Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball -- Preasymptotics, asymptotics, and tractability

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by  Jia Chen

APPROXIMATION NUMBERS OF SOBOLEV AND GEVREY
TYPE EMBEDDINGS ON THE SPHERE AND ON THE BALL –
7 PREASYMPTOTICS, ASYMPTOTICS, AND TRACTABILITY
1
0
2 JIACHEN,HEPINGWANG

n
a Abstract. In this paper, we investigate optimal linear approximations (n-
J approximationnumbers)oftheembeddingsfromtheSobolevspacesHr (r>

3 0)forvariousequivalentnormsandtheGevreytypespacesGα,β (α,β>0)on
1 thesphereSdandontheballBd,wheretheapproximationerrorismeasuredin
theL2-norm. Weobtainpreasymptotics,asymptotics,andstrongequivalences

] of the above approximation numbersas afunction inn andthe dimensiond.
A
We emphasis that all equivalence constants in the above preasymptotics and
C asymptotics are independent of the dimension d and n. As a consequence
we obtain that for the absolute error criterion the approximation problems
.
h Id :Hr →L2 areweaklytractable ifandonlyifr>1,notuniformlyweakly
t tractable, and do not suffer from the curse of dimensionality. We also prove
a
m that for any α,β > 0, the approximation problems Id : Gα,β → L2 are uni-
formly weakly tractable, not polynomially tractable, and quasi-polynomially
[ tractable ifandonlyifα≥1.

1
v
5
4 1. Introduction
5
3 Thispaperisdevotedtoinvestigatingthebehavioroftheapproximationnumbers
0 of embeddings of Sobolev spaces and Gevrey type spaces on the sphere Sd and on
. the ball Bd into L . The approximation numbers of a bounded linear operator
1 2
0 T :X →Y between two Banach spaces are defined as
7
a (T :X →Y):= inf sup kTx−Ax|Yk
1 n
rankA<nkx|Xk≤1
:
v = inf kT −A:X →Yk, n∈N ,
+
i rankA<n
X
where N = {1,2,3,...}, N = {0,1,2,3,...}. They describe the best approxi-
r +
a mation of T by finite rank operators. If X and Y are Hilbert spaces and T is
compact, then a (T) is the nth singular number of T. Also a (T) is the nth mini-
n n
mal worst-case error with respect to arbitrary algorithms and general information
in the Hilbert setting.
On the torus Td, there are many results concerning asymptotics of the approxi-
mationnumbers ofsmoothfunction spaces,see the monographs[21] by Temlyakov
and the references therein. However, the obtained asymptotics often hide depen-
dencies on the dimension d in the constants, and can only be seen after “waiting
exponentiallylong(n≥2d)”ifdislarge. Inordertoovercomethisdeficiency,Ku¨hn
and other authors obtained preasymptotics and asymptotics of the approximation
numbers of the classical isotropic Sobolev spaces, Sobolev spaces of dominating
Key words and phrases. Approximation number; Sobolev spaces; Gevrey type spaces,
preasymptotics andasymptotics;tractability.
1
2
mixed smoothness, periodic Gevrey spaces, and anisotropic Sobolev spaces (see
[10, 11, 9, 2]). Note that in these preasymptotics and asymptotics, the equivalence
constants are independent of the dimension d and n.
On the sphere Sd and on the ball Bd, the following two-sided estimates can be
found in [7] and [22] in a slightly more general setting:
(1.1) C (r,d)n−r/d ≤a (I :Hr(Sd)→L (Sd))≤C (r,d)n−r/d, n∈N ,
1 n d 2 2 +
and
(1.2) C (r,d)n−r/d ≤a (I :Hr(Bd)→L (Bd))≤C (r,d)n−r/d, n∈N ,
3 n d 2 4 +
where I is the identity (embedding) operator, Hr(Sd), Hr(Bd) are the Sobolev
d
spaces on the sphere Sd and on the ball Bd, the constants C (r,d),i = 1,2,3,4,
i
onlydepending onthe smoothnessindexr andthedimensiond,werenotexplicitly
determined.
In the paper we discuss preasymptotics and asymptotics of the approximation
numbers of the embeddings I of the Sobolev spaces Hr (r > 0) and the Gevrey
d
typespacesGα,β (α,β >0)onthesphereSd andontheballBd intoL . Weremark
2
that the Gevrey type spaces have a long history and have been used in numerous
problems related to partial differential equations.
Our main focus in this paper is to clarify, for arbitrary but fixed r > 0 and
α,β > 0, the dependence of these approximation numbers a (I ) on d. In fact, it
n d
is necessary to fix the norms on the spaces Hr on Sd and on Bd in advance, since
the constants C (r,d), i = 1,2,3,4 in (1.1) and (1.2) depend on the size of the
i
respective unit balls. Surprisingly, for a collection of quite natural norms of Hr
(see Sections 2.1 and 6.1),and for sufficiently large n, sayn≥2d, it turns outthat
the optimal constants decay polynomially in d, i.e.,
C (r,d)≍ C (r,d)≍ C (r,d)≍ C (r,d)≍ d−r,
1 r 2 r 3 r 4 r
where A ≍ B means that there exist two constants c and C which are called
the equivalence constants such that cA ≤ B ≤ CA, and ≍ indicates that the
r
equivalence constants depend only on r. This means that on Sd and on Bd, for
n≥2d,
a (I :Hr →L )≍ d−rn−r/d,
n d 2 r
where the equivalence constants areindependent ofd andn. We alsoshow that on
Sd and on Bd, for n≥2d,
ln(a (I :Gα,β →L ))≍ −βdαnα/d,
n d 2 α
where the equivalence constants depend only on α, but not on d and n.
Specially, we prove that the limits
2 r/d
lim nr/da (I :Hr(Sd)→L (Sd))=
n d 2
n→∞ d!
(cid:16) (cid:17)
and
1 r/d
lim nr/da (I :Hr(Bd)→L (Bd))=
n d 2
n→∞ d!
exist, having the same value for various norms. We also(cid:16)prov(cid:17)e that for 0<α <1,
−α/d −α/d
β >0, γ = 2 , γ˜ = 1 ,
d! d!
(cid:16) (cid:17) (cid:16) (cid:17)
lim eβγnα/da (I :Gα,β(Sd)→L (Sd))=1.
n d 2
n→∞
3
and
lim eβγ˜nα/da (I :Gα,β(Bd)→L (Bd))=1.
n d 2
n→∞
For small n,1 ≤ n ≤ 2d, we also determine explicitly how these approximation
numbers a (I ) behave preasymptotically. We emphasize that the preasymptotic
n d
behavior of a (I ) is completely different from its asymptotic behavior. For exam-
n d
ple, we show that
a (I :Hr,∗(Sd)→L (Sd))≍ a (I :Hr,∗(Bd)→L (Bd))
n d 2 r n d 2
1, n=1,
d−r/2, 2≤n≤d,
≍
r 
d−r/2 log(1+logdn) r/2, d≤n≤2d,
logn
and  (cid:16) (cid:17)
ln a (I :Gα,β(Sd)→L (Sd)) ≍ ln a (I :Gα,β(Bd)→L (Bd))
n d 2 α n d 2
(cid:0) (cid:1) (cid:0) 1, 1≤n(cid:1)≤d,
≍ −β α
α logn , d≤n≤2d,
( log(1+ d )
logn
(cid:16) (cid:17)
where the equivalence constants depend only on r or α, but not on d and n. Here
“∗” stands for a specific (but natural) norm in Hr on Sd and on Bd.
Finally we consider tractability results for the approximation problems of the
Sobolev embeddings and the Gevrey type embeddings on Sd and on Bd. Based
on the asymptotic and preasymptotic behavior of a (I : Hr → L ) and a (I :
n d 2 n d
Gα,β →L ), we show thatfor the absolute errorcriterionthe approximationprob-
2
lems I : Hr →L are weakly tractable if and only if r >1, not uniformly weakly
d 2
tractable, and do not suffer from the curse of dimensionality. We also prove that
foranyα,β >0,the approximationproblemsI :Gα,β →L areuniformly weakly
d 2
tractable,notpolynomially tractable,and quasi-polynomiallytractableif and only
if α≥1 and the exponent of quasi-polynomial tractability is
m
tqpol = sup , α≥1.
m∈N1+βmα
The paper is organized as follows. In Section 2.1 we give definitions of the
Sobolev spaces with various equivalent norms and the Gevrey type spaces on the
sphere. Section 2.2 is devoted to some basics on the approximation numbers on
the sphere. In Section 3, we study strong equivalence of the approximation num-
bers a (I : Hr(Sd) → L (Sd)) and a (I : Gα,β(Sd) → L (Sd)). Section 4 con-
n d 2 n d 2
tains results concerning preasymptotics and asymptotics of the above approxima-
tion numbers. Section 5 transfers our approximation results into the tractability
ones of the respective approximation problems. In the final Section 6, we ob-
tain the corresponding results on Bd such as strong equivalence, preasymptotics
and asymptotics of the approximation numbers a (I : Hr(Bd) → L (Bd)) and
n d 2
a (I :Gα,β(Bd)→L (Bd)),andtractabilityoftherespectiveapproximationprob-
n d 2
lems.
2. Preliminaries on the sphere
2.1. Sobolev spaces and Gevrey type spaces on the sphere.
Let Sd = {x ∈ Rd+1 : |x| = 1} (d ≥ 2) be the unit sphere of Rd+1 (with
|x| denoting the Euclidean norm in Rd+1) endowed with the rotationally invariant
4
measure σ normalized by dσ = 1. Denote by L (Sd) the collection of real
Sd 2
measurable functions f on Sd with finite norm
R
1
kf|L (Sd)k= |f(x)|2dσ(x) 2 <+∞.
2
(cid:16)ZSd (cid:17)
We denote by Hd the space of all sphericalharmonics of degree l on Sd. Denote
ℓ
byΠd+1(Sd)the setofsphericalpolynomialsonSd ofdegree≤m,whichisjustthe
m
setofpolynomialsofdegree≤m onRd+1 restrictedto Sd. It is wellknown(see [3,
Corollaries 1.1.4 and 1.1.5]) that the dimension of Hd is
ℓ
1, if ℓ=0,
(2.1) Z(d,ℓ):=dimHd =
ℓ (2ℓ+d−1)(ℓ+d−2)!, if ℓ=1,2,...,
( (d−1)! l!
and
(2m+d)(m+d−1)!
(2.2) C(d,m):=dimΠd+1(Sd)= , m∈N.
m m!d!
Let
{Y ≡Yd | k =1,...,Z(d,ℓ)}
ℓ,k ℓ,k
be a fixed orthonormal basis for Hd. Then
ℓ
{Y | k =1,...,Z(d,ℓ), ℓ=0,1,2,...}
ℓ,k
is an orthonormalbasis for the Hilbert space L (Sd). Thus any f ∈L (Sd) can be
2 2
expressed by its Fourier (or Laplace) series
∞ ∞ Z(d,ℓ)
f = H (f)= hf,Y iY ,
ℓ ℓ,k ℓ,k
ℓ=0 ℓ=0 k=1
X X X
Z(d,ℓ)
where Hd(f) = hf,Y iY , ℓ = 0,1,..., denote the orthogonal projections
ℓ ℓ,k ℓ,k
k=1
of f onto Hd, andP
ℓ
hf,Y i= f(x)Y (x)dσ(x)
ℓ,k ℓ,k
ZSd
are the Fourier coefficients of f. We have the following Parsevalequality:
∞ Z(d,ℓ)
1/2
kf|L (Sd)k= |hf,Y i|2 .
2 ℓ,k
(cid:16)Xℓ=0 kX=1 (cid:17)
Let Λ = {λ }∞ be a bounded sequence, and let TΛ be a multiplier operator
k k=0
on L (Sd) defined by
2
∞ ∞ Z(d,ℓ)
TΛ(f)= λ H (f)= λ hf,Y iY .
ℓ ℓ ℓ ℓ,k ℓ,k
ℓ=0 ℓ=0 k=1
X X X
Let SO(d+1) be the special rotation group of order d+1, i.e., the set of all
rotationonRd+1. Foranyρ∈SO(d+1)andf ∈L (Sd),wedefineρ(f)(x)=f(ρx).
2
It is well known (see [3, Proposition 2.2.9]) that a bounded linear operator T on
L (Sd) is a multiplier operator if and only if Tρ=ρT for any ρ∈SO(d+1).
2
5
Definition 2.1. Let Λ = {λ }∞ be a non-increasing positive sequence with
k k=0
lim λ =0. We define the multiplier space HΛ(Sd) by
k
k→∞
HΛ(Sd):= TΛf f ∈L (Sd) and kTΛf HΛ(Sd)k=kf|L (Sd)k<∞
2 2
n (cid:12) (cid:12) ∞ Z(d,ℓ) o
(cid:12) (cid:12) 1 1/2
:= f ∈L (Sd) kf HΛ(Sd)k:= |hf,Y i|2 <∞ .
2 λ2 ℓ,k
n (cid:12) (cid:12) (cid:16)Xℓ=0 ℓ Xk=1 (cid:17) o
Clearly, the multiplier s(cid:12)pace(cid:12)HΛ(Sd) is a Hilbert space with inner product
∞ Z(d,ℓ)
1
hf,giHΛ(Sd) = λ2 hf,Yℓ,kihg,Yℓ,ki.
ℓ=0 ℓ k=1
X X
We remark that Sobolev spaces and Gevrey type spaces on the sphere Sd are
special multiplier spaces whose definitions are given as follows.
Definition 2.2. Let r > 0 and (cid:3) ∈ {∗,+,#,−}. The Sobolev space Hr,(cid:3)(Sd) is
the collection of all f ∈L (Sd) such that
2
∞ Z(d,ℓ)
1/2
kf Hr,(cid:3)(Sd)k= (r(cid:3) )−2 |hf,Y i|2 <∞,
ℓ,d ℓ,k
(cid:12) (cid:16)Xℓ=0 Xk=1 (cid:17)
where (cid:12)
r∗ =(1+(ℓ(ℓ+d−1))r)−1/2, r+ =(1+ℓ(ℓ+d−1))−r/2,
ℓ,d ℓ,d
r# =(1+ℓ)−r, r− =(ℓ+(d−1)/2)−r, l=0,1,....
ℓ,d ℓ,d
If we set Λ(cid:3) = {r(cid:3) }∞ , then the Sobolev space Hr,(cid:3)(Sd) is just the multiplier
k,d k=0
space HΛ(cid:3)(Sd).
Remark 2.3. We note that the above four Sobolev norms are equivalent with the
equivalence constants depending on d and r. The natural Sobolev space on the
sphere is Hr,∗(Sd). The Sobolev space Hr,−(Sd) is given in [1, Definition 3.23].
Remark 2.4. Let △ be the Laplace-Beltrami operator on the sphere. It is well
known that the spaces Hd, ℓ = 0,1,2,..., are just the eigenspaces corresponding
ℓ
to the eigenvalues −ℓ(ℓ+d−1) of the operator△. Given s>0, we define the s-th
(fractional)order Laplace-Beltramioperator(−△)s on Sd in a distributional sense
by
H ((−△)s(f))=0, H ((−△)s(f))=(ℓ(ℓ+d−1))sH (f), ℓ=1,2,... ,
0 ℓ ℓ
where f is a distribution on Sd. We can define (I −△)s, (((d−1)/2)2I −△)s
analogously, where I is the identity operator. For r > 0 and f ∈ Hr(Sd), we have
the following equalities.
1/2
kf Hr,∗(Sd)k= kf|L (Sd)k2+k(−△)r/2f|L (Sd)k2 ,
2 2
(cid:12) (cid:16) 1/2 (cid:17)
kf (cid:12)Hr,+(Sd)k= k(I −△)r/2f|L (Sd)k2 ,
2
(cid:12) (cid:16) (cid:17) 1/2
kf(cid:12)Hr,−(Sd)k= k(((d−1)/2)2I−△)r/2f|L (Sd)k2 .
2
(cid:12) (cid:16) (cid:17)
(cid:12)
6
Definition 2.5. Let 0<α,β <∞. The Gevrey type space Gα,β(Sd) is the collec-
tion of all f ∈L (Sd) such that
2
∞ Z(d,ℓ)
1/2
kf Gα,β(Sd)k= e2βℓα |hf,Y i|2 <∞.
ℓ,k
(cid:12) (cid:16)Xℓ=0 kX=1 (cid:17)
If we set Λ =(cid:12) {e−βkα}∞ , then the Gevrey type space Gα,β(Sd) is just the
α,β k=0
multiplier space HΛα,β(Sd).
Remark 2.6. M. Gevrey [5] introduced in 1918 the classes of smooth functions on
Rd that are nowadays called Gevrey classes. They have played an important role
in study of partial differential equation. A standard reference on Gevrey spaces
is Rodinos book [18]. The Gevrey spaces on d-dimensional torus were introduced
and investigated in [9]. Our definition is a natural generalization from Rd to Sd.
When d= 1, the unit ball of Gα,β(Sd) recedes to the class Aτ,b with q = 2, b= α
q
and τ = β introduced in [21, p. 73]. When α = 1, the action of the Gevrey type
multiplier TΛ1,β on f ∈ L2(Sd) is just the Possion integral of f on the sphere (see
[3, pp. 34-35]).
2.2. Approximation numbers.
Let H and G be two Hilbert spaces and S be a compact linear operator from
H to G. The fact concerning the approximation numbers a (S : H → G) is well-
n
known, see e.g. K¨onig [8, Section 1.b], Pinkus [17, Theorem IV.2.2], and Novak
andWon´iakowski[12, Corollary4.12]. The followinglemma is asimple formofthe
above fact.
Let H be a separable Hilbert space, {e }∞ an orthonormal basis in H, and
k k=1
τ ={τ }∞ a sequence ofpositive numberswith τ ≥τ ≥···≥τ ≥···. LetHτ
k k=1 1 2 k
be a Hilbert space defined by
∞ |(x,e )|2 1/2
Hτ = x∈H | kx|Hk= k <∞ .
τ2
n (cid:16)Xk=1 k (cid:17) o
According to [17, Corollary 2.6] we have the following lemma.
Lemma 2.7. Let H,τ and Hτ be defined as above. Then
a (I :Hτ →H)=τ , n∈N .
n d n +
Inthesequel,wealwayssupposethatΛ={λ }∞ isanon-increasingpositive
k,d k=0
sequencewith lim λ =0. ItfollowsfromDefinition2.1thatthemutiplierspace
k,d
k→∞
HΛ(Sd) is of form Hτ =(L (Sd))τ with
2
{τ }∞ ={λ ,λ ,··· ,λ ,λ ,··· ,λ ,··· ,λ ,··· ,λ ,···},
k k=0 0,d 1,d 1,d 2,d 2,d k,d k,d
Z(d,1) Z(d,2) Z(d,k)
where Z(d,m) and C(d,|m) a{rze give}n|in (2.{1z) and}(2.2).|Acco{rzding t}o Lemma 2.7
we obtain
Theorem 2.8. For C(d,k−1)<n≤C(d,k), k =0,1,2,..., we have
a (TΛ :L (Sd)→L (Sd))=a (I :HΛ(Sd)→L (Sd))=λ ,
n 2 2 n d 2 k,d
where we set C(d,−1)=0.
7
Specially, let r >0, (cid:3)∈{∗,+,#,−},and 0<α,β <∞. Then for C(d,k−1)<
n≤C(d,k), k=0,1,2,..., we have
(2.3) a (I :Hr,(cid:3)(Sd)→L (Sd))=r(cid:3) ,
n d 2 k,d
and
(2.4) a (I :Gα,β(Sd)→L (Sd))=e−βkα,
n d 2
where the definitions of Hr,(cid:3)(Sd), r(cid:3) , (cid:3) ∈ {∗,+,#,−} are given in Definition
k,d
2.2, and the definition of Gα,β(Sd) is in Definition 2.5.
3. Strong equivalences of approximation numbers
This section is devoted to giving strong equivalence of the approximation num-
bers of the Sobolev embeddings and the Gevrey type embeddings on the sphere.
Theorem 3.1. Suppose that lim λ ks =1 for some s>0. Then
k,d
k→∞
2 s/d
(3.1) lim ns/da (I :HΛ(Sd)→L (Sd))= ,
n d 2
n→∞ d!
Specially, for r >0 and (cid:3)∈{∗,+,#,−}, we have (cid:16) (cid:17)
2 r/d
(3.2) lim nr/da (I :Hr,(cid:3)(Sd)→L (Sd))= .
n d 2
n→∞ d!
(cid:16) (cid:17)
Proof. For C(d,k−1)<n≤C(d,k), k =0,1,2,..., we have
a (I :HΛ(Sd)→L (Sd))=λ .
n d 2 k,d
It follows that
(C(d,k−1))s/dλ <ns/da (I :HΛ(Sd)→L (Sd))≤(C(d,k))s/dλ .
k,d n d 2 k,d
We recall from (2.2) that
(2k+d)(k+d−1)!
C(d,k)= ,
k!d!
which yields that
C(d,k) C(d,k−1) 2
lim = lim = .
k→∞ kd k→∞ kd d!
Since
lim (C(d,k−1))s/dλ = lim (C(d,k))s/dλ
k,d k,d
k→∞ k→∞
C(d,k) s/d 2 s/d
= lim ksλ = ,
k→∞ kd k,d d!
(cid:16) (cid:17) (cid:16) (cid:17)
we obtain (3.1). Theorem 3.1 is proved. (cid:3)
Remark 3.2. One can rephrase (3.2) as strong equivalences
2 r/d
a (I :Hr,(cid:3)(Sd)→L (Sd))∼n−r/d
n d 2
d!
for r > 0 and (cid:3) ∈ {∗,+,#,−}. The novelty of Theo(cid:16)rem(cid:17)s 3.1 is that they give
strong equivalences of a (I : Hr,(cid:3)(Sd) → L (Sd)) and provide asymptotically
n d 2
optimal constants, for arbitrary fixed d and r >0.
8
−α/d
Theorem 3.3. Let 0<α<1, β >0, and γ = 2 . Then we have
d!
(cid:16) (cid:17)
(3.3) lim eβγnα/da (I :Gα,β(Sd)→L (Sd))=1.
n d 2
n→∞
Proof. It follows from (2.4) that for C(d,k−1)<n≤C(d,k), k =0,1,2,...,
a ≡a (I :Gα,β(Sd)→L (Sd))=e−βkα.
n n d 2
Therefore, we have
(3.4) eβγ(C(d,k−1))α/de−βkα <eβγnα/da ≤eβγ(C(d,k))α/de−βkα.
n
Since for 0<α<1,
d d−1 j α
lim (γ(C(d,k))α/d−kα)= lim kα 1+ (1+ ) d −1 =0,
k→∞ k→∞ 2k k
(cid:16)(cid:16)(cid:0) (cid:1)jY=1 (cid:17) (cid:17)
we get
(3.5) lim eβγ(C(d,k))α/de−βkα =eβkl→im∞(γ(C(d,k))α/d−kα) =1.
n→∞
Similarly, we can show that
lim eβγ(C(d,k−1))α/de−βkα =1,
n→∞
which combining with (3.5) and (3.4), yields (3.3). Theorem 3.3 is proved. (cid:3)
Remark 3.4. One can rephrase (3.3) as a strong equivalence
a (I :Gα,β(Sd)→L (Sd))∼e−βγnα/d
n d 2
−α/d
for 0 < α < 1 and β > 0, where γ = 2 . The novelty of Theorems 3.3
d!
is that they give a strong equivalence of(cid:16)a ((cid:17)I : Gα,β(Sd) → L (Sd)) and provide
n d 2
asymptotically optimal constants, for arbitrary fixed d, 0<α<1, and β >0.
Remark 3.5. For α=1, we have
lim eβγ(C(d,k−1))α/de−βkα =eβ(d2−d1)2 6=eβ2d = lim eβγ(C(d,k))α/de−βkα,
n→∞ n→∞
which means that the strong equivalence
a (I :Gα,β(Sd)→L (Sd))∼e−βγnα/d
n d 2
does not hold. However, we have the weak equivalence
(3.6) a (I :Gα,β(Sd)→L (Sd))≍e−β(nd!/2)1/d,
n d 2
where the equivalence constants may depend on d, but not on n.
For α>1,thereseems evenno weakasymptoticsofa (I :Gα,β(Sd)→L (Sd))
n d 2
as (3.6).
9
4. Preasymptotics and asymptotics of the approximation numbers
This sectionisdevotedto givingpreasymptoticsandasymptoticsofthe approx-
imation numbers of the Sobolev embeddings and the Gevrey type embeddings on
the sphere.
Lemma 4.1. For m∈N and d∈N we have
+
(4.1)
max 1+ m d, 1+ d m ≤C(d,m)≤min ed 1+ m d, em 1+ d m .
d m d m
Proof. Wn(cid:0)e note t(cid:1)hat(cid:0) (cid:1) o n (cid:0) (cid:1) (cid:0) (cid:1) o
m+d m+d 2m+d m+d
≤C(d,m)= ≤2 .
d d m+d d
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Using the inequality (see [10, (3.6)])
m+d m d
≤ed−1(1+ ) ,
d d
(cid:18) (cid:19)
we get the upper estimate of C(d,m). Using the inequality (see [10, (3.5)])
m d d m m+d
max 1+ , 1+ ≤ ,
d m d
we get the lower estimateno(cid:0)f C(d,m(cid:1) ).(cid:0)Lemma(cid:1)4.o1 is(cid:18)proved.(cid:19) (cid:3)
Theorem 4.2. Let r >0. We have
1, n=1,
d−r/2, 2≤n≤d,
(4.2) an(Id :Hr,∗(Sd)→L2(Sd))≍d−r/2 log(1lo+glnodgn) r/2, d≤n≤2d,
(cid:16)d−rn−r/d, (cid:17) n≥2d,
where the equivalence constants dependonly on r, but not on d and n.
Proof. We have for n=1,
a (I :Hr,∗(Sd)→L (Sd))=1.
n d 2
For 2≤n≤C(d,1)=d+2, we have
a (I :Hr,∗(Sd)→L (Sd))=r∗ =(1+dr)−1/2 ≍d−r/2.
n d 2 1,d
This means that
d−r/2, 2≤n≤d,
an(Id :Hr,∗(Sd)→L2(Sd))≍d−r/2 ≍(d−r/2 log(1+lodgn) r/2, d≤n≤C(d,1).
logn
(cid:16) (cid:17)
For C(d,m−1)<n≤C(d,m), 2≤m≤d, we have
(4.3) a (I :Hr,∗(Sd)→L (Sd))=(1+(m(m+d−1))r)−1/2 ≍m−r/2d−r/2.
n d 2
By (4.1) we get that
2ed m
n≤C(d,m)≤em(1+d/m)m ≤ .
m
(cid:16) (cid:17)
It follows that
logn≤mlog(2ed/m),
10
which implies
logn
(4.4) m≥
log(2ed/m)
and
2ed 2ed 2ed 2ed
log ≥log =log( )−log log( ) .
logn mlog(2ed/m) m m
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Using the inequality x≥2logx for x≥2, we obtain
2ed 1 2ed
log ≥ log( ).
logn 2 m
(cid:16) (cid:17)
This combining with (4.4) yields
logn
(4.5) m≥ .
2log(2ed/(logn))
On other hand, it follows from (4.1) that
d m−1
n>C(d,m−1)≥ 1+ .
m−1
(cid:16) (cid:17)
This yields
logn
m−1≤ ≤logn.
log 1+ d
m−1
It follows that
(cid:0) (cid:1)
logn
m≤ +1,
log 2d
logn
which combining with (4.5), leads to (cid:0) (cid:1)
logn
(4.6) m≍ .
1+log d
logn
It follows from (4.3) that for C(d,m−1)<(cid:0)n≤C(cid:1)(d,m), 2≤m≤d,
log(1+ d ) r/2
a (I : Hr,∗(Sd)→L (Sd))≍m−r/2d−r/2 ≍d−r/2 logn .
n d 2
logn
(cid:16) (cid:17)
Note that
2d ≤C(d,d)≤(2e)d
and for 2d ≤n≤(2e)d,
logn
≍d and n−r/d ≍1.
log(1+ d )
logn
We obtain that for C(d,1)<n≤C(d,d),
log(1+ d ) r/2
a (I : Hr,∗(Sd)→L (Sd))≍d−r/2 logn
n d 2
logn
(cid:16) (cid:17)
d−r/2 log(1+logdn) r/2, d≤n≤2d,
≍ logn .
( (cid:16)d−rn−r/d, (cid:17) 2d ≤n≤C(d,d).
For C(d,m−1)<n≤C(d,m), m>d, we have
(4.7) a (I : Hr,∗(Sd)→L (Sd))=(1+(m(m+d−1))r)−1/2 ≍m−r.
n d 2