ON A PRODUCT-TYPE OPERATOR FROM MIXED

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES HAIYING LI AND ZHITAO GUO

A BSTRACT. The boundedness and compactness of a product-type operator DMu Cψ from mixed-norm spaces to Bloch-Orlicz spaces are characterized in this paper.

1. I NTRODUCTION Let D denote the unit disk in the complex plane C, H(D) the class of all analytic functions on D and N the set of nonnegative integers. A positive continuous function φ on [0,1) is called normal if there exist two positive numbers s and t with 0 < s < t, and δ ∈ [0, 1) such that (see [19]) φ(r) φ(r) is decreasing on [δ, 1), lim = 0; r→1 (1 − r)s (1 − r)s φ(r) φ(r) is increasing on [δ, 1), lim = ∞. t r→1 (1 − r)t (1 − r) For p, q ∈ (0, ∞) and φ normal, the mixed-norm space H(p, q, φ)(D) = H(p, q, φ) is the space of all functions f ∈ H(D) such that Z 1  p1 φp (r) p kf kH(p,q,φ) = Mq (f, r) dr < ∞, 1−r 0 where

 q1 Z 2π 1 iθ q Mq (f, r) = |f (re )| dθ . 2π 0 For 1 ≤ p, q < ∞, H(p, q, φ), equipped with the norm kf kH(p,q,φ) , is a Banach space, while for the other vales of p and q, k · kH(p,q,φ) is a quasinorm on H(p, q, φ), H(p, q, φ) 

α+1

is a Fr´echet space but not a Banach space. Note that if φ(r) = (1 − r) p , then H(p, q, φ) is equivalent to the weighted Bergman space Apα (D) = Apα defined for 0 < p < ∞ and α > −1, as the spaces of all f ∈ H(D) such that Z kf kpApα = (α + 1) |f (z)|p (1 − |z|2 )α dm(z) < ∞, D

where dm(z) = π1 rdrdθ is the normalized Lebesgue area measure on D ([8, 12, 18, 25, 27, 33, 35, 48, 51]). For more details on the mixed-norm space on various domains and operators on them, see, e.g., [1, 7, 10, 20, 22, 23, 24, 28, 29, 34, 36, 37, 38, 41, 42, 43, 44, 46, 47, 54]. 2000 Mathematics Subject Classification. Primary 47B33. Key words and phrases. A product-type operator, mixed-norm spaces, Bloch-Orlicz spaces, boundedness, compactness. 1

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HAIYING LI AND ZHITAO GUO

For every 0 < α < ∞, the α-Bloch space, denoted by B α , consists of all functions f ∈ H(D) such that sup(1 − |z|2 )α |f 0 (z)| < ∞. z∈D

B α is a Banach space under the norm kf kBα = |f (0)| + sup(1 − |z|2 )α |f 0 (z)|. z∈D

For α = 1 is obtained the Bloch space. α-Bloch space is introduced and studied by numerous authors. Recently, many authors studied different classes of Bloch-type spaces, where the typical weight function, ω(z) = 1 − |z|2 (z ∈ D) is replaced by a bounded continuous positive function µ defined on D. More precisely, a function f ∈ H(D) is called a µ-Bloch function, denoted by f ∈ B µ , if kf kµ = sup µ(z)|f 0 (z)| < ∞. z∈D α

Clearly, if µ(z) = ω(z) with α > 0, B µ is just the α-Bloch space B α . It is readily seen that B µ is a Banach space with the norm kf kBµ = |f (0)| + kf kµ . For some information on the Bloch, α-Bloch and Bloch-type spaces, as well as some operators on them see, e.g., [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 23, 25, 26, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55]. Recently, Fern´andez in [17] used Young’s functions to define the Bloch-Orlicz space. More precisely, let ϕ : [0, ∞) → [0, ∞) be a strictly increasing convex function such that ϕ(0) = 0 and limt→∞ ϕ(t) = ∞. The Bloch-Orlicz space associated with the function ϕ, denoted by B ϕ , is the class of all analytic functions f in D such that sup(1 − |z|2 )ϕ(λ|f 0 (z)|) < ∞ z∈D

for some λ > 0 depending on f . Also, since ϕ is convex, it is not hard to see that the Minkowski’s functional    0 f kf kϕ = inf k > 0 : Sϕ ≤1 k define a seminorm for B ϕ , which, in this case, is known as Luxemburg’s seminorm, where Sϕ (f ) = sup(1 − |z|2 )ϕ(|f (z)|) z∈D ϕ

We know that B is a Banach space with the norm kf kBϕ = |f (0)| + kf kϕ . We also have that the Bloch- Orlicz space is isometrically equal to µ-Bloch space, where 1 , z ∈ D. µ(z) = −1 1 ϕ ( 1−|z| 2) Thus for any f ∈ B ϕ , we have kf kBϕ = |f (0)| + sup µ(z)|f 0 (z)|. z∈D

It is well known that the differentiation operator D is defined by (Df )(z) = f 0 (z), f ∈ H(D). Let u ∈ H(D), then the multiplication operator Mu is defined by (Mu f )(z) = u(z)f (z), f ∈ H(D).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES

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Let ψ be an analytic self-map of D. The composition operator Cψ is defined by (Cψ f )(z) = f (ψ(z)), f ∈ H(D). Investigation of products of these and integral-type operators attracted a lot of attention recently (see, e.g., [2]-[49], [51]-[55]). For example, in [3] and [17], the authors investigated bounded superposition operators between Bloch-Orlicz and α-Bloch spaces and composition operators on Bloch-Orlicz type spaces. In [37] and [38], S. Stevi´c investigated extended Ces`aro operators between mixed-norm spaces and Bloch-type spaces and an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball. In [36] and [41], S. Stevi´c investigated an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces and weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. In [42] and [46], S. Stevi´c investigated an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball and weighted differentiation composition operators from the mixed-norm space to the nth weighted-type space on the unit disk. S. Stevi´c in [34] gave the properties of products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. In [47], S. Stevi´c investigated weighted radial operator from the mixed-norm space to the nth weighted-type space on the unit ball. In [54], X. Zhu studied extended Ces`aro operators from mixed-norm spaces to Zygmund type spaces. Motivated, among others, by these papers, we will study here the boundedness and compactness of the following operator, which is also a product-type one, (DMu Cψ f )(z) = u0 (z)f (ψ(z)) + u(z)ψ 0 (z)f 0 (ψ(z)), f ∈ H(D), from H(p, q, φ) to B ϕ . In what follows, µ(z) =

1

ϕ−1 (

, 1 1−|z|2 )

and we use the letter C to denote a positive constant whose value may change at each occurrence. 2. T HE B OUNDEDNESS AND COMPACTNESS OF DMu Cφ : H(p, q, φ) → Bϕ In this section, we will give our main results and proofs. In order to prove our main results, we need some auxiliary results. Our first lemma characterizes compactness in terms of sequential convergence. Since the proof is standard, it is omitted here (see, Proposition 3.11 in [4]). Lemma 1. Suppose u ∈ H(D), ψ is an analytic self-map of D, 0 < p, q < ∞ and φ is normal. Then the operator DMu Cψ : H(p, q, φ) → B ϕ is compact if and only if it is bounded and for each sequence {fn }n∈N which is bounded in H(p, q, φ) and converges to zero uniformly on compact subsets of D as n →∞, we have kDMu Cψ fn kBϕ → 0 as n → ∞. The following lemma can be found in [36]. Lemma 2. Assume 0 < p, q < ∞, ψ is normal and f ∈ H(p, q, φ). Then for every n ∈ N, there is a positive constant C independent of f such that |f (n) (z)| ≤

Ckf kH(p,q,φ) 1

φ(|z|)(1 − |z|2 ) q +n

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HAIYING LI AND ZHITAO GUO

Theorem 3. Let u ∈ H(D), ψ be an analytic self-map of D, 0 < p, q < ∞ and φ be normal. Then DMu Cψ : H(p, q, φ) → Bϕ is bounded if and only if k1 = sup z∈D

µ(z)|u00 (z)|

1 < ∞, φ(|ψ(z)|)(1 − |ψ(z)|2 ) q µ(z)|2u0 (z)ψ 0 (z) + u(z)ψ 00 (z)|

(1)

< ∞, 1 φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +1 µ(z)|u(z)||ψ 0 (z)|2 k3 = sup < ∞. 1 +2 z∈D φ(|ψ(z)|)(1 − |ψ(z)|2 ) q

k2 = sup

(2)

z∈D

(3)

Proof. Assume that (1), (2) and (3) hold. By Lemma 2, then we get C1 kf kH(p,q,φ)

|f (ψ(z))| ≤

,

1

|f 0 (ψ(z))| ≤

φ(|ψ(z)|)(1 − |ψ(z)|2 ) q C2 kf kH(p,q,φ)

|f 00 (ψ(z))| ≤

1

φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +1 C3 kf kH(p,q,φ) 1

,

φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +2

.

Then for each f ∈ H(p, q, φ) \ {0}, we have:  Sϕ

(DMu Cψ f )0 (z) Ckf kH(p,q,φ)



1

k1 φ(|ψ(z)|)(1 − |ψ(z)|2 ) q |f (ψ(z))| Cµ(z)kf kH(p,q,φ) z∈D   1 2 q +1 0 k2 φ(|ψ(z)|)(1 − |ψ(z)| ) |f (ψ(z))| + Cµ(z)kf kH(p,q,φ)   1 k3 φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +2 |f 00 (ψ(z))| + Cµ(z)kf kH(p,q,φ)   k1 C1 + k2 C2 + k3 C3 ≤ sup(1 − |z|2 )ϕ Cµ(z) z∈D    1 ≤ sup(1 − |z|2 )ϕ ϕ−1 =1 1 − |z|2 z∈D ≤ sup(1 − |z|2 )ϕ





where C is a constant such that C ≥ k1 C1 + k2 C2 + k3 C3 . Now, we can conclude that there exists a constant C such that kDMu Cψ f kBϕ ≤ Ckf kH(p,q,φ) for all f ∈ H(p, q, φ), so the product-type operator DMu Cψ : H(p, q, φ) → Bϕ is bounded. Conversely, suppose that DMu Cψ : H(p, q, φ) → B ϕ is bounded, i.e., there exists C > 0 such that kDMu Cψ f kB ϕ ≤ Ckf kH(p,q,φ) for all f ∈ H(p, q, φ). Taking the function f (z) = 1 ∈ H(p, q, φ), and kf kH(p,q,φ) ≤ C, then    00   00  u (z) |u (z)| (DMu Cψ f )0 (z) 2 = Sϕ = sup(1 − |z| )ϕ ≤ 1. Sϕ C C C z∈D It follows that sup µ(z)|u00 (z)| < ∞.

(4)

z∈D

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES

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Taking the function f (z) = z ∈ H(p, q, φ), and kf kH(p,q,φ) ≤ C, then   (DMu Cψ f )0 (z) Sϕ C   00 |u (z)ψ(z) + (2u0 (z)ψ 0 (z) + u(z)ψ 00 (z))| 2 = sup(1 − |z| )ϕ ≤ 1. C z∈D Hence sup µ(z)|u00 (z)ψ(z) + 2u0 (z)ψ 0 (z) + u(z)ψ 00 (z)| < ∞. z∈D

By (4) and the boundedness of ψ(z), we can see that sup µ(z)|2u0 (z)ψ 0 (z) + u(z)ψ 00 (z)| < ∞.

(5)

z∈D

Taking the function f (z) =

z2 2

∈ H(p, q, φ), similarly, we can get

sup µ(z)|u(z)||ψ 0 (z)|2 < ∞.

(6)

z∈D

For a fixed ω ∈ D, set fψ(ω) (z)

=

A(1 − |ψ(ω)|2 )t+1 1 q +t+1

+

B(1 − |ψ(ω)|2 )t+2 1

φ(|ψ(ω)|)(1 − ψ(ω)z) φ(|ψ(ω)|)(1 − ψ(ω)z) q +t+2 (1 − |ψ(ω)|2 )t+3 + , 1 φ(|ψ(ω)|)(1 − ψ(ω)z) q +t+3 where the constant t is from the definition of the normality of the function φ. Then supω∈D kfψ(ω) kH(p,q,φ) < ∞, and we have fψ(ω) (ψ(ω)) = 0 fψ(ω) (ψ(ω)) =

A+B+1 1

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q

,

(AM1 + BM2 + M3 )ψ(ω) 1

(7)

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q +1

, 2

00 fψ(ω) (ψ(ω)) =

(AM1 M2 + BM2 M3 + M3 M4 )ψ(ω) 1

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q +2

.

where Mi = 1q + t + i, i = 1, 2, 3, 4. To prove (1), we choose the corresponding function in (7) with M3 2M3 A= , B=− , M1 M2 and denote it by fψ(ω) , then we have fψ(ω) (ψ(ω)) =

P 1

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q

0 00 , fψ(ω) (ψ(ω)) = fψ(ω) (ψ(ω)) = 0,

(8)

M3 3 where P = M − 2M M2 + 1. 1 By the boundedness of DMu Cψ : H(p, q, φ) → B ϕ , we have kDMu Cψ fψ(ω) kBϕ ≤ C, then   (DMu Cψ fψ(ω) )0 (z) 1 ≥ Sϕ C   P |u00 (ω)| ≥ sup (1 − |ω|2 )ϕ , 1 w∈D Cφ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q

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from which we can get (1). To prove (2), we choose the corresponding function in (7) with A=

−2M2 − M1 M2 + M3 M4 M1 M2 − M3 M4 , B= , 2M2 2M2

and denote it by gψ(ω) , then we have 0 gψ(ω) (ψ(ω)) =

Eψ(ω) φ(|ψ(ω)|)(1 −

1

|ψ(ω)|2 ) q +1

00 , gψ(ω) (ψ(ω)) = gψ(ω) (ψ(ω)) = 0,

(9)

where E=

−2M1 M2 − M12 M2 + M1 M3 M4 M1 M2 − M3 M4 + + M3 . 2M2 2

By the boundedness of DMu Cψ : H(p, q, φ) → B ϕ , we have kDMu Cψ gψ(ω) kBϕ ≤ C, then   (DMu Cψ gψ(ω) )0 (z) 1 ≥ Sϕ C   |(DMu Cψ gψ(ω) )0 (ω)| ≥ sup (1 − |ω|2 )ϕ 1 C 2