An Estimate of the Essential Norm of

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

in 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,[email protected] 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 spaceB^αinduced 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, stoB^αis obtained.

1. Introduction

Throughout the paper,dvdenotes the Lebesegue measure on the unit ballB_nofCⁿnormalized so thatvBn 1,dσdenotes the normalized rotation invariant measure on the boundary∂B_n ofBn, andHBndenotes the class of all holomorphic functions onBn.

For a ∈ B_n, let gz, a log|ϕaz|⁻¹ be Green’s function on B_n with logarithmic singularity ata, where ϕ_a is the Mobius¨ transformation ofB_n with ϕ_a0 a, ϕ_aa 0 andϕaϕ⁻¹_a .

Let 0< p, s <∞,−n−1< q <∞, andqs >−1. We say thatfis a function ofFp, q, s iff∈HBnand

f_Fp,q,sf0

sup

a∈Bn

Bn

∇fz^p

1− |z|²_q

g^sz, advz _1/p

<∞, 1.1

where∇fz ∂f/∂z1, . . . , ∂f/∂z_ndenotes 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 onB_nis calledα-Bloch space onB_nand denoted byB^α. It is easy to prove thatB^αis a Banach space with the norm

f

B^α 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|²_αRfz, 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

H_z^1/2u, u :z∈Bn, u∈Cⁿ\ {0}

, 1.5

where∇fzu∇fz, u, andHzu, uis the Bergman metric defined by

H_zu, u n1 2

1− |z|²

|u|²|u, z|²

1− |z|²₂ forz∈B_n, 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

G_zu, 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|²_2α

|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 operatorC_φinduced byφis defined by

C_φf

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,

T_e 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 spaceH^pto H^q,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, stoB^αin 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 operatorsC_φto be compact forFp, q, stoB^α.

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, . . . , φ_nbe a holomorphic self-map ofB_nand letCφebe the essential norm of a bounded composition operatorC_φ :Fp, q, s → B^α; 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|²_α

1−φw²_n1q/p

G_φw

Rφw, Rφw_1/2

, 1.12

and when 0<n1q/p <1/2, G_φw

Rφw, Rφw

1−φw^22n1q/pRφw²Rφw, φw²; 1.13

when 0<n1q/p1/2, G_φw

Rφw, Rφw

1−φw²

log² 1

1−φw²Rφw²Rφw, φw²; 1.14

whenn1q/p >1/2, G_φw

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 ∈ B^α andw∈B_n,the estimate

fw≤cGαwf

B^α 2.1

holds, where the functionG_αhas 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⊂ B^n1q/pand there existsc >0 such that for allf∈Fp, q, s,f_B^n1q/p≤cfFp,q,s.

Lemma 2.3see3, Theorem 2. Let 0< α < ∞,f ∈ B^α. Thenfα,1,fα,2, andfα,3 are equivalent.

In 12, Zhou and Chen characterize the boundedness of weighted composition operatorWψ,φ betweenFp, q, sandB^α. 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 → B^αis 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|^2qg^sz, advz≤c, 2.3 for everyw∈B_n.

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|²_p−n−1

|1− z, w|^p g^sz, 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|²_p−n−1

|1− z, w|^pg^sz, 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 ofB_nto 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 ofB_n to a holomorphic functionf. By Weierstrass’s theorem we havef ∈ HBnand∂f_kj/∂z_l → ∂f/∂z_l for each l ∈ {1,2, . . . , n}on every compact subsets of Bn. It follows that ∇fkj → ∇f uniformly on compact subsets ofBn.

LetB_m{z∈Cⁿ:|z|<1−1/m} ⊂B_n m1,2, . . .; then

Bn

∇f^p1− |z|^2qg^sz, advz lim

m→∞

Bm

jlim→∞

∇fkj

^p1− |z|^2qg^sz, advz lim

m→∞ lim

j→∞

Bm

∇fkj

^p1− |z|^2qg^sz, advz.

2.6

ButfkjFp,q,s≤c, then

Bm

∇fkj

^p1− |z|^2qg^sz, advz≤c^p, 2.7

and therefore

Bn

∇f^p1− |z|^2qg^sz, advz≤c^p. 2.8

SofFp, q, s≤c^p, 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

∂z_jz ≤

√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 → B^αis 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≤g_w

Fp,q,s≤c₂; 3.1

ii{gw}converges to zero uniformly forzon compact subsets ofB_nwhen|φw| → 1;

iiithere is a constantc >0, for allw∈B_n,

1− |w|^2α∇ gw◦φ

w> cXw, w, 3.2

whereXw, wis the same asTheorem 1.1.

Proof. For allw∈B_nwith|φw|>

2/3, we supposeφw r_we₁,wherer_w|φw|,e₁is the vector1,0, . . . ,0.

Next we break the proof into two cases.

1Assume that

G_φw

Rφw, Rφw

≤2Rφw, φw².

Let

gwz z1−r_w 1−r_w²

1−r_wz₁^n1q/p1. 3.3

Then

∂g_wz

∂z1 1−r_w² 1−r_wz₁^n1q/p1

1 n1q p

z1−r_wrw

1−rwz1

,

∂gwz

∂z_k 0, k2, . . . , n.

3.4

Therefore

∇gwz 1−r_w²

|1−rwz1|^n1q/p1

1 n1q p

z1−rwrw

1−r_wz₁

≤

1 n1q p

1−r_w²

|1−r_wz₁|^n1q/p1.

3.5

ByLemma 2.5,g_w∈Fp, q, s, and there existsc₂>0 independent ofwsuch thatgw_Fp,q,s≤ c₂.

On the other hand, takingz0 z⁰₁,0, . . . ,0 rw,0, . . . ,0∈Bn; then

1− |z0|^2n1q/p∇gwz0

1− |r_w|²_n1q/p

1− |rw|^21/n1q/p1. 3.6

So

gw_B^n1q/pg_w0sup

z∈Bn

1− |z|^2n1q/p∇gwz≥1r_w³ −r_w>

⎛

⎝

2 3

⎞

⎠

3

. 3.7

ByLemma 2.2,g_w∈ B^n1q/p, andgw_Fp,q,s ≥cgw_B^n1q/p, we have

gw_Fp,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 ofB_n. This shows thatiandiihold.

By simple calculation it is easy to get thatGww, w<2; so byLemma 2.3we have

1− |w|^2α∇ g_w◦φ

w≥c1− |w|^2α∇gw

φw

Rφw

G_ww, w

≥c1− |w|^2α∇gw

φw

Rφw.

3.10

Notice that∇gwφw 1−r_w²/1−r_w^2n1q/pe1. Therefore, from our assumption, we get

1− |w|²_α∇ g_w◦φ

w≥c

1− |w|²_α rw

1−r_w²_n1q/pe₁r_wRφw

≥c

1− |w|²_α

1−r_w²_n1q/pRφw, φw

≥c

1− |w|²_α 1−r_w²_n1q/p

G_φw

Rφw, Rφw_1/2

cXw, w.

3.11

2Assume that

G_φw

Rφw, Rφw

>2Rφw, φw².

LetRφw ξ1, . . . , ξ_n^T. Forj 2, . . . , n,letθ_jargξ_j anda_je^−iθj ifξ_j/0, or leta_j 0 if ξ_j0.

In Casen1q/p >1/2, take

gwz a2z₂· · ·a_nz_n1−r_w^23/2

1−r_wz₁^n1q/p1 , 3.12

whererw|φw|. Then

∂gwz

∂z1

n1q /p1

r_w

1−r_w²3/2

1−r_wz₁^n1q/p2 a2z2· · ·anzn,

∂gwz

∂zk a_k

1−r_w²3/2

1−r_wz₁^n1q/p1, k2, . . . , n.

3.13

Therefore ∇gwz

∂g_wz

∂z₁ ²

∂g_wz

∂z₂

²· · · ∂g_wz

∂z₁ ²

n1q/p1²r_w²1−r_w²³|a2z₂· · ·a_nz_n|²

|1−r_wz₁|^2n1q/p2 n−11−r_w²³

|1−r_wz₁|^2n1q/p1

≤

n−1n1q/p1²r_w²1−r_w²³

|z2|²· · ·|zn|²

|1−r_wz₁|^2n1q/p2 n−11−r_w²³

|1−r_wz₁|^2n1q/p1

≤

n−11−r_w^23/2

|1−rwz1|^n1q/p1

n1q/p1²r_w²

1− |z1|²

|1−r_wz₁|² 1

≤

n−1 1−r_w²

|1−r_wz₁|^n1q/p1

1−r_w² r_w²

n1q

p 1

2

1/2

≤c 1−r_w²

|1−rwz1|^n1q/p1.

3.14 It follows fromLemma 2.5thatg_w∈Fp, q, s, and there existsc₂>0 independent ofwsuch that gw_Fp,q,s≤c₂.

On the other hand, taking

z0

z⁰₁ , . . . , z⁰_n

rw, 1

√2

1−r_w²,0, . . . ,0

, 3.15

then

|z0|² r_w² 1 2

1−r_w² 1

2

1r_w²

<1. 3.16

Thusz0∈Bn. Notice that 1≥rw≥

2/3 and byLemma 2.2we have g_wFp,q,s≥cg_wBn1q/p

≥c1− |z₀|^2n1q/p∇gwz0≥c

1− |z₀|²_n1q/p

∂g_wz0

∂z1

c1−r_w^2n1q/p

n1q/p1²r_w²1−r_w²³a2z⁰₂ · · ·anz⁰_n ² 1−rwz⁰₁ ^2n1q/p2

c1−r_w^2n1q/p

n1q/p1²r_w²1−r_w²³ 1/√

2

1−r_w² 1−r_w^22n1q/p2

c

n1q

p 1

r_w1−r_w^2−1/4

≥c₁.

3.17 By the discussion above we get thatc1≤ gw_Fp,q,s≤c2. At the same time, it is also clear that lim_rw→1|gwz| → 0; soiandiihold.

Next we show thatiiiholds. First, by3.13andφw rw,0, . . . ,0it is easy to get that

∇gw

φw

1−r_w^21/2

1−r_w^2n1q/p0, a2, . . . , a_n. 3.18

Notice thatRφw ξ1, . . . , ξn^T andaiξi|ξi|i2, . . . , n; so we have ∇gw

φw

Rφw 1−r_w^21/2

1−r_w^2n1q/p|ξ2|· · ·|ξn|. 3.19 Second, since|φw|>

2/3 andn1q/p >1/2, it is clear that

1−r_w^2n1q/pRφw>1−r²_w^1/2Rφw>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, G_φw

Rφw, Rφw

1−φw²Rφw²Rφw, φw². 3.23 So by our assumptionwe get

1− |φw|^21/2Rφw>

1 2

G_φwRφw, Rφw_1/2

, 3.24

and it follows that

1− |φw|^2n1q/pRφw>

1 2

G_φw

Rφw, Rφw_1/2

. 3.25

Combining3.19,3.22, and3.25, it follows fromG_ww, w<2 andLemma 2.3that

1− |w|^2α∇ g_w◦φ

w≥c1− |w|^2α∇gw

φw

Rφw

G_ww, w

≥c1− |w|^2α∇gw

φw

Rφw

c

1− |w|²α

1−r_w²_n1q/p1−r_w^21/2|ξ2|· · ·|ξn|

≥c

1− |w|²α

1−r_w²_n1q/p1−r_w^21/2

|ξ2|²· · ·|ξn|²

≥c

1− |w|²α

1−r_w^2n1q/p

1−r_w²_n1q/p

|ξ1|²· · ·|ξn|²

c

1− |w|²_α 1−r_w²_n1q/p

1−r_w²_n1q/pRφw

≥c

1− |w|²_α 1−r_w²_n1q/p

G_φw

Rφw, Rφw_1/2 .

3.26

This isiii.

In Casen1q/p1/2 ands > n, take

g_wz a2z₂· · ·a_nz_nlog⁻¹ 1

1−r_w²log² 1 1−rwz1

. 3.27

In Casen1q/p1/2 ands≤n, take

g_wz a2z₂· · ·a_nz_n

log 1 1−r_w²

_−2/px

log 1

1−r_wz_{1 12/px}

, 3.28

wherexis the one used inLemma 2.6.

In Case 0<n1q/p <1/2, take

gwz a2z2· · ·anzn

1− 1−rw^3/2 1−r_wz₁

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⁻¹_w is the desired function.

In fact, by∇fwz ∇gw◦U⁻¹_wz ∇gwzU⁻¹_wU⁻¹_w^Tand|zU_w⁻¹||z|, we have

Bn

∇fwz^p

1− |z|²_q

g^sz, advz

Bn

∇gwzU_w⁻¹ U_w⁻¹T

^p1− |z|^2qg^sz, advz

Bn

∇gwz^p1− |z|^2qg^sz, advz,

3.30

where in the last equation we use the linear coordinate translationzzU⁻¹_w and the fact that Fp, q, sis invariant undermobius¨ translation. So

f_w

Fp,q,sgw_Fp,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 g_w

Fp,q,s

, 3.32

wheregwzis defined asProposition 3.1. It is clear thatFwFp,q,s1 andFwzconverges to zero uniformly on compact subsets ofB_nwhen|φw| → 1. Suppose thatK:Fp, q, s → B^αis compact, thenKFwB^α → 0 uniformly forzin compact subsets ofB_nwhen|φw| → 1 in the following, it is clear that|φw| → 1 whenδ → 0; so we have

C_φ−K sup

f_Fp,q,s1

Cφ−Kf

B^α

≥ sup

f_Fp,q,s1

C_φf

B^α−Kf_Bα

≥ sup

distφw,∂Bn<δ

CφFw

B^α− KFw_B^α

≥ sup

distφw,∂Bn<δ

C_φF_w

B^α− sup

distφw,∂Bn<δKFw_Bα.

3.33

On the other hand, byiinProposition 3.1, for|φw|>

2/3 we get

sup

distφw,∂Bn<δ

gw◦φ

B^α

gw_Fp,q,s ≥ 1 c2

sup

distφw,∂Bn<δ

g_w◦φ

B^α

≥ 1

c₂ sup

distφw,∂Bn<δ

sup

z∈Bn

1− |z|^2α∇ g_w◦φ

z

≥ 1

c₂ sup

distφw,∂Bn<δ1− |w|^2α∇ g_w◦φ

w.

3.34

ByiiiinProposition 3.1, when|φw|>

2/3 we have 1− |w|²α∇

g_w◦φ

w≥c·Xw, w. 3.35

Therefore

Cφ−K≥ c

c2 sup

dist^φw,∂Bn<δXw, w− sup

dist^φw,∂Bn<δKFw_B^α. 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 C_φ

e infC_φ−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 ofB_n. 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,K_mis compact onFp, q, s.

iiiIfCφ :Fp, q, s → B^αis bounded, thenCφKmf ∈ B^αandCφKm :Fp, q, s → B^αis compact.

ivI−K_m ≤2.

v I−K_mftends to zero uniformly on compact subsets ofB_n, whenm → ∞.

Proof. iSincef ∈HBn, there exists aM >0only depending onfsuch that ∂f

∂zk

m−1

m w

≤M, k1, . . . , n, 3.40

wherez z1, . . . , z_n m−1/mw1, . . . , w_n; 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|^2qg^sw, advw<∞. 3.42

So

Bn

∇Kmfw^p1− |w|^2qg^sw, advw

≤

m−1

m nM

_p

Bn

1− |w|^2qg^sw, 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 ofB_nto holomorphic function∂f/∂w_i. So whenjis large enough, for any >0,z∈E₁{m−1/mz:z∈B_n}, andl1, . . . , n, we have

∂ f_j−f

∂z_l z

< . 3.44

So whenj → ∞, we get

sup

w∈Bn

∇

K_mf_j−K_mf

wsup

w∈Bn

∇

f_j−fm−1

m w

≤sup

z∈E1

m−1 m

n l1

∂ fj−f

∂z_l z

≤ m−1 m n.

3.45

Therefore Kmfj−Kmf

Fp,q,sfj0−f0sup

a∈Bn

Bn

∇Kmfj−Kmfw^p1−|w|^2qg^sw, advw

≤f_j0−f0

m−1

m n

_p sup

a∈Bn

Bn

1− |w|^2qg^sw, advw

≤fj0−f0c m−1

m n

_p

−→0.

3.46 This shows that{Kmf_j}converges togK_mf ∈Fp, q, s. Soiiholds.

iiiBy iand the fact thatCφ is bounded, the former is obvious. Byiiand noting thatC_φis bounded, we get thatC_φK_mis compact.

ivFirst, for allf ∈ B^n1q/p, we haveI−Kmf0 0; therefore I−K_mf

B^n1q/p sup

w∈Bn

1− |w|^2n1q/p∇!

I−K_mf"

w

≤sup

w∈Bn

1− |w|^2n1q/p∇fw∇ K_mf

w

≤f

B^n1q/pm−1 m sup

w∈Bn

# 1−

m−1 m w

²

$_n1q/p

∇f m−1

m w

≤2f

B^n1q/p,

3.47 which implies thatI−Km ≤2.

vFor any compact subsetE⊂B_n, there existsr 0< r < 1such thatE⊂ rB_n ⊂B_n. On the other hand, for allz∈E, writer_m m−1/m:

I−K_mfzfz−f_mz fz−frmz

₁

rm

d dt

ftz dt

₁

rm

n k1

∂f

∂w_ktz·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 ofB_n. 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∈B_n : 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 f_Fp,q,s, then byLemma 2.2andivofProposition 3.2 we have

C_φ

e ≤C_φ−C_φK_m C_φI−K_m sup

fF1

C_φI−K_mf

B^α

sup

f_F1

sup

w∈Bn

1− |w|^2α∇!

I−Kmf◦φ"

w!I−Kmf"

φ0

sup

f_F1

⎧⎪

⎨

⎪⎩sup

w∈Bn

Xw, w

1−φw²_n1q/p∇!

I−Kmf◦φ"

w

G_φw

Rφw, Rφw

!I−K_mf"

φ0

⎫⎪

⎬

⎪⎭

≤c₂sup

f_F1

sup

w∈Bn

Xw, w

1−φw²_n1q/p∇!

I−Kmf"

φw

!I−Kmf"

φ0

≤c2I−Kmsup

w∈G1

Xw, w

c₂ sup

f_F1

sup

w∈G2

Xw, w

1−φw²_n1q/p∇!

I−K_mf"

φw

c₂ sup

fF1

!I−K_mf"

φ0

≤c₂sup

w∈G1

Xw, w III.

3.51

ByvofProposition 3.2we know thatI−Kmfzconverges to zero uniformly on G₂, and soI−K_mfφwalso converges to zero uniformly onG₂for every fixedf. Next we prove that for anyw∈G2andfF 1,I, II → 0 whenm → ∞andδ → 0.

Since

!I−K_mf"

φ0

f

φ0

−f m−1

m φ0

, 3.52

letFt ftφ0 1−tm−1/mφ0. Thus

!I−K_mf"

φ0

₁

0

Ftdt

≤ ₁

0 n k1

∂f

∂ζk

tφ0 1−tm−1

m φ0

φ_k0−m−1

m φ_k0 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 ⊂ B^n1q/p,1− |z|^2n1q/p|∇fz| ≤ f_B^n1q/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, . . . , z_n; then

I c2 sup

f_F1sup

w∈G2

Xw, w

1− |z|²_n1q/p∇!

I−Kmf"

z c₂ sup

f_F1

sup

w∈G2

Xw, w

1− |z|²_n1q/p

∇fz− m−1 m ∇f

m−1

m z

≤c2 sup

f_F1sup

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

f_F1sup

w∈G2

Xw, w1− |z|^2n1q/p

∇fz− ∇f m−1

m z

c2

m sup

f_F1sup

w∈G2

Xw, w

# 1−

m−1 m z

²

$_n1q/p

∇f m−1

m z

≤c2 sup

f_F1sup

w∈G2

Xw, w ⁿ

l1

∂f

∂z_lz− ∂f

∂z_l

m−1

m z

c₂

msup

f1sup

w∈G2

Xw, wf

B^n1q/p

I1I2.

3.56 ByLemma 2.4we get sup_w∈G

2Xw, w<∞, and noticing thatf_B^n1q/p ≤c, so it is easy to get thatI2 → 0 whenm → ∞.

ForI₁, first we have ∂f

∂zlz− ∂f

∂zl

1− 1

m

z

∂f

∂z_lz− ∂f

∂z_l

1− 1 m

z₁, z₂, . . . , z_n

∂f

∂z_l

1− 1 m

z₁, z₂, . . . , z_n

− · · · ∂f

∂zl

1− 1

m

z₁, . . . ,

1− 1 m

z_n−1, z_n

− ∂f

∂zl

1− 1

m

z

≤ ⁿ

j2

∂f

∂zl

1− 1

m

z₁, . . . ,

1− 1 m

z_j−1, z_j, . . . , z_n

−∂f

∂z_l

1− 1 m

z1, . . . ,

1− 1

m

zj, z_j1, . . . , zn

∂f

∂zlz1, z₂, . . . , z_n− ∂f

∂zl

1− 1

m

z₁, z₂, . . . , z_n ⁿ

j2

_zj

1−1/mzj

∂²f

∂z_l∂z_j

1− 1 m

z₁, . . . ,

1− 1

m

z_j−1, ζ, z_j1, . . . , z_n

dζ

_z1

1−1/mz1

∂²f

∂z_l∂z₁ζ, z2, . . . , z_ndζ

≤ 1 m

n j1

sup

z∈G₂

∂²f

∂z_l∂z_jz 1

m

n j1

sup

w∈G2

∂²f

∂z_l∂z_jz .

3.57