Composition Operators Mapping Logarithmic Bloch Functions into Hardy Space

This paper is to characterize the class of holomorphic selfmap f of the open complex unit disc D for which the composition operator induced byfmaps logarithmic Bloch space boundedly into Hardy spaceHp. Main result of this paper is Theorem 2 whose primitive form is as follows. Theorem 1. If f is a holomorphic self-map ofD and if 0 < p < ∞, −1/2 < α < ∞, then the following are equivalent: (i) h ∘f ∈ H2p/(1+2α) for all holomorphic h onD satisfying


Introduction
This paper is to characterize the class of holomorphic selfmap  of the open complex unit disc  for which the composition operator induced by  maps logarithmic Bloch space boundedly into Hardy space   .Main result of this paper is Theorem 2 whose primitive form is as follows.
Theorem 1.If  is a holomorphic self-map of  and if 0 <  < ∞, −1/2 <  < ∞, then the following are equivalent: (i) ℎ ∘  ∈  2/(1+2) for all holomorphic ℎ on  satisfying When  = 0 the above equivalence is known.We refer to [1][2][3], wherein the results are investigated in  and in the ball of C  , respectively.Also, the case  < 1 + 2 was considered in [4], so the main case under consideration is  ≥ 1 + 2.In the latter case, a different approach, based on duality, is used.
Note that Theorem 1 not only characterizes the composition operators mapping logarithmic Bloch functions into the Hardy space but also introduces a kind of -function.The result will be stated precisely and more extensively in Section 3.
The restriction of the range of  is essential.If  < −1/2, then it reduces to a trivial result.If  = −1/2, then it corresponds to another space instead of Hardy space.These will be treated separately in the last section.

Hardy Space and Hyperbolic Hardy Class.
Let  be the unit disc of the complex plane.For 0 <  < ∞ and for || subharmonic in , we denote Then the right side limit is monotone increasing.And by definition, the Hardy space   =   () consists of holomorphic  in  for which ‖‖  < ∞, while the Yamashita hyperbolic Hardy class   consists of holomorphic self-map  of  for which ‖()‖  < ∞.Here () denotes the hyperbolic distance of  and 0 in , namely, Though   is not a linear space, it has, as hyperbolic counterparts, many properties analogous to those of   . 2

Journal of Function Spaces
For holomorphic self-maps  of , we let following the notation of Yamashita and let Then they have the following basic properties: are subharmonic f unctions for any  > 0 and  # are automorphism invariant in the sense that for any  ∈ , where Here and throughout,   means   ([0, 2]), Δ denotes the Laplacian: Δ = 4( 2 /),  ≲  means that  is bounded by a positive uniform constant times , and  ≈  means that either both sides are zero or the quotient / lies between two positive uniform constants.We refer to [1,[5][6][7] for (6)∼ (11).For a general theory of   and   , we refer to [8][9][10] and [7,11,12], respectively.
We pay attention to the absence of the square root in the defining of G in ( 17) when we compare it to that of Gℎ in (15).The difference actually explains lots of known parallelism (see [11]) between  2 and  1 .Suggested by (11), we define for Note that G 0  = G.Of course, main objective of introducing ( 19) is to establish an equivalence as (18).

Main Result Revisited.
Equipped with the notions introduced in Section 2, Theorem 1 can be stated as the first part of the following.

More on G-Function
Equivalence.For (20), we in fact can prove more extensively the following: we define for −1/2 <  < ∞ and 0 ≤  < ∞ that To cover all of our results stated up to here, it is sufficient to prove (24) and ( 21).This will be done in Section 5 after stating preparatory lemmas in Section 4.

Preparatory Lemmas
We describe some lemmas, whose proof will be deferred to Section 6, that will be used in proving our main theorem.Lemma 5. Let  be a holomorphic self-map of  and −2 ≤  < ∞.Then, for any positive  the function is subharmonic in .Furthermore, for 0 ≤  < ∞,   (,   ) is an increasing function of  and where (, ) is the Poisson kernel: The subharmonicity of   in Lemma 5 gives the following, where  = .

Proof of Main Results
Let  be a holomorphic self-map of  with (0) = 0. We are sufficient to show (24) and (21).We assume  is not constant because there's nothing left to prove when  is a constant.Let us denote for simplicity by  the boundary of  and  the arc length measure on  normalized to be () = ||.(24).For notational clearance we prove only for  = 0.But replacing G   and   , respectively, by G ,  and  +   , it is easy to check that the proof below works for general  in the same way.

Proof of
To show ‖G  ‖   ≲ ‖  ‖  , we divide it into two cases:  ≤ 1 and  > 1.

Proof of
where   =   () is the Rademacher function (see [8,23]) defined by Then ℎ  is holomorphic in  and by (30) there is a positive constant   such that

Remarks and Acknowledgement
In view of our main result, Theorem for any .