# Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

Authors: Flavia Colonna; Songxiao Li
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The logarithmic Bloch spaceℬlog 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 Hp (with 1≤p≤∞) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping Hp 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.


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## Body

## 1. Introduction

LetX 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 symbolsψandφ from X to Y is the operator Wψ,φ with range in Y defined by(1.1)Wψ,φf=MψCφf=ψ(f∘φ),forf∈X,
where Mψ denotes the multiplication operator with symbol ψ, and Cφ denotes the composition operator with symbol φ.LetH(𝔻) be the set of analytic functions on 𝔻={z∈ℂ:|z|<1}. For 0<p<∞ the Hardy spaceHp is the space consisting of all f∈H(𝔻) such that(1.2)‖f‖Hpp=sup⁡0<r<1∫02π|f(reiθ)|pdθ2π<∞.
Let H∞ denote the space of all f∈H(𝔻) for which ∥f∥∞=supz∈𝔻|f(z)|<∞.TheBloch spaceℬ on the open unit disk 𝔻 is the Banach space consisting of the analytic functions f on 𝔻 such that(1.3)‖f‖β=sup⁡z∈D(1-|z|2)|f′(z)|<∞.
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.Thelittle Bloch space, denoted by ℬ0, 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 ℬ0 is a closed separable subspace of ℬ. The interested reader is referred to [1] for more information on the Bloch space.Thelogarithmic Bloch spaceℬlog is defined as the set of functions f on 𝔻 such that(1.4)‖f‖=sup⁡z∈D(1-|z|2)log⁡21-|z|2|f′(z)|<∞.
It is a Banach space under the norm defined by ∥f∥ℬlog=|f(0)|+∥f∥. Clearly, if f∈ℬlog, then lim|z|→1(1-|z|2)|f′(z)|=0, so ℬlog is a subset of the little Bloch space.Thelittle logarithmic Bloch space, denoted by ℬlog,0, is defined as the subspace of ℬlog whose elements f satisfy the condition(1.5)lim⁡|z|→1(1-|z|2)log⁡21-|z||f′(z)|=0.The spaceℬlog arises in connection to the study of certain operators with symbol. Arazy [2] proved that the multiplication operator Mψ is bounded on the Bloch space if and only if ψ∈ℬlog∩H∞. In [3], Brown and Shields extended this result to the little Bloch space.The spaceℬlog also arises in the study of Hankel operators on the Bergman one space. The Bergman space A1 on 𝔻 is defined to be the set of analytic functions f on 𝔻 whose modulus is Lebesgue integrable over 𝔻.The Hankel operatorHf on A1 is defined as Hfg=(I-P)(f¯g), where I is the identity operator, and P is the standard Bergman projection from L1 into A1. In [4], Attele showed that Hf is bounded on A1 if and only if f∈ℬlog.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 ℬlog. He also introduced a class of Banach spaces Qlogp (p>0) closely related to ℬlog and studied the Taylor coefficients of the functions in ℬlog. In [7], Ye characterized the bounded and the compact weighted composition operators on the little logarithmic Bloch space ℬlog,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 spaceHp (with 1≤p≤∞) to the logarithmic Bloch space ℬlog as well as to its subspace ℬlog,0. The paper consists of five sections. Specifically, in Section 2, we consider the bounded weighted composition operators mapping H∞ into ℬlog and ℬlog,0. In particular, we show that(1.6)‖Wψ,φ‖~sup⁡n∈N∪{0}‖ψφn‖Blog⁡,
where the notation A~B stands for c1A≤B≤c2A, for some positive constants c1 and c2. In Section 3, we look at the issue of compactness of such operators.In Section4, we characterize the bounded and the compact weighted composition operators mapping Hp into ℬlog in the case when 1≤p<∞. Finally, in Section 5, we study the operators mapping Hp into ℬlog,0.


## 2. Boundedness ofWψ,φ from H∞ into ℬlog and ℬlog,0

In the following theorem, we give two characterizations of boundedness when the operator mapsH∞ into ℬlog.Theorem 2.1.
Letψ be an analytic function on  𝔻, and let  φ be an analytic self-map of  𝔻. The following statements are equivalent. (a)
The operatorWψ,φ:H∞→ℬlog is bounded.(b)
supn∈ℕ∪{0}∥ψφn∥ℬlog<∞.(c)
ψ∈ℬlog and σψ,φ=supz∈𝔻((1-|z|2)|ψ(z)φ′(z)|/1-|φ(z)|2)log(2/(1-|z|2))<∞.Proof.
(a)⇒ (b). For n∈ℕ, the function pn(z)=zn is bounded and ∥pn∥∞=1. Therefore, if Wψ,φ is bounded, then ∥ψφn∥ℬlog≤∥Wψ,φ∥.
(b)⇒ (c) Let C be an upper bound for ∥ψφn∥ℬlog, n≥0. Taking n=0, we deduce that ∥ψ∥ℬlog≤C, so ψ∈ℬlog.
ForN∈ℕ and n≥2, define the sets
(2.1)EN={z∈D:|φ(z)|≤1-1N},Δn={z∈D:1-1n-1≤|φ(z)|≤1-1n}.
Fix an integerN>2, and z∈𝔻. For z∈EN, by the product rule, we have
(2.2)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2≤(1-|z|2)(|(ψφ)′(z)|+|ψ′(z)φ(z)|)1-(1-1/N)2log⁡21-|z|2≤11-(1-1/N)2(‖ψφ‖+‖ψ‖)≤2C1-(1-1/N)2.
In the proof of Theorem 2 of [10], it was shown that
(2.3)inf⁡z∈Δnn|φ(z)|n-1(1-|φ(z)|)≥1e.
For|φ(z)|>1-1/N, there exists n>N such that z∈Δn. So
(2.4)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2≤(1-|z|2)|ψ(z)nφ(z)n-1φ′(z)|(1-|φ(z)|)n|φ(z)|n-1log⁡21-|z|2≤e[(1-|z|2)log⁡21-|z|2|(ψφn)′(z)|+(1-|z|2)log⁡21-|z|2|ψ′(z)φ(z)n|]≤2eC.
From (2.2) and (2.4), we deduce that σψ,φ is finite.
(c)⇒ (a) Let f∈H∞ with ∥f∥∞≤1 and pick z∈𝔻. Then
(2.5)(1-|z|2)log⁡21-|z|2|ψ′(z)f(φ(z))|≤‖ψ‖,
and, since ∥f∥β≤∥f∥∞≤1,
(2.6)(1-|z|2)log⁡21-|z|2|ψ(z)f′(φ(z))φ′(z)|=(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2(1-|φ(z)|2)|f′(φ(z))|≤(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2.
Thus, by (2.5) and (2.6), we deduce that ∥Wψ,φf∥ℬlog≤∥ψ∥ℬlog+σψ,φ, completing the proof.We next turn our attention to the weighted composition operators mapping into the little logarithmic Bloch space.Theorem 2.2.
Letψ be an analytic function on  𝔻, and let  φ be an analytic self-map of  𝔻. The following statements are equivalent. (a)
The operatorWψ,φ:H∞→ℬlog,0 is bounded.(b)
For each integern≥0,ψφn∈ℬlog,0 and supn∈ℕ∥ψφn∥ℬlog<∞.(c)
ψ∈ℬlog,0 and lim|z|→1((1-|z|2)|ψ(z)φ′(z)|/(1-|φ(z)|2))log(2/(1-|z|2))=0.Proof.
(a)⇒ (b) is proved as in the case of the operator mapping into ℬlog.
(b)⇒ (c) Suppose that (b) holds. If EN=𝔻 for some integer N>1, then for all z∈𝔻, we have
(2.7)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2≤11-(1-1/N)2(1-|z|2)|ψ(z)φ′(z)|log⁡21-|z|2≤11-(1-1/N)2(1-|z|2)(|(ψφ)′(z)|+|ψ′(z)φ(z)|)log⁡21-|z|2≤11-(1-1/N)2(1-|z|2)|(ψφ)′(z)|log⁡21-|z|2+11-(1-1/N)2(1-|z|2)|ψ′(z)|log⁡21-|z|2⟶0,as|z|⟶1.
If EN is properly contained in 𝔻, then for z∈𝔻∖EN, arguing as in the proof of (b) ⇒ (c) in Theorem 2.1, there exists k≥N such that z∈Δk, so that
(2.8)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2≤(1-|z|2)|ψ(z)kφ(z)k-1φ′(z)|(1-|φ(z)|)k|φ(z)|k-1log⁡21-|z|2≤e[(1-|z|2)log⁡21-|z|2|(ψφk)′(z)|+(1-|z|2)log⁡21-|z|2|ψ′(z)|]=I+II,
where I=e(1-|z|2)log(2/(1-|z|2))|(ψφk)′(z)| and II=e(1-|z|2)log(2/(1-|z|2))|ψ′(z)|.
By the assumption ofψφn∈ℬlog,0, we have
(2.9)I≤esup⁡n≥k(1-|z|2)log⁡21-|z|2|(ψφn)′(z)|⟶0
as |z|→1. On the other hand, since ψ∈ℬlog,0, II→0 as |z|→1. Therefore,
(2.10)lim⁡|z|→1(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2=0.
(c)⇒ (a) Assume that (c) holds. To prove that Wψ,φ is bounded, it suffices to show that Wψ,φf∈ℬlog,0 for each f∈H∞, since the boundedness of the operator can be shown as in the proof of Theorem 2.1. Since ψ∈ℬlog,0, for f∈H∞ and z∈𝔻, we have
(2.11)(1-|z|2)|ψ′(z)f(φ(z))|log⁡21-|z|2≤(1-|z|2)|ψ′(z)|log⁡21-|z|2‖f‖∞⟶0,
as |z|→1. On the other hand, by (2.6),
(2.12)(1-|z|2)|ψ(z)f′(φ(z))φ′(z)|log⁡21-|z|2≤(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2‖f‖∞⟶0,
as |z|→1. Hence,
(2.13)(1-|z|2)|(Wψ,φf)′(z)|log⁡21-|z|2⟶0
as |z|→1, completing the proof.In Section3, we shall prove that all bounded weighted composition operators from H∞ into ℬlog,0 are compact.


## 3. Compactness ofWψ,φ from H∞ into ℬlog and ℬlog,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.
Letψ be analytic on  𝔻,  φ an analytic self-map of  𝔻,  1≤p≤∞. The operator  Wψ,φ:Hp→ℬlog is compact if and only if for any bounded sequence  {fn}n∈ℕ in  Hp which converges to zero uniformly on compact subsets of  𝔻, we have  ∥Wψ,φfn∥ℬlog→0 as  n→∞.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 inℬlog,0 is compact if and only if it is bounded and satisfies the following:
(3.1)lim⁡|z|→1sup⁡f∈K(1-|z|2)|f′(z)|log⁡21-|z|2=0.We now introduce two one-parameter families of functions which will be used to characterize the compactness of the operators under consideration.Fixa∈𝔻 and, for z∈𝔻, define(3.2)fa(z)=(1-|a|21-a¯z)1/2,ga(z)=1-|a|21-a¯z.Theorem 3.3.
Letψ be analytic on  𝔻,  φ an analytic self-map of  𝔻, and assume that  Wψ,φ:H∞→ℬlog is bounded. Then the following conditions are equivalent:  (a)
Wψ,φ:H∞→ℬlog is compact.(b)
lim|φ(w)|→1∥Wψ,φfφ(w)∥ℬlog=0 and lim|φ(w)|→1∥Wψ,φgφ(w)∥ℬlog=0.(c)
lim|φ(w)|→1((1-|w|2)|ψ(w)φ′(w)|/1-|φ(w)|2)log(2/(1-|w|2))=0 and lim|φ(w)|→1(1-|w|2)|ψ′(w)|log(2/(1-|w|2))=0.(d)
limn→∞∥ψφn∥ℬlog=0 and lim|φ(w)|→1(1-|w|2)|ψ′(w)|log(2/(1-|w|2))=0.Proof.
We begin by showing that (a), (b), and (c) are equivalent.
(a)⇒(b) Suppose that Wψ,φ is compact and that {wn} is a sequence in 𝔻 such that |φ(wn)|→1 as n→∞. Since the sequences {fφ(wn)} and {gφ(wn)} are bounded in H∞ and converge to 0 uniformly on compact subsets of 𝔻, by Lemma 3.1, it follows that ∥Wψ,φfφ(wn)∥ℬlog→0 and ∥Wψ,φgφ(wn)∥ℬlog→0 as n→∞.
(b)⇒(c) Assume that (b) holds. Fix w∈𝔻. A straightforward calculation shows that
(3.3)(ψ(fφ(w)∘φ))′(w)=ψ′(w)+ψ(w)φ(w)¯φ′(w)2(1-|φ(w)|2),(ψ(gφ(w)∘φ))′(w)=ψ′(w)+ψ(w)φ(w)¯φ′(w)1-|φ(w)|2.
Eliminating ψ′(w), we obtain that
(3.4)ψ(w)φ(w)¯φ′(w)2(1-|φ(w)|2)=(ψ(gφ(w)∘φ))′(w)-(ψ(fφ(w)∘φ))′(w).
Thus, for |φ(w)|>r∈(0,1),
(3.5)(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2log⁡21-|w|2≤2r(‖Wψ,φfφ(w)‖Blog⁡+‖Wψ,φgφ(w)‖Blog⁡).
Taking the limit as |φ(w)|→1, we deduce that
(3.6)(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2log⁡21-|w|2⟶0.
On the other hand, using (3.3), we obtain that
(3.7)(1-|w|2)|ψ′(w)|log⁡21-|w|2≤‖Wψ,φgφ(w)‖Blog⁡+(1-|w|2)|ψ(w)φ(w)φ′(w)|1-|φ(w)|2log⁡21-|w|2.
Taking the limit as |φ(w)|→1, we obtain that
(3.8)(1-|w|2)|ψ′(w)|log⁡21-|w|2⟶0,
proving (c).
(c)⇒(a) Suppose that (c) holds. Let {fn} be a bounded sequence in H∞ converging to 0 uniformly on compact subsets of 𝔻. Set C=supn∈ℕ∥fn∥∞. Then, given ε>0, there exists r∈(0,1) such that for |φ(w)|>r,
(3.9)(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2log⁡21-|w|2<ε2C,(1-|w|2)|ψ′(w)|log⁡21-|w|2<ε2C.
Then, for w∈𝔻, noting that (1-|φ(w)|2)|fn′(φ(w))|≤∥fn∥∞≤C, we obtain that
(3.10)(1-|w|2)|(ψ(fn∘φ))′(w)|log⁡21-|w|2≤(1-|w|2)(|ψ′(w)fn(φ(w))|+|ψ(w)fn′(φ(w))φ′(w)|)log⁡21-|w|2≤(1-|w|2)|ψ′(w)|‖fn‖∞log⁡21-|w|2+(1-|w|2)(1-|φ(w)|2)|ψ(w)fn′(φ(w))φ′(w)|1-|φ(w)|2log⁡21-|w|2≤C[(1-|w|2)|ψ′(w)|+(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2]log⁡21-|w|2.
Thus, for |φ(w)|>r, we have
(3.11)(1-|w|2)|(ψ(fn∘φ))′(w)|log⁡21-|w|2<ε.
On the other hand, for |φ(w)|≤r,
(3.12)(1-|w|2)|(ψ(fn∘φ))′(w)|log⁡21-|w|2≤‖ψ‖|fn(φ(w))|+σψ,φ|fn′(φ(w))|.
Thus, by the uniform convergence to 0 of fn and fn′ on compact sets, we see that (3.11) holds also in this case for n sufficiently large. Hence, ∥ψ(fn∘φ)∥≤ε for n sufficiently large. Since |ψ(0)fn(φ(0))|→0, we deduce that ∥ψ(fn∘φ)∥ℬlog→0 as n→∞. Therefore, Wψ,φ is compact.
(a)⇒ (d) Suppose that Wψ,φ is compact. Since the sequence {pn} defined by pn(z)=zn, z∈𝔻 is bounded in H∞ and converges to 0 uniformly on compact subsets, by Lemma 3.1, it follows that ∥ψφn∥ℬlog=∥Wψ,φpn∥ℬlog→0 as n→∞. The second condition in (d) follows from the equivalence of (a) and (c).
(d)⇒ (c) Fix ε>0 and choose N>2 such that ∥ψφn∥ℬlog<ε/2 for all n≥N and (1-|w|2)|ψ′(w)|log(2/(1-w|2))<ε/2 for |φ(w)|>1-1/N.
For|φ(w)|>1-1/N, there exists n>N such that z∈Δn. Using the product rule, we may write the following
(3.13)ψ(w)nφ(w)n-1φ′(w)=(ψφn)′(w)-ψ′(w)φ(w)n,
so that
(3.14)(1-|w|2)|ψ(w)nφ(w)n-1φ′(w)|log⁡21-|w|2≤‖ψφn‖Blog⁡+(1-|w|2)|ψ′(w)φ(w)n|log⁡21-|w|2.
The left-hand side of (3.14) can be written as
(3.15)(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2(1-|φ(w)|2)n|φ(w)|n-1log⁡21-|w|2,
which, by (2.3), is bounded below by
(3.16)1e(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2log⁡21-|w|2.
Thus, from (3.14), we deduce that
(3.17)1e(1-|w|2)|ψ(w)φ′(w)|1-|φ(w)|2log⁡21-|w|2≤‖ψφn‖Blog⁡+(1-|w|2)|ψ′(w)|log⁡21-|w|2<ε,
proving (c). The equivalence of statements (a)–(d) is now established.Next, we characterize the compact weighted composition operators fromH∞ into ℬlog,0.Theorem 3.4.
Letψ be an analytic function on  𝔻, and let  φ be an analytic self-map of  𝔻. The following statements are equivalent. (a)
The operatorWψ,φ:H∞→ℬlog,0 is compact.(b)
For each integern≥0, ψφn∈ℬlog,0 and limn→∞∥ψφn∥ℬlog=0.(c)
ψ∈ℬlog,0 and(3.18)lim⁡|z|→1(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2=0.Proof.
(a)⇒ (b) is immediate.
(b)⇒ (c) It suffices to show that if (b) holds, then
(3.19)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2⟶0
as |z|→1. Fix ε>0 and let N>2 be an integer such that ∥ψφn∥ℬlog<ε/e for all n≥N. Observe that, since for z∈𝔻,
(3.20)|ψ(z)φ′(z)|≤|(ψφ)′(z)|+|ψ′(z)φ(z)|≤|(ψφ)′(z)|+|ψ′(z)|,
and, by assumption, the functions ψφ and ψ are in ℬlog,0, we have
(3.21)lim⁡|z|→1(1-|z|2)|ψ(z)φ′(z)|log⁡21-|z|2=0.
Therefore, there is r∈(0,1), such that for |z|>r,
(3.22)(1-|z|2)|ψ(z)φ′(z)|log⁡21-|z|2<(2N-1)εN2.
Thus, if z∈EN, and |z|>r, then
(3.23)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2≤N22N-1(1-|z|2)|ψ(z)φ′(z)|log⁡21-|z|2<ε.
On the other hand, if z∉EN, then there exists n>N such that z∈Δn, so, as shown in the proof of (d) implies (c) of Theorem 3.3, we have
(3.24)(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2≤e[‖ψφn‖Blog⁡+(1-|z|2)|ψ′(z)|log⁡21-|z|2]<ε+e(1-|z|2)|ψ′(z)|log⁡21-|z|2⟶ε,
as |z|→1. Since ε is arbitrary, the result follows.
(c)⇒ (a) Let {fn} be a bounded sequence in H∞ converging to 0 uniformly on compact subsets, and let C=supn∈ℕ∥fn∥∞. We wish to show that Wψ,φfn∈ℬlog,0 and ∥Wψ,φfn∥ℬlog→0 as n→∞. As shown in the proof of (c) implies (a) of Theorem 3.3, for z∈𝔻 and n∈ℕ,
(3.25)(1-|z|2)|(Wψ,φfn)′(z)|log⁡21-|z|2≤C(1-|z|2)|ψ′(z)|log⁡21-|z|2+C(1-|z|2)|ψ(z)φ′(z)|1-|φ(z)|2log⁡21-|z|2⟶0,
as |z|→1. Thus, Wψ,φfn∈ℬlog,0. The convergence to 0 of ∥Wψ,φfn∥ℬlog is proved as in the case of the operator mapping into ℬlog.From Theorems2.2 and 3.4, we obtain the following result.Corollary 3.5.
Letψ be an analytic function on  𝔻, and let  φ be an analytic self-map of  𝔻. The following statements are equivalent. (a)
The operatorWψ,φ:H∞→ℬlog,0 is bounded.(b)
The operatorWψ,φ:H∞→ℬlog,0 is compact.(c)
For each integern≥0, ψφn∈ℬlog,0 and limn→∞∥ψφn∥ℬlog=0.(d)
ψ∈ℬlog,0 and lim|z|→1((1-|z|2)|ψ(z)φ′(z)|/(1-|φ(z)|2))log(2/(1-|z|2))=0.In the special cases whenφ is the identity, respectively, ψ is identically 1, we obtain the following results.Corollary 3.6.
Letψ be analytic on  𝔻. The following statements are equivalent: (a)
Mψ:H∞→ℬlog is bounded,(b)
Mψ:H∞→ℬlog,0 is bounded,(c)
ψ is identically 0.Corollary 3.7.
Letφ be an analytic self map of  𝔻. Then the following statements are equivalent: (a)
Cφ:H∞→ℬlog is bounded,(b)
supn∈ℕ∥φn∥ℬlog<∞,(c)
supz∈𝔻((1-|z|2)|φ′(z)|/(1-|φ(z)|2))log(2/(1-|z|2))<∞.Corollary 3.8.
Letφ be an analytic self map of  𝔻. Then the following statements are equivalent: (a)
Cφ:H∞→ℬlog is compact,(b)
limn→∞∥φn∥ℬlog=0,(c)
lim|φ(z)|→1((1-|z|2)|φ′(z)|/(1-|φ(z)|2))log(2/(1-|z|2))=0.Corollary 3.9.
Letφ be an analytic self map of  𝔻. Then the following statements are equivalent: (a)
Cφ:H∞→ℬlog,0 is bounded,(b)
Cφ:H∞→ℬlog,0 is compact,(c)
φ∈ℬlog,0 and limn→∞∥φn∥ℬlog=0,(d)
lim|z|→1((1-|z|2)|φ′(z)|/(1-|φ(z)|2))log(2/(1-|z|2))=0.


## 4.Wψ,φ from Hp(1≤p<∞) into ℬlog

We begin this section with two useful point evaluation estimates that will be needed to prove our results.Lemma 4.1 (See [11]).
Let0<p<∞. Then for any  f∈Hp,  z∈𝔻,
(4.1)|f(z)|≤‖f‖Hp(1-|z|2)1/p.Lemma 4.2 (See [13]).
Let0<p<∞. Then for any  f∈Hp,  z∈𝔻,
(4.2)|f′(z)|≤C∥f∥Hp(1-|z|2)1+1/p.Fix1≤p<∞ and a∈𝔻. For z∈𝔻, define the functions(4.3)fa(z)=(1-|a|2)2-1/p(1-a¯z)2,ga(z)=(1-|a|2)2(1-a¯z)2+1/p.
Then fa,ga∈Hp and the norms ∥fa∥Hp and ∥ga∥Hp are bounded by constants only dependent of p. In addition, a straightforward calculation shows that(4.4)fa(a)=ga(a)=1(1-|a|2)1/p,fa′(a)=2a¯(1-|a|2)1+1/p,ga′(a)=(2+1/p)a¯(1-|a|2)1+1/p.We use these two families of functions to characterize the bounded and the compact weighted composition operators fromHp to ℬlog.Theorem 4.3.
Let1≤p<∞,  ψ analytic on  𝔻 and let  φ be an analytic self-map of  𝔻. Then the following conditions are equivalent: (a)
Wψ,φ:Hp→ℬlog is bounded,(b)
ψφ∈ℬlog, A=supw∈𝔻∥Wψ,φfφ(w)∥ℬlog<∞ and  B=supw∈𝔻∥Wψ,φgφ(w)∥ℬlog<∞,(c)
(4.5)xψ,φ=sup⁡z∈D(1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2<∞,yψ,φ=sup⁡z∈D(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2<∞.Proof.
(a)⇒ (b) Assume that Wψ,φ:Hp→ℬlog is bounded. Then ψφ∈ℬlog and for each w∈𝔻,
(4.6)‖Wψ,φfφ(w)‖Blog⁡≤‖Wψ,φ‖‖fφ(w)‖Hp≤C‖Wψ,φ‖,‖Wψ,φgφ(w)‖Blog⁡≤‖Wψ,φ‖‖gφ(w)‖Hp≤C‖Wψ,φ‖,
for some constant C, so A and B are finite.
(b)⇒ (c) Suppose that ψφ∈ℬlog, and the quantities A and B are finite. From (4.4), for w∈𝔻, we have
(4.7)(ψ(fφ(w)∘φ))′(w)=ψ′(w)(1-|φ(w)|2)1/p+2ψ(w)φ′(w)φ(w)¯(1-|φ(w)|2)1+1/p,
whence
(4.8)(1-|w|2)|ψ′(w)|(1-|φ(w)|2)1/plog⁡21-|w|2≤(1-|w|2)log⁡21-|w|2|(ψ(fφ(w)∘φ))′(w)|+2(1-|w|2)|ψ(w)φ′(w)φ(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2≤‖Wψ,φfφ(w)‖Blog⁡+2(1-|w|2)|ψ(w)φ′(w)φ(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2≤A+2(1-|w|2)|ψ(w)φ′(w)φ(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2.
Moreover,
(4.9)(ψ(gφ(w)∘φ))′(w)=ψ′(w)(1-|φ(w)|2)1/p+(2+1/p)ψ(w)φ′(w)φ(w)¯(1-|φ(w)|2)1+1/p.
Therefore, subtracting (4.7) from (4.9) and taking the modulus, we obtain
(4.10)1p|ψ(w)φ′(w)φ(w)|(1-|φ(w)|2)1+1/p≤|(ψ(fφ(w)∘φ))′(w)|+|(ψ(gφ(w)∘φ))′(w)|,
which yields that
(4.11)(1-|w|2)|ψ(w)φ′(w)φ(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2≤p(A+B).
Consequently, from (4.8), we deduce that
(4.12)(1-|w|2)|ψ′(w)|(1-|φ(w)|2)1/plog⁡21-|w|2≤(1+2p)A+2pB.
Taking the supremum over all w∈𝔻, we see that xψ,φ is finite.
Fixr∈(0,1). If |φ(w)|>r, then from (4.11) we have
(4.13)(1-|w|2)|ψ(w)φ′(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2<p(A+B)r.
On the other hand, since ψφ∈ℬlog, if |φ(w)|≤r, then,
(4.14)(1-|w|2)|ψ(w)φ′(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2≤(1-|w|2)|(ψφ)′(w)|(1-r2)1+1/plog⁡21-|w|2+(1-|w|2)|ψ′(w)φ(w)|(1-r2)(1-|φ(w)|2)1/plog⁡21-|w|2≤‖ψφ‖(1-r2)1+1/p+xψ,φ1-r2<∞.
Taking the supremum over all w∈𝔻, it follows that yψ,φ is finite as well.
(c)⇒ (a) Suppose that xψ,φ and yψ,φ are finite. For arbitrary z in 𝔻 and f∈Hp, by Lemmas 4.1 and 4.2, we have
(4.15)(1-|z|2)log⁡21-|z|2|(Wψ,φf)′(z)|≤(1-|z|2)log⁡21-|z|2(|ψ′(z)||f(φ(z))|+|f′(φ(z))||ψ(z)φ′(z)|)≤((1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2+C(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2)‖f‖Hp≤(xψ,φ+Cyψ,φ)‖f‖Hp.
Taking the supremum over allz∈𝔻 and applying Lemma 4.1, we obtain that
(4.16)‖Wψ,φf‖Blog⁡=|ψ(0)f(φ(0))|+‖Wψ,φf‖≤(|ψ(0)|(1-|φ(0)|2)1/p+xψ,φ+Cyψ,φ)‖f‖Hp.
The boundedness of the operator Wψ,φ:Hp→ℬlog follows by taking the supremum over all f∈Hp.Theorem 4.4.
Let1≤p<∞,  ψ analytic on  𝔻,  φ an analytic self-map of  𝔻, and assume that  Wψ,φ:Hp→ℬlog is bounded. Then the following conditions are equivalent: (a)
Wψ,φ:Hp→ℬlog is compact,(b)
lim|φ(w)|→1∥Wψ,φfφ(w)∥ℬlog=0 and lim|φ(w)|→1∥Wψ,φgφ(w)∥ℬlog=0,(c)
(4.17)lim⁡|φ(w)|→1(1-|w|2)|ψ(w)φ′(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2=0,lim⁡|φ(w)|→1(1-|w|2)|ψ′(w)|(1-|φ(w)|2)1/plog⁡21-|w|2=0.Proof.
(a)⇒(b) Suppose that Wψ,φ:Hp→ℬlog is compact. Let {wn} be a sequence in 𝔻 such that limn→∞|φ(wn)|=1. Observe that the sequences {fφ(wn)} and {gφ(wn)} are bounded in Hp and converge to 0 uniformly on compact subsets of 𝔻. By Lemma 3.1, it follows that ∥Wψ,φfφ(wn)∥ℬlog→0 and ∥Wψ,φgφ(wn)∥ℬlog→0 as n→∞, proving (b).
(b)⇒(c) Assume that the limits in (b) are 0. Using the inequality (4.10), we obtain that
(4.18)(1-|w|2)|ψ(w)φ′(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2≤p(‖Wψ,φfφ(w)‖Blog⁡+‖Wψ,φgφ(w)‖Blog⁡)|φ(w)|⟶0
as |φ(w)|→1. Moreover, using (4.8), we deduce that
(4.19)(1-|w|2)|ψ′(w)|(1-|φ(w)|2)1/plog⁡21-|w|2⟶0
as |φ(w)|→1.
(c)⇒(a) Suppose that (c) holds. Let {fn} be a bounded sequence in Hp converging to 0 uniformly on compact subsets of 𝔻. Set C=supn∈ℕ∥fn∥Hp. Then, given ε>0, there exists r∈(0,1) such that
(4.20)(1-|w|2)|ψ(w)φ′(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2<ε2C,(1-|w|2)|ψ′(w)|(1-|φ(w)|2)1/plog⁡21-|w|2<ε2C,
for |φ(w)|>r. Therefore, again by Lemmas 4.1 and 4.2, and (4.20), for |φ(w)|>r, we have
(4.21)(1-|w|2)log⁡21-|w|2|(ψ(fn∘φ))′(w)|≤(1-|w|2)log⁡21-|w|2(|ψ′(w)fn(φ(w))|+|ψ(w)fn′(φ(w))φ′(w)|)≤‖fn‖Hp((1-|w|2)|ψ′(w)|(1-|φ(w)|2)1/plog⁡21-|w|2+(1-|w|2)|ψ(w)φ′(w)|(1-|φ(w)|2)1+1/plog⁡21-|w|2)<ε.
On the other hand, for |φ(w)|≤r, by the uniform convergence to 0 of fn and fn′ on compact sets, we have
(4.22)(1-|w|2)log⁡21-|w|2|(ψ(fn∘φ))′(w)|≤xψ,φ|fn(φ(w))|+yψ,φ|f′(φ(w))|⟶0,
as n→∞. Since |ψ(0)fn(φ(0))|→0, we conclude that ∥Wψ,φfn∥ℬlog→0 as n→∞. Consequently, by Lemma 3.1, the operator Wψ,φ is compact.As a consequence of Theorems4.3 and 4.4, noting that for w∈𝔻,(4.23)(Cφgφ(w))′(w)=12(2+1p)(Cφfφ(w))′(w)=(2+1/p)φ′(w)φ(w)¯(1-|φ(w)|2)1+1/p,
we obtain the following characterizations of the bounded and the compact composition operators from Hp into ℬlog.Corollary 4.5.
Letφ be an analytic self-map of  𝔻, and  1≤p<∞. The following statements are equivalent: (a)
Cφ:Hp→ℬlog is bounded,(b)
supw∈𝔻∥Cφfφ(w)∥ℬlog<∞,(c)
supw∈𝔻∥Cφgφ(w)∥ℬlog<∞,(d)
supw∈𝔻((1-|w|2)|φ′(w)|/(1-|φ(w)|2)1+1/p)log(2/(1-|w|2))<∞.Corollary 4.6.
Letφ be an analytic self-map of  𝔻, and  1≤p<∞. If  Cφ:Hp→ℬlog is bounded, then the following statements are equivalent: (a)
Cφ:Hp→ℬlog is compact,(b)
lim|φ(w)|→1∥Cφfφ(w)∥ℬlog=0,(c)
lim|φ(w)|→1∥Cφgφ(w)∥ℬlog=0,(d)
lim|φ(w)|→1((1-|w|2)|φ′(w)|/(1-|φ(w)|2)1+1/p)log(2/(1-|w|2))=0.


## 5.Wψ,φ from Hp(1≤p<∞) into ℬlog,0

In this section, we characterize the boundedness and the compactness of the weighted composition operatorsWψ,φ:Hp→ℬlog,0. Arguing as in the proof of Lemma 4.2 of [14], we easily get the following two lemmas.Lemma 5.1.
Suppose thatφ is an analytic self-map of the unit disk,  ψ analytic on  𝔻, and  1≤p<∞. Then,
(5.1)lim⁡|z|→1(1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2=0
if and only if ψ∈ℬlog,0 and
(5.2)lim⁡|φ(z)|→1(1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2=0.Lemma 5.2.
Suppose thatφ is an analytic self-map of the unit disk,  ψ analytic on  𝔻, and  1≤p<∞. Then,
(5.3)lim⁡|z|→1(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2=0
if and only if lim|z|→1(1-|z|)2log(2/(1-|z|2))|ψ(z)φ′(z)|=0 and
(5.4)lim⁡|φ(z)|→1(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2=0.The proof of the following theorem is a straightforward adaptation of the proof of Theorem  4.4 in [14]. We omit the details.Theorem 5.3.
Letφ be an analytic self-map of the unit disk,  ψ∈H(𝔻), and  1≤p<∞. Then, the operator  Wψ,φ:Hp→ℬlog,0 is bounded if and only if  Wψ,φ:Hp→ℬlog is bounded,  ψ∈ℬlog,0, and
(5.5)lim⁡|z|→1(1-|z|2)log⁡21-|z|2|ψ(z)φ′(z)|=0.We are now ready to prove the main result of this section.Theorem 5.4.
Suppose thatφ is an analytic self-map of the unit disk,  ψ∈H(𝔻), and  1≤p<∞. Then,  Wψ,φ:Hp→ℬlog,0 is compact if and only if
(5.6)lim⁡|z|→1(1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2=0,lim⁡|z|→1(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2=0.Proof.
First, we assume thatWψ,φ:Hp→ℬlog,0 is compact, and hence bounded. By Theorem 5.3, it follows that ψ∈ℬlog,0 and
(5.7)lim⁡|z|→1(1-|z|)2log⁡21-|z|2|ψ(z)φ′(z)|=0.
Since by assumption Wψ,φ:Hp→ℬlog is compact, from Theorem 4.4, we know that
(5.8)lim⁡|φ(z)|→1(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2=0,lim⁡|φ(z)|→1(1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2=0.
Applying Lemmas 5.1 and 5.2 yields the desired result.
Conversely, suppose that conditions (5.6) hold. From the proof of Theorem 4.3, we have that
(5.9)(1-|z|2)log⁡21-|z|2|(Wψ,φf)′(z)|≤((1-|z|2)|ψ′(z)|(1-|φ(z)|2)1/plog⁡21-|z|2+C(1-|z|2)|ψ(z)φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2)‖f‖Hp.
Taking the supremum over all f∈Hp such that ∥f∥Hp≤1, then letting |z|→1, we obtain that
(5.10)lim⁡|z|→1sup⁡∥f∥Hp≤1(1-|z|2)log⁡21-|z|2|(Wψ,φ(f))′(z)|=0,
from which, by Lemma 3.2, we deduce that the operator Wψ,φ:Hp→ℬlog,0 is compact.From Theorems5.3 and 5.4, we obtain the following corollary.Corollary 5.5.
Suppose thatφ is an analytic self-map of the unit disk and  1≤p<∞. Then, the following statements hold. (i)
Cφ:Hp→ℬlog,0 is bounded if and only if Cφ:Hp→ℬlog is bounded and φ∈ℬlog,0.(ii)
Cφ:Hp→ℬlog,0 is compact if and only if
(5.11)lim⁡|z|→1(1-|z|2)|φ′(z)|(1-|φ(z)|2)1+1/plog⁡21-|z|2=0.



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*Source: 454820-2012-05-16.xml*