© Hindawi Publishing Corp. COMPOSITION OPERATORS FROM THE BLOCH SPACE INTO THE SPACES QT

Suppose that φ(z) is an analytic self-map of the unit disk Δ. We consider the boundedness of the composition operator Cφ from Bloch space ℬ into the spaces QT (QT,0) defined by a nonnegative, nondecreasing function T(r) on 0≤r<∞.


Introduction.
Let ∆ = {z : |z| < 1} be the unit disk of complex plane C and let H(∆) be the space of all analytic functions in ∆.For a ∈ ∆, Green's function with logarithmic singularity at a ∈ ∆ is denoted by g(z, a) = log |(1 − āz)/ (a − z)|.For 0 < p < ∞, the space Q p consists of all functions f analytic in ∆ for which where dA(z) is the Euclidean area element on ∆.Q p -spaces have been investigated by many authors (cf.[1,2,3,9]).We know that Q 1 = BMOA, the space of all analytic functions of bounded mean oscillation (cf.[4]).Further, the spaces Q p are the same for each p ∈ (1, ∞), and each space equals to the Bloch space Ꮾ, which is a Banach space with the norm (1.2) Recently, we introduced a new space Q T (cf.[5,10]) by a nondecreasing function T (r ) on 0 ≤ r < ∞ as follows.
Definition 1.1.Let T (r ) ≡ 0 be a nonnegative, nondecreasing function on then f is said to belong to Q T ,0 .
For 0 < p < ∞, if we take T (r ) = r p , the space Q T coincides with the space Q p .We note that Q T ⊂ Ꮾ for all nondecreasing functions T .We have previously shown that Q T = Q p under certain growth conditions on T (r ) (cf. [10]).
In the present paper, first we give some basic properties of Q T spaces, some of which are also new for the special case Q T = Q p .For example, Q T is a Banach space with the norm f T defined by (1.5) Then we investigate the boundedness of the composition operators from the Bloch space Ꮾ into Q T or Q T ,0 .These results extend some previously known results (cf.[6,8]).
2. Basic properties of Q T spaces.We give the following propositions.
Proposition 2.1.The space Q T is a subspace of the Bloch space Ꮾ.
The proof of Proposition 2.1 can be found in [10].
Proposition 2.2.The space Q T is a Banach space with the norm defined in (1.5).
Proof.For f ∈ Q T and a ∈ ∆, define 2 T g(z, a) dA(z). (2.1) and then (2.3) (2.4) Therefore, Then by changing a variable w = φ a (z), we obtain (2.6) For r 0 , 0 < r 0 < 1, such that T (log(1/r 0 )) = 0, we have Then there is a constant M > 0 such that By the estimate (2.8) for a fixed r 0 ∈ (0, 1), we obtain that holds for all integral numbers n = 1, 2,.... Hence, there exist a subsequence {f n j (z)} of {f n (z)} and an analytic function f defined on the unit disk ∆ such that both {f n j (z)} and {f n j (z)} converge uniformly to f and f , respectively.The conditions here are such that both the sequence of functions and the sequence of derivatives converge since we know that {f n (z)} is bounded on compact subsets of ∆ by inequality (2.10).By Fatou's lemma, we get that holds for all a ∈ ∆, so that f ∈ Q T .By a similar reasoning, we can prove that The proof of Proposition 2.2 is complete.

Boundedness of composition operators.
Let ϕ(z) be an analytic selfmap of the unit disk ∆.Let the composition operator C ϕ induced by ϕ from H(∆) to itself be defined by C ϕ (f ) = f •ϕ for f ∈ H(∆).The boundednesses of composition operators from Ꮾ to itself and from Ꮾ to Q p have been studied in [6,8], respectively.In this paper, we consider the same problems for the general spaces Q T .Theorem 3.1.Let T (r ) ≡ 0 be a nonnegative, nondecreasing function on 0 ≤ r < ∞ and let ϕ be an analytic self-map of ∆.
Proof.Let (3.1) hold and let K 2 1 (K 1 > 0) be the supremum in (3.1).If f ∈ Ꮾ, then for all a ∈ ∆, we have where Conversely, assume that C ϕ : Ꮾ → Q T is bounded, there exists a constant K > 0 such that for each f ∈ Ꮾ, we have On the other hand, by a result in [7], there exist holds for all z ∈ ∆, so that Thus, the inequalities hold for all z, a ∈ ∆, which establishes (3.1).The proof of Theorem 3.1 is completed.
Remark 3.2.Note that if C ϕ : Ꮾ → Ꮾ, then (3.1) holds for any increasing function T satisfying Q T = Ꮾ.Indeed, we know that Q T = Ꮾ (see [5]) if and only if The Schwarz-Pick lemma guarantees that ((1 For the spaces Q T ,0 , we have the following results. Theorem 3.4.Let T (r ) be a nonnegative, nondecreasing function on 0 ≤ r < ∞ and let ϕ be an analytic self-map of ∆.Then C ϕ : Ꮾ → Q T ,0 is bounded if and only if Proof.Suppose C ϕ : Ꮾ → Q T ,0 is bounded.Using a way similar to the proof of Theorem 3.1, we choose functions which shows that (3.11) holds.Conversely, by Theorem 3.1, we know that We need only to prove that C ϕ f ∈ Q T ,0 for each f ∈ Ꮾ, and this follows from the inequality (3.15) The proof of Theorem 3.4 is completed.
Acknowledgments.The essential part of this paper was written when the authors were at the University of Joensuu, Finland, in 1998.This research is now also supported in part by grants from the National Natural Science Foundation of China no.10171058, Guangdong Province no.010446, and Guizhou Province of China.We would like to thank the referee for helpful comments.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
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