GENERALIZED COMPOSITION OPERATOR FROM BLOCH–TYPE ...

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. Peˇcari´c) 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

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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. Stevi´c and X. Zhu and studied in [8, 11, 12, 15, 16, 17, 18, 19, 22, 23, 24, 25]. S. Stevi´c [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 Stevi´c 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).

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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 Stevi´c 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

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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



∑

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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

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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

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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. Stevi´c 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:

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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. REFERENCES [1] C. C. C OWEN , B. D. M AC C LUER, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [2] D. D. C LAHANE , S. S TEVI C´ , Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl. Vol. 2006 (2006) 11. Article ID 61018. [3] Z. H U, Extended Ces`aro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131 (7) (2003) 2171–2179.

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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]

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