolds. W. Sickel and the firstnamed author found the necessary and sufficient conditions for compactnessof the embeddings of radial Besov and Triebel-Lizorkin spaces, cf. [21].Later the corresponding theory was developed for more general symmetryconditions, cf. [24] as well as for function spaces defined on Riemannianmanifolds, [22]. On manifolds the symmetry conditions are expressed interms of invariance with respect to the action of a compact group ofisometries.

If X is a connected d-dimensional Riemannian manifold and o is a fixedpoint of X , then we call a function f radial if its value at x depends onlyon a distance of x to the point o . Isotropic Riemannian manifolds seemsto be of special interest here since on such manifolds radial means invariantwith respect to the action of isotropy group of the point o . A Riemannianmanifold X is called isotropic if for each y ∈ X the linear isotropy groupacts transitively on the unit sphere in the tangent space TyX . It is wellknown that a noncompact Riemannian manifold is isotropic if and onlyif it is either an Euclidean space or a noncompact Riemannian globallysymmetric space of rank one.

The necessary and sufficient conditions for compactness of Sobolevembeddings for Besov and Sobolev function spaces defined on the Euclideanspace R

d with d ≥ 2, are the same as for the functions on thenoncompact Riemannian symmetric space of rank one cf. [21], [22]. Namelythe embeddings

RHs0p0

(X) ↪→ RHs1p1

(X) and RBs0p0,q0

(X) ↪→ RBs1p1,q1

(X)(1)

RHs0p0

(Rd) ↪→ RHs1p1

(Rd) and RBs0p0,q0

(Rd) ↪→ RBs1p1,q1

(Rd)(2)

are compact if and only if p0 < p1 and s0 − dp0

> s1 − dp1

.Asymptotic behaviour of entropy en and approximation numbers an of

the embeddings of radial functions on Rn was studied in [17] and [25]respectively. The behaviour of the entropy numbers in the case of symmetricspaces of rank one was described in [26].

Even though the necessary and sufficient conditions for compactness ofSobolev embedding are the same for Rd and X the asymptotic behaviour ofcorresponding approximation (as well as entropy) numbers is quite different.

We put 1p = 1

p0− 1

p1and t = min{p′0, p1} . The main result of the paper

describes the behaviour of the approximation sequence for embeddings (1)

L. Skrzypczak and B. Tomasz 29

as follows

ak∼

⎧⎪⎪⎨⎪⎪⎩k−(s0−s1− 1

p ) if 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞ ,

k−(s0−s1− 1p + 1

2− 1t ) if 1 ≤ p0 < 2 < p1 ≤ ∞ and s0 − s1 − 1

p ≥ 1t ,

k−(s0−s1− 1

p

)t2 if 1 ≤ p0 < 2 < p1 ≤ ∞ and s0 − s1 − 1

p < 1t .

Whereas, for the compact embeddings (2) we have

(3) ak ∼

⎧⎪⎪⎨⎪⎪⎩k− d−1

p if 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞ ,

k− d−1p − 1

2+ 1t if 1 ≤ p0 < 2 < p1 ≤ ∞ and d−1

p > 1t ,

k− d−1p

t2 if 1 ≤ p0 < 2 < p1 ≤ ∞ and d−1

p ≤ 1t .

In both cases (p0, p1) �= (1,∞), d ≥ 2 and an ∼ bn means that there areconstants c1, c2 > 0 such that inequalities c1an ≤ bn ≤ c2an hold for anyn ∈ N .

2. Preliminaries

2.1. Approximation numbers. We recall the definition of approximationnumbers and corresponding operator ideal quasi-norms, that will be usedwidely in the paper. We refer to books of B. Carl and I. Stephani [2] andA. Pietsch [20] for details, proofs and more information.

Let B0 and B1 be two complex Banach spaces and let T : B0 → B1 bea bounded linear operator. The k th approximation number ak(T ) of theoperator T : B0 → B1 is the infimum of all numbers ‖T − A‖ where A

runs over the collection of all continuous linear operators A : B0 → B1 ofrank smaller than k . So,

ak(T ) := inf{‖T −A‖ : A ∈ L(B0, B1), rank(A) < k},

where rank(A) denote the dimension of the range A(B0).Approximation numbers ak(T ) form a decreasing sequence with a1(T ) =‖T ‖ . If this sequence converges to zero then the operator T is compact.The opposite implication is generally not true. It may happen thatlimk→∞ ak(T ) > 0 for some compact T if B1 fails to have theapproximation property. The approximation numbers have in particularthe following properties:

• (additivity) an+k−1(T1 + T2) ≤ ak(T1) + an(T2),• (multiplicativity) an+k−1(T1T2) ≤ ak(T1)an(T2),• (rank property) an(T ) = 0 ⇔ rank (T ) < n .


Later on we use the notation of operator ideals, cf. [2] and [20] for details.Here we recall only just what we need for the proofs. Given a bounded linearoperator T and a positive real number s we put

L(a)s,∞(T ) := sup

k∈N

k1/s ak(T ) .

The expression L(a)s,∞(T ) is an example of an operator ideal quasi-norm This

means in particular that there exists a number 0 < � ≤ 1 such that

(4) L(a)s,∞

(∑j

Tj

)�

≤∑

j

L(a)s,∞(Tj)� ,

cf. H. Konig, [16, 1.c.5].The following lemma concerning approximation numbers of embeddings

of finite dimensional complex sequence spaces is essentially due to Gluskin[8] cf. also [6, Corollary 3.2.3] and [1]

Lemma 1. Let N, k ∈ N .

(i) If 1 ≤ p0 ≤ p1 ≤ 2 or 2 ≤ p0 ≤ p1 ≤ ∞ , then there is a positiveconstant C independent of N and k such that

ak

(id : �N

p0�→ �N

p1

)≤ C .

(ii) If 1 ≤ p0 < 2 < p1 ≤ ∞ , (p0, p1) �= (1,∞) , then there is a positiveconstant C independent of N and k such that

ak

(id : �N

p0�→ �N

p1

)≤ C

⎧⎪⎨⎪⎩1 if k ≤ N2/t ,

N1/tk−1/2 if N2/t < k ≤ N ,

0 if k > N ,

where 1t = max

(1p1

, 1p′0

).

Moreover if k ≤ N4 then in both cases we have an equivalence.

2.2. Function spaces, embeddings and traces. We recall the definitionof Sobolev spaces and Besov spaces on Riemannian symmetric space. Thereare several possible approaches. Here we present the definition in terms of aheat semi-group. We assume that the reader is familiar with the definitionand elementary properties of function from fractional Sobolev spaces Hs

p

and Besov spaces Bsp,q on R

n . All we need can be found in [29].Our notation related to symmetric spaces is standard and can be found,

for example, in [14] or [15]. Let G be noncompact connected semisimpleLie group with finite center of real rank one. Let K be its maximal compact


subgroup and let X = G/K be an associated symmetric spaces of dimensiond . The Killing form of G induces a G-invariant Riemannian metric on X .So X is simply connected homogeneous Riemannian manifolds and G actstransitively on X as a group of isometries. If o = eK then K is an isotropygroup of o . The assumption that G is a group of rank one implies that thegroup K acts transitively on any sphere S(o, r) = {x ∈ X : |x| = r} , |x|denotes the Riemannian distance from x to o .

Let Δ denote the Laplace operator on X . The heat semi-group Ht =etΔ , t ≥ 0, is a positive, symmetric, semigroup of contractions in Lp(X),1 ≤ p ≤ ∞ , such that Ht1 = 1. It is strongly continuous if p < ∞ andanalytical if 1 < p <∞ .

Let S1(X) denote the L1 -Schwartz space on X and S′1(X) be thecorresponding space of distributions. For convenience we put

H0,kf =k−1∑�=0

1�!

(−Δ)�H1f , f ∈ S′1(X) .

Definition 1. Let s ∈ R and d ≥ 2.(a) Let 1 < p <∞ . A fractional Sobolev space Hs

p(X) is defined by

Hsp(X) =

{f ∈ S′1(X) :

∥∥f |Hsp(X)

∥∥ :=∥∥(I −Δ)s/2f

∥∥p

< ∞}.

(b) Let 1 ≤ p, q ≤ ∞ and k > |s|2 . A Besov space Bs

p,q(X) is a space ofdistributions f ∈ S′1(X) such that

∥∥f |Bsp,q(X)

∥∥ :=∥∥H0,kf

∥∥p

+(∫ 1

0

t(k−s/2)q

∥∥∥∥ dk

dtkHtf

∥∥∥∥q

p

dt

t

)1/q

< ∞ .

Remark 1. (1) The definition of the Besov spaces is independent of k

up to norm equivalence, cf. [22]. One can give an equivalent norm for theSobolev spaces in term of heat semi- group, cf. ibidem.

(2) The spaces Hsp(X) and Bs

p,q(X) are Banach spaces. If s is a positiveinteger then the space Hs

p(X) coincides with the classical Sobolev spacedefined in term of gradient of order s .

(3) If s > 0 then ‖H0,kf‖p can be replaced by ‖f‖p in the definition ofBesov spaces.

(4) The above definition coincides with the definition of Hsp -Bs

p,q spaces ona Riemannian manifold with bounded geometry by the uniform localizationprinciple, cf. [22]. The last approach is due to H.Triebel, [30].


(5) The Besov spaces are real interpolation spaces of the Sobolev spaces.More precisely(Hs0

p (X) , Hs1p (X))θ,q = Bs

p,q(X) , s = θs0+(1−θ)s1 , 0 < θ < 1 , s0 �= s1 ,

if 1 < p < ∞ . In particular Hs2(X) = Bs

2,2(X) and the norms areequivalent.

(6) Most of the results, we quote for Sobolev spaces, is also true, mutatismutandis, for more general Triebel-Lizorkin function spaces F s

p,q .

Since the group K acts transitively on spheres centered at o , a functionis radial if and only if is invariant with respect to the action of K . So thenotation of radiality can be extended to distribution. The distribution f iscalled radial if

f(ϕk) = f(ϕ) , where ϕk(x) = ϕ(k · x) , k ∈ K, x ∈ X .

So for any possible s, p, q we can put

RBsp,q(X) =

{f ∈ Bs

p,q(X) : f is radial}.

The spaces RHsp(X) are defined in the similar way. The both spaces are

close subspaces of Bsp,q(X) and Hs

p(X) respectively, so they are Banachspaces.

The following theorem describing the compactness of the Sobolevembeddings was proved in [22]

Theorem 1. Let −∞ < s1, s0 < ∞ , 1 ≤ p0, p1, q0, q1 ≤ ∞ and d ≥ 2 .The embeddings

RBs0p0,q0

(X) ↪→ RBs1p1,q1

(X) , RHs0p0

(X) ↪→ RHs1p1

(X)

are compact if and only if p0 < p1 and s0 − dp0

> s1 − dp1

(with therestriction 1 < p0 < p1 <∞ for Sobolev spaces.)

We will calculate the approximation numbers of the above embeddings.For future use we need also some information on traces of radial Besov

spaces on geodesic rays starting at origin o . The traces can be described interms of weighted Besov spaces on the ray.

It can be proved that related trace and extension operators give us anisomorphisms of some radial and weighted Besov spaces, cf. [26].


To present this result in details we will need a positive weight functionvp ∈ C∞(R) such that

(5) vp(t) = expϑ|t|p

if |t| ≥ 1

(the exact behaviour near zero is not important). Here ϑ is a positiveconstant depending on X . It is sufficient for us to regard the weightedBesov spaces Bs

p,q(R, vp) with positive smoothness s > 0. They may bedefined as follows. The function f ∈ Lp(R) belongs to Bs

p,q(R, vp) if andonly if vpf ∈ Bs

p,q(R) moreover∥∥f |Bsp,q(R, vp)

∥∥ :=∥∥vpf |Bs

p,q(R)∥∥ .

Let γ : (−∞,∞)→ X be a geodesic parametrized by the arc length suchthat γ(0) = o . It should be clear that we can define a weighted Besov spaceon the geodesic by pulling back to the weighted Besov space defined on R ,i.e.

f ∈ Bsp,q(γ(−∞,∞), vp) if and only if f ◦ γ ∈ Bs

p,q(R, vp).

We are interested in a trace on γ((0,∞)). For f continuous on X we put

trf(t) = f(γ(t)) .

To describe the traces of radial function on geodesic ray out of the originwe need the following spaces.

Definition 2. Let 0 < τ <∞ , 1 ≤ p, q ≤ ∞ , and s > 0. Then we put

Bsp,q(γ[τ,∞), vp) := {f ∈ Bs

p,q(γ(−∞,∞), vp) : suppf ⊂ γ[τ,∞)},RBs

p,q(X \B(o, τ)) := {f ∈ RBsp,q(X) : suppf ⊂ {x ∈ X : |x| ≥ τ}}.

Theorem 2. Let 1 ≤ p, q ≤ ∞ , s > 0 and τ > 0 . There exist continuousoperators

tr : RBsp,q(X \B(o, τ)) �→ Bs

p,q(γ[τ,∞), vp),

ext : Bsp,q(γ[τ ;∞), vp) �→ RBs

p,q(X \B(o, τ)).

such that tr ◦ ext = id and ext ◦ tr = id.

The proof of this theorem can be found in [26], but it is similar to theproof of theorem about the traces on a straight line in the Euclidean case,cf. [17].


3. Approximation numbers of the Sobolev embeddings

Let us remind that 1p = 1

p0− 1

p1and t = min{p′0, p1} , as before. The

main result of the paper reads as follows

Theorem 3. Let s0, s1 ∈ R , 1 ≤ p0 < p1 ≤ ∞ (1 < p0 < p1 < ∞in the case of the Sobolev spaces), (p0, p1) �= (1,∞) , 1 ≤ q0, q1 ≤ ∞ ,s0 − d

p0> s1 − d

p1and d ≥ 2 . Then

(6) ak

(RBs0

p0,q0(X) ↪→ RBs1

p1,q1(X)

) ∼ k−κ

and

(7) ak

(RHs0

p0(X) ↪→ RHs1

p1(X)

) ∼ k−κ ,

where(8)

κ=

⎧⎪⎪⎨⎪⎪⎩s0 − s1 − 1

p if 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞s0 − s1 − 1

p + 12 − 1

t if 1 ≤ p0 < 2 < p1 ≤ ∞ and s0 − s1 − 1p ≥ 1

t(s0 − s1 − 1

p

)t2 if 1 ≤ p0 < 2 < p1 ≤ ∞ and s0 − s1 − 1

p < 1t .

Remark 2. (1) The formula (7) follows from (6) by elementaryembeddings and properties of approximation numbers. The proof of (6)is presented in Section 3.1.(2) The estimates of the approximation numbers described in the abovetheorem are independent of the dimension d of the underlying space X .This is in some contrast with the behaviour of the approximation numbersfor radial function spaces defined on R

d , where we have dependence on d inall cases as it was proved in [25], cf. (3). Moreover the asymptotic behaviourof the approximation numbers in (6) and (7) is the same as in the case ofbounded domains in one dimensional Euclidean space R , cf. [6].The same phenomenon occurs for entropy numbers ek of the embeddings(6) and (7). In this case we have

ek ∼ k−(s0−s1) .

as it was proved in [26].(3) The theorem holds also for Triebel-Lizorkin spaces F s

p,q(X) with 1 ≤p, q <∞ .

3.1 The proof of the main theorem. The proof of the main theorem isbased on the same idea as that for entropy numbers (see [26]). We dividethe origin space into two parts: the local part near origin o and the global


part out of the origin. Since X is the symmetric space of real rank onethe analysis of the local part can be reduced to corresponding embeddingsof function spaces defined on the Euclidean ball. On the other hand theanalysis of the global part can be reduced to embeddings of weightedfunction spaces on the real line due to the trace theorem cf. Theorem2.

So to prove Theorem 3 we divide the identity operator

id : RBs0p0,q0

(X) ↪→ RBs1p1,q1

(X)

into two parts in the following way. Let φ ∈ C∞0 (X), φ(x) = 1 for

x ∈ B(o, 12 ), supp φ ⊂ B(o, 1), we put id1f = φf , id2f = (1 − φ)f .

Then

id = id1 + id2

and in consequence by additivity of approximation numbers

(9) a2k(id) ≤ ak(id1) + ak(id2).

Moreover, it should be clear that

ak(id1) ∼ ak(Id1) and ak(id2) ∼ ak(Id2) ,

where

Id1 : RBs0

p0,q0

(B(o, 1)

)↪→ RB

s1

p1,q1

(B(o, 1)

)and

Id2 : RBs0p0,q0

(X \B(o, 1)

)↪→ RBs1

p1,q1

(X \B(o, 1)

).

Here the spaces RBs

p,q

(B(o, 1)

)are defined by

(10) RBs

p,q

(B(o, 1)

)=

{f ∈ RBs

p,q(X) : supp f ⊂ B(o, 1)}.

whereas the spaces RBsp,q

(X \ B(o, 1)

)were introduced in Definition 2.

Thus to estimate ak(id) it is sufficient to estimate ak(Id1) and ak(Id2).But the space described in (10) is isomorphic to the corresponding space

RBs

p,q

(Be(0, 1)

)defined on the unit ball Be(0, 1) ⊂ Rd of the Euclidean

space Rd , in the way similar to (10). Thus

(11) ak(Id1) ∼ ak

(RB

s0

p0,q0

(Be(0, 1)

)↪→ RB

s1

p1,q1

(Be(0, 1)

)).

The last approximation numbers are estimated from above in Subsection3.1.2


On the other hand, the space RBsipi,qi

(X \ B(o, 1)

)is isomorphic to the

space Bsipi,qi

(γ[1,∞), vpi

), cf. (5) and Theorem 2.

By standard arguments

ak

(Id2

) ∼ ak

(Bs0

p0,q0

(γ[1,∞), vp

)↪→ Bs1

p1,q1

(γ[1,∞)

))(12)

∼ ak

(Bs0

p0,q0

(R, vp

)↪→ Bs1

p1,q1

(R))

where 1p = 1

p0− 1

p1. Both the lower and upper estimates of the last

approximation numbers are known at least for specialist. Indeed, it followseasily from the estimates for polynomial weights, cf. Lemma 3 below andProposition 2. This estimates will finish the estimates of ak(id) from aboveas well as give us the estimates from below since we have the followingcommutative diagram:

RBs0p0,q0

(X \B(o, 1)) id−−−−→ RBs0p0,q0

(X)

Id2

⏐⏐� ⏐⏐�id

RBs1p1,q1

(X \B(o, 1))Tφ←−−−− RBs1

p1,q1(X) .

Here Tφ denotes an operator Tφ : f �→ (1− φ)f .

The approximation numbers an

(Bs0

p0,q0

(R, vp

)↪→ Bs1

p1,q1

(R))

aredescribed in Subsection 3.1.3. But first we need information aboutapproximation numbers of some weighted sequence spaces.

3.1.1. Approximation numbers of some sequence spaces. We willneed weighted spaces of double indexed sequences. Let 1 ≤ p, q ≤ ∞ . Letδ ≥ 0 and let w : N0 × N0 → R+ denote a weight function. In particularfor the positive α > 0 we put(13)wα(j, k) = (1 + k)α and wα(j, k) = (1 + 2−jk)α and vα(j, k) = exp(α2−jk)

We will work with the following weighted sequence spaces

(14) �q(2jδ�p(w)) =

{(sj,k)j,k : ‖sj,k|�q(2jδ�p(w))‖ =⎛⎝ ∞∑

j=0

( ∞∑k=0

2jδpw(j, k)|sj,k|p)q/p )1/q

<∞⎫⎬⎭

(with the usual modification if p = ∞ or q = ∞). If δ = 0 we will write�q(�p(w)).


In future considerations it will be useful to regard the following subspacesof the above sequence spaces:

�q(2jδ�γ2j

p (w)) =

{(sj,k)j,k ∈ �q(2jδ�p(w)) : sj,k = 0 if k > γ2j

}(15)

�q(2jδ �γ2j

p (w)) =

{(sj,k)j,k ∈ �q(2jδ�p(w)) : sj,k = 0 if k ≤ γ2j

},(16)

γ ∈ N . Now we formulate the main result of this subsection.

Lemma 2. Let 1 ≤ p0 < p1 ≤ ∞ , (p0, p1) �= (1,∞) , 1 ≤ q0, q1 ≤ ∞ ,α > 0 , δ > 0 and γ > 0 . Then for all k ∈ N the following estimate

ak(id : �q0(2jδ�γ2j

p0(wα))→ �q1(�

γ2j

p1(wα))) ∼ k−β

holds, where

β =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩δ + α

p for 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞ ,

δ + αp + 1

2 − 1t for 1 ≤ p0 < 2 < p1 ≤ ∞ and δ + α

p ≥ 1t ,

(δ + αp ) t

2 for 1 ≤ p0 < 2 < p1 ≤ ∞ and δ + αp < 1

t ,

and 1p = 1

p0− 1

p1, t = min{p1, p

′0} .

Proof. Step 1. Preparations. Let

Λ := {λ = (λj,�)j,� : λj,� ∈ C, j ∈ N0, 0 ≤ � ≤ γ2j}

and

B0 = �q0(2jδ�γ2j

p0(wα)) , B1 = �q1(�

γ2j

p1(wα)) .

Let Pj : Λ �→ Λ be the canonical projection onto j -level, i.e. forλ =

(λk,�

)we put

(Pjλ)� :={

λk,� if k = j,0 otherwise, � ∈ N0 .

Monotonicity arguments and elementary properties of the approximationnumbers yield

(17) ak

(Pj : B0 �→ B1

)≤ 2−jδ ak

(Id : �γ2j

p0(wα) �→ �γ2j

p1(wα)

),


and

ak

(Id : �γ2j

p0(wα) �→ �γ2j

p1(wα)

)= ak

(Dσ : �γ2j

p0�→ �γ2j

p1

)(18)

where Dσ is a diagonal operator defined by the sequence σ� = (1+�)−α/p .Step 2. The estimate from above.Substep 2.1. First we regard the case 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤∞ . Using (17) and (18) we find

(19) L(a)r,∞(Pj) ≤ c 2−jδ L(a)

r,∞(Dσ : �γ2j

p0�→ �γ2j

p1) .

We have

(20) ak

(Dσ : �γ2j

p0�→ �γ2j

p0

)= σk = (1 + k)−α/p,

cf. [20, p. 108]. Now using (20) and the elementary properties of approxi-mation numbers we get

k1r a2k−1

(Dσ : �γ2j

p0�→ �γ2j

p1

)≤ C k

1r −α

p ak

(id : �γ2j

p0�→ �γ2j

p1

).

In consequence

(21) L(a)r,∞

(Dσ : �γ2j

p0�→ �γ2j

p1

)≤ C L(a)

s,∞(

id : �γ2j

p0�→ �γ2j

p1

)if 1

s = 1r − α

p > 0. But Lemma 1 implies(22)

L(a)s,∞(id : �N

p0�→ �N

p1) ≤ C N

1s if 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞ .

Under the assumption 1/r > α/p we conclude from (21) and (22) that

(23) L(a)r,∞(Dσ : �γ2j

p0�→ �γ2j

p1) ≤ C 2−j( α

p − 1r ) ,

where the constant C depends on γ but is independent of j .Now, for given M ∈ N0 let

(24) P :=M∑

j=0

Pj and Q :=∞∑

j=M+1

Pj .

Hence (4), (19), (23) and (24) yield:(25)

L(a)r,∞(P )� ≤

M∑j=0

L(a)r,∞(Pj)� ≤ c1

M∑j=0

2−j�(δ− 1r + α

p ) ≤ c2 2−M�(δ− 1r + α

p ) ,


with the constant c2 independent of M , if 1r > α

p + δ . Hence for everyδ > 0 we have

a2M (P : B0 �→ B1) ≤ c3 2−M(δ+ αp ) .

In a similar way to (25) we obtain

L(a)r,∞(Q)� ≤ c1

∞∑j=M+1

2−j�(δ− 1r + α

p ) ≤ c22−M�(δ− 1r + α

p ) ,

if r > 0 such that αp < 1

r < δ + αp . Hence, for r s.t. α

p < 1r < δ + α

p andk = 2M we have

a2M (Q : B0 �→ B1) ≤ c3 2−M(δ+ αp ) .

This implies the estimate from above for 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤∞ .

Substep 2.2. Now let 1 ≤ p0 < 2 < p1 ≤ ∞ and let (p0, p1) �= (1,∞).First we prove that

L(a)r,∞(Dσ : �γ2j

p0→ �γ2j

p1)≤C 2j( 1

r − 12+ 1

t −αp )(26)

if1r

> max(12,α

p+

12− 1

t

),

L(a)r,∞(Dσ : �γ2j

p0→ �γ2j

p1)≤C 2j( 2

rt−αp )(27)

iftα

2p<

1r

<12

(that meansα

p<

1t).

We put N = γ2j . Let us choose K ∈ N such that 2K−1 ≤ N < 2K andlet Πi : �N

p0→ �N

p0be a projection defined in the following way

(Πiλ)� :={

λ� if 2i−1 ≤ � < min(2i, N),0 otherwise ,

for λ =(λ�

), i = 1, 2, . . . , K . Then

Dσ =K∑

i=1

Dσ ◦Πi .

Multiplicativity of the approximation numbers yields

ak

(Dσ ◦Πi : �N

p0→ �N

p1

)≤ 2−(i−1) α

p ak

(id : �2i−1

p0→ �2i−1

p1

), i ∈ N.


Let r be real positive number. Lemma 1 implies

L(a)r,∞(Dσ ◦Πi : �N

p0→ �N

p1)≤C 2(i−1)( 1

t + 1r − 1

2−αp ) if

12

<1r

,(28)

L(a)r,∞(Dσ ◦Πi : �N

p0→ �N

p1)≤C 2(i−1)( 2

rt−αp ) if 0 <

1r

<12

.(29)

We choose r such that 1r > 1

2 and that 1r − 1

2 + 1t − α

p > 0. The formulas(4) and (28) yield

L(a)r,∞(Dσ)� ≤

K∑i=1

L(a)r,∞(Dσ ◦Πi)�

≤ c1

K−1∑i=0

2−�i( αp − 1

t − 1r + 1

2 ) ≤ c2 2�K( 1r − 1

2+ 1t −α

p ).

The constant c2 is independent of K . This implies (26). In the similar wayif tα

2p < 1r < 1

2 , then the formulas (4) and (29) yield

L(a)r,∞(Dσ)� ≤

K∑i=1

L(a)r,∞(Dσ ◦Πi)�

≤ c1

K−1∑i=0

2−�i( αp− 2

rt) ≤ c2 2�K( 2

rt−α

p)

and the constant c2 is independent of K . This implies (27).Substep 2.3. Once more let 1 ≤ p0 < 2 < p1 ≤ ∞ , (p0, p1) �= (1,∞).

Using (26) instead of (23) we get

L(a)r,∞(P )� ≤

M∑j=0

L(a)r,∞(Pj)� ≤ c32−M�(δ− 1

r + αp + 1

2− 1t )

if1r

> max{

δ +α

p+

12− 1

t,12

},

and in consequence

a2M (P : B0 �→ B1) ≤ c3 2−M(δ+ αp + 1

2− 1t )(30)

for 1 ≤ p0 < 2 < p1 ≤ ∞ ,

(p0, p1) �= (1,∞).


In the similar way

L(a)r,∞(Q)� ≤ c4

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

2−M�(δ− 1r + α

p + 12− 1

t )

if max(

12 , α

p + 12 − 1

t

)< 1

r < δ + αp + 1

2 − 1t

2−M�(δ+ αp − 2

tr )

if tα2p < 1

r < min(

12 , t

2 (δ + αp )).

(31)

Now, (30) and the first case of (31) imply the estimates from above forδ + α

p > 1t .

For δ + αp ≤ 1

t we take k = [2M 2t ] . Then α

p < 1t so by the second case of

(31) we have

ak(Q : B0 �→ B1) ≤ c k− 1r 2−(δ+ α

p − 2tr ) ≤ c k− t

2 (δ+ αp ) .(32)

By standard argument the above estimates hold for any positive integer k .Moreover, (30) implies

ak(P : B0 �→ B1) ≤ C k−(δ+ αp + 1

2− 1t ) ≤ C k− t

2 (δ+ αp )(33)

since δ + αp ≤ 1

t . Now (32) and (33) give us the estimates from above inthe remaining case.

Step 3. We estimate the approximation numbers from below. The space�q1

(�γ2j

p1

)is isomorphic to �q1

(�γ2j

p1(wα)

)and the isomorphism is given by

the diagonal operator aj,i �→ (1+i)−α/p1aj,i . Moreover the inverse operatoraj,i �→ (1 + i)α/p1aj,i is an isomorphism of the space �q02jδ

(�γ2j

p0(wα)

)onto �q02jδ

(�γ2j

p0(wβ)

)with β = αp0

p . This implies the equivalence ofapproximation numbers

ak

(�q02

jδ(�γ2j

p0(wα)

)→�q1

(�γ2j

p1(wα)

)) ∼ ak

(�q02

jδ(�γ2j

p0(wβ)

)→�q1

(�γ2j

p1

)).

Thus we can regard the following commutative diagram

�γ2j

p0

R−−−−→ �q0

(2jδ�γ2j

p0(wβ)

)Id

⏐⏐� ⏐⏐�id

�γ2j

p1←−−−−

P�q1

(�γ2j

p1

)where an operator R : �γ2j

p0→ �q0

(2jδ�γ2j

p0(wβ)

)is given by

(Rλ)u,i :={

λi u = j,0 otherwise. u ∈ N0, 0 ≤ i ≤ γ2j .


and P is a projection. It should be clear that ‖R‖ ≤ c 2j(δ+ αp ) and ‖P‖ = 1.

So, by Lemma 1,(i) if 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞ then taking k = γ2j−2 we get

1 ≤ C ak

(Id : �γ2j

p0�→ �γ2j

p1

)≤ C 2j(δ+ α

p )ak

(id : �q02

jδ(�γ2j

p0(wα)

) �→ �q1

(�γ2j

p1(wα)

)).

(ii) If 1 ≤ p0 < 2 < p1 ≤ ∞ and δ+ αp ≥ 1

t then taking one more k = γ2j−2

we have

2−j( 12− 1

t ) ≤ C ak

(Id : �γ2j

p0�→ �γ2j

p1

)≤ C 2j(δ+ α

p )ak

(id : �q02

jδ(�γ2j

p0(wα)

) �→ �q1

(�γ2j

p1(wα)

)).

(iii) If 1 ≤ p0 < 2 < p1 ≤ ∞ and δ + αp < 1

t then we take k =[2j 2

t

]. So,

1 ≤ C ak

(Id : �γ2j

p0�→ �γ2j

p1

)≤ C 2j(δ+ α

p )ak

(id : �q0

(�γ2j

p0(wα)

) �→ �q12jδ(�γ2j

p1(wα)

)).

This finishes the proof. �

Lemma 3. Let 1 ≤ p0 < p1 ≤ ∞ , 1 ≤ q0, q1 ≤ ∞ , α, δ > 0 . Then thefollowing estimate

ak

(id : �q0(2

jδ�p0(vα))→ �q1(�p1)) ∼ k−κ,

holds, where

κ =

⎧⎪⎨⎪⎩δ if 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞,

δ + 12 − 1

t if 1 ≤ p0 < 2 < p1 ≤ ∞ and δ > 1t ,

δ t2 if 1 ≤ p0 < 2 < p1 ≤ ∞ and δ ≤ 1

t .

Proof. The estimates from above follows immediately from thecorresponding results for polynomial weights cf. [23]. Let us choose β > δ .Then

�q0(2jδ�p0(vα)) ↪→ �q0(2

jδ�p0(wβ)) ↪→ �q1(�p1)So, the results from in [23] implies

ak

(id : �q0(2

jδ�p0(vα))→ �q1(�p1)) ≤ Cak

(id : �q0(2

jδ�p0(wβ))→ �q1(�p1))

≤ C k−κ,

where κ is defined as before.


The estimates from below can be proved in the way similar to Step 3 inthe proof of the last lemma. We regard the following commutative diagram

�Np0

S−−−−→ �q0

(2jδ�p0(vα)

)id

⏐⏐� ⏐⏐�id

�Np1

T←−−−− �q1(�p1) ,

with

(Tλ)i = λj,i , i = 1, . . . , N and (Sη)k,i =

{ηi k = j and i ≤ N,

0 otherwise ;

Then

ak

(id : �N

p0−→ �N

p1

) ≤ C 2jδak

(id : �q0(2

jδ�p0(vα)) −→ �q1(�p1)).(34)

(i) For 1 ≤ p0 < p1 ≤ 2 or 2 ≤ p0 < p1 ≤ ∞ , and for 1 ≤ p0 < 2 < p1 ≤ ∞and δ > 1

t we take k = 2j−2 and N = 2j . So by Lemma 1 and (34) we get

C2−jδ ≤ ak

(id : �q0(2

jδ�p0(vα)) −→ �q1(�p1)),

C2−j( 12− 1

t +δ) ≤ ak

(id : �q0(2

jδ�p0(vα)) −→ �q1(�p1)),

respectively.(ii) For 1 ≤ p0 < 2 < p1 ≤ ∞ and δ ≤ 1

t we take k = [2j 2t ] and N as

above. Using once again (34) and Lemma 1 we get

Ck− δt2 ≤ C2−jδ ≤ ak

(id : �q0(2

jδ�p0(vα)) −→ �q1(�p1)). �

3.1.2. Approximation numbers for function spaces: local analysis.Now we apply the estimates for approximation numbers of embeddings ofsequence spaces to function spaces, first we regard the local part near origin.We use the Epperson-Frazier approach to radial Besov spaces and theirconstruction of the radial ϕ-transform, cf. [7]. We apply the constructionwith normalization described [17] that is different to the original one. Forany possible s and p Epperson and Frazier constructed two families of radialfunctions ϕ

(s,p)j,k ∈ S(Rd) and η

(s,p)j,k ∈ S(Rd), j = 0, 1, . . ., k = 1, 2, . . ., such

that any radial distribution f ∈ S′(Rn) can be decomposed into

f =∞∑

j=0

∞∑k=1

< f, ϕ(s,p)j,k > η

(s,p)j,k


(convergence in S′(Rn)), cf. [7]. Moreover the following theorem holds forradial Besov spaces.

Theorem 4 (Epperson-Frazier). Let s ∈ R and 1 ≤ p, q ≤ ∞ . Theoperators

S(s,p) : RBsp,q(R

n) −→ �q(�p(wd−1))and

T(s,p) : �q(�p(wd−1)) −→ RBsp,q(R

n)defined by

S(s,p)(f) = (< f, ϕ(s,p)j,k >)j,k , T(s,p)((sj,k+1)) =

∞∑j=0

∞∑k=0

sj,k η(s,p)j,k+1 .

are bounded. The operator T is a retraction, i.e. T ◦ S = id .

The last theorem and the next proposition reduce the estimates fromabove of approximation numbers of Sobolev embeddings of the functionspaces to the estimates of approximation numbers of embeddings of thesequence spaces defined in (14)-(16). The following lemma was proved in[26]

Lemma 4. Let 1 ≤ p0 < p1 ≤ ∞ , and 1 ≤ q0, q1 ≤ ∞ . Lets0− d

p0> s1 − d

p1, s0 > 0 and α ≥ d− 1 . We put δ = s0 − s1 + d( 1

p1− 1

p0)

and s = s1 + d( 1p0− 1

p1) . Then

S(s,p0) = S(s1,p1) = S , T(s,p0) = T(s1,p1) = T

and the following diagram is commutative

RBs0

p0,q0

(Be(0, 1)

)S−−−−→ �q0(2jδ�2j+2

p0(wd−1)) ⊕ �q0(2jδ �2j+2

p0(wα))

id

⏐⏐� ⏐⏐�id

RBs1

p1,q1

(Be(0, 1)

)T←−−−− �q1(�2j+2

p1(wd−1)) ⊕ �q1(�2j+2

p1(wα)) .

The last lemma has an important consequence. Since α is at our disposaland can be as large as we want, approximation numbers of the embeddingsof the above function spaces can be estimated from above by approximationnumbers of sequence spaces of the type �q(2jδ�2j+2

p (wd−1)).

Proposition 1. Suppose 1 ≤ p0 < p1 ≤ ∞ , 1 ≤ q0, q1 ≤ ∞ ands0 − s1 − d( 1

p0− 1

p1) > 0 . Then there exist a positive constant C such that

ak

(RB

s0

p0,q0

(Be(0, 1)

)→ RB

s1

p1,q1

(Be(0, 1)

))≤ Ck−κ ,


where κ is defined by (8).

Proof. By Lemma 4 we have

(35) ak

(id : RB

s0

p0,q0

(Be(0, 1)

)→ RB

s1

p1,q1

(Be(0, 1)

))≤ C ak

(Id : �q0(2

jδ�2j+2

p0(wd−1))⊕ �q0(2

jδ �2j+2

p0(wα))

→ �q1(�2j+2

p1(wd−1)) ⊕ �q1(�

2j+2

p1(wα))

).

We divide the operator Id into two parts

(id1(λ))j,� =

{λj,� � ≤ 2j+2

0 otherwise ,(id2(λ))j,� =

{λj,� � ≥ 2j+2 + 10 otherwise .

Then Id = id1 + id2 and

(36) a2k−1

(Id) ≤ ak

(id1

)+ ak

(id2

).

On the one hand Lemma 2 with δ = s0 − s1 − d( 1p0− 1

p1) and α = d − 1

asserts that(37)ak

(id1

) ≤ ak

(id : �q0(2

jδ�2j+2

p0(wd−1))→ �q1(�

2j+2

p1(wd−1))

)≤ Ck−κ,

where κ is given by (8). On the other hand if we take α such that(α + 1

)(1p0− 1

p1

)= s0 − s1 then α > d− 1 and

ak

(id2

) ≤ ak

(�q0(2

jδ �2j+2

p0(wα) ↪→ �q1(�

2j+2

p1(wα))

)(38)

≤ ak

(�q0(2

jδ�p0(wα/p) ↪→ �q1(�p1))

≤ Ck−κ ,

where the last inequality was proved in [23] with κ given by (8). Now thelemma follows from (35)-(38). �

3.1.3. Approximation numbers for function spaces: globalanalysis. We will need the estimates of approximation numbers of Sobolevembeddings of weighted function Besov spaces on R with exponentialweights. Let vp0 , vp1 , vp be the exponential weights defined by (5). Forα = ϑ

p , let vα be the weight for the sequence space (13)-(14). It was provedin [18] that the Besov space Bs0

p0,q0(R, vp) is isomorphic to the sequence


space �q0

(2jδ0�p0(vα)

), 0 < s0 <∞ and δ0 = s0 + 1

2 − 1p0

, cf. [18, Theorem1]. Thus for the approximation numbers we get

ak

(Bs0

p0,q0(R, vp0) ↪→ Bs1

p1,q1(R, vp1)

)(39)

∼ ak

(Bs0

p0,q0(R, vp) ↪→ Bs1

p1,q1(R)

)∼ ak

(�q0

(2jδ0�p0(vα)

)↪→ �q1

(2jδ1�p1

))∼ ak

(�q0

(2jδ�p0(vα)

)↪→ �q1

(�p1

))δ1 = s1 + 1

2 − 1p1

and δ = s0 − s1 + 1p1− 1

p0. Now (39), Lemma 3 and the

lift property for the Besov spaces imply the following proposition

Proposition 2. Suppose 1 ≤ p0 < p1 ≤ ∞ , 1 ≤ q0, q1 ≤ ∞ , α > 0 andδ = s0 − s1 − ( 1

p0− 1

p1) > 0 , then

ak

(Bs0

p0,q0(R, vp0) ↪→ Bs1

p1,q1(R, vp1)

)∼ k−κ ,

where κ is the same as in Lemma 3.

Now Theorem 3 follows from (9), (11), (12), Theorem 2, Proposition 1and Proposition 2.

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Faculty of Mathematics and Computer ScienceA. Mickiewicz UniversityUmultowska 8761-614 PoznanPoland(E-mail : [email protected])(E-mail : [email protected])

(Received : October 2005 )

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