J ournal of
Mathematical
I nequalities
Volume 6, Number 4 (2012), 523–532
doi:10.7153/jmi-06-50
GENERALIZED COMPOSITION OPERATOR FROM BLOCH–TYPE
SPACES TO MIXED–NORM SPACE ON THE UNIT BALL
L I Z HANG AND Z E -H UA Z HOU
(Communicated by J. Pečarić)
Abstract. Let H(B) be the space of all holomorphic functions on the unit ball B in CN , and
S(B) the collection of all holomorphic self-maps of B . Let ϕ ∈ S(B) and g ∈ H(B) with
g(0) = 0 , the generalized composition operator is defined by
Cϕg ( f )(z) =
1
0
ℜ f (ϕ (tz))g(tz)
dt
,
t
Here, we characterize the boundedness and compactness of the generalized composition operator acting from Bloch-type spaces Bω and Bω ,0 to mixed-norm space H(p,q, φ ) on the unit
ball B .
1. Introduction
Let H(B) be the class of all holomorphic functions on B, S(B) the collection of
all holomorphic self-maps of B, where B is the unit ball in the n -dimensional complex
space Cn . Let d σ be the normalized rotation invariant measure on the boundary S =
∂ B of B.
For f ∈ H(B), let
ℜ f (z) = ∇ f , z =
n
∂f
∑ z j ∂ z j (z)
j=1
be the radial derivative of f , ∇ f = ( ∂∂zf , · · · , ∂∂zfn ). If f ∈ H(B) with the Taylor expan1
sion
f (z) = ∑ aβ zβ ,
|β |0
then
ℜ f (z) =
∑
|β |0
|β |aβ zβ ,
Mathematics subject classification (2010): Primary: 47B38; secondary: 32A37, 32A38, 32H02,
47B33.
Keywords and phrases: Generalized composition operator, Bloch-type spaces, mixed-norm spaces.
The second author is corresponding author and supported in part by the National Natural Science Foundation of China
(Grant Nos. 10971153, 10671141).
c
, Zagreb
Paper JMI-06-50
523
524
L. Z HANG AND Z. H. Z HOU
β
β
where |β | = β1 + ... + βn, and zβ = z1 1 ...zn n .
A positive continuous function μ on [0, 1) is called normal (see [20]), if there
exist positive numbers s and t , 0 < s < t , and δ ∈ [0, 1) such that
μ (r)
μ (r)
(i) (1−r)
s is decreasing on [δ , 1), lim (1−r)s → 0;
(ii)
μ (r)
(1−r)t
r→1
μ (r)
is increasing on [δ , 1), lim (1−r)
t → ∞.
r→1
A normal function ω can be extended on whole B by ω (z) = ω (|z|). We recall
that the Bloch-type space Bω consists of all f ∈ H (B) such that
Bω ( f ) = sup ω (z)|ℜ f (z)| < ∞.
(1.1)
z∈B
The expression Bω ( f ) defines a semi-norm while the natural norm is given by
f Bω = | f (0)| + Bω ( f ).
This norm makes Bω into a Banach space.
Let Bω ,0 denote the subspace of Bω consisting of those f ∈ Bω for which
lim ω (z)|ℜ f (z)| = 0.
|z|→1
This space is called the little Bloch-type space.
In [21], the author showed that f ω is equivalent to | f (0)| + sup ω (z)|∇ f (z)|,
z∈B
and f ∈ Bω ,0 if and only if lim ω (z)|∇ f (z)| = 0 . For particular case where ω (r) =
|z|→1
(1 − r2)α see, for example, paper [2].
It is well-known that Bω ,0 is a closed subspace of Bω . When ω (r) = (1 − r2 )α ,
the induced spaces Bω and Bω ,0 are the α -Bloch space B α and the little α -Bloch
space B0α . In particular, for α = 1 , B 1 and B01 are the classical Bloch space B and
the classical little Bloch space B0 .
From now on if we say that a function φ : B → [0, ∞) is normal we will also
assume that it is radial on B, that is, φ (z) = φ (|z|), z ∈ B. In the rest of this paper we
always assume that ω and φ are normal on [0, 1).
For 0 < p, q < ∞ and φ normal, the mixed-norm space H(p, q, φ ) in B, consists
of all functions f ∈ H(B) such that
p
f H(p,q,
φ) =
1
0
Mqp ( f , r)
φ p (r)
dr < ∞,
1−r
(1.2)
where
Mq ( f , r) =
S
1/q
| f (rξ )|q d σ (ξ )
.
For p = q and φ (r) = (1 − r2 )(α +1)/p , α > −1 , the mixed-norm space is equivalent with the weighted Bergman space Aαp . In particular, H(p, q, φ ) is equivalent to the
Bergman space Lap if 0 < p = q < ∞ and φ (r) = (1 − r)1/p.
G ENERALIZED COMPOSITION OPERATOR
525
Every ϕ ∈ S(B) induces a composition operator Cϕ defined by Cϕ f = f ◦ ϕ for
f ∈ H (B) , z ∈ B. It is of interest to provide function-theoretic characterizations for
when ϕ induces a bounded or compact composition operator on various spaces. For
some results on composition operators, see, e.g. [1] and the references therein.
Let g ∈ H(D) and ϕ ∈ S(D), where D is the unit disk of C. The generalized
composition operator is defined by
(Cϕg f )(z) =
z
0
f (ϕ (ξ ))g(ξ )d ξ
for z ∈ D and f ∈ H(D). When g = ϕ , we see that this operator is essentially composition operator, since the following difference Cϕg −Cϕ is a constant. Therefore, Cϕg is a
generalization of the composition operator Cϕ . The generalized composition operator
was introduced in [4] and [9]. For related results and operators, see, e.g., [5, 6, 7] and
the references therein.
Let ϕ ∈ S(B) and g ∈ H(B) with g(0) = 0 . The following operator, so called,
generalized composition operator on the unit ball
(Cϕg f )(z) =
1
0
ℜ f (ϕ (tz))g(tz)
dt
, f ∈ H(B), z ∈ B.
t
was recently introduced by S. Stević and X. Zhu and studied in [8, 11, 12, 15, 16, 17,
18, 19, 22, 23, 24, 25]. S. Stević [9] characterized the equivalent conditions about the
boundedness and compactness of Cϕg acting from logarithmic Bloch-type space to the
mixed-norm space on the unit ball and obtained some conditions about Cϕg acting from
the mixed-norm space to weighted Bloch space on the disk or unit ball in [10, 12],
respectively. For a natural counterpart of operator Cϕg see operator Pϕg defined in [13].
The purpose of this paper is to characterize the boundedness and compactness of
the generalized composition operator Cϕg from the ω -Bloch spaces to the mixed-norms
pace H(p, q, φ ).
Throughout the rest of this paper, C will denote a positive constant, the exact value
of which will vary from one appearance to the next. The notation A B means that
there is a positive constant C such that B/C A CB.
2. Auxiliary results
Several auxiliary results, which are used in the proofs of the main results, are
quoted in this section.
The proof of the following lemma was given by Stević in [11, 15].
L EMMA 2.1. Assume that ϕ ∈ S(B), and g ∈ H(B) with g(0) = 0 . Then for
every f ∈ H(B) it holds
ℜCϕg ( f )(z) = ℜ f (ϕ (z))g(z).
526
L. Z HANG AND Z. H. Z HOU
L EMMA 2.2. ([3, Theorem 2]) Assume that 0 < p, q < ∞, φ is normal and m ∈
N . Then the following asymptotic relationship holds
1/p
1
1
m−1
p
φ p (r)
p
m
mp φ (r)
dr ∑ |grad j f (0)| +
dr
Mq ( f , r)
Mq (ℜ f , r)(1 − r)
1−r
1−r
0
0
j=0
(2.1)
α
for every f ∈ H(B), where |grad j f | = ∑ | ∂∂ zαf |.
|α |= j
The following lemma can be proved in a standard way (see, for example, Proposition 3.11 in [1]), we omit its proof.
L EMMA 2.3. Assume that ϕ ∈ S(B), and g ∈ H(B) with g(0) = 0 . Then Cϕg :
Bω → H(p, q, φ ) is compact if and only if Cϕg : Bω → H(p, q, φ ) is bounded and for
any bounded sequence { fk }k∈N in Bω which converges to zero uniformly on compact
subsets of B as k → ∞, we have Cϕg fk H(p,q,φ ) → 0 as k → ∞.
Lemma 2.3 also holds if space Bω is replaced by Bω ,0 .
The following lemma was obtained by Stević in [17].
L EMMA 2.4. ([17, Lemma 6]) Let ω be a normal weight, then there are N ∈ N,
r0 ∈ (0, 1) and functions f1 , · · · , fN ∈ Bω such that
|ℜ f1 (z)| + |ℜ f2 (z)| + · · · + |ℜ fN (z)| 1
ω (z)
for r0 |z| < 1 .
The next lemma is well-known.
L EMMA2.5. For 0 < p< ∞, there
is a positive constant C p depending on p and
p
n
n
p
n , such that ∑ xi C p ∑ xi , when xi ∈ (0, ∞), i ∈ {1, 2, · · · , n} .
i=1
i=1
3. Main results
In this section, we formulate and prove our main results. We use and modify some
ideas from papers [14, 16, 17].
T HEOREM 3.1. Assume that ϕ ∈ S(B), and g ∈ H(B) with g(0) = 0 . Then the
following statements are equivalent.
(a) Cϕg : Bω → H(p, q, φ ) is bounded.
(b) Cϕg : Bω ,0 → H(p, q, φ ) is bounded.
(c)
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr < ∞.
(3.1)
ω (ϕ (rξ ))
(1 − r)1−p
S
0
527
G ENERALIZED COMPOSITION OPERATOR
Proof. First note that the implication (a)⇒(b) is trivial.
(b) ⇒(c). Assume that (b) holds. Then for f ∈ Bω ,0 , by Lemma 2.1 and Lemma
2.2 with m = 1 , we obtain
1
φ p (r)
dr
1−r
0
1
φ p (r)
g
Mqp (ℜ(Cϕ f ), r)
dr
(1 − r)1−p
0
1
φ p (r)
=
Mqp (gℜ f ◦ ϕ , r)
dr
(1 − r)1−p
0
p/q
1 |ℜ f (ϕ (rξ ))g(rξ )|q d σ (ξ )
=
p
Cϕg f H(p,q,
φ) =
0
Mqp (Cϕg f , r)
S
φ p (r)
dr.
(1 − r)1−p
Then by Lemma 2.4, using the dilation functions
( f j )l (z) = f j (lz) ∈ Bω ,0 ,
j = 1, · · · , N,
N
where l > r0 > 0 . When |lz| > r0 , we have ∑ |(ℜ f j )l (z)| >
j=1
Lemma 2.5, we have
∞>
N
1
ω (lz) ,
thus according to
N
p
g
p
∑ Cϕg f j H(p,q,
φ ) ∑ Cϕ (( f j )l )H(p,q,φ )
j=1
N
1 j=1
p/q
|(ℜ f j )l (ϕ (rξ ))g(rξ )|q d σ (ξ )
φ p (r)
dr
(1 − r)1−p
S
j=1 0
p/q
1 N φ p (r)
q
=
|(ℜ
f
)
(
ϕ
(r
ξ
))g(r
ξ
)|
d
σ
(
ξ
)
dr
j
l
∑ S
(1 − r)1−p
0
j=1
p/q
1 N φ p (r)
q
C
|(ℜ f j )l (ϕ (rξ ))g(rξ )| d σ (ξ )
dr
∑
(1 − r)1−p
0
j=1 S
p/q
1 N
φ p (r)
q
q
=C
|(ℜ f j )l (ϕ (rξ ))| |g(rξ )| d σ (ξ )
dr
∑
(1 − r)1−p
0
S j=1
p/q
1 N
φ p (r)
( ∑ |(ℜ f j )l (ϕ (rξ ))|)q |g(rξ )|q d σ (ξ )
dr
C
(1 − r)1−p
0
S j=1
p/q
1 N
φ p (r)
q
q
C
( ∑ |(ℜ f j )l (ϕ (rξ ))|) |g(rξ )| d σ (ξ )
dr
(1 − r)1−p
E1 j=1
0
p/q
q
1 |g(rξ )|
φ p (r)
C
d σ (ξ )
dr
(1 − r)1−p
E1 ω (l ϕ (rξ ))
0
p/q
1 |l ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr,
C
ω (l ϕ (rξ ))
(1 − r)1−p
0
E1
∑
528
L. Z HANG AND Z. H. Z HOU
where E1 = {ξ ∈ S : r0 < |l ϕ (rξ )| < 1} . From the inequality above and by the monotone convergence theorem we get
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr < ∞.
ω (ϕ (rξ ))
(1 − r)1−p
|ϕ (rξ )|>r0
0
For the test functions g j (z) = z j ∈ Bω ,0 , j = 1, · · · , n . By Lemma 2.5 and |z|2 (|z1 | + · · · + |zn |)2 , we obtain
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr
ω (ϕ (rξ ))
(1 − r)1−p
|ϕ (rξ )|r0
0
q
p/q
1 n
|ϕ j (rξ )g(rξ )|
φ p (r)
d σ (ξ )
dr
∑
(1 − r)1−p
|ϕ (rξ )|r0 j=1 ω (ϕ (rξ ))
0
p/q
n 1 |ϕ j (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr
C∑
ω (ϕ (rξ ))
(1 − r)1−p
|ϕ (rξ )|r0
j=1 0
p/q
1 n 1
φ p (r)
q
C max
|ϕ j (rξ )g(rξ )| d σ (ξ )
dr
∑
0tr0 ω (t) j=1 0
(1 − r)1−p
|ϕ (rξ )|r0
n
g
C ∑ Cϕ f j H(p,q,φ ) .
j=1
So, it follows from Lemma 2.5 that
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr
ω (ϕ (rξ ))
(1 − r)1−p
0
S
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
=
+
d σ (ξ )
dr
ω (ϕ (rξ ))
(1 − r)1−p
|ϕ (rξ )|r0
|ϕ (rξ )|>r0
0
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr
C
ω (ϕ (rξ ))
(1 − r)1−p
|ϕ (rξ )|r0
0
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
+C
d σ (ξ )
dr < ∞.
ω (ϕ (rξ ))
(1 − r)1−p
0
|ϕ (rξ )|r0
Thus the condition (c) holds.
Next we prove (c) ⇒(a). Assume (c) holds. Then for any f ∈ Bω , by the former
calculation and the equivalence of norm on Bω , we obtain
p
Cϕg f H(p,q,
φ)
p/q
1 |ℜ f (ϕ (rξ ))g(rξ )|q d σ (ξ )
0
C
S
1 |ϕ (rξ )g(rξ )| q
0
C f ωp .
S
ω (ϕ (rξ ))
φ p (r)
dr
(1 − r)1−p
p/q
(ω (ϕ (rξ ))|∇ f (ϕ (rξ ))|) d σ (ξ )
q
φ p (r)
dr
(1 − r)1−p
529
G ENERALIZED COMPOSITION OPERATOR
Then we get (a). This completes the proof of this theorem.
Next we prove the necessary and sufficient conditions for the compactness of Cϕg .
T HEOREM 3.2. Assume that ϕ ∈ S(B), and g ∈ H(B) with g(0) = 0 . Then the
following statements are equivalent.
(d) Cϕg : Bω → H(p, q, φ ) is compact.
(e) Cϕg : Bω ,0 → H(p, q, φ ) is compact.
(f) (3.1) holds.
Proof. The implication (d) ⇒(e) is obvious.
(e) ⇒(f). Note that the compactness of Cϕg : Bω ,0 → H(p, q, φ ) implies the boundedness of Cϕg : Bω ,0 → H(p, q, φ ), the desired result (3.1) follows by Theorem 3.1.
Next we prove (f) ⇒(d). Assume that { fk }k∈N is a bounded sequence in Bω , say
by L , and fk → 0 uniformly on compact subsets of B as k → ∞.
From (3.1) we have that for any ε > 0 , there exists a constant δ ∈ (0, 1), such that
p/q
1 |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr < ε p .
(3.2)
ω (ϕ (rξ ))
(1 − r)1−p
δ
S
Let δ B = {w ∈ B, |w| δ }. By Lemma 2.1 and Lemma 2.2 with m = 1 we obtain
p
Cϕg fk H(p,q,
φ) =
=
1
0
Mqp (Cϕg fk , r)
1 δ 0
+
p/q
|ℜ fk (ϕ (rξ ))g(rξ )| d σ (ξ )
q
S
0
p/q
|ℜ fk (ϕ (rξ ))g(rξ )| d σ (ξ )
q
S
1 δ
φ p (r)
dr
1−r
S
φ p (r)
dr
(1 − r)1−p
φ p (r)
dr
(1 − r)1−p
p/q
|ℜ fk (ϕ (rξ ))g(rξ )|q d σ (ξ )
φ p (r)
dr
(1 − r)1−p
= J1 + J2.
First, we estimate J1 , by (3.1), we have
p/q
δ |ϕ (rξ )g(rξ )| q
φ p (r)
d σ (ξ )
dr < C.
ω (ϕ (rξ ))
(1 − r)1−p
0
S
then
J1 : =
δ 0
S
p
p/q
|ℜ fk (ϕ (rξ ))g(rξ )|q d σ (ξ )
max ω (ϕ (w)) sup |∇ fk (ϕ (w))| p
|w|δ
×
w∈δ B
δ |ϕ (rξ )g(rξ )| q
ω (ϕ (rξ ))
C sup |∇ fk (ϕ (w))| p .
0
w∈δ B
S
p/q
d σ (ξ )
φ p (r)
dr
(1 − r)1−p
φ p (r)
dr
(1 − r)1−p
530
L. Z HANG AND Z. H. Z HOU
Since ϕ (δ B) is a compact subset of B, and fk → 0 uniformly on the compact
subset of B as k → ∞, so ∂∂zf → 0 uniformly on the compact subset of B as k → ∞,
k
we have
lim sup |∇ fk (ϕ (w))| p = 0,
k→∞ w∈δ B
then
lim J1 C lim sup |∇ fk (ϕ (w))| p = 0.
k→∞
(3.3)
k→∞ w∈δ B
On the other hand by (3.2) and sup fk ω L, we have
k∈N
1 p/q
|ℜ fk (ϕ (rξ ))g(rξ )| d σ (ξ )
φ p (r)
dr
(1 − r)1−p
δ
S
p/q
p
1 |ϕ (rξ )g(rξ )| q
φ p (r)
C
d σ (ξ )
dr
sup
ω
(z)|∇
f
(z)|
ω (ϕ (rξ ))
(1 − r)1−p
δ
S
z∈B
J2 : =
q
Cε p fk ωp CL p ε p .
Then
p
p p
lim Cϕg fk H(p,q,
φ ) lim (J1 + J2 ) CL ε .
k→∞
(3.4)
k→∞
Since ε is an arbitrary positive number, from (3.4) we obtain
p
lim Cϕg fk H(p,q,
φ ) = 0.
k→∞
Using Lemma 2.3 we get (f). The proof of this theorem is complete.
4. Other two operators
If we use the radial derivative of some function k to instead of g in operators Cϕg ,
or let ϕ (z) = z, we can get the following three operators:
(Lkϕ f )(z) =
1
g
0
(L f )(z) =
ℜ f (ϕ (tz))ℜk(tz)
1
0
ℜ f (tz)g(tz)
dt
, f ∈ H(BN ), z ∈ BN ,
t
dt
, f ∈ H(BN ), z ∈ BN ,
t
and
(Lk f )(z) =
1
0
ℜ f (tz)ℜk(tz)
dt
, f ∈ H(BN ), z ∈ BN ,
t
S. Stević characterized the boundedness and compactness of Lg acting from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball in [14]. According
to the previous sections, we can obtain some results about these operators at once. We
state them as follows:
531
G ENERALIZED COMPOSITION OPERATOR
T HEOREM 4.1. Assume that ϕ ∈ S(B), and k ∈ H(B). Then the following statements are equivalent.
(a) Lkϕ : Bω → H(p, q, φ ) is bounded.
(b) Lkϕ : Bω ,0 → H(p, q, φ ) is bounded.
(c) Lkϕ : Bω → H(p, q, φ ) is compact.
(d) Lkϕ : Bω ,0 → H(p, q, φ ) is compact.
(e)
1 |ϕ (rξ )ℜk(rξ )| q
ω (ϕ (rξ ))
S
0
p/q
d σ (ξ )
φ p (r)
dr < ∞.
(1 − r)1−p
T HEOREM 4.2. Assume that g ∈ H(B) with g(0) = 0 . Then the following statements are equivalent.
(a) Lg : Bω → H(p, q, φ ) is bounded.
(b) Lg : Bω ,0 → H(p, q, φ ) is bounded.
(c) Lg : Bω → H(p, q, φ ) is compact.
(d) Lg : Bω ,0 → H(p, q, φ ) is compact.
(e)
1 S
0
p/q
|g(rξ )|q d σ (ξ )
r p φ p (r)
ω p (r)(1 − r)1−p
dr < ∞.
T HEOREM 4.3. Assume that k ∈ H(B). Then the following statements are equivalent.
(a) Lk : Bω → H(p, q, φ ) is bounded.
(b) Lk : Bω ,0 → H(p, q, φ ) is bounded.
(c) Lk : Bω → H(p, q, φ ) is compact.
(d) Lk : Bω ,0 → H(p, q, φ ) is compact.
(e)
1 0
S
p/q
|ℜk(rξ )| d σ (ξ )
q
r p φ p (r)
dr < ∞.
ω p (r)(1 − r)1−p
Acknowledgement. The authors would like to thank the referees for the useful
comments and suggestions which improved the presentation of this paper.
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(Received October 26, 2011)
Li Zhang
Department of Mathematics
Tianjin University
Tianjin 300072, P. R. China
e-mail: [email protected]
Ze-Hua Zhou
Department of Mathematics
Tianjin University
Tianjin 300072, P. R. China
e-mail: [email protected]
Journal of Mathematical Inequalities
www.ele-math.com
[email protected]