Weighted composition operators from the Besov spaces into the weighted-type space Hμ∞

J. Math. Anal. Appl. 402 (2013) 594–611

Contents lists available at SciVerse ScienceDirect

Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa

Weighted composition operators from the Besov spaces into the weighted-type space Hµ∞ Flavia Colonna a , Maria Tjani b,∗ a

George Mason University, Department of Mathematical Sciences, Fairfax, VA 22030, USA

b

University of Arkansas, Department of Mathematical Sciences, Fayetteville, AR 72701, USA

article

abstract

info

Let µ denote a positive continuous function on the open unit disk D. In this work, we characterize the bounded weighted composition operators from the analytic Besov spaces Bp (1 ≤ p < ∞) into the weighted-type space Hµ∞ consisting of the analytic functions on D such that supz ∈D µ(z )|f (z )| < ∞ and determine their operator norms. We also determine the essential norm of the bounded weighted composition operators acting on the Dirichlet space and obtain explicit estimates when p = 1. In the general case when the operator maps Bp into Hµ∞ , we derive an approximation of the essential norm that yields a characterization of the compact weighted composition operators. Finally, we derive characterizations of the bounded and the compact weighted composition operators from the spaces of antiderivatives of functions in Bp and BMOA into the α -Bloch spaces. © 2013 Elsevier Inc. All rights reserved.

Article history: Received 5 July 2012 Available online 1 February 2013 Submitted by Richard M. Timoney Keywords: Weighted composition operator Besov space Dirichlet space Weighted-type space BMOA Hardy space Bloch space

1. Introduction Let D be the open unit disk in C and denote by H (D) the space of analytic functions on D. Let ψ, ϕ ∈ H (D) be fixed analytic functions on D with ϕ(D) ⊆ D. The linear operator Wψ,ϕ defined by Wψ,ϕ f = ψ · (f ◦ ϕ),

f ∈ H (D),

is called the weighted composition operator with symbols ψ and ϕ . For p ∈ (1, ∞), the analytic Besov space is the set Bp of all f ∈ H (D) for which bpp (f ) =



|f ′ (z )|p (1 − |z |2 )p−2 dA(z ) < ∞, D

where dA is the normalized area measure on D. The correspondence f → ∥f ∥Bp = |f (0)| + bp (f ) defines a norm which yields a Banach space structure on Bp . In particular, B2 is the classical Dirichlet space D equipped with an equivalent norm. In this paper, when dealing with D , instead of the Dirichlet norm



∥f ∥ = |f (0)| + 2

1/2 1/2  ∞  2 2 |f (z )| dA(z ) = |a0 | + n|a| ,



′

2

D

n =1

n for f (z ) = n=0 an z , we shall use the Besov norm ∥f ∥B2 , which we shall denote by ∥ · ∥D . When a function f in D fixes the origin, the Dirichlet norm and the Besov norm of f coincide.

∞

∗

Corresponding author. E-mail addresses: [email protected] (F. Colonna), [email protected] (M. Tjani).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2013.01.037

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

595

The spaces Bp are Möbius invariant in the sense that the seminorm bp is preserved under composition of automorphisms of D. Note that by Theorem 9 in [19], the functions in Bp satisfy the following Lipschitz-type condition: there is a constant C > 0 only dependent on p such that for all f ∈ Bp ,

|f (z ) − f (w)| ≤ C ∥f ∥Bp ρ(z , w)1−1/p ,

for z , w ∈ D,

(1)

where ρ(z , w) denotes the hyperbolic distance between z and w . In particular,

|f (z ) − f (0)| ≤ C ∥f ∥Bp



1 2

1 + |z |

log

1−1/p

1 − |z |

for z ∈ D.

(2)

The Bloch space B , defined as the Banach space of the analytic functions f on D such that

βf = sup(1 − |z |2 )|f ′ (z )| < ∞ z ∈D

with norm ∥f ∥B = |f (0)| + βf , is widely regarded as the limit of Bp as p → ∞, since for 1 < p < ∞, an analytic function f belongs to Bp if and only if the function z → (1 −|z |2 )|f ′ (z )| is in Lp (dλ), where dλ is the conformally invariant area measure dA(z )

dλ(z ) =

(1 − |z |2 )2

.

Also, the functions f ∈ B satisfy the Lipschitz condition

|f (z ) − f (w)| ≤ ∥f ∥B ρ(z , w),

z , w ∈ D,

and hence the growth condition

|f (z ) − f (0)| ≤

1 2

∥f ∥B log

1 + |z | 1 − |z |

,

z ∈ D,

(3)

which corresponds to (2) with p = ∞. By the Schwarz–Pick lemma, the Bloch space is Möbius invariant. The analytic Besov space B1 is defined as the collection of functions f on D which may be represented as f (z ) =

∞ 

an Lλn (z ),

z ∈ D,

n =1

where {an } ∈ ℓ1 , λn ∈ D for n ∈ N, and for λ, z ∈ D, Lλ ( z ) =

λ−z 1 − λz

.

The norm in B1 is defined as

 ∞

∥f ∥B1 = inf

| an | : f ( z ) =

∞ 

n =1



an Lλn (z ), z ∈ D .

n =1

It is immediate from the above definition that B1 is Möbius invariant and that the functions in B1 are bounded. In fact, for ∞ 1 f = k=1 ak Lλk ∈ B1 , with {ak } ∈ ℓ and λk ∈ D, k ∈ N,

∥f ∥∞ ≤

∞ 

|ak |∥Lλk ∥∞ =

k=1

∞ 

|ak |,

k=1

having denoted by ∥f ∥∞ the supremum norm of f . Taking the infimum over all such sequences {ak } in the above representation of f , we obtain

∥f ∥∞ ≤ ∥f ∥B1 .

(4)

Moreover, as noted in [3], the functions in B1 can be extended continuously to the closure of D. The space B1 was studied in detail in [4], where it was shown that B1 is the smallest Möbius invariant space, which is why it also bears the name of minimal Möbius invariant space. In [4,18] it was shown that f ∈ B1 if and only if b(f ) =



|f ′′ (z )|dA(z ) < ∞. D

In addition, ∥f ∥B1 ,∗ := max{|f (0)|, |f ′ (0)|, b(f )} defines a non-Möbius-invariant norm on B1 (see e.g. [5]) equivalent to ∥·∥B1 . An important space of analytic functions, which is widely considered as a Möbius invariant version of the Hardy Hilbert space H 2 , is the space BMOA of analytic functions of bounded mean oscillation, defined as the collection of functions

596

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

f ∈ H (D) such that

∥f ∥∗ = sup ∥f ◦ La − f (a)∥H 2 < ∞, a∈D

where ∥g ∥2H 2 = lim

 2π

r →1

0

|f (reiθ )|2 dθ . The BMOA-norm is defined as

∥f ∥BMOA = |f (0)| + ∥f ∥∗ . It is well known (e.g., see Lemma 1.4 in [16]) that for each p ≥ 1, Bp ⊆ BMOA ⊆ B , and the inclusions are proper and continuous. In fact, as shown in [10], if f ∈ BMOA, then βf ≤ ∥f ∥∗ , so

∥f ∥B ≤ ∥f ∥BMOA . An interesting analogy between BMOA and the Bloch space is that the functions in BMOA have the same growth rate as the Bloch functions. In fact, for all f ∈ BMOA and z ∈ D, by (3),

|f (z )| ≤ |f (0)| + ∥f ∥β ρ(0, z ) ≤ |f (0)| + ∥f ∥∗ ρ(0, z ). For a fixed positive continuous function µ on D, the weighted-type Banach space Hµ∞ is defined as the collection of functions f ∈ H (D) such that ∥f ∥Hµ∞ = sup µ(z )|f (z )| < ∞. z ∈D

If µ is identically 1, we adopt the standard notation H ∞ for Hµ∞ and ∥f ∥∞ for ∥f ∥H ∞ . In Theorem 3.1 of [7], it was shown that a weighted composition operator from B into H ∞ is bounded (respectively, compact) if and only if it is bounded (respectively, compact) as an operator from BMOA into H ∞ . This equivalence, which, as it will be shown in Section 2, easily extends to the weighted composition operators mapping B and BMOA into the weighted-type Banach space Hµ∞ , prompted us to investigate whether the weighted composition operators from the Besov spaces Bp (with 1 ≤ p < ∞) into Hµ∞ share this property. For each of the spaces X = B , BMOA and Bp , we define the function ωX on D by

ωX (z ) = sup{|f (z )| : f ∈ X , f (0) = 0, ∥f ∥X ≤ 1}, where ∥ · ∥X is the norm in X . An immediate consequence of (3) and Theorem 1 in [6] is that

ωB (z ) =

1 2

log

1 + |z | 1 − |z |

,

z ∈ D.

(5)

The function ωX plays an important role in characterizing the boundedness and the compactness of the operators under consideration in this work. As expected, the function ωX is related to the growth rate of the functions in X , and thus, for the Besov spaces Bp , whose growth rate is p-dependent, the bounded and the compact weighted composition operators mapping into Hµ∞ will be characterized in terms of a quantity depending on p, the hyperbolic distance ρ , and the symbols ψ and ϕ . Thus, the equivalence that holds for the bounded and the compact operators into Hµ∞ acting on the Bloch space and BMOA does not extend to the Besov spaces. 1.1. Organization of the paper After presenting in Section 2 the results concerning the equivalence between the bounded (respectively, compact) weighted composition operators from BMOA into Hµ∞ to those from B into Hµ∞ , in Section 3, we characterize the bounded weighted composition operators Wψ,ϕ from Besov spaces into Hµ∞ and determine the norm of such operators in terms of ωBp (z ). We also give an approximation of the operator norm for a general p ∈ [1, ∞) in terms of the symbols ψ and ϕ and obtain another exact expression for p = 1, 2. In Section 4, we obtain an approximate formula for the essential norm of the bounded weighted composition operators acting on a general Besov space Bp (with 1 ≤ p < ∞). In the case when p = 1 we obtain essential norm estimates, and in the case p = 2 we determine the precise value of the essential norm. We deduce a characterization of the compact operators Wψ,ϕ from Bp to Hµ∞ . For p = ∞ and µ = 1, our characterizations of boundedness and compactness of the weighted composition operators from Bp to Hµ∞ correspond to those for the weighted composition operators from B to H ∞ given in Theorems 6.1 and 6.2 of [12]. Analogous characterizations of the bounded and the compact weighted composition operators from the Bloch space to Hµ∞ in the general setting of a bounded homogeneous domain in Cn were given in [1]. In the one-dimensional setting, when p = ∞ and in the presence of a weight µ, our formula for the operator norm given in Section 3 corresponds to the formula obtained in [14]. In [2], Allen and the first author obtained a similar formula for the norm of weighted composition operator from the Bloch space into H ∞ in the setting of a bounded homogeneous domain in Cn . The essential norm estimates obtained in Theorem 6 correspond to those obtained in Theorem 3.4 of [15] for p = ∞ in the unit disk setting. In Section 5, we highlight the results for the component operators Mψ and Cϕ defined as Mψ (f ) = ψ f and Cϕ (f ) = f ◦ ϕ . In Section 6, we consider the case when the weights µ are of the form µ(z ) = (1 − |z |2 )α , z ∈ D and α > −1, which are known in the literature as Bergman weights. After defining the Besov (respectively, BMOA) Zygmund-type space Zp

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

597

(respectively, Z∗ ) as the subset of H (D) whose elements are antiderivatives of functions in Bp (respectively, BMOA) and recalling the definition of α -Bloch spaces Bα , we note that for α > 0 and µ(z ) = (1 − |z |2 )α , the space Hµ∞ can be interpreted as the space whose elements are derivatives of functions in Bα . Thus, aided by the earlier results, we obtain characterizations of the bounded and the compact weighted composition operators from Zp and Z∗ into the α -Bloch spaces. Finally, in Section 7, we summarize our results for the component operators with some specific choices of weights. Throughout this paper we shall adopt the convention of denoting by C a positive constant whose value may change at each occurrence. For positive numbers A and B, the notation A ≍ B means that there exist positive numbers C1 and C2 such that C1 A ≤ B ≤ C2 A. 2. Boundedness and compactness of Wψ,ϕ : BMOA → Hµ∞ For z ∈ D, let

ωBMOA (z ) = sup{|f (z )| : f ∈ BMOA, f (0) = 0, ∥f ∥BMOA ≤ 1}, ωB (z ) = sup{|f (z )| : f ∈ B , f (0) = 0, ∥f ∥B ≤ 1}. Lemma 1. For all a ∈ D, ωBMOA (a) ≍ ωB (a) = ρ(0, a). Proof. It is clear that ωBMOA (0) = 0 = ρ(0, 0). So fix a ∈ D \ {0}. Let f ∈ BMOA with f (0) = 0 and ∥f ∥BMOA ≤ 1. Then f ∈ B and ∥f ∥B ≤ ∥f ∥BMOA ≤ 1. Thus, by (5), ωBMOA (a) ≤ ωB (a) = ρ(0, a). On the other hand, by Theorem 5 of [9], the function fa ( z ) =

1 2

log

|a| + az , |a| − az

for z ∈ D,

is in BMOA, fixes 0 and has norm bounded above by a constant M. It follows that ρ(0, a) = |f (a)| ≤ M ωBMOA (a). Hence, 

ωBMOA (a) ≍ ρ(0, a).

We now give a characterization of the bounded and the compact weighted composition operators from BMOA into Hµ∞ . Theorem 1. For ψ, ϕ ∈ H (D), with ϕ(D) ⊆ D and µ a positive continuous function on D, the following propositions are equivalent: (a) (b) (c) (d)

Wψ,ϕ : BMOA → Hµ∞ is bounded. ψ ∈ Hµ∞ and supz ∈D µ(z )|ψ(z )|ωBMOA (ϕ(z )) < ∞. ψ ∈ Hµ∞ and supz ∈D µ(z )|ψ(z )|ρ(0, ϕ(z )) < ∞. Wψ,ϕ : B → Hµ∞ is bounded.

Proof. The proof of the equivalence of (a), (c) and (d) is similar to the proof of Theorem 3.1 in [7] and will be omitted. The equivalence of (b) and (c) follows from Lemma 1.  Theorem 2. Let ψ, ϕ ∈ H (D), with ϕ(D) ⊆ D, and let µ be a positive continuous function on D such that Wψ,ϕ : BMOA → Hµ∞ is bounded. The following propositions are equivalent: (a) (b) (c) (d)

Wψ,ϕ : BMOA → Hµ∞ is compact. lim|ϕ(z )|→1 µ(z )|ψ(z )|ωBMOA (ϕ(z )) = 0. lim|ϕ(z )|→1 µ(z )|ψ(z )|ρ(0, ϕ(z )) = 0. Wψ,ϕ : B → Hµ∞ is compact.

Proof. The proof of the equivalence of (a), (c) and (d) is analogous to that of Theorem 3.2 in [7]. The equivalence of (b) and (c) is an immediate consequence of Lemma 1.  The equivalence of (c) and (d) in Theorems 1 and 2 was noted in [1] and, under the assumption of a radial normal weight

µ, in [14].

Remark 1. Observe that in Theorem 2 above, in Corollary 2, as well as in Theorems 8 and 10 below, if ∥ϕ∥∞ < 1, then Wψ,ϕ is a compact operator if and only if it is a bounded operator. 3. Boundedness of Wψ,ϕ : Bp → Hµ∞ and the operator norm For z ∈ D, let

ω1 (z ) = sup{|f (z )| : f ∈ B1 , ∥f ∥B1 ≤ 1}, and for 1 < p < ∞, define

ωp (z ) = ωBp (z ) = sup{|f (z )| : f ∈ Bp , f (0) = 0, ∥f ∥Bp ≤ 1}.

598

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

Lemma 2. (a) For f ∈ B1 and z ∈ D,

|f (z )| ≤ ω1 (z )∥f ∥B1 . (b) For 1 < p < ∞, f ∈ Bp and z ∈ D,

|f (z )| ≤ |f (0)| + ωp (z )bp (f ). f

Proof. Part (a) is immediate if f is identically 0. For f not identically 0, define g = ∥f ∥ . Then g ∈ B1 and ∥g ∥B1 = 1; so for B1 z ∈ D, we have

|f (z )| = |g (z )|∥f ∥B1 ≤ ω1 (z )∥f ∥B1 . Statement (b) is trivial if f is constant. So assume f is nonconstant. Then bp (f ) ̸= 0, so the function g defined by g =

f − f (0) bp (f )

is in Bp , g (0) = 0, and ∥g ∥Bp = bp (g ) = 1. Thus, for z ∈ D,

|f (z ) − f (0)| = |g (z )|bp (f ) ≤ ωp (z )bp (f ). Therefore, (b) follows at once from the triangle inequality.



We now determine ω1 . Lemma 3. For z ∈ D, ω1 (z ) = 1. Proof. Fix z ∈ D and let a be an arbitrary element of D. Then the function La ∈ B1 , and by (4) and the definition of B1 -norm, we have

|La (z )| ≤ ∥La ∥B1 ≤ 1. Thus, by the definition of ω1 , |La (z )| ≤ ω1 (z ). Taking the supremum over all a ∈ D, we obtain 1 ≤ ω1 (z ). On the other hand, since for each f ∈ B1 , |f (z )| ≤ ∥f ∥∞ ≤ ∥f ∥B1 , it follows that ω1 (z ) ≤ 1, proving the result.  For 1 ≤ p < ∞, ψ, ϕ ∈ H (D), with ϕ(D) ⊆ D, define Qψ,ϕ,µ,p = sup µ(z )|ψ(z )|ωp (ϕ(z )). z ∈D

In particular, by Lemma 3, Qψ,ϕ,µ,1 = ∥ψ∥Hµ∞ .

(6)

To prove the main theorem in this section, we shall need the following result (see Lemma 4.2.2 of [18]). Lemma 4. For z ∈ D and t > −1,

 D

(1 − |w|2 )t 1 dA(w) ≍ log , |1 − w z |t +2 1 − | z |2

as |z | → 1.

Theorem 3. Suppose that 1 ≤ p < ∞, ψ ∈ H (D) and ϕ is an analytic self-map of D. Then the following statements are equivalent. (a) Wψ,ϕ : Bp → Hµ∞ is bounded.



1 (b) ψ ∈ Hµ∞ and βψ,ϕ,µ,p = supz ∈D µ(z )|ψ(z )| log 1−|ϕ( z )|2 ∞ (c) ψ ∈ Hµ and Qψ,ϕ,µ,p < ∞.

1−1/p

< ∞.

Moreover, under any of the equivalent conditions (a), (b) and (c), the operator norm of Wψ,ϕ is given by

  ∥Wψ,ϕ ∥ = max ∥ψ∥Hµ∞ , Qψ,ϕ,µ,p .

(7)

Note that for p = 1 condition (b) reduces to just ψ ∈ Hµ∞ and ∥Wψ,ϕ ∥ = ∥ψ∥Hµ∞ . Proof. (a) ⇒ (b) Suppose Wψ,ϕ : Bp → Hµ∞ is bounded. Then, ψ = Wψ,ϕ 1 ∈ Hµ∞ , which proves (b) for p = 1. For 1 < p < ∞ and a ∈ D such that ϕ(a) ̸= 0, define the function fϕ(a) on D by fϕ(a) (z ) =

 log

1 1 − |ϕ(a)|2

−1/p log

1 1 − ϕ(a)z

,

z ∈ D.

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

599

By Lemma 4, we see that bp (fϕ(a) ) =



−1/p 

1

log

1 − |ϕ(a)|2

(1 − | z | )

2 p−2

D

  1/p  ϕ(a) p     dA(z )  1 − ϕ(a)z 

is bounded by a constant independent of a if for some r ∈ (0, 1) sufficiently close to 1, |ϕ(a)| > r. Thus, fϕ(a) ∈ Bp , fϕ(a) (0) = 0 and Lr = supa∈D,|ϕ(a)|>r ∥fϕ(a) ∥Bp < ∞. Thus, by the boundedness of Wψ,ϕ , if |ϕ(a)| > r,

 µ(a)|ψ(a)| log

1−1/p

1 1 − |ϕ(a)|2

≤ ∥Wψ,ϕ fϕ(a) ∥Hµ∞ ≤ Lr ∥Wψ,ϕ ∥.

(8)

If |ϕ(a)| ≤ r, then

 µ(a)|ψ(a)| log

1

1−1/p

1 − |ϕ(a)|2

 ≤ µ(a)|ψ(a)| log

1−1/p

1 1 − r2

≤ C ∥ψ∥Hµ∞ .

(9)

From (8) and (9), taking the supremum over all a ∈ D, we obtain (b) in the case when 1 < p < ∞. (b) ⇒ (c) In the case p = 1 there is nothing to prove. So assume 1 < p < ∞. Let f ∈ Bp with f (0) = 0 and ∥f ∥Bp ≤ 1. By 1 (2), since log 1−|w| ≍ log 1−|w| 2 as |w| approaches 1, for r ∈ (0, 1) large enough and for all z ∈ D such that |ϕ(z )| > r, 1+|w|



µ(z )|ψ(z )||f (ϕ(z ))| ≤ C ∥f ∥Bp µ(z )|ψ(z )| log

1

1−1/p

1 − |ϕ(z )|2

≤ C ∥f ∥Bp βψ,ϕ,µ,p .

On the other hand, for |ϕ(z )| ≤ r,



µ(z )|ψ(z )||f (ϕ(z ))| ≤ C ∥f ∥Bp µ(z )|ψ(z )| log

1+r

1−1/p

1−r

≤ C ∥f ∥Bp ∥ψ∥Hµ∞ .

Taking the supremum over all f ∈ Bp such that f (0) = 0 and ∥f ∥Bp ≤ 1, we obtain

 C βψ,ϕ,µ,p µ(z )|ψ(z )|ωp (ϕ(z )) ≤ C ∥ψ∥ ∞ Hµ

if |ϕ(z )| > r , if |ϕ(z )| ≤ r .

(10)

Hence, from (10), taking the supremum over all z ∈ D, we deduce Qψ,ϕ,µ,p ≤ C max{∥ψ∥Hµ∞ , βψ,ϕ,µ,p },

(11)

which proves that Qψ,ϕ,µ,p is finite. (c) ⇒ (a) Assume (c) holds with p = 1 and let f ∈ B1 . For z ∈ D, using part (a) of Lemma 2, we have

µ(z )|ψ(z )f (ϕ(z ))| ≤ µ(z )|ψ(z )|ω1 (ϕ(z ))∥f ∥B1 ≤ Qψ,ϕ,µ,1 ∥f ∥B1 . Taking the supremum over all z ∈ D, we see that Wψ,ϕ : B1 → Hµ∞ is bounded and ∥Wψ,ϕ f ∥Hµ∞ ≤ Qψ,ϕ,µ,1 ∥f ∥B1 . Therefore, by (6),

∥Wψ,ϕ ∥ ≤ Qψ,ϕ,µ,1 = ∥ψ∥Hµ∞ .

(12)

Next, let 1 < p < ∞ and f ∈ Bp . For z ∈ D, using part (b) of Lemma 2, and recalling that |f (0)| = ∥f ∥Bp − bp (f ), we obtain

µ(z )|ψ(z )f (ϕ(z ))| ≤ µ(z )|ψ(z )|(|f (0)| + ωp (ϕ(z )) bp (f )) ≤ ∥ψ∥Hµ∞ (∥f ∥Bp − bp (f )) + Qψ,ϕ,µ,p bp (f ) ≤ max{∥ψ∥Hµ∞ , Qψ,ϕ,µ,p }∥f ∥Bp . Taking the supremum over all z ∈ D, we deduce that Wψ,ϕ : Bp → Hµ∞ is bounded and

∥Wψ,ϕ ∥ ≤ max{∥ψ∥Hµ∞ , Qψ,ϕ,µ,p },

(13)

completing the proof of the boundedness of Wψ,ϕ and the equivalence of (a), (b) and (c). Suppose Wψ,ϕ is bounded. Since ∥1∥Bp = 1, then

∥ψ∥Hµ∞ ≤ ∥Wψ,ϕ ∥. Moreover, if 1 < p < ∞, for each f ∈ Bp with ∥f ∥Bp ≤ 1 and each z ∈ D, we have

∥Wψ,ϕ ∥ ≥ ∥Wψ,ϕ f ∥Hµ∞ ≥ µ(z )|ψ(z )f (ϕ(z ))|.

(14)

600

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

Taking the supremum over all f ∈ Bp such that f (0) = 0 and ∥f ∥Bp ≤ 1 and all z ∈ D, we obtain the lower estimate Qψ,ϕ,µ,p ≤ ∥Wψ,ϕ ∥. Therefore, max{∥ψ∥Hµ∞ , Qψ,ϕ,µ,p } ≤ ∥Wψ,ϕ ∥.

(15)

Formula (7) follows at once from (12) and (14) for p = 1, and from (13) and (15) for 1 < p < ∞.



In the proposition below we obtain asymptotic estimates on ωp . In the special case of p = 2 (i.e. B2 = D , the Dirichlet space), we determine its value precisely. For 1 < p < ∞ and z ∈ D, define



ℓp (z ) = log

1−1/p

1 1 − |z |2

.

Proposition 1. For 2 ≤ p < ∞ there exist constants C1 and C2 such that for each z ∈ D, C1 ℓp (z ) ≤ ωp (z ) ≤ C2 ρ(z , 0)1−1/p .

(16)

For 1 ≤ p < ∞ and for z sufficiently close to 1,

ωp (z ) ≍ ℓp (z ). Furthermore, for all z ∈ D,

 ω2 (z ) = log

 12

1 1 − | z |2

.

(17)

Proof. Fix p such that 2 ≤ p < ∞ and z ∈ D. First observe that by (2), there exists a positive constant C such that for all f ∈ Bp , with f (0) = 0 and ∥f ∥Bp ≤ 1, |f (z )| ≤ C ρ(z , 0)1−1/p . Next, for z ∈ D \ {0}, let



fz (w) =

log

−1/p

1

log

1 − | z |2

1 1 − zw

,

w ∈ D.

Then fz ∈ Bp , fz (0) = 0, and fz (z ) = ℓp (z ). Fixing R ∈ (0, 1), for 0 < |z | ≤ R,

−1/p  1/p  (1 − |w|2 )p−2 1 dA (w) ∥fz ∥Bp = |z | log 1 − | z |2 |1 − z w|p  −1/p D  1/p 1 1 2 p−2 ≤ |z | log ( 1 − |w| ) dA (w) 1 − | z |2 1−R D  −1/p −1/p 1 (p − 1) |z | log . = 1−R 1 − |z |2

(18)

Expanding log 1−|1z |2 about 0, we see that

 |z | log

−1/p

1 1 − |z |2

= |z |1−2/p (1 + O(|z |2 ))1/p .

(19)

Therefore,

 lim |z | log

z →0

1

−1/p

1 − | z |2

= δp,2 ,

having denoted by δp,2 the Kronecker delta. Hence, for each r ∈ (0, 1) there exists C > 0 such that if 0 < |z | < r, then

 −1/p |z | log 1−|1z |2 ≤ C , which by (18) yields ∥fz ∥Bp ≤ C .

On the other hand, as shown in the proof of Theorem 3, there exists R ∈ (0, 1) such that for |z | > R, sup|z |>R ∥fz ∥Bp < ∞. Therefore, we have shown that L = sup ∥fz ∥Bp < ∞. z ∈D\{0}

By the definition of ωp , it follows that for all z ∈ D \ {0},

ℓp (z ) = |fz (z )| ≤ L ωp (z ). Since ℓp (0) = 0 = ωp (0), we deduce the lower estimate in (16) with C1 = 1L . The upper estimate in (16) follows by (2). The upper estimate ωp (z ) ≤ C ℓp (z ) for z sufficiently close to the unit circle follows from (2) and the approximation log 1−|z | ≍ log 1−|1z |2 as |z | approaches 1, proving that ωp (z ) ≍ ℓp (z ) in this case. 1+|z |

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

To prove (17) note that for f ∈ D with f (z ) = Kz (w) = log

1

,

1 − zw

∞

n=1

an z n , we have ∥f ∥2D =

∞

n =1

601

n|an |2 and f (z ) = ⟨f , Kz ⟩, where

w ∈ D.



To conclude, use ∥Kz ∥D = log 1−|1z |2

1/2

.



Remark 2. In the case 1 < p < 2, our test functions fz do not lead to the lower estimate in (16) because the norm of fz is not bounded in z for z near 0. In fact, from (19) it is easy to see that the following estimate holds for 1 < p < 2: C1 |z |1/p

 log

1−1/p

1 1 − |z |2

≤ ωp (z )

for |z | small. Observe that for any p ∈ [1, ∞), if Wψ,ϕ : Bp → Hµ∞ is bounded, then by (8) and (9),

βψ,ϕ,µ,p ≤ C ∥Wψ,ϕ ∥,

while by (11),

∥Wψ,ϕ ∥ ≤ C max{∥ψ∥Hµ∞ , βψ,ϕ,µ,p }. Thus, from Theorem 3 and Proposition 1, we deduce the following result. Corollary 1. For 1 ≤ p < ∞, ψ ∈ H (D), ϕ an analytic self-map of D such that Wψ,ϕ : Bp → Hµ∞ is bounded,





∥Wψ,ϕ ∥ ≍ max ∥ψ∥

∞ Hµ

, sup µ(z )|ψ(z )| log z ∈D

1

1−1/p 

1 − |ϕ(z )|2

.

In the special case p = 2 (when Bp is the Dirichlet space),





∥Wψ,ϕ ∥ = max ∥ψ∥Hµ∞ , sup µ(z )|ψ(z )| log z ∈D

1

1/2 

1 − |ϕ(z )|2

,

and in particular, if ϕ(D) ⊆ {z ∈ D : |z | < (1 − 1/e)1/2 }, then ∥Wψ,ϕ ∥ = ∥ψ∥Hµ∞ . 4. The essential norm and compactness In this section, we obtain an approximation of the essential norm of the bounded weighted composition operators from Bp into Hµ∞ in the general case 1 ≤ p < ∞, which will allow us to characterize the compact operators. For 0 < r < 1, let Tr denote the linear operator mapping an analytic function f on D to the function fr (z ) = f (rz ),

for z ∈ D.

To prove the main results of the section, we shall need the following lemma. Lemma 5. Let 1 ≤ p < ∞. (a) For each r ∈ (0, 1), the operator Tr is compact on Bp . (b) For each r ∈ (0, 1),

 sup sup log ∥f ∥Bp ≤1 z ∈D

1 + |z |

−(1−1/p)

1 − |z |

|((I − Tr )f )(z )| < ∞.

(c) For each s ∈ (0, 1) and each ε > 0, there exists r ∈ (0, 1) such that sup sup |((I − Tr )f )(z )| < ε. ∥f ∥Bp ≤1 |z |≤s

Proof. First note that for r ∈ (0, 1), Tr is compact on Bp . Indeed, by Lemma 3.7 of [17], it suffices to show that if {fk } is a bounded sequence in Bp converging to 0 uniformly on compact subsets of D, then ∥Tr fk ∥Bp → 0 as k → ∞. Thus compactness follows by uniform convergence on compacta of fk′ and fk (and fk′′ in the case p = 1). Next, let r ∈ (0, 1), f ∈ Bp , and z ∈ D. For 1 < p < ∞, from (1) we have

|f (z ) − f (rz )| ≤ C ∥f ∥Bp ≤ C ∥f ∥Bp

(1 + |z |)(1 − r |z |) log (1 − |z |)(1 + r |z |)  1−1/p 1 + |z | log . 1 − |z | 

1−1/p

602

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

Thus, dividing by the logarithmic term in the last display and taking the supremum over z ∈ D and over all f ∈ Bp with ∥f ∥Bp ≤ 1, we obtain

 sup sup log ∥f ∥Bp ≤1 z ∈D

1 + |z |

−(1−1/p)

|((I − Tr )f )(z )| ≤ C .

1 − |z |

(20)

For p = 1, we have

|f (z ) − f (rz )| ≤ 2∥f ∥∞ ≤ 2∥f ∥B1 . Therefore, sup sup |((I − Tr )f )(z )| ≤ 2.

(21)

∥f ∥B1 ≤1 z ∈D

To prove (c), note first that (2) together with Montel’s theorem implies that the unit ball of Bp is relatively compact in the topology of uniform convergence on compact subsets of D, as therefore is {f ′ : f ∈ Bp , ∥f ∥Bp ≤ 1}. Then (c) follows easily via f (z ) − f (rz ) =

z

f ′ (ζ ) dζ .

rz



Remark 3. In the case of the Dirichlet space, the constant C in (20) logarithmic term can be ∞can ben taken to be 1 and the ∞ n n replaced with log 1−|1z |2 . Indeed, for r ∈ (0, 1) and f ∈ D , f (z ) = n=0 an z , f (z ) − f (rz ) = n=1 an (1 − r )z . So,



|f (z ) − f (rz )| = |⟨f − fr , Kz ⟩| ≤ ∥f − fr ∥D ∥Kz ∥D = ∥f − fr ∥D log Noting that ∥f − fr ∥2D =

 sup sup log ∥f ∥D ≤1 z ∈D

∞

n =1

n|an |2 (1 − r n )2 ≤

−1/2

1 1 − | z |2

∞

n=1

1 1 − |z |2

 21

.

n|an |2 = ∥f ∥2D , we obtain

|f (z ) − f (rz )| ≤ 1,

as claimed. We shall now give explicit essential norm estimates in the case p = 1. Theorem 4. Let ψ and ϕ be analytic functions on D with ϕ(D) ⊆ D, µ a positive continuous function on D, and suppose Wψ,ϕ : B1 → Hµ∞ is bounded. Then 1 2

A(ψ, ϕ) ≤ ∥Wψ,ϕ ∥e ≤ 2A(ψ, ϕ),

where A(ψ, ϕ) = lims→1 sup|ϕ(z )|>s µ(z )|ψ(z )|. Proof. Since the result is clear for ψ = 0, we assume ψ is not identically 0. To prove the upper estimate, fix ε > 0 and s ∈ (0, 1). Since Wψ,ϕ is bounded, ψ ∈ Hµ∞ . By Lemma 5, we may choose r ∈ (0, 1) such that sup sup |(I − Tr )f (z )| < ∥f ∥B1 ≤1 |z |≤s

ε . ∥ψ∥Hµ∞

(22)

By the boundedness of Wψ,ϕ : B1 → Hµ∞ and the compactness of Tr : B1 → B1 , the operator Wψ,ϕ Tr : B1 → Hµ∞ is also compact. Thus, by (21) and (22),

∥Wψ,ϕ ∥e ≤ ∥Wψ,ϕ − Wψ,ϕ Tr ∥B1 →Hµ∞ = sup ∥Wψ,ϕ (I − Tr )f ∥Hµ∞ ∥f ∥B1 ≤1

= sup sup µ(z )|ψ(z )||(I − Tr )f (ϕ(z ))| ∥f ∥B1 ≤1 z ∈D

sup µ(z )|ψ(z )||(I − Tr )f (ϕ(z ))|

≤ sup

∥f ∥B1 ≤1 |ϕ(z )|≤s

sup µ(z )|ψ(z )||(I − Tr )f (ϕ(z ))|

+ sup

∥f ∥B1 ≤1 |ϕ(z )|>s

≤ ∥ψ∥Hµ∞ sup

sup |(I − Tr )f (ϕ(z ))|

∥f ∥B1 ≤1 |ϕ(z )|≤s

+ sup µ(z )|ψ(z )| sup sup |(I − Tr )f (ϕ(z ))| |ϕ(z )|>s

∥f ∥B1 ≤1 z ∈D

< ε + 2 sup µ(z )|ψ(z )|. |ϕ(z )|>s

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

603

If ∥ϕ∥∞ < 1 and since ε is arbitrary, we see that ∥Wψ,ϕ ∥e = 0 and the operator Wψ,ϕ is compact. If ∥ϕ∥∞ = 1 and since ε is arbitrary, the upper estimate follows by letting s → 1. Let {an } be a sequence in D such that |ϕ(an )| → 1 and A(ψ, ϕ) = lim µ(an )|ψ(an )|. n→∞

For each n ∈ N, define fn (z ) =

1 − |ϕ(an )| 1 − ϕ(an )z

,

z ∈ D.

Then {fn } converges to 0 uniformly on compact subsets of D and a straightforward calculation shows that fn =

1



1 + |ϕ(an )|



1 − ϕ(an )Lϕ(an ) .

Thus, fn ∈ B1 and ∥fn ∥B1 ≤ 1. Since clearly ∥fn ∥∞ = 1, from (4) we also have 1 = ∥fn ∥∞ ≤ ∥fn ∥B1 . If T is any compact operator from B1 to Hµ∞ , then ∥Tfn ∥Hµ∞ → 0 as n → ∞. Therefore,

∥Wψ,ϕ − T ∥ ≥ lim sup ∥(Wψ,ϕ − T )fn ∥Hµ∞ n→∞

≥ lim sup ∥Wψ,ϕ fn ∥Hµ∞ n→∞

≥ lim µ(an )|ψ(an )| n→∞

1 1 + |ϕ(an )|

=

1 2

A(ψ, ϕ).

Taking the infimum over all compact operators T : B1 → Hµ∞ , we obtain the desired lower estimate.



We next provide a precise formula for the essential norm of the bounded weighted composition operator from the Dirichlet space into Hµ∞ . Theorem 5. Let ψ be analytic on D and ϕ an analytic self-map of D such that Wψ,ϕ : D → Hµ∞ is bounded. Then

 ∥Wψ,ϕ ∥e = lim sup µ(z )|ψ(z )| log

1/2

1 1 − |ϕ(z )|2

s→1 |ϕ(z )|>s

.

Proof. Since the result is clear for ψ = 0, we assume ψ is not identically 0. Define



A(ψ, ϕ) = lim sup µ(z )|ψ(z )| log s→1 |ϕ(z )|>s

1 1 − |ϕ(z )|2

1/2

.

We begin by showing the upper estimate

∥Wψ,ϕ ∥e ≤ A(ψ, ϕ).

(23)

Fix η > 0 and s ∈ (0, 1). Since Wψ,ϕ is bounded, ψ ∈ Hµ∞ . Choose r ∈ (0, 1) as in Lemma 5 corresponding to ε =

η/∥ψ∥Hµ∞ . Moreover, the boundedness of Wψ,ϕ : D → Hµ∞ and the compactness of Tr as an operator on D imply the compactness of Wψ,ϕ Tr as an operator from D to Hµ∞ . Arguing as in the proof of Theorem 4, we see that

∥Wψ,ϕ ∥e ≤ η + I ,

(24)

where by Remark 3, I =

sup

sup µ(z )|ψ(z )||(I − Tr )f (ϕ(z ))|

∥f ∥D ≤1 |ϕ(z )|>s

 ≤ sup µ(z )|ψ(z )| log |ϕ(z )|>s

 × sup sup log ∥f ∥D ≤1 z ∈D

1 − |ϕ(z )|2

−1/2

1 1 − |ϕ(z )|2



≤ sup µ(z )|ψ(z )| log |ϕ(z )|>s

1/2

1

1 1 − |ϕ(z )|2

|(I − Tr )f (ϕ(z ))|

1/2

.

(25)

604

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

Therefore, from (24) and (25), we obtain

 ∥Wψ,ϕ ∥e ≤ η + sup µ(z )|ψ(z )| log |ϕ(z )|>s

1/2

1 1 − |ϕ(z )|2

.

If ∥ϕ∥∞ < 1 and since η is arbitrary, we see that ∥Wψ,ϕ ∥e = 0 and the operator is compact. If ∥ϕ∥∞ = 1 and since η is arbitrary, letting s → 1, we obtain (23). We next prove the lower estimate

∥Wψ,ϕ ∥e ≥ A(ψ, ϕ).

(26)

Since when ∥ϕ∥∞ < 1 the operator is compact, we shall assume ∥ϕ∥∞ = 1. Let {an } be a sequence in D such that |ϕ(an )| → 1 and



1/2

1

A(ψ, ϕ) = lim µ(an )|ψ(an )| log

1 − |ϕ(an )|2

n→∞

.

For each n ∈ N, let us define fn ( z ) =



−1/2

1

log

1

log

1 − |ϕ(an )|2

1 − ϕ(an )z

,

z ∈ D.

Then {fn } is a sequence in D converging uniformly on compact subsets of D and ∥fn ∥D = 1. Thus, if T is any compact operator from D into Hµ∞ , then ∥Tfn ∥Hµ∞ → 0 as n → ∞. Therefore,

∥Wψ,ϕ − T ∥ ≥ lim sup ∥(Wψ,ϕ − T )fn ∥Hµ∞ n→∞

≥ lim sup ∥Wψ,ϕ fn ∥Hµ∞ n→∞

 ≥ lim µ(an )|ψ(an )| log n→∞

1/2

1 1 − |ϕ(an )|2

.

Taking the infimum over all compact operators T : D → Hµ∞ , we obtain (26). The conclusion now follows at once from (23) and (26).  Theorem 6. Let 1 ≤ p < ∞, ψ and ϕ analytic functions on D with ϕ(D) ⊆ D, µ a positive continuous function on D, and suppose Wψ,ϕ : Bp → Hµ∞ is bounded. Then

1−1/p  1 + |ϕ(z )| . ∥Wψ,ϕ ∥e ≍ lim sup µ(z )|ψ(z )| log s→1 |ϕ(z )|>s 1 − |ϕ(z )| Proof. Since the result is clear for ψ = 0, we assume ψ is not identically 0. We begin by proving that

∥Wψ,ϕ ∥e ≤ CA(ψ, ϕ),

(27)

for some constant C > 0, where



A(ψ, ϕ) = lim sup µ(z )|ψ(z )| log s→1 |ϕ(z )|>s

1 + |ϕ(z )|

1−1/p

1 − |ϕ(z )|

.

Fix ε > 0. Proceeding as in the proof for the cases of the Dirichlet space and B1 and using Lemma 5, we obtain

 1−1/p 1 + |ϕ(z )| ∥Wψ,ϕ ∥e < ∥ψ∥Hµ∞ ε + C sup µ(z )|ψ(z )| log . 1 − |ϕ(z )| |ϕ(z )|>s If ∥ϕ∥∞ < 1 and since ε is arbitrary, we see that ∥Wψ,ϕ ∥e = 0 and the operator Wψ,ϕ is compact. If ∥ϕ∥∞ = 1 and since ε is arbitrary, letting s → 1, we obtain (27). We next prove that ∥Wψ,ϕ ∥e ≥ C A(ψ, ϕ), for some constant C > 0, arguing as in the proof of Theorem 5 with the sequence {fn } replaced by gn (z ) =

 log

1 + |ϕ(an )|2

−1/p log

1 − |ϕ(an )|

2

1 + |ϕ(an )|2 1 − ϕ(an )z

z ∈ D,

where {an } is a sequence in D such that √1 ≤ |ϕ(an )| → 1 and 2



A(ψ, ϕ) = lim µ(an )|ψ(an )| log n→∞

1 + |ϕ(an )| 1 − |ϕ(an )|

1−1/p

.

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

605

p

By Lemma 4, we have ∥gn ∥Bp ≍ 1 if |ϕ(an )| is sufficiently close to 1. Moreover, {gn } converges to 0 uniformly on compact subsets of D. Thus, if T is a compact operator from Bp into Hµ∞ , then ∥Tgn ∥Hµ∞ → 0 as n → ∞. Therefore, if L is an upper bound on ∥gn ∥Bp ,

∥Wψ,ϕ − T ∥ ≥ ≥ ≥

1 L 1

lim sup ∥(Wψ,ϕ − T )gn ∥Hµ∞ n→∞

L

lim sup ∥Wψ,ϕ gn ∥Hµ∞

1



n→∞

1 + |ϕ(an )|2

lim µ(an )|ψ(an )| log L n→∞ 1 − |ϕ(an )|2

1−1/p

.

(28)

√

Note that for x ∈ [1/ 2, 1), 1 + x2 1 − x2

 ≥

1+x

1/2

1−x

. √

Thus, by the assumption |ϕ(an )| ≥ 1/ 2, (28) yields

∥Wψ,ϕ − T ∥ ≥



1

lim µ(an )|ψ(an )| log

21−1/p L n→∞

1 + |ϕ(an )| 1 − |ϕ(an )|

1−1/p

.

Taking the infimum over all compact operators T : Bp → Hµ∞ , we obtain ∥Wψ,ϕ ∥e ≥ proof. 

1 21−1/p L

A(ψ, ϕ), completing the

We deduce the following characterization of the compact weighted composition operators from the Besov spaces into Hµ∞ . Corollary 2. Let 1 ≤ p < ∞, ψ be analytic on D and ϕ an analytic self-map of D such that Wψ,ϕ : Bp → Hµ∞ is bounded. Then Wψ,ϕ : Bp → Hµ∞ is compact if and only if



lim sup µ(z )|ψ(z )| log

s→1 |ϕ(z )|>s

1 + |ϕ(z )| 1 − |ϕ(z )|

1−1/p

= 0.

In particular, if ϕ(D) is a compact subset of D, then Wψ,ϕ : Bp → Hµ∞ is compact if and only if it is bounded. 5. Component operators ∞ ∞ Define Hµ, 0 to be the subspace of Hµ whose elements f satisfy the condition

lim µ(z )|f (z )| = 0.

|z |→1

From the earlier results, we obtain the following corollaries for the multiplication and the composition operators from the Besov spaces to Hµ∞ , which to the best of our knowledge, have not appeared in the literature. Since, as observed above, the operators Wψ,ϕ and Cϕ are necessarily compact when the range of ϕ is relatively compact in D, we shall set aside this case. Corollary 3. Let ψ be analytic on D. Then (a) Mψ : BMOA → Hµ∞ is bounded if and only if Mψ : B → Hµ∞ is bounded if and only if sup µ(z )|ψ(z )|ρ(0, z ) < ∞. z ∈D

(b) Mψ : BMOA → Hµ∞ is compact if and only if Mψ : B → Hµ∞ is compact if and only if lim µ(z )|ψ(z )|ρ(0, z ) = 0. |z |→1

In particular, Mψ : BMOA → H ∞ is bounded or compact if and only if Mψ : B → H ∞ is bounded or compact if and only if ψ is identically 0. Corollary 4. Let ψ be analytic on D. Then (a) Mψ : B1 → Hµ∞ is bounded if and only if ψ ∈ Hµ∞ . ∞ (b) Mψ : B1 → Hµ∞ is compact if and only if ψ ∈ Hµ, 0. ∞ In particular, Mψ : B1 → H is bounded if and only if ψ ∈ H ∞ and Mψ : B1 → H ∞ is compact if and only if ψ is identically 0.

606

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

Corollary 5. Let ψ be analytic on D and 1 < p < ∞. Then (a) Mψ : Bp → Hµ∞ is bounded if and only if



sup µ(z )|ψ(z )| log z ∈D

1 + |z |

1−1/p

1 − |z |

< ∞.

(b) Mψ : Bp → Hµ∞ is compact if and only if



lim µ(z )|ψ(z )| log

|z |→1

1 + |z |

1−1/p

1 − |z |

= 0.

In particular, for 1 < p < ∞, the only bounded or compact multiplication operator from Bp into H ∞ has symbol identically 0. Corollary 6. Let ϕ be an analytic self-map of D. Then (a) Cϕ : BMOA → Hµ∞ is bounded if and only if sup µ(z ) log z ∈D

1 + |ϕ(z )| 1 − |ϕ(z )|

< ∞.

(b) Cϕ : BMOA → Hµ∞ is compact if and only if Cϕ is bounded and lim

|ϕ(z )|→1

µ(z ) log

1 + |ϕ(z )| 1 − |ϕ(z )|

= 0.

In particular, the composition operator Cϕ : BMOA → H ∞ is bounded if and only if it is compact if and only if ∥ϕ∥∞ < 1. Corollary 7. Let ϕ be an analytic self-map of D. Then (a) Cϕ : B1 → Hµ∞ is bounded if and only if µ is bounded. (b) Cϕ : B1 → Hµ∞ is compact if and only if lim|ϕ(z )|→1 µ(z ) = 0. In particular, the composition operator Cϕ : B1 → H ∞ is bounded for any choice of the symbol ϕ and is compact if and only if ∥ϕ∥∞ < 1. Corollary 8. Let ϕ be an analytic self-map of D and 1 < p < ∞. Then (a) Cϕ : Bp → Hµ∞ is bounded if and only if



sup µ(z ) log z ∈D

1

1−1/p

1 − |ϕ(z )|2

< ∞.

(b) Cϕ : Bp → Hµ∞ is compact if and only if Cϕ is bounded and

1−1/p  1 + |ϕ(z )| = 0. µ(z ) log |ϕ(z )|→1 1 − |ϕ(z )| lim

In particular, the composition operator Cϕ : Bp → H ∞ is bounded if and only if it is compact if and only if ∥ϕ∥∞ < 1. 6. The special case of the Bergman weights Recall from the Introduction that for α > −1, the Bergman weights are defined by

µ(z ) = (1 − |z |2 )α , for z ∈ D. Since for −1 < α < 0, the space Hµ∞ is trivial, we shall only consider the case α ≥ 0. With this choice of µ, for α > 0, the space Hµ∞ is known as the growth space A−α (see [11]) which is a non-separable Banach space with norm

∥f ∥−α = sup(1 − |z |2 )α |f (z )|. z ∈D

−1

The space A is also known as the Bers space. Theorems 1–6 and Corollaries 1 and 2 yield characterizations of the bounded and the compact weighted composition operators and descriptions of their operator norm and their essential norm from BMOA and the Besov spaces into the growth spaces A−α . Recall that the Zygmund space Z consists of the functions F ∈ H (D) such that F ′ ∈ B with norm

∥F ∥Z = |F (0)| + ∥F ′ ∥B .

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

607

It is well-known (see e.g. [8], Theorem 5.2) that the Zygmund space is contained in the disk algebra. Similarly, we may consider the BMOA Zygmund-type space Z∗ consisting of the analytic functions F on D such that F ′ ∈ BMOA with norm

∥F ∥Z∗ = |F (0)| + ∥F ′ ∥BMOA . For 1 ≤ p < ∞, let Zp denote the space of analytic functions F on D such that F ′ ∈ Bp with norm ∥F ∥Zp = |F (0)| + ∥F ′ ∥Bp , which we refer to as the Besov Zygmund-type space. Since Bp and BMOA are contained in the Bloch space, it follows that the spaces Zp and Z∗ are subsets of Z, and hence they are contained in the disk algebra. For α > 0, the α -Bloch space Bα is a Banach space of analytic functions f with norm

∥f ∥Bα = |f (0)| + sup(1 − |z |2 )α |f ′ (z )| < ∞. z ∈D

By Theorem 5.1 in [8], if 0 < α < 1, then Bα can be identified with the analytic Lipschitz space Lip1−α , the space of analytic functions f on D satisfying the Lipschitz condition

|f (z ) − f (w)| ≤ C |z − w|1−α , for some constant C > 0 (depending on f ) and all z , w ∈ D. Moreover, by Theorem 5.5 in [8], if α > 1, then Bα can be identified with the space Hµ∞ where µ(z ) = (1 − |z |2 )α−1 . Thus, the earlier results in this paper yield characterizations of the bounded and the compact weighted composition operators from BMOA and Bp into the space Bα , which we do not state explicitly. Our earlier results will also allow us to characterize the bounded and the compact weighted composition operators from the Besov and the BMOA Zygmund-type spaces Zp and Z∗ into the space Bα . Lemma 6. (a) For 1 ≤ p < ∞ there exists a positive constant C such that if F ∈ Zp , then

  t  p+1   (i) F (z ) − F z  ≤ C ∥F ∥Zp (1 − |z |) 2p , for all t ∈ (0, 1), z ∈ D \ {0}, |z | (ii) |F (z )| ≤ C ∥F ∥Zp , for all z ∈ D. (b) There exists a positive constant C such that if F ∈ Z∗ , then   t  1   z  ≤ C ∥F ∥Z∗ (1 − |z |) 2 , for all t ∈ (0, 1), z ∈ D \ {0}, (iii) F (z ) − F |z | (iv) |F (z )| ≤ C ∥F ∥Z∗ , for all z ∈ D. Proof. Suppose first that 1 < p < ∞ and let F ∈ Zp and z ∈ D \ {0}. Then by (2),

  t    z  = F (z ) − F |z |

  

t

|z |

1

 

zF ′ (zs)ds

≤ |F ′ (0)|(1 − |z |) + |z |

1



|z |

|F ′ (sz ) − F ′ (0)|ds

1

≤ ∥F ∥Zp (1 − |z |) + C ∥F ∥Zp |z |

1



|z |

log

1

Since for all t ∈ (0, 1],

2



1− 1p

1 − s|z |

ds.

√

√ t log

2 t

≤

2 2 , e

we see that

 1   t  1−p |z |   z  ≤ ∥F ∥Zp (1 − |z |) + C ∥F ∥Zp |z | (1 − s|z |) 2p ds F (z ) − F |z | 1 ≤ ∥F ∥Zp (1 − |z |) + C ∥F ∥Zp (1 − |z |) ≤ C ∥F ∥Zp (1 − |z |)

p+1 2p

p+1 2p

,

where C is a constant that depends only on p. In the case p = 1, using (4), we have

  t    z  ≤ F (z ) − F |z | ≤

  

t

  t   |zF ′ (zs)|ds ≤ |z |∥F ′ ∥B1 −1 |z | 1 ∥F ′ ∥B1 (1 − |z |) ≤ ∥F ∥Z1 (1 − |z |). |z |

This proves (i). To prove (ii), let t → 0 in (i) to get |F (z ) − F (0)| ≤ C ∥F ∥Zp , and hence |F (z )| ≤ |F (0)| + |F (z ) − F (0)| ≤ (1 + C )∥F ∥Zp . To prove (b) note that by (3), since BMOA is contained in B and for f ∈ BMOA, ∥f ∥B ≤ ∥f ∥BMOA , the proof for the case of Z∗ is similar to the above. 

608

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

Theorem 7. Let 1 ≤ p < ∞, α > 0, ψ, ϕ ∈ H (D) with ϕ(D) ⊆ D. Then the following statements are equivalent: (a) Wψ,ϕ : Zp → Bα is bounded. (b) ψ ∈ Bα and Wψϕ ′ ,ϕ : Bp → A−α is bounded.



1 (c) ψ ∈ Bα , ψϕ ′ ∈ A−α and sup(1 − |z |2 )α |ψ(z )ϕ ′ (z )| log 1−|ϕ( z )|2 z ∈D

1−1/p

< ∞.

Proof. The equivalence of (b) and (c) is an immediate consequence of Theorem 3. Therefore, it suffices to show that (a) and (b) are equivalent. (a) ⇒ (b) Assume (a) holds. Then ψ = Wψ,ϕ 1 ∈ Bα and ψϕ = Wψ,ϕ id ∈ Bα , having denoted by id the identity map of D. Let f ∈ Bp and let F be the antiderivative of f that fixes 0. Then F ∈ Zp and ∥F ∥Zp = ∥f ∥Bp . Using Lemma 6, for z ∈ D, we have

|ψ(z )ϕ ′ (z )f (ϕ(z ))| = |ψ(z )ϕ ′ (z )F ′ (ϕ(z ))| = |(ψ(F ◦ ϕ))′ (z ) − ψ ′ (z )F (ϕ(z ))| ≤ |(Wψ,ϕ F )′ (z )| + C |ψ ′ (z )|∥F ∥Zp . Thus, multiplying by (1 − |z |2 )α , taking the supremum over D, and using the boundedness of Wψ,ϕ : Zp → Bα , we obtain

∥Wψϕ ′ ,ϕ f ∥−α ≤ ∥Wψ,ϕ F ∥Bα + C ∥ψ∥Bα ∥F ∥Zp ≤ (∥Wψ,ϕ ∥Zp →Bα + C ∥ψ∥Bα )∥F ∥Zp , = (∥Wψ,ϕ ∥Zp →Bα + C ∥ψ∥Bα )∥f ∥Bp . (b) ⇒ (a) Suppose (b) holds. Let F ∈ Zp . Then, by Lemma 6, for z ∈ D, (1 − |z |2 )α |(Wψ,ϕ F )′ (z )| ≤ (1 − |z |2 )α |ψ ′ (z )F (ϕ(z ))| + (1 − |z |2 )α |ψ(z )ϕ ′ (z )F ′ (ϕ(z ))| ≤ C (1 − |z |2 )α |ψ ′ (z )|∥F ∥Zp + ∥Wψϕ ′ ,ϕ F ′ ∥−α . Therefore, by the boundedness of Wψϕ ′ ,ϕ : Bp → A−α applied to F ′ ,

∥Wψ,ϕ F ∥Bα ≤ |ψ(0)F (ϕ(0))| + C sup(1 − |z |2 )α |ψ ′ (z )|∥F ∥Zp + ∥Wψϕ ′ ,ϕ F ′ ∥−α z ∈D

≤ C ∥ψ∥Bα ∥F ∥Zp + ∥Wψϕ ′ ,ϕ ∥Bp →A−α ∥F ′ ∥Bp ≤ (C ∥ψ∥Bα + ∥Wψϕ ′ ,ϕ ∥Bp →A−α )∥F ∥Zp , completing the proof.



We next wish to characterize the bounded weighted composition operators from Zp to Bα which are compact. We shall make use of the following lemmas. Lemma 7. For 1 ≤ p < ∞, every sequence in Zp bounded in norm has a subsequence which converges uniformly in D to a function in Zp . Proof. Suppose {Fn } is bounded in Zp by some positive constant M. By Lemma 6, {Fn } is uniformly bounded in D. Therefore by Montel’s Theorem, there exists a sequence {nk } in N, with n1 < n2 < · · ·, such that Fnk → F uniformly on compact subsets of D, where F is analytic in D. By Fatou’s Theorem and since Fn is a bounded sequence in Zp ,

 p ∥F ∥Zp − |F (0)| = ∥F ′ ∥pBp ≤ lim inf k→∞



|Fn′ k (z )|p (1 − |z |2 )p−2 dA(z ) D

≤ lim inf ∥Fnk ∥pZp < ∞. k→∞

Thus F ∈ Zp , and hence it is in the disk algebra. Let Gnk (z ) := Fnk (z ) − F (z ), and fix ε > 0. Pick t ∈ (0, 1) such that p+1

(1 − t ) 2p < ε. By Lemma 6, for t < |z | < 1,   t  p+1   z  ≤ CM (1 − t ) 2p < C ε. Gnk (z ) − Gnk |z |      Therefore, supt <|z |<1 |Gnk (z )| ≤ supz ∈D Gnk |zt | z  + C ε , whence   t    sup |Gnk (z )| ≤ sup |Gnk (z )| + supGnk z  + C ε. |z | z ∈D z ∈D |z |≤t Since Fnk → F uniformly in compact subset of D, the above shows that Fnk converges uniformly to F in all of D and hence in all of D. 

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611

609

Below is a compactness criterion whose proof, which we provide for completeness, is similar to that of Lemma 3.8 in [17]. Lemma 8. Let X be a Banach space that is continuously contained in the disk algebra, and let Y be any Banach space of analytic functions on D. Suppose that: 1. The point-evaluation functionals on Y are continuous. 2. For every sequence {Fn } in the unit ball of X there exists F ∈ X and a subsequence {Fnj } such that Fnj → F uniformly on D. 3. The operator T : X → Y is continuous if X has the supremum norm and Y is given the topology of uniform convergence on compact sets. Then, T : X → Y is a compact operator if and only if, given a bounded sequence {Fn } in X such that Fn → 0 uniformly on D, then ∥TFn ∥Y → 0 as n → ∞. Proof. Suppose that T : X → Y is a compact operator. Let {Fn } be a sequence in the unit ball of X such that Fn → 0 uniformly in D. If the conclusion is false then there exists an ε > 0 and a subsequence {Fnj } such that ∥TFnj ∥Y ≥ ε for all j ∈ N. Since {Fn } is a bounded sequence and T is a compact operator, we can find a further subsequence {Fnj k } and F ∈ Y such ∥TFnj − F ∥Y → 0 as k → ∞. Since the point-evaluation functionals on Y are continuous, if z ∈ D then there exists a k constant Cz (depending only on z) such that

|(TFnjk − F )(z )| ≤ Cz ∥TFnjk − F ∥ and therefore TFnj − F → 0 pointwise; by our hypothesis statement (3) and since Fn → 0 uniformly in D, we also have k that TFn → 0 pointwise on D. Thus, we must have F (z ) = 0, for all z ∈ D and ∥TFnj ∥Y → 0 as k → ∞. We arrived at a k contradiction and therefore our conclusion is valid. Conversely, let {Fn } be a bounded sequence in X . Without loss of generality, we may assume ∥Fn ∥X ≤ 1. By our hypothesis statement (2), there exists an F ∈ X and a subsequence {Fnj } such that Fnj → F uniformly on D. Using our hypothesis for the bounded sequence {Fnj − F }, we obtain ∥TFnj − TF ∥Y → 0. Thus T : X → Y is a compact operator.  Remark 4. The proof of the necessity in Lemma 8 only uses statements (1) and (3), while the proof of the sufficiency only uses statement (2). Note that the hypotheses in Lemma 8 are satisfied for Y = Bα by (3) for the case α = 1 and by the remarks preceding Lemma 6 for the general α . Then the following result is a direct consequence of Lemmas 6 and 8. Lemma 9. Let 1 ≤ p < ∞ and α > 0. If T is a bounded linear operator from Zp into Bα , then T is compact if and only if ∥TFn ∥Bα → 0 as n → ∞ for any sequence {Fn } in Zp bounded in norm which converges to 0 uniformly in D. Theorem 8. Let 1 ≤ p < ∞, α > 0, ψ, ϕ ∈ H (D) with ϕ(D) ⊆ D. If Wψ,ϕ : Zp → Bα is bounded, then the following statements are equivalent: (a) Wψ,ϕ : Zp → Bα is compact. (b) Wψϕ ′ ,ϕ : Bp → A−α is compact. (c)



1+|ϕ(z )|

sup (1 − |z |2 )α |ψ(z )ϕ ′ (z )| log 1−|ϕ(z )|

|ϕ(z )|→1

1−1/p

= 0.

Proof. The equivalence of (b) and (c) is an immediate consequence of Corollary 2. Therefore, it suffices to show that (a) and (b) are equivalent. (a) ⇒ (b) Assume (a) holds. Let {fn } be a bounded sequence in Bp converging to 0 uniformly on compact subsets of D. For each n ∈ N, let Fn be the antiderivative of fn fixing 0. Then Fn ∈ Zp and converges to 0 uniformly on compact subsets of D. Since ∥Fn ∥Zp = ∥fn ∥Bp , which is bounded, by Lemma 7, some subsequence {Fnk } converges uniformly in D and its limit is 0. By the hypothesis and Lemma 9, it follows that ∥Wψ,ϕ Fnk ∥Bα → 0 as n → ∞. It follows that sup(1 − |z |2 )α |ψ(z )ϕ ′ (z )fnk (ϕ(z ))| ≤ ∥Wψ,ϕ Fnk ∥Bα + sup(1 − |z |2 )α |ψ ′ (z )Fnk (ϕ(z ))| → 0, z ∈D

z ∈D

as n → ∞. Therefore, ∥Wψϕ ′ ,ϕ fnk ∥−α → 0. By Lemma 3.7 of [17], we conclude that Wψϕ ′ ,ϕ : Bp → A−α is a compact operator. (b) ⇒ (a) Suppose (b) holds. Let {Fn } be a sequence in Zp bounded in norm and converging to 0 uniformly in D. By Lemma 9, it suffices to show that ∥Wψ,ϕ Fn ∥Bα → 0 as n → ∞. Since {Fn′ } is a bounded sequence in Bp converging to 0 uniformly on compact subsets of D, by Lemma 3.7 of [17], we have

∥Wψ,ϕ Fn ∥Bα ≤ |ψ(0)Fn (ϕ(0))| + sup(1 − |z |2 )α |ψ ′ (z )||Fn (ϕ(z ))| z ∈D

+ sup(1 − |z |2 )α |ψ(z )ϕ ′ (z )Fn′ (ϕ(z ))| ≤ |ψ(0)Fn (ϕ(0))| z ∈D

+ ∥ψ∥Bα max |Fn (ϕ(z ))| + ∥Wψϕ ′ ,ϕ Fn′ ∥−α → 0, as n → ∞.  z ∈D

The following two results follow from Theorems 1 and 2 using the same arguments adopted in the proofs of Theorems 7 and 8.

610

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611 Table 1 On the boundedness of Mψ : Bp , BMOA → Hµ∞ .

µ(z )

B1

1

ψ ∈H

(1 − |z |2 )α

ψ ∈ A−a

e

−

e

1+|z | 1−|z |

1+|z |

α

BMOA or B

ψ =0 

supz ∈D |ψ(z )|(1 − |z |2 )α log 1−|z |

|ψ(z )| ≤ Ce

1+|z | 1−|z |

log 1−|z |

Bp ∞

1+|z | 1−|z |

supz ∈D |ψ(z )|e

−

1+|z | 1−|z |



1+|z |

log 1−|z |

1+|z |

1− 1p

1− 1p

ψ =0 <∞

<∞

supz ∈D |ψ(z )|(1−|z |2 )α log 1−|z | < ∞ 1+|z |

supz ∈D |ψ(z )|e

ψ =0

ψ =0

ψ =0

ψ =0

ψ =0

ψ =0

−

1+|z | 1−|z |

log 1−|z | < ∞ 1+|z |

Table 2 On the boundedness of Cϕ : Bp , BMOA → Hµ∞ .

µ(z )

B1

1

All ϕ

∥ϕ∥∞ < 1

∥ϕ∥∞ < 1

(1 − |z |2 )α

All ϕ

|ϕ(z )| ≤

|ϕ(z )| ≤

ϕ=0

|ϕ(z )| ≤

ϕ=0

ϕ=0

e

1 − 1−| z|



log

1+|z | 1−|z |

α

Bp

BMOA or B 2 −α q −1 eM (1−|z | ) 2 −α q +1 eM (1−|z | ) q(1−|z |)−1 Me e −1 q(1−|z |)−1 eMe +1

|ϕ(z )| ≤

2 −α −1 eM (1−|z | ) 2 −α eM (1−|z | ) +1 (1−|z |)−1 Me e −1 −1 ( 1 −| z |) eMe +1

ϕ=0

Theorem 9. Let α > 0, ψ, ϕ ∈ H (D) with ϕ(D) ⊆ D. Then the following statements are equivalent: (a) Wψ,ϕ : Z∗ → Bα is bounded. (b) Wψ,ϕ : Z → Bα is bounded. (c) ψ ∈ Bα and Wψϕ ′ ,ϕ : BMOA → A−α is bounded. (d) ψ ∈ Bα and Wψϕ ′ ,ϕ : B → A−α is bounded. 1+|ϕ(z )| (e) ψ ∈ Bα , ψϕ ′ ∈ A−α and sup(1 − |z |2 )α |ψ(z )ϕ ′ (z )| log 1−|ϕ(z )| < ∞. z ∈D

Theorem 10. Let α > 0, ψ, ϕ ∈ H (D) with ϕ(D) ⊆ D. If Wψ,ϕ is bounded as an operator from Z∗ or Z into Bα , then the following statements are equivalent: (a) Wψ,ϕ : Z∗ → Bα is compact. (b) Wψ,ϕ : Z → Bα is compact. (c) Wψϕ ′ ,ϕ : BMOA → A−α is compact. (d) Wψϕ ′ ,ϕ : B → A−α is compact. 1+|ϕ(z )| (e) lim (1 − |z |2 )α |ψ(z )ϕ ′ (z )| log 1−|ϕ(z )| = 0. |ϕ(z )|→1

The equivalence of (b) and a condition equivalent to (e) in Theorems 9 and 10 was shown in [13] in the special case when

α = 1.

7. Examples We conclude the paper by giving in Tables 1 and 2 some examples of weights µ and the corresponding conditions on the symbols that guarantee boundedness of the component operators. As the conditions for compactness are the corresponding ‘‘little-oh’’ versions, separate tables will not be displayed. − 1+|z |

1+z

If µ = e 1+|z | and ψ = e 1−z or any disk automorphism, then Mψ : B1 → Hµ∞ is a bounded operator. In Table 2, the constant q is the conjugate index of p. Acknowledgment We wish to express our gratitude to the referee for his/her insightful comments and suggestions for the improvement of the manuscript. References [1] R.F. Allen, Weighted composition operators from the Bloch space to weighted Banach spaces of holomorphic functions on a bounded homogeneous domain, Preprint. [2] R.F. Allen, F. Colonna, Weighted composition operators on the Bloch space of a bounded homogeneous domain, Oper. Theory: Advances and Applications 202 (2009) 11–37. [3] J. Arazy, S.D. Fisher, Some aspects of the minimal, Möbius-invariant space of analytic functions on the unit disc, in: Interpolation spaces and allied topics in analysis (Lund, 1983), in: Lecture Notes in Math., vol. 1070, Springer, Berlin, 1984, pp. 24–44. [4] J. Arazy, S.D. Fisher, J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985) 110–145. [5] O. Blasco, Composition operators on the minimal space invariant under Möbius transformations, in: Complex and harmonic analysis, DEStech Publ. Inc., Lancaster, PA, 2007, pp. 157–166.

F. Colonna, M. Tjani / J. Math. Anal. Appl. 402 (2013) 594–611 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

611

F. Colonna, The Bloch constant of bounded analytic functions, J. London Math. Soc. (2) 36 (no. 1) (1987) 95–101. F. Colonna, Weighted composition operators between H ∞ and BMOA, Bull. Korean Math. Soc. 50 (1) (2013) 185–200. P.L. Duren, Theory of H p Spaces, Academic press, New York, 1970. D. Girela, Integral means and BMOA-norms of logarithms of univalent functions, J. Lond. Math. Soc. 33 (no. 1) (1986) 117–132. D. Girela, Analytic functions of bounded mean oscillation, in: Complex Function Spaces (Mekrijärvi, 1999), in: Univ. Joensuu Dept. Math. Rep. Ser., No. 4, 2001, pp. 61–170. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, in: Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. T. Hosokawa, K. Izuchi, S. Ohno, Topological structure of the space of weighted composition operators on H ∞ , Integr. Equ. Oper. Theory 53 (2005) 509–526. S. Li, S. Stević, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput. 206 (2008) 825–831. S. Stević, Norm of weighted composition operators from Bloch space to Hµ∞ on the unit ball, Ars. Combin. 88 (2008) 125–127. S. Stević, Essential norms of weighted composition operators from the a-Bloch space to a weighted-type space on the unit ball, Abstr. Appl. Anal. (2008) 11. Art. ID 279691. M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Doctoral dissertation, Michigan State University, 1996. M. Tjani, Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355 (2003) 4683–4698. K. Zhu, Operator Theory on Function Spaces, in: Pure and Applied Mathematics, vol. 139, New York and Basel, 1990. K. Zhu, Analytic Besov spaces, J. Math. Anal. Appl. 157 (1991) 318–336.