Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

The logarithmic Bloch space Blog is the Banach space of analytic functions on the open unit disk D whose elements f satisfy the condition ‖f‖ supz∈D 1 − |z|2 log 2/ 1 − |z|2 |f ′ z | < ∞. In this work we characterize the bounded and the compact weighted composition operators from the Hardy space H with 1 ≤ p ≤ ∞ into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping H into the little logarithmic Bloch space defined as the subspace of Blog consisting of the functions f such that lim|z|→ 1 1 − |z|2 log 2/ 1 − |z|2 |f ′ z | 0.


Introduction
Let X and Y be Banach spaces of analytic functions on a domain Ω in C, ψ an analytic function on Ω, and let ϕ be an analytic function mapping Ω into itself.The weighted composition operator with symbols ψ and ϕ from X to Y is the operator W ψ,ϕ with range in Y defined by where M ψ denotes the multiplication operator with symbol ψ, and C ϕ denotes the composition operator with symbol ϕ.Let H D be the set of analytic functions on D {z ∈ C : |z| < 1}.For 0 < p < ∞ the Hardy space H p is the space consisting of all f ∈ H D such that The Bloch norm is given by f B |f 0 | f β .Using the Schwarz-Pick lemma, it is easy to see that the Hardy space H ∞ is contained in B and f β ≤ f ∞ .The inclusion is proper, as the function f z log 1 z / 1 − z shows.The little Bloch space, denoted by B 0 , is defined as the set of the analytic functions f on D such that lim |z| → 1 1 − |z| 2 |f z | 0. It is well known that B 0 is a closed separable subspace of B. The interested reader is referred to 1 for more information on the Bloch space.
The logarithmic Bloch space B log is defined as the set of functions f on D such that It is a Banach space under the norm defined by f B log |f 0 | f .Clearly, if f ∈ B log , then lim |z| → 1 1 − |z| 2 |f z | 0, so B log is a subset of the little Bloch space.
The little logarithmic Bloch space, denoted by B log,0 , is defined as the subspace of B log whose elements f satisfy the condition lim The space B log arises in connection to the study of certain operators with symbol.Arazy 2 proved that the multiplication operator M ψ is bounded on the Bloch space if and only if ψ ∈ B log ∩ H ∞ .In 3 , Brown and Shields extended this result to the little Bloch space.
The space B log also arises in the study of Hankel operators on the Bergman one space.The Bergman space A 1 on D is defined to be the set of analytic functions f on D whose modulus is Lebesgue integrable over D.
The Hankel operator H f on A 1 is defined as H f g I − P fg , where I is the identity operator, and P is the standard Bergman projection from The study of operators with symbol on the logarithmic Bloch space began with the characterizations of the bounded and the compact composition operators given in 5 by Yoneda.In 6 , Galanopoulos extended these results to the weighted composition operators on B log .He also introduced a class of Banach spaces Q p log p > 0 closely related to B log and studied the Taylor coefficients of the functions in B log .In 7 , Ye characterized the bounded and the compact weighted composition operators on the little logarithmic Bloch space B log,0 .See 8, 9 for the study of the weighted composition operators on Bloch spaces and weighted Bloch spaces.
In this paper, we characterize the bounded and the compact weighted composition operators from the Hardy space H p with 1 ≤ p ≤ ∞ to the logarithmic Bloch space B log as well as to its subspace B log,0 .The paper consists of five sections.Specifically, in Section 2, we consider the bounded weighted composition operators mapping H ∞ into B log and B log,0 .In particular, we show that where the notation A ∼ B stands for c 1 A ≤ B ≤ c 2 A, for some positive constants c 1 and c 2 .In Section 3, we look at the issue of compactness of such operators.
In Section 4, we characterize the bounded and the compact weighted composition operators mapping H p into B log in the case when 1 ≤ p < ∞.Finally, in Section 5, we study the operators mapping H p into B log,0 .

Boundedness of
In the following theorem, we give two characterizations of boundedness when the operator maps H ∞ into B log .
Theorem 2.1.Let ψ be an analytic function on D, and let ϕ be an analytic self-map of D. The following statements are equivalent.
Proof. a ⇒ b .For n ∈ N, the function p n z z n is bounded and p n ∞ 1.Therefore, if W ψ,ϕ is bounded, then ψϕ n B log ≤ W ψ,ϕ .
b ⇒ c Let C be an upper bound for ψϕ n B log , n ≥ 0. Taking n 0, we deduce that ψ B log ≤ C, so ψ ∈ B log .
For N ∈ N and n ≥ 2, define the sets

2.1
Fix an integer N > 2, and z ∈ D. For z ∈ E N , by the product rule, we have In the proof of Theorem 2 of 10 , it was shown that inf
We next turn our attention to the weighted composition operators mapping into the little logarithmic Bloch space.Proof.a ⇒ b is proved as in the case of the operator mapping into B log .b ⇒ c Suppose that b holds.If E N D for some integer N > 1, then for all z ∈ D, we have

2.7
If where By the assumption of ψϕ n ∈ B log,0 , we have as |z| → 1.On the other hand, since ψ ∈ B log,0 , II → 0 as |z| → 1.Therefore, Journal of Function Spaces and Applications c ⇒ a Assume that c holds.To prove that W ψ,ϕ is bounded, it suffices to show that W ψ,ϕ f ∈ B log,0 for each f ∈ H ∞ , since the boundedness of the operator can be shown as in the proof of Theorem 2.1.Since ψ ∈ B log,0 , for f ∈ H ∞ and z ∈ D, we have as |z| → 1.On the other hand, by 2.6 , as |z| → 1, completing the proof.
In Section 3, we shall prove that all bounded weighted composition operators from H ∞ into B log,0 are compact.

Compactness of W ψ,ϕ from H ∞ into B log and B log,0
The following criterion for compactness follows by a standard argument similar, for example, to that outlined in Proposition 3.11 of 11 .Lemma 3.1.Let ψ be analytic on D, ϕ an analytic self-map of D, 1 ≤ p ≤ ∞.The operator W ψ,ϕ : H p → B log is compact if and only if for any bounded sequence {f n } n∈N in H p which converges to zero uniformly on compact subsets of D, we have The proof of the following result is similar to the proof of Lemma 1 of 12 .Hence we omit it.

Lemma 3.2. A closed set K in B log,0 is compact if and only if it is bounded and satisfies the following:
We now introduce two one-parameter families of functions which will be used to characterize the compactness of the operators under consideration.
Fix a ∈ D and, for z ∈ D, define Theorem 3.3.Let ψ be analytic on D, ϕ an analytic self-map of D, and assume that W ψ,ϕ : H ∞ → B log is bounded.Then the following conditions are equivalent: Proof.We begin by showing that a , b , and c are equivalent.a ⇒ b Suppose that W ψ,ϕ is compact and that {w n } is a sequence in D such that |ϕ w n | → 1 as n → ∞.Since the sequences {f ϕ w n } and {g ϕ w n } are bounded in H ∞ and converge to 0 uniformly on compact subsets of D, by Lemma 3.1, it follows that

3.3
Eliminating ψ w , we obtain that Taking the limit as |ϕ w | → 1, we deduce that

Journal of Function Spaces and Applications
On the other hand, using 3.3 , we obtain that

3.7
Taking the limit as |ϕ w | → 1, we obtain that c ⇒ a Suppose that c holds.Let {f n } be a bounded sequence in H ∞ converging to 0 uniformly on compact subsets of D. Set C sup n∈N f n ∞ .Then, given ε > 0, there exists r ∈ 0, 1 such that for |ϕ w | > r, 3.9 Then, for w ∈ D, noting that 1

3.10
Journal of Function Spaces and Applications 9 Thus, for |ϕ w | > r, we have On the other hand, for |ϕ w | ≤ r, Thus, by the uniform convergence to 0 of f n and f n on compact sets, we see that 3.11 holds also in this case for n sufficiently large.Hence,

3.14
The left-hand side of 3.14 can be written as

3.23
On the other hand, if z / ∈ E N , then there exists n > N such that z ∈ Δ n , so, as shown in the proof of d implies c of Theorem 3.3, we have

3.24
as |z| → 1.Since ε is arbitrary, the result follows.c ⇒ a Let {f n } be a bounded sequence in H ∞ converging to 0 uniformly on compact subsets, and let C sup n∈N f n ∞ .We wish to show that W ψ,ϕ f n ∈ B log,0 and W ψ,ϕ f n B log → 0 as n → ∞.As shown in the proof of c implies a of Theorem 3.3, for z ∈ D and n ∈ N, as |z| → 1.Thus, W ψ,ϕ f n ∈ B log,0 .The convergence to 0 of W ψ,ϕ f n B log is proved as in the case of the operator mapping into B log .
From Theorems 2.2 and 3.4, we obtain the following result.In the special cases when ϕ is the identity, respectively, ψ is identically 1, we obtain the following results.Corollary 3.6.Let ψ be analytic on D. The following statements are equivalent: Corollary 3.7.Let ϕ be an analytic self map of D. Then the following statements are equivalent: Corollary 3.8.Let ϕ be an analytic self map of D. Then the following statements are equivalent: Corollary 3.9.Let ϕ be an analytic self map of D. Then the following statements are equivalent:

W ψ,ϕ from
We begin this section with two useful point evaluation estimates that will be needed to prove our results.

4.2
Fix 1 ≤ p < ∞ and a ∈ D. For z ∈ D, define the functions Then f a , g a ∈ H p and the norms f a H p and g a H p are bounded by constants only dependent of p.In addition, a straightforward calculation shows that f a a g a a 1

4.4
We use these two families of functions to characterize the bounded and the compact weighted composition operators from H p to B log .Theorem 4.3.Let 1 ≤ p < ∞, ψ analytic on D and let ϕ be an analytic self-map of D. Then the following conditions are equivalent:

4.5
Proof. a ⇒ b Assume that W ψ,ϕ : H p → B log is bounded.Then ψϕ ∈ B log and for each w ∈ D,

4.8
Moreover, Therefore, subtracting 4.7 from 4.9 and taking the modulus, we obtain 1 p ψ w ϕ w ϕ w Consequently, from 4.8 , we deduce that Taking the supremum over all w ∈ D, we see that x ψ,ϕ is finite.On the other hand, since ψϕ ∈ B log , if |ϕ w | ≤ r, then,

4.14
Taking the supremum over all w ∈ D, it follows that y ψ,ϕ is finite as well.c ⇒ a Suppose that x ψ,ϕ and y ψ,ϕ are finite.For arbitrary z in D and f ∈ H p , by Lemmas 4.1 and 4.2, we have

4.15
Taking the supremum over all z ∈ D and applying Lemma 4.1, we obtain that

4.16
The boundedness of the operator W ψ,ϕ : H p → B log follows by taking the supremum over all f ∈ H p .

Journal of Function Spaces and Applications
Theorem 4.4.Let 1 ≤ p < ∞, ψ analytic on D, ϕ an analytic self-map of D, and assume that W ψ,ϕ : H p → B log is bounded.Then the following conditions are equivalent: 4.17 c ⇒ a Suppose that c holds.Let {f n } be a bounded sequence in H p converging to 0 uniformly on compact subsets of D. Set C sup n∈N f n H p .Then, given ε > 0, there exists r ∈ 0, 1 such that On the other hand, for |ϕ w | ≤ r, by the uniform convergence to 0 of f n and f n on compact sets, we have

W ψ,ϕ from
In this section, we characterize the boundedness and the compactness of the weighted composition operators W ψ,ϕ : H p → B log,0 .Arguing as in the proof of Lemma 4.2 of 14 , we easily get the following two lemmas.The proof of the following theorem is a straightforward adaptation of the proof of Theorem 4.4 in 14 .We omit the details.

5.9
Taking the supremum over all f ∈ H p such that f H p ≤ 1, then letting |z| → Spaces and Applications Let H ∞ denote the space of all f ∈ H D for which f ∞ sup z∈D |f z | < ∞.The Bloch space B on the open unit disk D is the Banach space consisting of the analytic functions f on D such that f β sup z∈D 1 − |z| 2 f z < ∞. 1.3

4 . 6 for
some constant C, so A and B are finite.b ⇒ c Suppose that ψϕ ∈ B log , and the quantities A and B are finite.From 4.4 , for w ∈ D, we have ψ f ϕ w • ϕ w

Journal of Function Spaces and Applications 15 Fix r ∈ 0, 1 . 1 − |w| 2 ψ w ϕ w 1
If |ϕ w | > r, then from 4.11 we have Proof. a ⇒ b Suppose that W ψ,ϕ : H p → B log is compact.Let {w n } be a sequence in D such that lim n → ∞ |ϕ w n | 1. Observe that the sequences {f ϕ w n } and {g ϕ w n } are bounded in H p and converge to 0 uniformly on compact subsets of D. By Lemma 3.1, it follows that W ψ,ϕ f ϕ w n B log → 0 and W ψ,ϕ g ϕ w n B log → 0 as n → ∞, proving b .b ⇒ c Assume that the limits in b are 0. Using the inequality 4.10 , we obtain that 1 − |w| 2 ψ w ϕ w 2 ≤ p W ψ,ϕ f ϕ w B log W ψ,ϕ g ϕ w B log |ϕ w | > r.Therefore, again by Lemmas 4.1 and 4.2, and 4.20 , for |ϕ w | > r, we have 1
p into B log .Corollary 4.5.Let ϕ be an analytic self-map of D, and 1 ≤ p < ∞.The following statements are equivalent: