Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 132970, 22 pages
doi:10.1155/2010/132970
Research Article
An Estimate of the Essential Norm of
a Composition Operator from Fp, q, s to Bα 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, zehuazhou2003@yahoo.com.cn
Received 29 June 2009; Revised 9 January 2010; Accepted 17 February 2010
Academic Editor: Michel C. Chipot
Copyright q 2010 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.
Let Bn be the unit ball of Cn and φ φ1 , . . . , φn a holomorphic self-map of Bn . Let 0 < p, s < ∞,
−n−1 < q < ∞, qs > −1, α > 0, and let Cφ be the composition operator between the space Fp, q, s
and α-Bloch space Bα induced by φ. This paper gives an estimate of the essential norm of Cφ . As a
consequence, a necessary and sufficient condition for the composition operator Cφ to be compact
from Fp, q, s to Bα is obtained.
1. Introduction
Throughout the paper, dv denotes the Lebesegue measure on the unit ball Bn of Cn normalized
so that vBn 1, dσ denotes the normalized rotation invariant measure on the boundary ∂Bn
of Bn , and HBn denotes the class of all holomorphic functions on Bn .
For a ∈ Bn , let gz, a log |ϕa z|−1 be Green’s function on Bn with logarithmic
singularity at a, where ϕa is the Möbius transformation of Bn with ϕa 0 a, ϕa a 0
and ϕa ϕ−1
a .
Let 0 < p, s < ∞, −n − 1 < q < ∞, and q s > −1. We say that f is a function of Fp, q, s
if f ∈ HBn and
fFp,q,s
f0 sup
a∈Bn
q
∇fzp 1 − |z|2 g s z, advz
Bn
where ∇fz ∂f/∂z1 , . . . , ∂f/∂zn denotes the complex gradient of f.
1/p
< ∞,
1.1
2
Journal of Inequalities and Applications
For α > 0, we say that f ∈ HBn is an α-Bloch function on Bn , if
α f sup 1 − |z|2 ∇fz < ∞.
α,1
1.2
z∈Bn
The class of all α-Bloch functions on Bn is called α-Bloch space on Bn and denoted by Bα . It is
easy to prove that Bα is a Banach space with the norm
f α f0 f .
B
α,1
1.3
When α 1, we obtain the classical Bloch functions and Bloch space.
It is proved by Yang and Ouyang 1 that the norm fα,1 is equivalent to the norm
α f sup 1 − |z|2 Rfz,
α,2
1.4
z∈Bn
where Rfz ∇fzz ∇fz, z is the inner product of ∇fz and z. For α 1, Timoney
2 proved that the above two norms are equivalent to the third norm:
f sup
1,3
∇fzu
Hz1/2 u, u
1.5
: z ∈ Bn , u ∈ C \ {0} ,
n
where ∇fzu ∇fz, u, and Hz u, u is the Bergman metric defined by
2
1
−
|z|
|u|2 |u, z|2
n1
Hz u, u 2
2
1 − |z|2
for z ∈ Bn , u ∈ Cn \ {0}.
1.6
On this basis, Zhang and Xu 3 defined another norm fα,3 as follows:
α
1 − |z|2 ∇fz, u
f sup
,
α,3
{Gz u, u}1/2
u∈Cn \{0}
1.7
z∈Bn
where
⎧
⎪
1 − |z|2 |u|2 |u, z|2 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨
2
Gz u, u 1 − |z|2 |u|2 log2
|u, z|2 ,
2
⎪
⎪
1 − |z|
⎪
⎪
⎪
2α
⎪
⎪
⎩ 1 − |z|2 |u|2 |u, z|2 ,
α>
1
,
2
α
1
,
2
0<α<
1.8
1
.
2
They proved that this norm is equivalent to fα,1 and fα,2 for any α > 0. We give their
result as Lemma 2.3 in this paper. For more details, we recommend the readers refer to 3.
Journal of Inequalities and Applications
3
Let φz φ1 z, . . . , φn z be a holomorphic self-map of Bn ; the composition
operator Cφ induced by φ is defined by
Cφ f 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 to 2, 4–7.
Another hot topic is the essential norm of composition operators. First, we recall that
the essential norm of a continuous linear operator T is the distance from T to the compact
operators, that is,
T e inf T − K : K is compact .
1.10
Notice that T e 0 if and only if T is compact, so that estimates on T e lead to conditions
for T to 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 space H p to
H q , p > q, in one or several variables. In 9, Montes-Rodrı́guez 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 in 10, 11. This paper, with fundamental ideas of the proof
following Zhou and Shi, gives an estimate of composition operator from Fp, q, s to Bα in 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
operators Cφ to be compact for Fp, q, s to Bα .
In the following, we will use the symbols c, c1 , and c2 to denote a finite positive number
which does not depend on variables z, a, w and 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. Let φ φ1 , φ2 , . . . , φn be a holomorphic self-map of Bn and let Cφ e be the essential
norm of a bounded composition operator Cφ : Fp, q, s → Bα ; then there are c1 , c2 > 0, independent
of w, such that
c1 lim
sup
δ → 0 distφw,∂B <δ
n
Xw, w ≤ Cφ e ≤ c2 lim
sup
δ → 0 distφw,∂B <δ
n
Xw, w,
1.11
where
α
1 − |w|2
1/2
Xw, w ,
2 n1q/p Gφw Rφw, Rφw
1 − φw
1.12
4
Journal of Inequalities and Applications
and when 0 < n 1 q/p < 1/2,
2 2n1q/p Rφw2 Rφw, φw 2 ;
Gφw Rφw, Rφw 1 − φw 1.13
when 0 < n 1 q/p 1/2,
2 Gφw Rφw, Rφw 1 − φw log2
2 2
1
2 Rφw Rφw, φw ;
1 − φw
1.14
when n 1 q/p > 1/2,
2 2 2
Gφw Rφw, Rφw 1 − φw Rφw Rφw, φw .
1.15
2. Some Lemmas
In order to prove the main result, we will give some lemmas first.
Lemma 2.1 see 12, Lemma 2.2. Let α > 0. Then there is a constant c > 0, and for all f ∈ Bα
and w ∈ Bn , the estimate
fw ≤ cGα wf Bα
2.1
holds, where the function Gα has been defined as follows.
i If 0 < α < 1, then Gα w 1.
ii If α 1, then Gα w ln4/1 − |w|2 .
iii If α > 1, then Gα w 1/1 − |w|2 α−1 .
Lemma 2.2 see 12, Lemma 2.1. If 0 < p, s < ∞, −n−1 < q < ∞, qs > −1, then Fp, q, s ⊂
Bn1q/p and there exists c > 0 such that for all f ∈ Fp, q, s, fBn1q/p ≤ cfFp,q,s .
Lemma 2.3 see 3, Theorem 2. Let 0 < α < ∞, f ∈ Bα . Then fα,1 , fα,2 , and fα,3 are
equivalent.
In 12, Zhou and Chen characterize the boundedness of weighted composition
operator Wψ,φ between Fp, q, s and Bα . Take ψ 1 in 12, Theorem 1.2, page 902 and by
similar proof we can get the following lemma.
Lemma 2.4. For 0 < p, s < ∞, −n − 1 < q < ∞, q s > −1, α > 0, let φ be a holomorphic self-map
of Bn . Then Cφ : Fp, q, s → Bα is bounded if and only if
sup Xw, w < ∞,
w∈Bn
where Xw, w has been defined at 1.12.
2.2
Journal of Inequalities and Applications
5
Lemma 2.5 see 12, Lemma 2.5. For 0 < p, s < ∞, −n − 1 < q < ∞, q s > −1, there exists
c > 0 such that
sup
a∈Bn
Bn
1 − |w|2
|1 − z, w|
p
q
n1qp
1 − |z|2 g s z, advz ≤ c,
2.3
for every w ∈ Bn .
Lemma 2.6 see 12, Lemma 2.7. Suppose 0 < p, s < ∞ and s p > n, then one has the
following.
i If s > n, then there is a constant c > 0, for all w ∈ Bn
sup
a∈Bn
Bn
1
log
1 − |z2 |
p−n−1
p 1 − |z|2
−p 1
log
g s z, advz < c.
1 − z, w |1 − z, w|p
2.4
ii If s ≤ n, then when one chooses x which satisfies max{1, n/p} < x < n/n − s, (if n s,
just let x > max{1, n/p}), then
sup
a∈Bn
Bn
p−n−1
2/x 1 − |z|2
−2/x 1
1
log
g s z, advz < c.
log
1 − z, w |1 − z, w|p
1 − |z2 |
2.5
Lemma 2.7. If {fk } is a bounded sequence in Fp, q, s, then there exists a subsequence {fkj } of {fk }
which converges uniformly on compact subsets of Bn to a holomorphic function f ∈ Fp, q, s.
Proof. Choose a bounded sequence {fk } from Fp, q, s with fk Fp,q,s ≤ c. By Lemma 2.1,
{fk } is uniformly bounded on compact subsets of Bn . By Montel’s theorem, we may extract
subsequence {fkj } which converges uniformly on compact subsets of Bn to a holomorphic
function f. By Weierstrass’s theorem we have f ∈ HBn and ∂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 of Bn .
Let Bm {z ∈ Cn : |z| < 1 − 1/m} ⊂ Bn m 1, 2, . . .; then
p
∇f 1 − |z|2 q g s z, advz lim
m → ∞
Bn
p
q
lim ∇fkj 1 − |z|2 g s z, advz
Bm j → ∞
lim
p
q
∇fkj 1 − |z|2 g s z, advz.
lim
m → ∞ j → ∞
2.6
Bm
But fkj Fp,q,s ≤ c, then
p
q
∇fkj 1 − |z|2 g s z, advz ≤ cp ,
Bm
2.7
6
Journal of Inequalities and Applications
and therefore
p
∇f 1 − |z|2 q g s z, advz ≤ cp .
Bn
2.8
So fF p, q, s ≤ cp , which implies f ∈ Fp, q, s.
Lemma 2.8 see 10, 11, Lemma 2.6. Let Ω be a domain in Cn , f ∈ HΩ. If a compact set K and
its neighborhood G satisfy K ⊂ G ⊂⊂ Ω and ρ distK, ∂G > 0, then
√
∂f
n
supfz
sup
z ≤
ρ z∈G
z∈K ∂zj
j 1, . . . , n .
2.9
3. The Proof of Theorem 1.1
To obtain the lower estimate we first prove the following proposition.
Proposition 3.1. If Cφ : Fp, q, s → Bα is bounded, then for all w ∈ Bn which satisfies |φw| >
2/3, there is a function gw ∈ Fp, q, s such that
i there exists c1 , c2 > 0, independent of w, such that
c1 ≤ gw Fp,q,s ≤ c2 ;
3.1
ii {gw } converges to zero uniformly for z on compact subsets of Bn when |φw| → 1;
iii there is a constant c > 0, for all w ∈ Bn ,
α 1 − |w|2 ∇ gw ◦ φ w > cXw, w,
3.2
where Xw, w is the same as Theorem 1.1.
Proof. For all w ∈ Bn with |φw| > 2/3, we suppose φw rw e1 , where rw |φw|, e1 is
the vector 1, 0, . . . , 0.
Next we break the proof into two cases.
1 Assume that
2
Gφw Rφw, Rφw ≤ 2 Rφw, φw .
Let
gw z 2
z1 − rw 1 − rw
1 − rw z1 n1q/p1
.
3.3
Journal of Inequalities and Applications
7
Then
2
n 1 q z1 − rw rw
1 − rw
∂gw z
1
,
p
1 − rw z1
∂z1
1 − rw z1 n1q/p1
∂gw z
0,
∂zk
3.4
k 2, . . . , n.
Therefore
1 n 1 q z1 − rw rw n1q/p1 p
1 − rw z1 |1 − rw z1 |
2
n1q
1 − rw
≤ 1
.
p
|1 − rw z1 |n1q/p1
∇gw z 2
1 − rw
3.5
By Lemma 2.5, gw ∈ Fp, q, s, and there exists c2 > 0 independent of w such that gw Fp,q,s ≤
c2 .
On the other hand, taking z0 z01 , 0, . . . , 0 rw , 0, . . . , 0 ∈ Bn ; then
1 − |z0 |2 n1q/p
1/n1q/p
∇gw z0 1 − |rw |2
1 − |rw |2 1.
n1q/p 3.6
So
gw Bn1q/p
⎛ ⎞3
2⎠
n1q/p
3
∇gw z ≥ 1 rw
gw 0 sup1 − |z|2 − rw > ⎝
.
3
z∈Bn
3.7
By Lemma 2.2, gw ∈ Bn1q/p , and gw Fp,q,s ≥ cgw Bn1q/p , we have
gw Fp,q,s
⎛ ⎞3
2⎠
≥ c⎝
c1 .
3
3.8
By the discussion above we get
c1 ≤ gw Fp,q,s ≤ c2 .
3.9
At the same time, for fixed z ∈ Bn , it is clear that limrw → 1 |gw z| → 0 uniformly for z on
compact subsets of Bn . This shows that i and ii hold.
8
Journal of Inequalities and Applications
By simple calculation it is easy to get that Gw w, w < 2; so by Lemma 2.3 we have
α
1 − |w|2 ∇gw φw Rφw
1 − |w| ∇ gw ◦ φ w ≥ c
Gw w, w
2 α
3.10
α
≥ c1 − |w|2 ∇gw φw Rφw.
2
2 n1q/p
Notice that ∇gw φw 1 − rw
/1 − rw
e1 . Therefore, from our assumption ,
we get
α 1 − |w|2 ∇ gw ◦ φ w ≥ c
α
1 − |w|2
e1 rw Rφw
n1q/p
2
rw 1 − rw
α
1 − |w|2
Rφw, φw
≥ c
2 n1q/p
1 − rw
3.11
α
1 − |w|2
1/2
≥ c
Gφw Rφw, Rφw
n1q/p
2
1 − rw
cXw, w.
2 Assume that
2
Gφw Rφw, Rφw > 2 Rφw, φw .
Let Rφw ξ1 , . . . , ξn T . For j 2, . . . , n, let θj arg ξj and aj e−iθj if ξj /
0, or let aj 0 if
ξj 0.
In Case n 1 q/p > 1/2, take
3/2
gw z 2
a2 z2 · · · an zn 1 − rw
1 − rw z1 n1q/p1
,
3.12
Journal of Inequalities and Applications
9
where rw |φw|. Then
∂gw z
∂z1
2 3/2
n 1 q /p 1 rw 1 − rw
1 − rw z1 n1q/p2
∂gw z
∂zk
1
a2 z2 · · · an zn ,
3.13
2 3/2
ak 1 − rw
,
− rw z1 n1q/p1
k 2, . . . , n.
Therefore
∇gw z 2
∂gw z 2 ∂gw z 2
· · · ∂gw z ∂z ∂z ∂z 1
2
1
2 3
n 1 q/p 12 r 2 1 − r 2 3 |a2 z2 · · · an zn |2
n − 11 − rw
w
w
2n1q/p2
2n1q/p1
|1 − rw z1 |
|1 − rw z1 |
n−1n1q/p12 r 2 1−r 2 3 |z |2 · · ·|z |2
2 3
2
n
w
w
n − 11 − rw
≤
2n1q/p2
2n1q/p1
|1 − rw z1 |
|1 − rw z1 |
≤
2 3/2 n
n − 11 − rw
|1 − rw z1 |n1q/p1
2
1 − |z1 |2
1 q/p 12 rw
|1 − rw z1 |2
1
2 1/2
2
n − 1 1 − rw
n1q
2
2
1 ≤
1 − rw rw
n1q/p1 p
|1 − rw z1 |
≤c
2
1 − rw
|1 − rw z1 |n1q/p1
.
3.14
It follows from Lemma 2.5 that gw ∈ Fp, q, s, and there exists c2 > 0 independent of w such
that gw Fp,q,s ≤ c2 .
On the other hand, taking
z0 0
0
z1 , . . . , zn
1
2
rw , √
1 − rw , 0, . . . , 0 ,
2
3.15
then
2
|z0 |2 rw
1
1
2
2
1 − rw
1 rw
< 1.
2
2
3.16
10
Journal of Inequalities and Applications
Thus z0 ∈ Bn . Notice that 1 ≥ rw ≥
2/3 and by Lemma 2.2 we have
gw ≥ cgw Bn1q/p
Fp,q,s
n1q/p ∂gw z0 2
∇gw z0 ≥ c 1 − |z0 |
∂z
1
2 n1q/p ≥ c1 − |z0 | n 1 q/p 12 r 2 1 − r 2 3 z0 · · · a z0 2
a
2
n
n
w
w
2
2 n1q/p c1 − rw
2n1q/p2
0 1 − rw z1 n 1 q/p 12 r 2 1 − r 2 3 1/√2
2
1 − rw
w
w
2 n1q/p c1 − rw 2 2n1q/p2
1 − rw
n1q
2 −1/4
1 rw 1 − rw
c
p
≥ c1 .
3.17
By the discussion above we get that c1 ≤ gw Fp,q,s ≤ c2 . At the same time, it is also clear that
limrw → 1 |gw z| → 0; so i and ii hold.
Next we show that iii holds. First, by 3.13 and φw rw , 0, . . . , 0 it is easy to get
that
∇gw φw 2
1 − rw
2
1 − rw
1/2
n1q/p
0, a2 , . . . , an .
3.18
Notice that Rφw ξ1 , . . . , ξn T and ai ξi |ξi | i 2, . . . , n; so we have
∇gw φw Rφw Second, since |φw| >
2
1 − rw
2
1 − rw
2
1 − rw
1/2
n1q/p
|ξ2 | · · · |ξn |.
3.19
2/3 and n 1 q/p > 1/2, it is clear that
2 1/2 Rφw > 1 − rw
Rφw > φw, Rφw,
n1q/p 3.20
and it follows that
2 2
3 1 − φw
|ξ2 | · · · |ξn |2 > |ξ1 |.
3.21
Journal of Inequalities and Applications
11
Then
|ξ2 |2 · · · |ξn |2 ≥
1 2
|ξ1 | · · · |ξn |2 .
2
3.22
On the other hand, when n 1 q/p > 1/2,
2
2 2 Gφw Rφw, Rφw 1 − φw Rφw Rφw, φw .
3.23
So by our assumption we get
Rφw >
2 1/2 1 − |φw| 1/2
1
,
Gφw Rφw, Rφw
2
3.24
and it follows that
Rφw >
2 n1q/p 1 − |φw| 1/2
1
.
Gφw Rφw, Rφw
2
3.25
Combining 3.19, 3.22, and 3.25, it follows from Gw w, w < 2 and Lemma 2.3 that
α
1 − |w|2 ∇gw φw Rφw
1 − |w| ∇ gw ◦ φ w ≥ c
Gw w, w
α
≥ c1 − |w|2 ∇gw φw Rφw
α
1 − |w|2
2 1/2
c
1 − rw
|ξ2 | · · · |ξn |
2 n1q/p
1 − rw
α
1 − |w|2
2 1/2
≥ c
1
−
r
|ξ2 |2 · · · |ξn |2
w
2 n1q/p
1 − rw
α
1 − |w|2
n1q/p 2
1
−
r
≥c
|ξ1 |2 · · · |ξn |2
w
2 n1q/p
1 − rw
α
1 − |w|2
n1q/p 2
Rφw
1
−
r
c
w
2 n1q/p
1 − rw
α
1 − |w|2
1/2
≥ c
.
Gφw Rφw, Rφw
n1q/p
2
1 − rw
2 α
This is iii.
3.26
12
Journal of Inequalities and Applications
In Case n 1 q/p 1/2 and s > n, take
gw z a2 z2 · · · an zn log−1
1
1
log2
.
2
1 − rw z1
1 − rw
3.27
In Case n 1 q/p 1/2 and s ≤ n, take
gw z a2 z2 · · · an zn log
1
2
1 − rw
−2/px log
1
1 − rw z1
12/px
,
3.28
where x is the one used in Lemma 2.6.
In Case 0 < n 1 q/p < 1/2, take
1 − rw 3/2
.
gw z a2 z2 · · · an zn 1 −
1 − rw z1 n 1 q /p 1
3.29
According to Lemmas 2.5 and 2.6, and the discussion of the case of n 1 q/p > 1/2, we
can see that the functions above are just what we want.
In the general situation, or when φw /
|φw|e1 , we use the unitary transformation
−1
is
Uw which satisfies the equation φw rw e1 Uw , where rw |φw|. Then fw gw ◦ Uw
the desired function.
−1
−1
−1 T
−1
z ∇gw zUw
Uw
and |zUw
| |z|, we have
In fact, by ∇fw z ∇gw ◦ Uw
q
∇fw zp 1 − |z|2 g s z, advz
Bn
Bn
T p
q
−1
−1
∇gw zUw
Uw 1 − |z|2 g s z, advz
3.30
∇gw zp 1 − |z|2 q g s z, advz,
Bn
−1
where in the last equation we use the linear coordinate translation z zUw
and the fact that
Fp, q, s is invariant under möbius translation. So
fw gw Fp,q,s .
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 prove Theorem 1.1. We begin by proving the lower estimate.
Let
gw z
,
Fw z gw Fp,q,s
3.32
Journal of Inequalities and Applications
13
where gw z is defined as Proposition 3.1. It is clear that Fw Fp,q,s 1 and Fw z converges
to zero uniformly on compact subsets of Bn when |φw| → 1. Suppose that K : Fp, q, s →
Bα is compact, then KFw Bα → 0 uniformly for z in compact subsets of Bn when |φw| → 1
in the following, it is clear that |φw| → 1 when δ → 0; so we have
Cφ − K ≥
≥
≥
Cφ − Kf sup
fFp,q,s 1
Bα
Cφ f α − Kf α
B
B
sup
fFp,q,s 1
sup
distφw,∂Bn <δ
sup
distφw,∂Bn <δ
3.33
Cφ Fw α − KFw α
B
B
Cφ Fw α −
B
sup
distφw,∂Bn <δ
On the other hand, by i in Proposition 3.1, for |φw| >
KFw Bα .
2/3 we get
gw ◦ φ α
1
B
gw ◦ φ α
≥
sup
sup
B
c2 distφw,∂Bn <δ
distφw,∂Bn <δ gw Fp,q,s
≥
1
α sup
sup1 − |z|2 ∇ gw ◦ φ z
c2 distφw,∂Bn <δ z∈Bn
≥
1
α sup
1 − |w|2 ∇ gw ◦ φ w.
c2 distφw,∂Bn <δ
By iii in Proposition 3.1, when |φw| >
1 − |w|2
3.34
2/3 we have
α ∇ gw ◦ φ w ≥ c · Xw, w.
3.35
Therefore
Cφ − K ≥ c
sup
Xw, w −
sup
KFw Bα .
c2 distφw,∂B <δ
distφw,∂Bn <δ
n
3.36
Let δ → 0, we get
Cφ − K ≥ c lim
sup
Xw, w.
c2 δ → 0 distφw,∂Bn <δ
3.37
14
Journal of Inequalities and Applications
It follows from the definition of Cφ e that
Cφ inf Cφ − K : K is compact
e
≥
c
lim
sup
Xw, w
c2 δ → 0 distφw,∂Bn <δ
c1 lim
sup
δ → 0 distφw,∂B <δ
n
3.38
Xw, w.
This is the lower estimate.
To obtain the upper estimate in Theorem 1.1 we first prove the following proposition.
Proposition 3.2. Let φ be a holomorphic self-map of Bn . For m 2, 3, . . . one defines the operators as
follows:
Km fw f
m−1
w ,
m
f ∈ HBn , w ∈ Bn .
3.39
Then the operators Km have the following properties.
i For all f ∈ HBn , Km f ∈ Fp, q, s.
ii For fixed m, Km is compact on Fp, q, s.
iii If Cφ : Fp, q, s → Bα is bounded, then Cφ Km f ∈ Bα and Cφ Km : Fp, q, s → Bα is
compact.
iv I − Km ≤ 2.
v I − Km f tends to zero uniformly on compact subsets of Bn , when m → ∞.
Proof. i Since f ∈ HBn , there exists a M > 0 only depending on f such that
∂f m − 1
≤ M,
w
∂z
m
k
k 1, . . . , n,
3.40
where z z1 , . . . , zn m − 1/mw1 , . . . , wn ; therefore
n ∇ Km f w ≤ m − 1 ∂f m − 1 w ≤ m − 1 nM.
m k1 ∂zk
m
m
3.41
By Lemma 2.5 we have
q
1 − |w|2 g s w, advw < ∞.
Bn
3.42
Journal of Inequalities and Applications
15
So
∇Km fwp 1 − |w|2 q g s w, advw
Bn
≤
m−1
nM
m
3.43
p 2 q s
1 − |w| g w, advw < ∞.
Bn
This shows that Km f ∈ 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
function f ∈ Fp, q, s uniformly on compact subsets of Bn and {∂fj /∂wi } i 1, . . . , n also
converges uniformly on compact subsets of Bn to holomorphic function ∂f/∂wi . So when j is
large enough, for any > 0, z ∈ E1 {m − 1/mz : z ∈ Bn }, and l 1, . . . , n, we have
∂ fj − f z < .
∂zl
3.44
So when j → ∞, we get
m−1
sup ∇ Km fj − Km f w sup ∇ fj − f
w m
w∈Bn
w∈Bn
m − 1 n ∂ fj − f
≤ sup
z
∂zl
z∈E1 m l1
≤
3.45
m−1
n.
m
Therefore
Km fj −Km f fj 0−f0 sup
Fp,q,s
a∈Bn
∇Km fj −Km fwp 1−|w|2 q g s w, advw
Bn
p
m−1
q
n sup
1 − |w|2 g s w, advw
m
a∈Bn Bn
p
m−1
n −→ 0.
≤ fj 0 − f0 c
m
≤ fj 0 − f0 3.46
This shows that {Km fj } converges to g Km f ∈ Fp, q, s. So ii holds.
iii By i and the fact that Cφ is bounded, the former is obvious. By ii and noting
that Cφ is bounded, we get that Cφ Km is compact.
16
Journal of Inequalities and Applications
iv First, for all f ∈ Bn1q/p , we have I − Km f0 0; therefore
!
"
I − Km f n1q/p sup 1 − |w|2 n1q/p ∇ I − Km f w
B
w∈Bn
≤ sup 1 − |w|2 ∇fw ∇ Km f w
n1q/p w∈Bn
≤ f Bn1q/p
#
$n1q/p m − 1 2
m−1
∇f m − 1 w sup 1 − w
m w∈Bn
m
m
≤ 2f Bn1q/p ,
3.47
which implies that I − Km ≤ 2.
v For any compact subset E ⊂ Bn , there exists r 0 < r < 1 such that E ⊂ rBn ⊂ Bn .
On the other hand, for all z ∈ E, write rm m − 1/m:
I − Km fz fz − fm z
fz − frm z
1 d
ftz dt
rm dt
1 n ∂f
tz · zk dt
rm k1 ∂wk
≤
n
k1
3.48
1 ∂f
dt.
tz
∂w
rm
k
When t ∈ rm , 1, |tz| t|z| < |z| < r for all z ∈ E. But ∂f/∂wk w is bounded uniformly on
rBn ; therefore for all z ∈ E, |∂f/∂wk tz| ≤ M. So when m → ∞, we have
I − Km fz ≤ nM1 − rm −→ 0.
3.49
Thus I −Km f tends to zero uniformly on compact subsets of Bn . The proof is completed.
Let us now return to the proof of the upper estimate.
First, for some δ > 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
Journal of Inequalities and Applications
17
Then G1 ∪ G2 Bn and G2 is a compact set of Bn , and z φw ∈ G2 if and only if w ∈ G2 .
For any f ∈ Fp, q, s, write fF fFp,q,s , then by Lemma 2.2 and iv of Proposition 3.2
we have
Cφ ≤ Cφ − Cφ Km e
Cφ I − Km sup Cφ I − Km f Bα
fF 1
sup
fF 1
!
!
"
"
sup 1 − |w| ∇ I − Km f ◦ φ w I − Km f φ0 2 α
w∈Bn
⎧
⎪
⎨
2 n1q/p !
"
∇ I − Km f ◦ φ w
1 − φw
sup sup Xw, w
⎪
fF 1⎩w∈Bn
Gφw Rφw, Rφw
⎫
⎪
!
"
⎬
I − Km f φ0 ⎪
⎭
≤ c2 sup
fF 1
3.51
2 n1q/p !
"
∇ I −Km f φw
sup Xw, w 1− φw
w∈Bn
!
"
I −Km f φ0 ≤ c2 I − Km sup Xw, w
w∈G1
2 n1q/p !
"
∇ I − Km f φw c2 sup sup Xw, w 1 − φw
fF 1 w∈G2
!
"
c2 sup I − Km f φ0 fF 1
≤ c2 sup Xw, w I II.
w∈G1
By v of Proposition 3.2 we know that I − Km fz converges to zero uniformly on G2 ,
and so I − Km fφw also converges to zero uniformly on G2 for every fixed f. Next we
prove that for any w ∈ G2 and fF 1, I, II → 0 when m → ∞ and δ → 0.
Since
!
"
I − Km f φ0 f φ0 − f m − 1 φ0 ,
m
3.52
18
Journal of Inequalities and Applications
let Ft ftφ0 1 − tm − 1/mφ0. Thus
!
"
1 I − Km f φ0 F tdt
0
≤
1
0
∂f
m−1
m−1
φk 0 −
φk 0 dt
∂ζ tφ0 1 − t m φ0
m
k
k1
n
1 1
m−1
≤ n ∇f tφ0 1 − t
φ0 · φ0dt
m
m
0
1 n
∇f tφ0 1 − t m − 1 φ0 dt.
≤
m 0
m
3.53
Since f ∈ Fp, q, s ⊂ Bn1q/p , 1 − |z|2 n1q/p |∇fz| ≤ fBn1q/p ≤ c, we get |∇fz| ≤
c1 − |z|2 −n1q/p . On the other hand, when 0 < t < 1, we have
#
2 $−n1q/p
−n1q/p
m−1
φ0
1 − tφ0 1 − t
≤ 1 − φ0
.
m
3.54
So
!
"
I − Km f φ0 ≤ c n
m
1 #
0
2 $−n1q/p
m
−
1
1 − tφ0 1 − t
φ0
dt
m
−n1q/p
n
≤c
−→ 0 m −→ ∞.
1 − φ0
m
Let m → ∞; we get II → 0.
Let w ∈ G2 and φw z z1 , . . . , zn ; then
n1q/p !
" ∇ I − Km f z
I c2 sup sup Xw, w 1 − |z|2
fF 1 w∈G2
n1q/p m−1
m − 1 2
c2 sup sup Xw, w 1 − |z|
z ∇fz − m ∇f
m
fF 1 w∈G2
m − 1 z ∇fz − ∇f
m
2 n1q/p ≤ c2 sup sup Xw, w1 − |z| fF 1 w∈G2
c2
m − 1 2 n1q/p sup sup Xw, w1 − |z| z ∇f
m fF 1 w∈G2
m
3.55
Journal of Inequalities and Applications
19
m − 1 2 n1q/p ≤ c2 sup sup Xw, w1 − |z| z ∇fz − ∇f
m
fF 1 w∈G2
#
$n1q/p m − 1 2
c2
∇f m − 1 z sup sup Xw, w 1 − z
m fF 1 w∈G2
m
m
≤ c2 sup sup Xw, w
fF 1 w∈G2
∂f
∂f m − 1 z
−
z
∂z
∂zl
m
l
l1
n
c2
sup sup Xw, wf Bn1q/p
m f1 w∈G2
I1 I2 .
3.56
By Lemma 2.4 we get supw∈G2 Xw, w < ∞, and noticing that fBn1q/p ≤ c, so it is easy to
get that I2 → 0 when m → ∞.
For I1 , first we have
∂f
∂f
1
z 1
−
−
z
∂z
∂zl
m
l
∂f
∂f
∂f
1
1
z1 , z2 , . . . , zn z1 , z2 , . . . , zn
1−
1−
z −
∂zl
∂zl
m
∂zl
m
∂f
∂f
1
1
1
− ··· 1−
1−
z1 , . . . , 1 −
zn−1 , zn −
z ∂zl
m
m
∂zl
m
n 1
1
∂f
z1 , . . . , 1 −
zj−1 , zj , . . . , zn
≤
1−
∂z
m
m
l
j2
1
1
1−
z1 , . . . , 1 −
zj , zj1 , . . . , zn m
m
∂f
∂f
1
z1 , z2 , . . . , zn 1−
z1 , z2 , . . . , zn −
∂zl
∂zl
m
n zj
∂2 f
1
1
z1 , . . . , 1 −
zj−1 , ζ, zj1 , . . . , zn dζ
1−
∂z
∂z
m
m
l
j
1−1/mzj
j2
∂f
−
∂zl
z1
∂2 f
ζ, z2 , . . . , zn dζ
1−1/mz1 ∂zl ∂z1
∂2 f
1 n
≤
sup
z
m j1 z∈G ∂zl ∂zj
2
∂2 f
1 n
sup z.
m j1 w∈G2 ∂zl ∂zj
3.57
20
Journal of Inequalities and Applications
Denote G3 : {z ∈ Bn : distz, ∂Bn > δ/2}, then G2 ⊂ G3 ⊂⊂ Bn . Since distG2 , ∂G3 δ/2, by
Lemma 2.8, when z ∈ G2 i.e, w ∈ G2 we get
√
2n n
∂f
∂f
∂f
1
≤
.
z
sup
1
−
−
z
z
∂z
∂zl
m
mδ z∈G3 ∂zl
l
3.58
On the other hand, it follows from fBn1q/p ≤ c that
sup1 − |z|2 ∇fz ≤ fBn1q/p ≤ c.
n1q/p z∈G3
3.59
By 3.59 and the definition of G3 , we get
#
2 $−n1q/p
δ
sup ∇fz ≤ c 1 −
.
2
z∈G3
3.60
#
2 $−n1q/p
∂f
δ
sup
.
z ≤ sup ∇fz ≤ c 1 −
2
z∈G3 ∂zl
z∈G3
3.61
Therefore
Combining 3.58 and 3.61, we have
2 $−n1q/p
δ
1−
·
2
#
I1 ≤ c2 c sup sup Xw, w ·
fF 1 w∈G2
√
2 $−n1q/p
δ
2n2 n
.
1−
·
2
mδ
#
c2 c sup sup Xw, w ·
fF 1 w∈G2
√
2n n
mδ
l1
n
3.62
By Lemma 2.4, supw∈G2 Xw, w < ∞, and so limm → ∞ I1 0.
Now, let m → ∞ and δ → 0; we get the upper estimate:
Cφ ≤ c2 lim
e
sup
δ → 0 distφw,∂B <δ
n
So, the proof of Theorem 1.1 is finished.
Xw, w.
3.63
Journal of Inequalities and Applications
21
4. Two Corollaries
Lemma 2.2 tells us that Fp, q, s ⊂ Bn1q/p , as in the similar discussion of Theorem 1.1; so
we can get an estimate of the essential norm of a composition operator between Bloch-type
spaces. That is the following corollary.
Corollary 4.1. Let α, β > 0, let φ φ1 , φ2 , . . . , φn be a holomorphic self-map of Bn , and let Cφ e
be the essential norm of a bounded composition operator Cφ : Bβ → Bα . Then there are c1 , c2 > 0,
independent of w, such that
c1 lim
sup
δ → 0 distφw,∂B <δ
n
Xw, w ≤ Cφ e ≤ c2 lim
sup
δ → 0 distφw,∂B <δ
n
Xw, w.
4.1
Remark 4.2. In Corollary 4.1, the quantity Xw, w is similar to Theorem 1.1, but we need to
substitute n 1 q/p with β.
It is well known that T e 0 if and only if T is compact; so the estimate on Cφ e
leads to conditions for Cφ to be compact. From Theorem 1.1 we get the following corollary.
Corollary 4.3. Let φ φ1 , φ2 , . . . , φn be a holomorphic self-map of Bn . Then a bounded composition
operator Cφ : Fp, q, s → Bα is compact if and only if
lim
sup
δ → 0 distφw,∂B <δ
n
Xw, w 0,
4.2
where Xw, w is the same as Theorem 1.1.
Acknowledgments
The authors are very grateful to the Editor Professor Chipot and referee for many helpful
suggestions and interesting comments about the paper. This work was supported in part by
the National Natural Science Foundation of China Grant nos. 10971153 and 10671141.
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