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 đđž 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 Š 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 đđž spaces are discussed in this paper.
From [1], we know that đđž â B if
1. Introduction and Motivation Let Î be the open unit disk in the complex plane and let đť(Î) be the class of all analytic functions on Î. Let đđ´(đ§) be the Euclidean area element on Î. The Bloch space B on Î is the space of all analytic functions đ on Î such that óľ¨ óľ¨ óľ¨ óľŠóľŠ óľŠóľŠ 2 óľ¨ ó¸ óľŠóľŠđóľŠóľŠB = óľ¨óľ¨óľ¨đ (0)óľ¨óľ¨óľ¨ + sup (1 â |đ§| ) óľ¨óľ¨óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨óľ¨ < â. đ§âÎ
(1)
Under the above norm, B is a Banach space. Let B0 denote the subspace of B consisting of those đ â B for which (1 â |đ§|2 )đó¸ |đ§| â 0 as |đ§| â 1. This space is called the little Bloch space. Throughout this paper, we assume that đž : [0, â) â [0, â) is a nondecreasing and right-continuous function. A function đ â đť(Î) is said to belong to đđž space (see [1]) if óľ¨óľ¨ ó¸ óľ¨óľ¨ óľ¨óľ¨đ (đ§)óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < â, óľ¨ óľ¨ Î
óľŠóľŠ óľŠóľŠ2 óľŠóľŠđóľŠóľŠđž = sup ⏠đâÎ
(2)
where đ(đ§, đ) is the Green function with logarithmic singularity at đ; that is, đ(đ§, đ) = log(1/|đđ (đ§)|) (đđ is a conformal automorphism defined by đđ (đ§) = (đ â đ§)/(1 â đđ§) for đ â Î). đđž is a Banach space under the norm óľ¨ óľŠ óľŠ óľ¨ óľŠóľŠ óľŠóľŠ2 óľŠóľŠđóľŠóľŠđđž = óľ¨óľ¨óľ¨đ (0)óľ¨óľ¨óľ¨ + óľŠóľŠóľŠđóľŠóľŠóľŠđž .
(3)
1/đ
âŤ
0
đž (â log đ) đ đđ < â.
(4)
Let đ denote a nonconstant analytic self-map of Î. Associated with đ is the composition operator đśđ defined by đśđ (đ) = đ â đ for đ â đť(Î). 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 đˇ be the differentiation operator on đť(Î); then we have đˇđ(đ§) = đó¸ (đ§). For đ â đť(Î), the products of differentiation and composition operators đˇđśđ and đśđ đˇ are defined by ó¸
đˇđśđ (đ) = (đ â đ) = đó¸ (đ) đó¸ , đśđ đˇ (đ) = đó¸ (đ) ,
(5) đ â đť (Î) .
Operators đśđ đˇ as well as some other products of linear operators were studied, for example, in [3â9] (see also the references therein). Recall that a linear operator đ : đ â đ is said to be bounded if there exists a constant đ > 0 such that âđ(đ)âđ â¤
2
Journal of Function Spaces
đâđâđ for all maps đ â đ. And đ â đ is compact if it takes bounded sets in đ to sets in đ which have compact closure. For Banach spaces đ and đ of đť(Î), đ is compact from đ to đ if and only if for each sequence {đĽđ } in đ; the sequence {đđĽđ } â đ contains a subsequence converging to some limit in đ. Considering the definition of đđž spaces and đđž â B with some conditions, it is difficult to study the operator đśđ đˇ from Bloch spaces to đđž spaces. In this paper, some sufficient and necessary conditions for the boundedness and compactness of this operator are given.
suppose that đśđ đˇ : B â đđž is bounded. Fix đ¤ â Î and assume that đ(đ¤) ≠ 0. Consider the function đ defined by đđ¤ (đ§) =
đóľ¨ óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ sup (1 â |đ§|2 ) óľ¨óľ¨óľ¨óľ¨đ(đ) óľ¨óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đ (0)óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨óľ¨đó¸ (0)óľ¨óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨óľ¨đó¸ ó¸ (0)óľ¨óľ¨óľ¨óľ¨ đ§âÎ
óľ¨ óľ¨ óľŠ óľŠ + â
â
â
+ óľ¨óľ¨óľ¨óľ¨đ(đâ1)(0) óľ¨óľ¨óľ¨óľ¨ â óľŠóľŠóľŠđóľŠóľŠóľŠB ,
đđ¤ó¸
(6)
(đ = 1, 2, . . .) .
đž (â log đ) đ đđ < +â.
(7)
(1 â đ (đ¤)đ§)
2
3
,
(11)
Thus
Note that óľ¨óľ¨ ó¸ ó¸ óľ¨ óľ¨óľ¨đđ¤ (đ (đ¤))óľ¨óľ¨óľ¨ = óľ¨ óľ¨
(b) đśđ đˇ : B0 â đđž is bounded. (c) óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ sup ⏠đž (đ (đ§, đ)) đđ´ (đ§) < â. 4 Î (1 â óľ¨óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨2 ) đâÎ óľ¨ óľ¨
(c) â (a). Suppose that (c) holds. For any đ§ â Î and đ(đ§) â B, we have óľŠ2 óľŠóľŠ óľ¨ óľ¨2 óľŠóľŠđśđ đˇđ (đ§)óľŠóľŠóľŠ = sup ⏠óľ¨óľ¨óľ¨đó¸ (đ)ó¸ óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľŠđž đâÎ Î óľ¨ óľŠ óľ¨
2
(đó¸ (đ§)) sup ⏠đž (đ (đ§, đ)) đđ´ (đ§) 4 Î (1 â óľ¨óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨2 ) đâÎ óľ¨ óľ¨
óľ¨ óľ¨ 2 óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ . óľ¨2 2 óľ¨ (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ )
(14)
(15)
For this function đđ¤ and this point đ¤ we have óľ¨óľ¨ óľ¨ óľ¨óľ¨(đśđ đˇđđ¤ )ó¸ (đ¤)óľ¨óľ¨óľ¨ = óľ¨óľ¨óľ¨óľ¨đó¸ (đ¤) đó¸ ó¸ (đ (đ¤))óľ¨óľ¨óľ¨óľ¨ đ¤ óľ¨óľ¨ óľ¨óľ¨ óľ¨ óľ¨ óľ¨óľ¨ ó¸ óľ¨óľ¨ óľ¨óľ¨ 2 óľ¨ 2 óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ = óľ¨ . óľ¨2 2 óľ¨ (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ )
(8)
Proof.
óľ¨ óľ¨2 = sup ⏠óľ¨óľ¨óľ¨óľ¨(đó¸ (đ§)) đó¸ ó¸ (đ)óľ¨óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) Î đâÎ
(12)
for all đ§ â Î, đđ¤ â B. Furthermore, it is clear that đđ¤ â B0 , since óľ¨2 2 óľ¨óľ¨ óľ¨ óľ¨ óľ¨óľ¨ óľ¨óľ¨ (13) óľ¨óľ¨đđ¤ (0)óľ¨óľ¨óľ¨ = óľ¨óľ¨óľ¨óľ¨(1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ ) óľ¨óľ¨óľ¨óľ¨ ⤠4. óľ¨ óľ¨ óľŠ óľŠ đ1 = sup {óľŠóľŠóľŠđđ¤ óľŠóľŠóľŠB : đ¤ â Î} ⤠12.
(a) đśđ đˇ : B â đđž is bounded.
â¤
(đ§) = 2đ (đ¤)
óľ¨2 óľ¨ (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ )
8 ⤠1 â |đ§|
Then the following statements are equivalent:
óľŠóľŠ óľŠóľŠ2 óľŠóľŠđóľŠóľŠB
(10)
óľ¨2 2 óľ¨ óľ¨2 2 óľ¨ 2 (1 + óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ ) óľ¨óľ¨ 2 (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ ) óľ¨óľ¨ ó¸ óľ¨óľ¨đđ¤ (đ§)óľ¨óľ¨ â¤ óľ¨ (1 â óľ¨óľ¨đ (đ¤)óľ¨óľ¨ |đ§|)3 â¤ óľ¨ 1 â |đ§| óľ¨óľ¨ óľ¨óľ¨
Theorem 2. Let đ be an analytic self-map of Î. Suppose đž is a nondecreasing and right-continuous function on [0, +â) such that 0
,
for đ§ â Î; then
Lemma 1 (see [10]). If all đ â B, then
1/đ
2
(1 â đ (đ¤)đ§)
for đ§ â Î. Since
2. The Boundedness
âŤ
óľ¨2 2 óľ¨ (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ )
So óľ¨2 óľ¨ óľ¨ óľ¨4 4 óľ¨óľ¨óľ¨óľ¨đó¸ (đ¤)óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ sup ⏠đž (đ (đ§, đ)) đđ´ (đ§) 4 Î (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨2 ) đâÎ óľ¨ óľ¨
(17)
óľŠ óľŠ2 óľŠ óľŠ2 ⤠óľŠóľŠóľŠóľŠđśđ đˇđđ¤ óľŠóľŠóľŠóľŠđž ⤠đ1 óľŠóľŠóľŠóľŠđśđ đˇóľŠóľŠóľŠóľŠBâđ < +â,
(9)
< +â.
(16)
đž
for all đ¤ â Î. Then we can imply sup ⏠đâÎ
|đ(đ¤)|>1/2
Thus (a) holds.
óľ¨2 óľ¨óľ¨ ó¸ óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨2 4 óľ¨óľ¨ (1 â óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ ) óľ¨2 óľ¨óľ¨ ó¸ óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨óľ¨4 (18) óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨óľ¨2 4 óľ¨óľ¨ (1 â óľ¨óľ¨đ (đ¤)óľ¨óľ¨ )
(a) â (b). It is obvious.
⤠16 sup âŹ
(b) â (c). Assume (b) holds; that is, there exists a constant đś such that âđśđ đˇđ(đ§)âđž ⤠đśâđâB for all đ â B0 . Conversely,
óľŠ óľŠ2 óľŠ óľŠ2 ⤠óľŠóľŠóľŠóľŠđśđ đˇđđ¤ óľŠóľŠóľŠóľŠđž ⤠đ1 óľŠóľŠóľŠóľŠđśđ đˇóľŠóľŠóľŠóľŠBâđ < +â.
đâÎ
|đ(đ¤)|>1/2
đž
Journal of Function Spaces
3
On the other hand, we note the functions đ(đ§) ⥠đ§2 , which belong to B0 , and we get óľ¨2 óľ¨ 2 sup ⏠óľ¨óľ¨óľ¨óľ¨đó¸ (đ§)óľ¨óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < â. Î đâÎ Then sup ⏠đâÎ
|đ(đ¤)|đ| óľ¨
đâÎ
â
đž (đ (đ§, đ)) đđ´ (đ§) (20)
óľ¨2 óľ¨óľ¨ óľ¨óľ¨((đśđ đˇđđĄ )ó¸ â (đśđ đˇđ)) (đ§)óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨ óľ¨ |đ(đ§)>đ|
+ sup ⏠đâÎ
(26)
â
đž (đ (đ§, đ)) đđ´ (đ§)
⤠â.
óľ¨2 óľ¨óľ¨ ó¸ ó¸ óľ¨óľ¨đđĄ (đ (đ§)) đó¸ (đ§)óľ¨óľ¨óľ¨ óľ¨ óľ¨ |đ(đ§)>đ|
⤠sup âŹ
So, we get óľ¨2 óľ¨óľ¨ ó¸ óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ óľ¨ óľ¨ sup ⏠đž (đ (đ§, đ)) đđ´ < â. 4 Î (1 â óľ¨óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨2 ) đâÎ óľ¨ óľ¨
đâÎ
(21)
sup ⏠đâÎ
Î
đâÎ
ó¸
đ (đ§) đž (đ (đ§, đ)) đđ´ < â, óľ¨2 4 óľ¨ (1 â óľ¨óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨ )
(22)
The following lemma can be proved similarly to [11]. Lemma 3. Let đ be an analytic self-map of Î. Then đśđ đˇ : B â đđž (or B0 â đđž ) is compact if and only if đśđ đˇ : B â đđž (or B0 â đđž ) is bounded and for any bounded sequence (đđ )đâđ in B which converges to zero uniformly on compact subsets of Î; one has âđśđ đˇđđ âđđž â 0 đđ đ â â. Lemma 4. Let đ be an analytic self-map of Î. Suppose đž is a nondecreasing and right-continuous function on [0, +â) such that 0
đž (â log đ) đ đđ < +â.
(23)
If đśđ đˇ : B(B0 ) â đđž is compact, then for any đ > 0 there exists a đż, 0 < đż < 1 such that, for all đ in đ¸, óľ¨2 óľ¨óľ¨ óľ¨óľ¨(đśđ đˇđ)ó¸ (đ§)óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < đ óľ¨óľ¨ óľ¨ |đ(đ§)>đ| óľ¨
sup âŹ
+ đ.
óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľŠóľŠ ó¸ ó¸ óľŠóľŠ2 óľ¨ óľ¨ óľŠóľŠđđĄ óľŠóľŠ sup âŹ óľŠ óľŠâ đâÎ |đ(đ§)>đ| óľ¨óľ¨đ (đ§)óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < đ. (27)
3. The Compactness
1/đ
óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨ óľ¨ |đ(đ§)>đ|
Then, we prove that for that given đ > 0 and âđđĄó¸ ó¸ â2â > 0 there exists a đż â (0, 1) such that if đż < đ < 1,
for all đ§ â Î. This completes the proof of this theorem.
âŤ
óľŠ óľŠ2 â
đž (đ (đ§, đ)) đđ´ (đ§) + đ ⤠óľŠóľŠóľŠóľŠđđĄó¸ ó¸ óľŠóľŠóľŠóľŠâ â
sup âŹ
By the arbitrary of đ¤, we have
đâÎ
óľ¨2 óľ¨óľ¨ óľ¨óľ¨(đśđ đˇđ)ó¸ (đ§)óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨óľ¨ óľ¨ |đ(đ§)>đ| óľ¨
sup âŹ
⤠sup âŹ
óľ¨2 óľ¨óľ¨ ó¸ óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨2 4 óľ¨óľ¨ (1 â óľ¨óľ¨đ (đ¤)óľ¨óľ¨óľ¨ )
4 4 óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) ⤠( ) sup âŹ óľ¨ 3 đâÎ |đ(đ¤)| 0 and âđđĄó¸ ó¸ â2â > 0, there exists an đ â N such that óľ¨2 óľ¨ óľ¨2 óľ¨ óľŠóľŠ ó¸ ó¸ óľŠóľŠ2 óľŠóľŠđđĄ óľŠóľŠ sup ⏠đ2 (đ â 1)2 óľ¨óľ¨óľ¨đđâ2 (đ§)óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨đó¸ (đ§)óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨ óľŠ óľŠâ đâđż Î â
đž (đ (đ§, đ)) đđ´ (đ§) < đ, whenever đ ⼠đ. Hence, for 0 < đ < 1, óľ¨2 óľ¨ óľ¨2 óľ¨ đ2 (đ â 1)2 sup ⏠óľ¨óľ¨óľ¨óľ¨đđâ2 (đ§)óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨óľ¨đó¸ (đ§)óľ¨óľ¨óľ¨óľ¨ Î đâđż â
đž (đ (đ§, đ)) đđ´ (đ§) ⼠đ2 (đ â 1)2 â
sup ⏠đâÎ
|đ(đ§)>đ|
óľ¨óľ¨ đâ2 óľ¨óľ¨2 óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨ óľ¨ 2
(29) 2
holds whenever đż < đ < 1, where đ¸ is the unit ball of B(B0 ).
â
đž (đ (đ§, đ)) đđ´ (đ§) ⼠đ (đ â 1)
Proof. For đ â B0 , let đđĄ (đ§) = đ(đĄđ§) (0 < đĄ < 1). Then đđĄ â B0 , and đđĄ â đ uniformly on compact subsets of Î as đĄ â 1. Since đśđ đˇ is compact, â(đśđ đˇđđĄ â đśđ đˇđ)(đ§)â â 0 as đĄ â 1. That is, for given đ > 0, there exists đĄ â (0, 1) such that
â
đ2(đâ2) sup âŹ
óľ¨óľ¨2 óľ¨óľ¨ ó¸ sup ⏠óľ¨óľ¨óľ¨((đśđ đˇđđĄ ) â (đśđ đˇđ)) (đ§)óľ¨óľ¨óľ¨ óľ¨ óľ¨ Î đâÎ â
đž (đ (đ§, đ)) đđ´ (đ§) < đ.
(28)
đâÎ
óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ |đ(đ§)>đ|
â
đž (đ (đ§, đ)) đđ´ (đ§) . Therefore, for đ > (đ2 â đ)â1/(đâ2) ,
(25)
óľŠóľŠ ó¸ ó¸ óľŠóľŠ2 óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨ óľŠóľŠđđĄ óľŠóľŠ sup âŹ óľ¨ óľŠ óľŠâ đâÎ |đ(đ§)>đ| óľ¨óľ¨đ (đ§)óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < đ. (30)
4
Journal of Function Spaces
Thus we have already proved that, for any đ > 0 and đ â B0 , there exists a đż = đż(đ, đ) such that óľ¨2 óľ¨óľ¨ ó¸ ó¸ óľ¨óľ¨đ (đ (đ§)) đó¸ (đ§)óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨ óľ¨ |đ(đ§)>đ|
sup ⏠đâÎ
(31)
0 and đ â đ¸, we can find đđ (1 ⤠đ ⤠đ) satisfying óľ¨óľ¨ óľ¨2 óľ¨óľ¨((đśđ đˇđđ )ó¸ â (đśđ đˇđ)) (đ§)óľ¨óľ¨óľ¨ óľ¨ óľ¨óľ¨ |đ(đ§)>đ| óľ¨
sup ⏠đâÎ
(32)
On the other hand, if max1â¤đâ¤đ đżđ (đ, đđ ) = đż < đ < 1, we have from the previous observation that, for all đ = 1, 2, . . . , đ, óľ¨óľ¨ óľ¨2 óľ¨óľ¨(đśđ đˇđđ )ó¸ (đ§)óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < đ. (33) sup âŹ óľ¨ óľ¨óľ¨ óľ¨ |đ(đ§)>đ| đâÎ
đâÎ
(34)
holds whenever đż < đ < 1. The proof is complete. Theorem 5. Let đ be an analytic self-map of Î. Suppose đž is a nondecreasing and right-continuous function on [0, +â) such that âŤ
1/đ
0
đž (â log đ) đ đđ < +â.
(35)
Then the following statements are equivalent: (a) đśđ đˇ : B â đđž is compact. (b) đśđ đˇ : B0 â đđž is compact. (c) đśđ đˇ : B â đđž is bounded: lim sup ⏠đĄâ1 đâÎ
|đ(đ§)|>đĄ
óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨2 4 óľ¨óľ¨ (1 â óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨ )
|đ(đ§)|>đĄ0
đâÎ
óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < đ. óľ¨2 4 óľ¨óľ¨ (1 â óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨ )
(37)
Let đ = {đ§ â Î : |đ(đ§)| ⤠đĄ0 }; then we have óľŠ2 óľŠóľŠ óľŠóľŠđśđ đˇđđ óľŠóľŠóľŠ óľŠ óľŠ óľ¨ óľ¨2 = sup ⏠óľ¨óľ¨óľ¨óľ¨đó¸ (đ§) đđó¸ ó¸ (đ (đ§))óľ¨óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) Î đâÎ óľ¨ óľ¨2 = sup ⏠óľ¨óľ¨óľ¨óľ¨đó¸ (đ§) đđó¸ ó¸ (đ (đ§))óľ¨óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) đ đâÎ óľ¨óľ¨ ó¸ óľ¨2 óľ¨óľ¨đ (đ§) đđó¸ ó¸ (đ (đ§))óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨ óľ¨ Î/đ
+ sup âŹ
óľ¨2 óľ¨ óľŠ óľŠ óľŠ óľŠ2 ⤠sup {óľ¨óľ¨óľ¨óľ¨đđó¸ ó¸ (đ (đ§))óľ¨óľ¨óľ¨óľ¨ : đ§ â đ} óľŠóľŠóľŠđ (đ§)óľŠóľŠóľŠđž + óľŠóľŠóľŠđđ óľŠóľŠóľŠB â
sup âŹ
Î/đ
đâÎ
(38)
óľ¨óľ¨ ó¸ óľ¨óľ¨2 óľ¨óľ¨đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) óľ¨2 4 óľ¨óľ¨ (1 â óľ¨óľ¨đ (đ§)óľ¨óľ¨óľ¨ )
óľ¨ óľ¨2 óľŠ óľŠ óľŠ óľŠ2 ⤠sup {óľ¨óľ¨óľ¨óľ¨đđó¸ ó¸ (đ (đ§))óľ¨óľ¨óľ¨óľ¨ : đ§ â đ} óľŠóľŠóľŠđ (đ§)óľŠóľŠóľŠđž + óľŠóľŠóľŠđđ óľŠóľŠóľŠB
By the triangle inequality we obtain that óľ¨2 óľ¨óľ¨ óľ¨óľ¨(đśđ đˇđ)ó¸ (đ§)óľ¨óľ¨óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) < đ óľ¨óľ¨ óľ¨ óľ¨ |đ(đ§)>đ|
sup âŹ
đâÎ
â
đž (đ (đ§, đ)) đđ´ (đ§) < đ.
sup âŹ
in đđž norm. By the assumption, for any đ > 0, there exists a đĄ0 â (0, 1) such that
â
đ. By đśđ đˇ is bounded, we know that đ â đđž . It follows that âđśđ đˇđđ â â 0 since that sup{|đđó¸ ó¸ (đ(đ§))|2 : đ§ â đ} â 0 as đ â â. By Lemma 1, we can obtain that đśđ đˇ : B â đđž is compact. (a) â (b). It is obvious. (b) â (c). Suppose that đśđ đˇ : B0 â đđž is compact. Then it is clear that đśđ đˇ : B0 â đđž is bounded. We know that đđ (đ§) = (1/2) log(1/(1 â đâđđ đ§)) â B for all đ â [0, 2đ). Choose a sequence {đ đ } in Î which converges to 1 as đ â â, and let đđ,đ = đđ (đ đ đ§) for đ â N. Thus đđ,đ in đ¸ for all đ â N and đ â [0, 2đ), where đ¸ is the unit ball of B0 . By Lemma 4, for any đ > 0 sup ⏠đâÎ
(36)
= 0. Proof. (c) â (a). Assume (c) holds. Without loss of generality, let {đđ }đâđ be a sequence in đ¸ which converges to 0 uniformly on compact subsets of Î, as đ â +â, where đ¸ is the unit ball of B. By Cauchyâs estimate, we know that {đđó¸ ó¸ }đâđ also converges to 0 uniformly on compact subsets of Î. For the sufficiency we will be verifying that {đśđ đˇđđ } converges to 0
|đ(đ§)|>đĄ
óľ¨óľ¨ 2 ó¸ óľ¨óľ¨2 óľ¨óľ¨đ đ đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) 4 âđđ (1 â đ đ đ đ (đ§))
(39)
đĄ
óľ¨óľ¨ 2 ó¸ óľ¨óľ¨2 óľ¨óľ¨đ đ đ (đ§)óľ¨óľ¨ óľ¨ óľ¨ đž (đ (đ§, đ)) đđ´ (đ§) 4 (40) (1 â đâđđ đ đ đ (đ§))
= 0.
Thus, we obtain (c) by integrating, with respect to đ, the Fubini theorem, the Poisson formula, and the Fatouâs lemma.
Journal of Function Spaces
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.
References [1] M. Ess´en and H. Wulan, âOn analytic and meromorphic functions and spaces of Qđž -type,â Illinois Journal of Mathematics, vol. 46, no. 4, pp. 1233â1258, 2002. [2] S. Ohno, âWeighted composition operators between đťâ and the Bloch space,â Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555â563, 2001. [3] S. Ohno, âProducts of composition and differentiation between Hardy spaces,â Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 235â243, 2006. [4] S. Ohno, âProducts of differentiation and composition on Bloch spaces,â Bulletin of the Korean Mathematical Society, vol. 46, no. 6, pp. 1135â1140, 2009. [5] S. Li and S. Stevi´c, âComposition followed by differentiation from mixed-norm spaces to đź-Bloch spaces,â Sbornik Mathematics, vol. 199, no. 12, pp. 1847â1857, 2008. [6] S. Li and S. Stevi´c, âComposition followed by differentiation between Bloch type spaces,â Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 195â205, 2007. [7] S. Li and S. Stevi´c, âComposition followed by differentiation between H â and đź-Bloch spaces,â Houston Journal of Mathematics, vol. 35, no. 1, pp. 327â340, 2009. [8] S. Stevi´c, âProducts of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces,â Siberian Mathematical Journal, vol. 50, no. 4, pp. 726â736, 2009. [9] S. Stevi´c, âProducts of composition and differentiation operators on the weighted Bergman space,â Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 16, no. 4, pp. 623â635, 2009. [10] K. H. Zhu, âBlock type spaces of analytic functions,â The Rocky Mountain Journal of Mathematics, vol. 23, no. 3, pp. 1143â1177, 1993. [11] K. Zhu and M. Stessin, âComposition operators induced by symbols defined on a polydisk,â Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 815â829, 2006.
5
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014