Research Article Products of Composition and Differentiation ...

Hindawi Publishing Corporation Journal of Function Spaces Volume 2016, Article ID 5084794, 5 pages http://dx.doi.org/10.1155/2016/5084794

Research Article Products of Composition and Differentiation Operators from Bloch into 𝑄𝐾 Spaces Shunlai Wang and Taizhong Zhang School of Mathematics and Statistics, Nanjing University of Information Science and Technology, 219 Ning Six Road, Pukou District, Nanjing, Jiangsu 210044, China Correspondence should be addressed to Shunlai Wang; wsl [email protected] Received 2 December 2015; Accepted 9 February 2016 Academic Editor: Carlo Bardaro Copyright © 2016 S. Wang and T. Zhang. 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. The boundedness and compactness of the product of differentiation and composition operators from Bloch spaces into 𝑄𝐾 spaces are discussed in this paper.

From [1], we know that 𝑄𝐾 ⊆ B if

1. Introduction and Motivation Let Δ be the open unit disk in the complex plane and let 𝐻(Δ) be the class of all analytic functions on Δ. Let 𝑑𝐴(𝑧) be the Euclidean area element on Δ. The Bloch space B on Δ is the space of all analytic functions 𝑓 on Δ such that 󵄨 󵄨 󵄨 󵄩󵄩 󵄩󵄩 2 󵄨 󸀠 󵄩󵄩𝑓󵄩󵄩B = 󵄨󵄨󵄨𝑓 (0)󵄨󵄨󵄨 + sup (1 − |𝑧| ) 󵄨󵄨󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨󵄨 < ∞. 𝑧∈Δ

(1)

Under the above norm, B is a Banach space. Let B0 denote the subspace of B consisting of those 𝑓 ∈ B for which (1 − |𝑧|2 )𝑓󸀠 |𝑧| → 0 as |𝑧| → 1. This space is called the little Bloch space. Throughout this paper, we assume that 𝐾 : [0, ∞) → [0, ∞) is a nondecreasing and right-continuous function. A function 𝑓 ∈ 𝐻(Δ) is said to belong to 𝑄𝐾 space (see [1]) if 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨𝑓 (𝑧)󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < ∞, 󵄨 󵄨 Δ

󵄩󵄩 󵄩󵄩2 󵄩󵄩𝑓󵄩󵄩𝐾 = sup ∬ 𝑎∈Δ

(2)

where 𝑔(𝑧, 𝑎) is the Green function with logarithmic singularity at 𝑎; that is, 𝑔(𝑧, 𝑎) = log(1/|𝜑𝑎 (𝑧)|) (𝜑𝑎 is a conformal automorphism defined by 𝜑𝑎 (𝑧) = (𝑎 − 𝑧)/(1 − 𝑎𝑧) for 𝑎 ∈ Δ). 𝑄𝐾 is a Banach space under the norm 󵄨 󵄩 󵄩 󵄨 󵄩󵄩 󵄩󵄩2 󵄩󵄩𝑓󵄩󵄩𝑄𝐾 = 󵄨󵄨󵄨𝑓 (0)󵄨󵄨󵄨 + 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐾 .

(3)

1/𝑒

∫

0

𝐾 (− log 𝑟) 𝑟 𝑑𝑟 < ∞.

(4)

Let 𝜑 denote a nonconstant analytic self-map of Δ. Associated with 𝜑 is the composition operator 𝐶𝜑 defined by 𝐶𝜑 (𝑓) = 𝑓 ∘ 𝜑 for 𝑓 ∈ 𝐻(Δ). The problem of characterizing the boundedness and compactness of composition operators on many Banach spaces of analytic functions has attracted lots of attention recently, for example, [2] and the reference therein. Let 𝐷 be the differentiation operator on 𝐻(Δ); then we have 𝐷𝑓(𝑧) = 𝑓󸀠 (𝑧). For 𝑓 ∈ 𝐻(Δ), the products of differentiation and composition operators 𝐷𝐶𝜑 and 𝐶𝜑 𝐷 are defined by 󸀠

𝐷𝐶𝜑 (𝑓) = (𝑓 ∘ 𝜑) = 𝑓󸀠 (𝜑) 𝜑󸀠 , 𝐶𝜑 𝐷 (𝑓) = 𝑓󸀠 (𝜑) ,

(5) 𝑓 ∈ 𝐻 (Δ) .

Operators 𝐶𝜑 𝐷 as well as some other products of linear operators were studied, for example, in [3–9] (see also the references therein). Recall that a linear operator 𝑇 : 𝑋 → 𝑌 is said to be bounded if there exists a constant 𝑀 > 0 such that ‖𝑇(𝑓)‖𝑌 ≤

2

Journal of Function Spaces

𝑀‖𝑓‖𝑋 for all maps 𝑓 ∈ 𝑋. And 𝑋 → 𝑌 is compact if it takes bounded sets in 𝑋 to sets in 𝑌 which have compact closure. For Banach spaces 𝑋 and 𝑌 of 𝐻(Δ), 𝑇 is compact from 𝑋 to 𝑌 if and only if for each sequence {𝑥𝑛 } in 𝑋; the sequence {𝑇𝑥𝑛 } ∈ 𝑌 contains a subsequence converging to some limit in 𝑌. Considering the definition of 𝑄𝐾 spaces and 𝑄𝐾 ⊆ B with some conditions, it is difficult to study the operator 𝐶𝜑 𝐷 from Bloch spaces to 𝑄𝐾 spaces. In this paper, some sufficient and necessary conditions for the boundedness and compactness of this operator are given.

suppose that 𝐶𝜑 𝐷 : B → 𝑄𝐾 is bounded. Fix 𝑤 ∈ Δ and assume that 𝜑(𝑤) ≠ 0. Consider the function 𝑓 defined by 𝑓𝑤 (𝑧) =

𝑛󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 sup (1 − |𝑧|2 ) 󵄨󵄨󵄨󵄨𝑓(𝑛) 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (0)󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑓󸀠 (0)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑓󸀠󸀠 (0)󵄨󵄨󵄨󵄨 𝑧∈Δ

󵄨 󵄨 󵄩 󵄩 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨󵄨𝑓(𝑛−1)(0) 󵄨󵄨󵄨󵄨 ≈ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩B ,

𝑓𝑤󸀠

(6)

(𝑛 = 1, 2, . . .) .

𝐾 (− log 𝑟) 𝑟 𝑑𝑟 < +∞.

(7)

(1 − 𝜑 (𝑤)𝑧)

2

3

,

(11)

Thus

Note that 󵄨󵄨 󸀠󸀠 󵄨 󵄨󵄨𝑓𝑤 (𝜑 (𝑤))󵄨󵄨󵄨 = 󵄨 󵄨

(b) 𝐶𝜑 𝐷 : B0 → 𝑄𝐾 is bounded. (c) 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 sup ∬ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < ∞. 4 Δ (1 − 󵄨󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨2 ) 𝑎∈Δ 󵄨 󵄨

(c) ⇒ (a). Suppose that (c) holds. For any 𝑧 ∈ Δ and 𝑓(𝑧) ∈ B, we have 󵄩2 󵄩󵄩 󵄨 󵄨2 󵄩󵄩𝐶𝜑 𝐷𝑓 (𝑧)󵄩󵄩󵄩 = sup ∬ 󵄨󵄨󵄨𝑓󸀠 (𝜑)󸀠 󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄩𝐾 𝑎∈Δ Δ 󵄨 󵄩 󵄨

2

(𝜑󸀠 (𝑧)) sup ∬ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 4 Δ (1 − 󵄨󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨2 ) 𝑎∈Δ 󵄨 󵄨

󵄨 󵄨 2 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 . 󵄨2 2 󵄨 (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 )

(14)

(15)

For this function 𝑓𝑤 and this point 𝑤 we have 󵄨󵄨 󵄨 󵄨󵄨(𝐶𝜑 𝐷𝑓𝑤 )󸀠 (𝑤)󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨𝜑󸀠 (𝑤) 𝑓󸀠󸀠 (𝜑 (𝑤))󵄨󵄨󵄨󵄨 𝑤 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 2 󵄨 2 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 = 󵄨 . 󵄨2 2 󵄨 (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 )

(8)

Proof.

󵄨 󵄨2 = sup ∬ 󵄨󵄨󵄨󵄨(𝜑󸀠 (𝑧)) 𝑓󸀠󸀠 (𝜑)󵄨󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) Δ 𝑎∈Δ

(12)

for all 𝑧 ∈ Δ, 𝑓𝑤 ∈ B. Furthermore, it is clear that 𝑓𝑤 ∈ B0 , since 󵄨2 2 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 (13) 󵄨󵄨𝑓𝑤 (0)󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨(1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 ) 󵄨󵄨󵄨󵄨 ≤ 4. 󵄨 󵄨 󵄩 󵄩 𝑀1 = sup {󵄩󵄩󵄩𝑓𝑤 󵄩󵄩󵄩B : 𝑤 ∈ Δ} ≤ 12.

(a) 𝐶𝜑 𝐷 : B → 𝑄𝐾 is bounded.

≤

(𝑧) = 2𝜑 (𝑤)

󵄨2 󵄨 (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 )

8 ≤ 1 − |𝑧|

Then the following statements are equivalent:

󵄩󵄩 󵄩󵄩2 󵄩󵄩𝑓󵄩󵄩B

(10)

󵄨2 2 󵄨 󵄨2 2 󵄨 2 (1 + 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 ) 󵄨󵄨 2 (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 ) 󵄨󵄨 󸀠 󵄨󵄨𝑓𝑤 (𝑧)󵄨󵄨 ≤ 󵄨 (1 − 󵄨󵄨𝜑 (𝑤)󵄨󵄨 |𝑧|)3 ≤ 󵄨 1 − |𝑧| 󵄨󵄨 󵄨󵄨

Theorem 2. Let 𝜑 be an analytic self-map of Δ. Suppose 𝐾 is a nondecreasing and right-continuous function on [0, +∞) such that 0

,

for 𝑧 ∈ Δ; then

Lemma 1 (see [10]). If all 𝑓 ∈ B, then

1/𝑒

2

(1 − 𝜑 (𝑤)𝑧)

for 𝑧 ∈ Δ. Since

2. The Boundedness

∫

󵄨2 2 󵄨 (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 )

So 󵄨2 󵄨 󵄨 󵄨4 4 󵄨󵄨󵄨󵄨𝜑󸀠 (𝑤)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 sup ∬ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 4 Δ (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨2 ) 𝑎∈Δ 󵄨 󵄨

(17)

󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩󵄩𝐶𝜑 𝐷𝑓𝑤 󵄩󵄩󵄩󵄩𝐾 ≤ 𝑀1 󵄩󵄩󵄩󵄩𝐶𝜑 𝐷󵄩󵄩󵄩󵄩B→𝑄 < +∞,

(9)

< +∞.

(16)

𝐾

for all 𝑤 ∈ Δ. Then we can imply sup ∬ 𝑎∈Δ

|𝜑(𝑤)|>1/2

Thus (a) holds.

󵄨2 󵄨󵄨 󸀠 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨2 4 󵄨󵄨 (1 − 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 ) 󵄨2 󵄨󵄨 󸀠 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨󵄨4 (18) 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨󵄨2 4 󵄨󵄨 (1 − 󵄨󵄨𝜑 (𝑤)󵄨󵄨 )

(a) ⇒ (b). It is obvious.

≤ 16 sup ∬

(b) ⇒ (c). Assume (b) holds; that is, there exists a constant 𝐶 such that ‖𝐶𝜑 𝐷𝑓(𝑧)‖𝐾 ≤ 𝐶‖𝑓‖B for all 𝑓 ∈ B0 . Conversely,

󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩󵄩𝐶𝜑 𝐷𝑓𝑤 󵄩󵄩󵄩󵄩𝐾 ≤ 𝑀1 󵄩󵄩󵄩󵄩𝐶𝜑 𝐷󵄩󵄩󵄩󵄩B→𝑄 < +∞.

𝑎∈Δ

|𝜑(𝑤)|>1/2

𝐾


3

On the other hand, we note the functions 𝑓(𝑧) ≡ 𝑧2 , which belong to B0 , and we get 󵄨2 󵄨 2 sup ∬ 󵄨󵄨󵄨󵄨𝜑󸀠 (𝑧)󵄨󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < ∞. Δ 𝑎∈Δ Then sup ∬ 𝑎∈Δ

|𝜑(𝑤)|𝑟| 󵄨

𝑎∈Δ

⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) (20)

󵄨2 󵄨󵄨 󵄨󵄨((𝐶𝜑 𝐷𝑓𝑡 )󸀠 − (𝐶𝜑 𝐷𝑓)) (𝑧)󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 |𝜑(𝑧)>𝑟|

+ sup ∬ 𝑎∈Δ

(26)

⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧)

≤ ∞.

󵄨2 󵄨󵄨 󸀠󸀠 󵄨󵄨𝑓𝑡 (𝜑 (𝑧)) 𝜑󸀠 (𝑧)󵄨󵄨󵄨 󵄨 󵄨 |𝜑(𝑧)>𝑟|

≤ sup ∬

So, we get 󵄨2 󵄨󵄨 󸀠 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 󵄨 󵄨 sup ∬ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 < ∞. 4 Δ (1 − 󵄨󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨2 ) 𝑎∈Δ 󵄨 󵄨

𝑎∈Δ

(21)

sup ∬ 𝑎∈Δ

Δ

𝑎∈Δ

󸀠

𝜑 (𝑧) 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 < ∞, 󵄨2 4 󵄨 (1 − 󵄨󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨 )

(22)

The following lemma can be proved similarly to [11]. Lemma 3. Let 𝜑 be an analytic self-map of Δ. Then 𝐶𝜑 𝐷 : B → 𝑄𝐾 (or B0 → 𝑄𝐾 ) is compact if and only if 𝐶𝜑 𝐷 : B → 𝑄𝐾 (or B0 → 𝑄𝐾 ) is bounded and for any bounded sequence (𝑓𝑛 )𝑛∈𝑁 in B which converges to zero uniformly on compact subsets of Δ; one has ‖𝐶𝜑 𝐷𝑓𝑛 ‖𝑄𝐾 → 0 𝑎𝑠 𝑛 → ∞. Lemma 4. Let 𝜑 be an analytic self-map of Δ. Suppose 𝐾 is a nondecreasing and right-continuous function on [0, +∞) such that 0

𝐾 (− log 𝑟) 𝑟 𝑑𝑟 < +∞.

(23)

If 𝐶𝜑 𝐷 : B(B0 ) → 𝑄𝐾 is compact, then for any 𝜖 > 0 there exists a 𝛿, 0 < 𝛿 < 1 such that, for all 𝑓 in 𝐸, 󵄨2 󵄨󵄨 󵄨󵄨(𝐶𝜑 𝐷𝑓)󸀠 (𝑧)󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜖 󵄨󵄨 󵄨 |𝜑(𝑧)>𝑟| 󵄨

sup ∬

+ 𝜀.

󵄨󵄨 󸀠 󵄨󵄨2 󵄩󵄩 󸀠󸀠 󵄩󵄩2 󵄨 󵄨 󵄩󵄩𝑓𝑡 󵄩󵄩 sup ∬ 󵄩 󵄩∞ 𝑎∈Δ |𝜑(𝑧)>𝑟| 󵄨󵄨𝜑 (𝑧)󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀. (27)

3. The Compactness

1/𝑒

󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨 󵄨 |𝜑(𝑧)>𝑟|

Then, we prove that for that given 𝜀 > 0 and ‖𝑓𝑡󸀠󸀠 ‖2∞ > 0 there exists a 𝛿 ∈ (0, 1) such that if 𝛿 < 𝑟 < 1,

for all 𝑧 ∈ Δ. This completes the proof of this theorem.

∫

󵄩 󵄩2 ⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) + 𝜀 ≤ 󵄩󵄩󵄩󵄩𝑓𝑡󸀠󸀠 󵄩󵄩󵄩󵄩∞ ⋅ sup ∬

By the arbitrary of 𝑤, we have

𝑎∈Δ

󵄨2 󵄨󵄨 󵄨󵄨(𝐶𝜑 𝐷𝑓)󸀠 (𝑧)󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨󵄨 󵄨 |𝜑(𝑧)>𝑟| 󵄨

sup ∬

≤ sup ∬

󵄨2 󵄨󵄨 󸀠 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨2 4 󵄨󵄨 (1 − 󵄨󵄨𝜑 (𝑤)󵄨󵄨󵄨 )

4 4 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) ≤ ( ) sup ∬ 󵄨 3 𝑎∈Δ |𝜑(𝑤)| 0 and ‖𝑓𝑡󸀠󸀠 ‖2∞ > 0, there exists an 𝑁 ∈ N such that 󵄨2 󵄨 󵄨2 󵄨 󵄩󵄩 󸀠󸀠 󵄩󵄩2 󵄩󵄩𝑓𝑡 󵄩󵄩 sup ∬ 𝑛2 (𝑛 − 1)2 󵄨󵄨󵄨𝜑𝑛−2 (𝑧)󵄨󵄨󵄨 󵄨󵄨󵄨𝜑󸀠 (𝑧)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩∞ 𝑎∈𝛿 Δ ⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀, whenever 𝑛 ≥ 𝑁. Hence, for 0 < 𝑟 < 1, 󵄨2 󵄨 󵄨2 󵄨 𝑁2 (𝑁 − 1)2 sup ∬ 󵄨󵄨󵄨󵄨𝜑𝑁−2 (𝑧)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜑󸀠 (𝑧)󵄨󵄨󵄨󵄨 Δ 𝑎∈𝛿 ⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) ≥ 𝑁2 (𝑁 − 1)2 ⋅ sup ∬ 𝑎∈Δ

|𝜑(𝑧)>𝑟|

󵄨󵄨 𝑁−2 󵄨󵄨2 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨 󵄨󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨 󵄨 2

(29) 2

holds whenever 𝛿 < 𝑟 < 1, where 𝐸 is the unit ball of B(B0 ).

⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) ≥ 𝑁 (𝑁 − 1)

Proof. For 𝑓 ∈ B0 , let 𝑓𝑡 (𝑧) = 𝑓(𝑡𝑧) (0 < 𝑡 < 1). Then 𝑓𝑡 ∈ B0 , and 𝑓𝑡 → 𝑓 uniformly on compact subsets of Δ as 𝑡 → 1. Since 𝐶𝜑 𝐷 is compact, ‖(𝐶𝜑 𝐷𝑓𝑡 − 𝐶𝜑 𝐷𝑓)(𝑧)‖ → 0 as 𝑡 → 1. That is, for given 𝜀 > 0, there exists 𝑡 ∈ (0, 1) such that

⋅ 𝑟2(𝑁−2) sup ∬

󵄨󵄨2 󵄨󵄨 󸀠 sup ∬ 󵄨󵄨󵄨((𝐶𝜑 𝐷𝑓𝑡 ) − (𝐶𝜑 𝐷𝑓)) (𝑧)󵄨󵄨󵄨 󵄨 󵄨 Δ 𝑎∈Δ ⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜖.

(28)

𝑎∈Δ

󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 |𝜑(𝑧)>𝑟|

⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) . Therefore, for 𝑟 > (𝑁2 − 𝑁)−1/(𝑁−2) ,

(25)

󵄩󵄩 󸀠󸀠 󵄩󵄩2 󵄨󵄨 󸀠 󵄨󵄨2 󵄨 󵄩󵄩𝑓𝑡 󵄩󵄩 sup ∬ 󵄨 󵄩 󵄩∞ 𝑎∈Δ |𝜑(𝑧)>𝑟| 󵄨󵄨𝜑 (𝑧)󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀. (30)

4


Thus we have already proved that, for any 𝜀 > 0 and 𝑓 ∈ B0 , there exists a 𝛿 = 𝛿(𝜀, 𝑓) such that 󵄨2 󵄨󵄨 󸀠󸀠 󵄨󵄨𝑓 (𝜑 (𝑧)) 𝜑󸀠 (𝑧)󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨 󵄨 |𝜑(𝑧)>𝑟|

sup ∬ 𝑎∈Δ

(31)

0 and 𝑓 ∈ 𝐸, we can find 𝑓𝑘 (1 ≤ 𝑘 ≤ 𝑚) satisfying 󵄨󵄨 󵄨2 󵄨󵄨((𝐶𝜑 𝐷𝑓𝑘 )󸀠 − (𝐶𝜑 𝐷𝑓)) (𝑧)󵄨󵄨󵄨 󵄨 󵄨󵄨 |𝜑(𝑧)>𝑟| 󵄨

sup ∬ 𝑎∈Δ

(32)

On the other hand, if max1≤𝑘≤𝑚 𝛿𝑘 (𝜀, 𝑓𝑘 ) = 𝛿 < 𝑟 < 1, we have from the previous observation that, for all 𝑘 = 1, 2, . . . , 𝑚, 󵄨󵄨 󵄨2 󵄨󵄨(𝐶𝜑 𝐷𝑓𝑘 )󸀠 (𝑧)󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀. (33) sup ∬ 󵄨 󵄨󵄨 󵄨 |𝜑(𝑧)>𝑟| 𝑎∈Δ

𝑎∈Δ

(34)

holds whenever 𝛿 < 𝑟 < 1. The proof is complete. Theorem 5. Let 𝜑 be an analytic self-map of Δ. Suppose 𝐾 is a nondecreasing and right-continuous function on [0, +∞) such that ∫

1/𝑒

0

𝐾 (− log 𝑟) 𝑟 𝑑𝑟 < +∞.

(35)

Then the following statements are equivalent: (a) 𝐶𝜑 𝐷 : B → 𝑄𝐾 is compact. (b) 𝐶𝜑 𝐷 : B0 → 𝑄𝐾 is compact. (c) 𝐶𝜑 𝐷 : B → 𝑄𝐾 is bounded: lim sup ∬ 𝑡→1 𝑎∈Δ

|𝜑(𝑧)|>𝑡

󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨2 4 󵄨󵄨 (1 − 󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨 )

|𝜑(𝑧)|>𝑡0

𝑎∈Δ

󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀. 󵄨2 4 󵄨󵄨 (1 − 󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨 )

(37)

Let 𝑇 = {𝑧 ∈ Δ : |𝜑(𝑧)| ≤ 𝑡0 }; then we have 󵄩2 󵄩󵄩 󵄩󵄩𝐶𝜑 𝐷𝑓𝑛 󵄩󵄩󵄩 󵄩 󵄩 󵄨 󵄨2 = sup ∬ 󵄨󵄨󵄨󵄨𝜑󸀠 (𝑧) 𝑓𝑛󸀠󸀠 (𝜑 (𝑧))󵄨󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) Δ 𝑎∈Δ 󵄨 󵄨2 = sup ∬ 󵄨󵄨󵄨󵄨𝜑󸀠 (𝑧) 𝑓𝑛󸀠󸀠 (𝜑 (𝑧))󵄨󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 𝑇 𝑎∈Δ 󵄨󵄨 󸀠 󵄨2 󵄨󵄨𝜑 (𝑧) 𝑓𝑛󸀠󸀠 (𝜑 (𝑧))󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨 󵄨 Δ/𝑇

+ sup ∬

󵄨2 󵄨 󵄩 󵄩 󵄩 󵄩2 ≤ sup {󵄨󵄨󵄨󵄨𝑓𝑛󸀠󸀠 (𝜑 (𝑧))󵄨󵄨󵄨󵄨 : 𝑧 ∈ 𝑇} 󵄩󵄩󵄩𝜑 (𝑧)󵄩󵄩󵄩𝐾 + 󵄩󵄩󵄩𝑓𝑛 󵄩󵄩󵄩B ⋅ sup ∬

Δ/𝑇

𝑎∈Δ

(38)

󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 󵄨2 4 󵄨󵄨 (1 − 󵄨󵄨𝜑 (𝑧)󵄨󵄨󵄨 )

󵄨 󵄨2 󵄩 󵄩 󵄩 󵄩2 ≤ sup {󵄨󵄨󵄨󵄨𝑓𝑛󸀠󸀠 (𝜑 (𝑧))󵄨󵄨󵄨󵄨 : 𝑧 ∈ 𝑇} 󵄩󵄩󵄩𝜑 (𝑧)󵄩󵄩󵄩𝐾 + 󵄩󵄩󵄩𝑓𝑛 󵄩󵄩󵄩B

By the triangle inequality we obtain that 󵄨2 󵄨󵄨 󵄨󵄨(𝐶𝜑 𝐷𝑓)󸀠 (𝑧)󵄨󵄨󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀 󵄨󵄨 󵄨 󵄨 |𝜑(𝑧)>𝑟|

sup ∬

𝑎∈Δ

⋅ 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) < 𝜀.

sup ∬

in 𝑄𝐾 norm. By the assumption, for any 𝜀 > 0, there exists a 𝑡0 ∈ (0, 1) such that

⋅ 𝜀. By 𝐶𝜑 𝐷 is bounded, we know that 𝜑 ∈ 𝑄𝐾 . It follows that ‖𝐶𝜑 𝐷𝑓𝑛 ‖ → 0 since that sup{|𝑓𝑛󸀠󸀠 (𝜑(𝑧))|2 : 𝑧 ∈ 𝑇} → 0 as 𝑛 → ∞. By Lemma 1, we can obtain that 𝐶𝜑 𝐷 : B → 𝑄𝐾 is compact. (a) ⇒ (b). It is obvious. (b) ⇒ (c). Suppose that 𝐶𝜑 𝐷 : B0 → 𝑄𝐾 is compact. Then it is clear that 𝐶𝜑 𝐷 : B0 → 𝑄𝐾 is bounded. We know that 𝑓𝜃 (𝑧) = (1/2) log(1/(1 − 𝑒−𝑖𝜃 𝑧)) ∈ B for all 𝜃 ∈ [0, 2𝜋). Choose a sequence {𝜆 𝑛 } in Δ which converges to 1 as 𝑛 → ∞, and let 𝑓𝜃,𝑛 = 𝑓𝜃 (𝜆 𝑛 𝑧) for 𝑛 ∈ N. Thus 𝑓𝜃,𝑛 in 𝐸 for all 𝑛 ∈ N and 𝜃 ∈ [0, 2𝜋), where 𝐸 is the unit ball of B0 . By Lemma 4, for any 𝜀 > 0 sup ∬ 𝑎∈Δ

(36)

= 0. Proof. (c) ⇒ (a). Assume (c) holds. Without loss of generality, let {𝑓𝑛 }𝑛∈𝑁 be a sequence in 𝐸 which converges to 0 uniformly on compact subsets of Δ, as 𝑛 → +∞, where 𝐸 is the unit ball of B. By Cauchy’s estimate, we know that {𝑓𝑛󸀠󸀠 }𝑛∈𝑁 also converges to 0 uniformly on compact subsets of Δ. For the sufficiency we will be verifying that {𝐶𝜑 𝐷𝑓𝑛 } converges to 0

|𝜑(𝑧)|>𝑡

󵄨󵄨 2 󸀠 󵄨󵄨2 󵄨󵄨𝜆 𝑛 𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 4 −𝑖𝜃 (1 − 𝑒 𝜆 𝑛 𝜑 (𝑧))

(39)

𝑡

󵄨󵄨 2 󸀠 󵄨󵄨2 󵄨󵄨𝜆 𝑛 𝜑 (𝑧)󵄨󵄨 󵄨 󵄨 𝐾 (𝑔 (𝑧, 𝑎)) 𝑑𝐴 (𝑧) 4 (40) (1 − 𝑒−𝑖𝜃 𝜆 𝑛 𝜑 (𝑧))

= 0.

Thus, we obtain (c) by integrating, with respect to 𝜃, the Fubini theorem, the Poisson formula, and the Fatou’s lemma.


Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This paper was supported in part by NSFC, Approval no. 41174165, the graduate student scientific research innovation project of Jiangsu Province of China, Approval no. KYLX15 0882, the Natural Science Research Program of the Education Department of Jiangsu Province of China, Approval no. 07KJB110069, and a grant of NUIST, Approval no. 20080290.

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