Volume 2010, Article ID 132970,22pages doi:10.1155/2010/132970
Research Article
An Estimate of the Essential Norm of
a Composition Operator from F p, q, s to B
αin the Unit Ball
Hong-Gang Zeng and Ze-Hua Zhou
Department of Mathematics, Tianjin University, Tianjin 300072, China
Correspondence should be addressed to Ze-Hua Zhou,zehuazhou2003@yahoo.com.cn Received 29 June 2009; Revised 9 January 2010; Accepted 17 February 2010
Academic Editor: Michel C. Chipot
Copyrightq2010 H.-G. Zeng and Z.-H. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetBnbe the unit ball ofCnandφ φ1, . . . , φna holomorphic self-map ofBn.Let 0< p, s <∞,
−n−1< q <∞,qs >−1,α >0, and letCφbe the composition operator between the spaceFp, q, s andα-Bloch spaceBαinduced byφ. This paper gives an estimate of the essential norm ofCφ. As a consequence, a necessary and sufficient condition for the composition operatorCφto be compact fromFp, q, stoBαis obtained.
1. Introduction
Throughout the paper,dvdenotes the Lebesegue measure on the unit ballBnofCnnormalized so thatvBn 1,dσdenotes the normalized rotation invariant measure on the boundary∂Bn ofBn, andHBndenotes the class of all holomorphic functions onBn.
For a ∈ Bn, let gz, a log|ϕaz|−1 be Green’s function on Bn with logarithmic singularity ata, where ϕa is the Mobius¨ transformation ofBn with ϕa0 a, ϕaa 0 andϕaϕ−1a .
Let 0< p, s <∞,−n−1< q <∞, andqs >−1. We say thatfis a function ofFp, q, s iff∈HBnand
fFp,q,sf0
sup
a∈Bn
Bn
∇fzp
1− |z|2q
gsz, advz 1/p
<∞, 1.1
where∇fz ∂f/∂z1, . . . , ∂f/∂zndenotes the complex gradient off.
Forα >0, we say thatf∈HBnis anα-Bloch function onBn, if f
α,1sup
z∈Bn
1− |z|2α∇fz<∞. 1.2
The class of allα-Bloch functions onBnis calledα-Bloch space onBnand denoted byBα. It is easy to prove thatBαis a Banach space with the norm
f
Bα f0f
α,1. 1.3
Whenα1, we obtain the classical Bloch functions and Bloch space.
It is proved by Yang and Ouyang1that the normfα,1is equivalent to the norm f
α,2sup
z∈Bn
1− |z|2αRfz, 1.4
whereRfz ∇fzz∇fz, zis the inner product of∇fzandz. Forα1, Timoney 2proved that the above two norms are equivalent to the third norm:
f
1,3sup ∇fzu
Hz1/2u, u :z∈Bn, u∈Cn\ {0}
, 1.5
where∇fzu∇fz, u, andHzu, uis the Bergman metric defined by
Hzu, u n1 2
1− |z|2
|u|2|u, z|2
1− |z|22 forz∈Bn, u∈Cn\ {0}. 1.6
On this basis, Zhang and Xu3defined another normfα,3as follows:
f
α,3 sup
u∈Cn\{0}
z∈Bn
1− |z|2α∇fz, u
{Gzu, u}1/2 , 1.7
where
Gzu, u
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
1− |z|2
|u|2|u, z|2, α > 1 2,
1− |z|2
|u|2log2 2
1− |z|2 |u, z|2, α 1 2,
1− |z|22α
|u|2|u, z|2, 0< α < 1 2.
1.8
They proved that this norm is equivalent tofα,1 andfα,2 for anyα > 0. We give their result asLemma 2.3in this paper. For more details, we recommend the readers refer to3.
Let φz φ1z, . . . , φnz be a holomorphic self-map of Bn; the composition operatorCφinduced byφis defined by
Cφf
z f
φz
. 1.9
In recent years, many specialists have devoted themselves to the research of composition operators which includes boundedness, compactness, and spectra. Concerning these results, we also recommend the interested readers refer to2,4–7.
Another hot topic is the essential norm of composition operators. First, we recall that the essential norm of a continuous linear operatorT is the distance fromT to the compact operators, that is,
Te inf
T−K:K is compact
. 1.10
Notice thatTe 0 if and only ifT is compact, so that estimates onTelead to conditions forTto be compact.
In 1987, J. H. Shapiro calculated the essential norm of a composition operator on Hilbert spaces of analytic functions Hardy and weighted Bergman spaces in terms of natural counting functions associated with φ. In 8, Gorkin and MacCluer obtained the estimates for the essential norm of a composition operator acting from the Hardy spaceHpto Hq,p > q, in one or several variables. In9, Montes-Rodr´ıguez gave the exact essential norm of a composition operator on the Bloch space in the disc. After that, Zhou and Shi generalized Alfonso’s result to the polydisc in10,11. This paper, with fundamental ideas of the proof following Zhou and Shi, gives an estimate of composition operator fromFp, q, stoBαin the unit ball. In addition, we get a similar estimate of composition operators between different Bloch type spaces and obtain some necessary and sufficient conditions for the composition operatorsCφto be compact forFp, q, stoBα.
In the following, we will use the symbolsc,c1, andc2to denote a finite positive number which does not depend on variablesz, a, wand may depend on some norms and parameters p, q, s, n, α, x, f, and so forth, not necessarily the same at each occurrence.
Our main result is the following.
Theorem 1.1. Letφ φ1, φ2, . . . , φnbe a holomorphic self-map ofBnand letCφebe the essential norm of a bounded composition operatorCφ :Fp, q, s → Bα; then there arec1, c2>0, independent ofw, such that
c1lim
δ→0 sup
distφw,∂Bn<δXw, w≤Cφ
e≤c2lim
δ→0 sup
distφw,∂Bn<δXw, w, 1.11
where
Xw, w
1− |w|2α
1−φw2n1q/p
Gφw
Rφw, Rφw1/2
, 1.12
and when 0<n1q/p <1/2, Gφw
Rφw, Rφw
1−φw22n1q/pRφw2Rφw, φw2; 1.13
when 0<n1q/p1/2, Gφw
Rφw, Rφw
1−φw2
log2 1
1−φw2Rφw2Rφw, φw2; 1.14
whenn1q/p >1/2, Gφw
Rφw, Rφw
1−φw2Rφw2Rφw, φw2. 1.15
2. Some Lemmas
In order to prove the main result, we will give some lemmas first.
Lemma 2.1see12, Lemma 2.2. Letα > 0. Then there is a constantc > 0,and for allf ∈ Bα andw∈Bn,the estimate
fw≤cGαwf
Bα 2.1
holds, where the functionGαhas been defined as follows.
iIf 0< α <1, thenGαw 1.
iiIfα1, thenGαw ln4/1− |w|2. iiiIfα >1, thenGαw 1/1− |w|2α−1.
Lemma 2.2see12, Lemma 2.1. If 0< p, s <∞,−n−1< q <∞,qs >−1, thenFp, q, s⊂ Bn1q/pand there existsc >0 such that for allf∈Fp, q, s,fBn1q/p≤cfFp,q,s.
Lemma 2.3see3, Theorem 2. Let 0< α < ∞,f ∈ Bα. Thenfα,1,fα,2, andfα,3 are equivalent.
In 12, Zhou and Chen characterize the boundedness of weighted composition operatorWψ,φ betweenFp, q, sandBα. Takeψ 1 in12, Theorem 1.2, page 902and by similar proof we can get the following lemma.
Lemma 2.4. For 0< p, s <∞,−n−1< q <∞,qs >−1,α >0, letφbe a holomorphic self-map ofBn. ThenCφ:Fp, q, s → Bαis bounded if and only if
sup
w∈Bn
Xw, w<∞, 2.2
whereXw, whas been defined at1.12.
Lemma 2.5see12, Lemma 2.5. For 0< p, s <∞,−n−1< q <∞,qs >−1, there exists c >0 such that
sup
a∈Bn
Bn
1− |w|2p
|1− z, w|n1qp1− |z|2qgsz, advz≤c, 2.3 for everyw∈Bn.
Lemma 2.6 see12, Lemma 2.7. Suppose 0 < p, s < ∞and sp > n, then one has the following.
iIfs > n, then there is a constantc >0, for allw∈Bn
sup
a∈Bn
Bn
log 1 1− |z2|
−p
log 1 1− z, w
p
1− |z|2p−n−1
|1− z, w|p gsz, advz< c. 2.4
iiIfs≤n, then when one choosesxwhich satisfies max{1, n/p}< x < n/n−s, (ifns, just letx >max{1, n/p}), then
sup
a∈Bn
Bn
log 1 1− |z2|
−2/x
log 1 1− z, w
2/x
1− |z|2p−n−1
|1− z, w|pgsz, advz< c. 2.5 Lemma 2.7. If{fk}is a bounded sequence inFp, q, s, then there exists a subsequence{fkj}of{fk} which converges uniformly on compact subsets ofBnto a holomorphic functionf∈Fp, q, s.
Proof. Choose a bounded sequence{fk}fromFp, q, swithfkFp,q,s ≤ c.ByLemma 2.1, {fk}is uniformly bounded on compact subsets ofBn. By Montel’s theorem, we may extract subsequence{fkj}which converges uniformly on compact subsets ofBn to a holomorphic functionf. By Weierstrass’s theorem we havef ∈ HBnand∂fkj/∂zl → ∂f/∂zl for each l ∈ {1,2, . . . , n}on every compact subsets of Bn. It follows that ∇fkj → ∇f uniformly on compact subsets ofBn.
LetBm{z∈Cn:|z|<1−1/m} ⊂Bn m1,2, . . .; then
Bn
∇fp1− |z|2qgsz, advz lim
m→∞
Bm
jlim→∞
∇fkj
p1− |z|2qgsz, advz lim
m→∞ lim
j→∞
Bm
∇fkj
p1− |z|2qgsz, advz.
2.6
ButfkjFp,q,s≤c, then
Bm
∇fkj
p1− |z|2qgsz, advz≤cp, 2.7
and therefore
Bn
∇fp1− |z|2qgsz, advz≤cp. 2.8
SofFp, q, s≤cp, which impliesf ∈Fp, q, s.
Lemma 2.8see10,11, Lemma 2.6. LetΩbe a domain inCn, f ∈HΩ.If a compact setKand its neighborhoodGsatisfyK⊂G⊂⊂ΩandρdistK, ∂G>0,then
sup
z∈K
∂f
∂zjz ≤
√n ρ sup
z∈G
fz j1, . . . , n
. 2.9
3. The Proof of Theorem 1.1
To obtain the lower estimate we first prove the following proposition.
Proposition 3.1. IfCφ :Fp, q, s → Bαis bounded, then for allw ∈Bn which satisfies|φw|>
2/3, there is a functiongw∈Fp, q, ssuch that
ithere existsc1, c2 >0, independent ofw, such that c1≤gw
Fp,q,s≤c2; 3.1
ii{gw}converges to zero uniformly forzon compact subsets ofBnwhen|φw| → 1;
iiithere is a constantc >0, for allw∈Bn,
1− |w|2α∇ gw◦φ
w> cXw, w, 3.2
whereXw, wis the same asTheorem 1.1.
Proof. For allw∈Bnwith|φw|>
2/3, we supposeφw rwe1,whererw|φw|,e1is the vector1,0, . . . ,0.
Next we break the proof into two cases.
1Assume that
Gφw
Rφw, Rφw
≤2Rφw, φw2.
Let
gwz z1−rw 1−rw2
1−rwz1n1q/p1. 3.3
Then
∂gwz
∂z1 1−rw2 1−rwz1n1q/p1
1 n1q p
z1−rwrw
1−rwz1
,
∂gwz
∂zk 0, k2, . . . , n.
3.4
Therefore
∇gwz 1−rw2
|1−rwz1|n1q/p1
1 n1q p
z1−rwrw
1−rwz1
≤
1 n1q p
1−rw2
|1−rwz1|n1q/p1.
3.5
ByLemma 2.5,gw∈Fp, q, s, and there existsc2>0 independent ofwsuch thatgwFp,q,s≤ c2.
On the other hand, takingz0 z01,0, . . . ,0 rw,0, . . . ,0∈Bn; then
1− |z0|2n1q/p∇gwz0
1− |rw|2n1q/p
1− |rw|21/n1q/p1. 3.6
So
gwBn1q/pgw0sup
z∈Bn
1− |z|2n1q/p∇gwz≥1rw3 −rw>
⎛
⎝
2 3
⎞
⎠
3
. 3.7
ByLemma 2.2,gw∈ Bn1q/p, andgwFp,q,s ≥cgwBn1q/p, we have
gwFp,q,s≥c
⎛
⎝
2 3
⎞
⎠
3
c1. 3.8
By the discussion above we get
c1≤ gwFp,q,s≤c2. 3.9
At the same time, for fixedz ∈ Bn, it is clear that limrw→1|gwz| → 0 uniformly forzon compact subsets ofBn. This shows thatiandiihold.
By simple calculation it is easy to get thatGww, w<2; so byLemma 2.3we have
1− |w|2α∇ gw◦φ
w≥c1− |w|2α∇gw
φw
Rφw
Gww, w
≥c1− |w|2α∇gw
φw
Rφw.
3.10
Notice that∇gwφw 1−rw2/1−rw2n1q/pe1. Therefore, from our assumption, we get
1− |w|2α∇ gw◦φ
w≥c
1− |w|2α rw
1−rw2n1q/pe1rwRφw
≥c
1− |w|2α
1−rw2n1q/pRφw, φw
≥c
1− |w|2α 1−rw2n1q/p
Gφw
Rφw, Rφw1/2
cXw, w.
3.11
2Assume that
Gφw
Rφw, Rφw
>2Rφw, φw2.
LetRφw ξ1, . . . , ξnT. Forj 2, . . . , n,letθjargξj andaje−iθj ifξj/0, or letaj 0 if ξj0.
In Casen1q/p >1/2, take
gwz a2z2· · ·anzn1−rw23/2
1−rwz1n1q/p1 , 3.12
whererw|φw|. Then
∂gwz
∂z1
n1q /p1
rw
1−rw23/2
1−rwz1n1q/p2 a2z2· · ·anzn,
∂gwz
∂zk ak
1−rw23/2
1−rwz1n1q/p1, k2, . . . , n.
3.13
Therefore ∇gwz
∂gwz
∂z1 2
∂gwz
∂z2
2· · · ∂gwz
∂z1 2
n1q/p12rw21−rw23|a2z2· · ·anzn|2
|1−rwz1|2n1q/p2 n−11−rw23
|1−rwz1|2n1q/p1
≤
n−1n1q/p12rw21−rw23
|z2|2· · ·|zn|2
|1−rwz1|2n1q/p2 n−11−rw23
|1−rwz1|2n1q/p1
≤
n−11−rw23/2
|1−rwz1|n1q/p1
n1q/p12rw2
1− |z1|2
|1−rwz1|2 1
≤
n−1 1−rw2
|1−rwz1|n1q/p1
1−rw2 rw2
n1q
p 1
2
1/2
≤c 1−rw2
|1−rwz1|n1q/p1.
3.14 It follows fromLemma 2.5thatgw∈Fp, q, s, and there existsc2>0 independent ofwsuch that gwFp,q,s≤c2.
On the other hand, taking
z0
z01 , . . . , z0n
rw, 1
√2
1−rw2,0, . . . ,0
, 3.15
then
|z0|2 rw2 1 2
1−rw2 1
2
1rw2
<1. 3.16
Thusz0∈Bn. Notice that 1≥rw≥
2/3 and byLemma 2.2we have gwFp,q,s≥cgwBn1q/p
≥c1− |z0|2n1q/p∇gwz0≥c
1− |z0|2n1q/p
∂gwz0
∂z1
c1−rw2n1q/p
n1q/p12rw21−rw23a2z02 · · ·anz0n 2 1−rwz01 2n1q/p2
c1−rw2n1q/p
n1q/p12rw21−rw23 1/√
2
1−rw2 1−rw22n1q/p2
c
n1q
p 1
rw1−rw2−1/4
≥c1.
3.17 By the discussion above we get thatc1≤ gwFp,q,s≤c2. At the same time, it is also clear that limrw→1|gwz| → 0; soiandiihold.
Next we show thatiiiholds. First, by3.13andφw rw,0, . . . ,0it is easy to get that
∇gw
φw
1−rw21/2
1−rw2n1q/p0, a2, . . . , an. 3.18
Notice thatRφw ξ1, . . . , ξnT andaiξi|ξi|i2, . . . , n; so we have ∇gw
φw
Rφw 1−rw21/2
1−rw2n1q/p|ξ2|· · ·|ξn|. 3.19 Second, since|φw|>
2/3 andn1q/p >1/2, it is clear that
1−rw2n1q/pRφw>1−r2w1/2Rφw>φw, Rφw, 3.20 and it follows that
3
1−φw2
|ξ2|2· · ·|ξn|2
>|ξ1|. 3.21
Then
|ξ2|2· · ·|ξn|2≥ 1 2
|ξ1|2· · ·|ξn|2
. 3.22
On the other hand, whenn1q/p >1/2, Gφw
Rφw, Rφw
1−φw2Rφw2Rφw, φw2. 3.23 So by our assumptionwe get
1− |φw|21/2Rφw>
1 2
GφwRφw, Rφw1/2
, 3.24
and it follows that
1− |φw|2n1q/pRφw>
1 2
Gφw
Rφw, Rφw1/2
. 3.25
Combining3.19,3.22, and3.25, it follows fromGww, w<2 andLemma 2.3that
1− |w|2α∇ gw◦φ
w≥c1− |w|2α∇gw
φw
Rφw
Gww, w
≥c1− |w|2α∇gw
φw
Rφw
c
1− |w|2α
1−rw2n1q/p1−rw21/2|ξ2|· · ·|ξn|
≥c
1− |w|2α
1−rw2n1q/p1−rw21/2
|ξ2|2· · ·|ξn|2
≥c
1− |w|2α
1−rw2n1q/p
1−rw2n1q/p
|ξ1|2· · ·|ξn|2
c
1− |w|2α 1−rw2n1q/p
1−rw2n1q/pRφw
≥c
1− |w|2α 1−rw2n1q/p
Gφw
Rφw, Rφw1/2 .
3.26
This isiii.
In Casen1q/p1/2 ands > n, take
gwz a2z2· · ·anznlog−1 1
1−rw2log2 1 1−rwz1
. 3.27
In Casen1q/p1/2 ands≤n, take
gwz a2z2· · ·anzn
log 1 1−rw2
−2/px
log 1
1−rwz1 12/px
, 3.28
wherexis the one used inLemma 2.6.
In Case 0<n1q/p <1/2, take
gwz a2z2· · ·anzn
1− 1−rw3/2 1−rwz1
n1q /p
1
. 3.29
According to Lemmas2.5and2.6, and the discussion of the case ofn1q/p >1/2, we can see that the functions above are just what we want.
In the general situation, or whenφw/|φw|e1,we use the unitary transformation Uwwhich satisfies the equationφw rwe1Uw, whererw |φw|. Thenfw gw◦U−1w is the desired function.
In fact, by∇fwz ∇gw◦U−1wz ∇gwzU−1wU−1wTand|zUw−1||z|, we have
Bn
∇fwzp
1− |z|2q
gsz, advz
Bn
∇gwzUw−1 Uw−1T
p1− |z|2qgsz, advz
Bn
∇gwzp1− |z|2qgsz, advz,
3.30
where in the last equation we use the linear coordinate translationzzU−1w and the fact that Fp, q, sis invariant undermobius¨ translation. So
fw
Fp,q,sgwFp,q,s. 3.31
Then we can prove the same result in the same way, and we omit the details here.
Now, we are ready to proveTheorem 1.1. We begin by proving the lower estimate.
Let
Fwz gwz gw
Fp,q,s
, 3.32
wheregwzis defined asProposition 3.1. It is clear thatFwFp,q,s1 andFwzconverges to zero uniformly on compact subsets ofBnwhen|φw| → 1. Suppose thatK:Fp, q, s → Bαis compact, thenKFwBα → 0 uniformly forzin compact subsets ofBnwhen|φw| → 1 in the following, it is clear that|φw| → 1 whenδ → 0; so we have
Cφ−K sup
fFp,q,s1
Cφ−Kf
Bα
≥ sup
fFp,q,s1
Cφf
Bα−KfBα
≥ sup
distφw,∂Bn<δ
CφFw
Bα− KFwBα
≥ sup
distφw,∂Bn<δ
CφFw
Bα− sup
distφw,∂Bn<δKFwBα.
3.33
On the other hand, byiinProposition 3.1, for|φw|>
2/3 we get
sup
distφw,∂Bn<δ
gw◦φ
Bα
gwFp,q,s ≥ 1 c2
sup
distφw,∂Bn<δ
gw◦φ
Bα
≥ 1
c2 sup
distφw,∂Bn<δ
sup
z∈Bn
1− |z|2α∇ gw◦φ
z
≥ 1
c2 sup
distφw,∂Bn<δ1− |w|2α∇ gw◦φ
w.
3.34
ByiiiinProposition 3.1, when|φw|>
2/3 we have 1− |w|2α∇
gw◦φ
w≥c·Xw, w. 3.35
Therefore
Cφ−K≥ c
c2 sup
distφw,∂Bn<δXw, w− sup
distφw,∂Bn<δKFwBα. 3.36
Letδ → 0, we get
Cφ−K≥ c c2lim
δ→0 sup
distφw,∂Bn<δXw, w. 3.37
It follows from the definition ofCφethat Cφ
e infCφ−K:K is compact
≥ c c2lim
δ→0 sup
distφw,∂Bn<δXw, w c1lim
δ→0 sup
distφw,∂Bn<δXw, w.
3.38
This is the lower estimate.
To obtain the upper estimate inTheorem 1.1we first prove the following proposition.
Proposition 3.2. Letφbe a holomorphic self-map ofBn. Form2,3, . . .one defines the operators as follows:
Kmfw f m−1
m w
, f∈HBn, w∈Bn. 3.39
Then the operatorsKmhave the following properties.
iFor allf ∈HBn, Kmf∈Fp, q, s.
iiFor fixedm,Kmis compact onFp, q, s.
iiiIfCφ :Fp, q, s → Bαis bounded, thenCφKmf ∈ BαandCφKm :Fp, q, s → Bαis compact.
ivI−Km ≤2.
v I−Kmftends to zero uniformly on compact subsets ofBn, whenm → ∞.
Proof. iSincef ∈HBn, there exists aM >0only depending onfsuch that ∂f
∂zk
m−1
m w
≤M, k1, . . . , n, 3.40
wherez z1, . . . , zn m−1/mw1, . . . , wn; therefore ∇
Kmf
w≤ m−1 m
n k1
∂f
∂zk
m−1
m w
≤ m−1
m nM. 3.41
ByLemma 2.5we have
Bn
1− |w|2qgsw, advw<∞. 3.42
So
Bn
∇Kmfwp1− |w|2qgsw, advw
≤
m−1
m nM
p
Bn
1− |w|2qgsw, advw<∞.
3.43
This shows thatKmf∈Fp, q, s.
ii Choose a bounded sequence {fj} from Fp, q, s. By Lemma 2.7, we know that there exists a subsequence of {fj} we still denote it by {fj} here which converges to a functionf ∈Fp, q, suniformly on compact subsets ofBnand{∂fj/∂wi}i1, . . . , nalso converges uniformly on compact subsets ofBnto holomorphic function∂f/∂wi. So whenjis large enough, for any >0,z∈E1{m−1/mz:z∈Bn}, andl1, . . . , n, we have
∂ fj−f
∂zl z
< . 3.44
So whenj → ∞, we get
sup
w∈Bn
∇
Kmfj−Kmf
wsup
w∈Bn
∇
fj−fm−1
m w
≤sup
z∈E1
m−1 m
n l1
∂ fj−f
∂zl z
≤ m−1 m n.
3.45
Therefore Kmfj−Kmf
Fp,q,sfj0−f0sup
a∈Bn
Bn
∇Kmfj−Kmfwp1−|w|2qgsw, advw
≤fj0−f0
m−1
m n
p sup
a∈Bn
Bn
1− |w|2qgsw, advw
≤fj0−f0c m−1
m n
p
−→0.
3.46 This shows that{Kmfj}converges togKmf ∈Fp, q, s. Soiiholds.
iiiBy iand the fact thatCφ is bounded, the former is obvious. Byiiand noting thatCφis bounded, we get thatCφKmis compact.
ivFirst, for allf ∈ Bn1q/p, we haveI−Kmf0 0; therefore I−Kmf
Bn1q/p sup
w∈Bn
1− |w|2n1q/p∇!
I−Kmf"
w
≤sup
w∈Bn
1− |w|2n1q/p∇fw∇ Kmf
w
≤f
Bn1q/pm−1 m sup
w∈Bn
# 1−
m−1 m w
2
$n1q/p
∇f m−1
m w
≤2f
Bn1q/p,
3.47 which implies thatI−Km ≤2.
vFor any compact subsetE⊂Bn, there existsr 0< r < 1such thatE⊂ rBn ⊂Bn. On the other hand, for allz∈E, writerm m−1/m:
I−Kmfzfz−fmz fz−frmz
1
rm
d dt
ftz dt
1
rm
n k1
∂f
∂wktz·zkdt
≤ n
k1
1
rm
∂f
∂wktz dt.
3.48
Whent∈rm,1,|tz|t|z|<|z|< rfor allz∈E. But∂f/∂wkwis bounded uniformly on rBn; therefore for allz∈E,|∂f/∂wktz| ≤M. So whenm → ∞, we have
I−Kmfz≤nM1−rm−→0. 3.49
ThusI−Kmftends to zero uniformly on compact subsets ofBn. The proof is completed.
Let us now return to the proof of the upper estimate.
First, for someδ >0 we denote that G1:
w∈Bn : dist
φw, ∂Bn
< δ , G2:
w∈Bn : dist
φw, ∂Bn
≥δ , G2 :{z∈Bn : distz, ∂Bn≥δ}.
3.50
ThenG1∪G2 BnandG2is a compact set ofBn, andz φw∈G2 if and only ifw ∈G2. For anyf ∈Fp, q, s, writefF fFp,q,s, then byLemma 2.2andivofProposition 3.2 we have
Cφ
e ≤Cφ−CφKm CφI−Km sup
fF1
CφI−Kmf
Bα
sup
fF1
sup
w∈Bn
1− |w|2α∇!
I−Kmf◦φ"
w!I−Kmf"
φ0
sup
fF1
⎧⎪
⎨
⎪⎩sup
w∈Bn
Xw, w
1−φw2n1q/p∇!
I−Kmf◦φ"
w
Gφw
Rφw, Rφw
!I−Kmf"
φ0
⎫⎪
⎬
⎪⎭
≤c2sup
fF1
sup
w∈Bn
Xw, w
1−φw2n1q/p∇!
I−Kmf"
φw
!I−Kmf"
φ0
≤c2I−Kmsup
w∈G1
Xw, w
c2 sup
fF1
sup
w∈G2
Xw, w
1−φw2n1q/p∇!
I−Kmf"
φw
c2 sup
fF1
!I−Kmf"
φ0
≤c2sup
w∈G1
Xw, w III.
3.51
ByvofProposition 3.2we know thatI−Kmfzconverges to zero uniformly on G2, and soI−Kmfφwalso converges to zero uniformly onG2for every fixedf. Next we prove that for anyw∈G2andfF 1,I, II → 0 whenm → ∞andδ → 0.
Since
!I−Kmf"
φ0
f
φ0
−f m−1
m φ0
, 3.52
letFt ftφ0 1−tm−1/mφ0. Thus
!I−Kmf"
φ0
1
0
Ftdt
≤ 1
0 n k1
∂f
∂ζk
tφ0 1−tm−1
m φ0
φk0−m−1
m φk0 dt
≤n 1
0
∇f
tφ0 1−tm−1
m φ0
· 1
mφ0dt
≤ n m
1
0
∇f
tφ0 1−tm−1
m φ0
dt.
3.53
Sincef ∈ Fp, q, s ⊂ Bn1q/p,1− |z|2n1q/p|∇fz| ≤ fBn1q/p ≤ c, we get|∇fz| ≤ c1− |z|2−n1q/p. On the other hand, when 0< t <1, we have
# 1−
tφ0 1−tm−1
m φ0
2
$−n1q/p
≤1−φ0−n1q/p. 3.54
So
!I−Kmf"
φ0≤cn m
1
0
# 1−
tφ0 1−tm−1
m φ0
2
$−n1q/p dt
≤cn m
1−φ0−n1q/p−→0 m−→ ∞.
3.55
Letm → ∞; we getII → 0.
Letw∈G2andφw z z1, . . . , zn; then
I c2 sup
fF1sup
w∈G2
Xw, w
1− |z|2n1q/p∇!
I−Kmf"
z c2 sup
fF1
sup
w∈G2
Xw, w
1− |z|2n1q/p
∇fz− m−1 m ∇f
m−1
m z
≤c2 sup
fF1sup
w∈G2
Xw, w1− |z|2n1q/p
∇fz− ∇f m−1
m z
c2
m sup
fF1
sup
w∈G2
Xw, w1− |z|2n1q/p ∇f
m−1
m z
≤c2 sup
fF1sup
w∈G2
Xw, w1− |z|2n1q/p
∇fz− ∇f m−1
m z
c2
m sup
fF1sup
w∈G2
Xw, w
# 1−
m−1 m z
2
$n1q/p
∇f m−1
m z
≤c2 sup
fF1sup
w∈G2
Xw, w n
l1
∂f
∂zlz− ∂f
∂zl
m−1
m z
c2
msup
f1sup
w∈G2
Xw, wf
Bn1q/p
I1I2.
3.56 ByLemma 2.4we get supw∈G
2Xw, w<∞, and noticing thatfBn1q/p ≤c, so it is easy to get thatI2 → 0 whenm → ∞.
ForI1, first we have ∂f
∂zlz− ∂f
∂zl
1− 1
m
z
∂f
∂zlz− ∂f
∂zl
1− 1 m
z1, z2, . . . , zn
∂f
∂zl
1− 1 m
z1, z2, . . . , zn
− · · · ∂f
∂zl
1− 1
m
z1, . . . ,
1− 1 m
zn−1, zn
− ∂f
∂zl
1− 1
m
z
≤ n
j2
∂f
∂zl
1− 1
m
z1, . . . ,
1− 1 m
zj−1, zj, . . . , zn
−∂f
∂zl
1− 1 m
z1, . . . ,
1− 1
m
zj, zj1, . . . , zn
∂f
∂zlz1, z2, . . . , zn− ∂f
∂zl
1− 1
m
z1, z2, . . . , zn n
j2
zj
1−1/mzj
∂2f
∂zl∂zj
1− 1 m
z1, . . . ,
1− 1
m
zj−1, ζ, zj1, . . . , zn
dζ
z1
1−1/mz1
∂2f
∂zl∂z1ζ, z2, . . . , zndζ
≤ 1 m
n j1
sup
z∈G2
∂2f
∂zl∂zjz 1
m
n j1
sup
w∈G2
∂2f
∂zl∂zjz .
3.57