CUBO A Mathematical Journal Vol.19, No¯ 01, (39–51). March 2017 Maximal functions and properties of the weighted composition operators acting on the Korenblum, α-Bloch and α-Zygmund spaces Gabriel M. Antón Marval1, René E. Castillo2, Julio C. Ramos-Fernández3 1 Area de Matemáticas, Universidad Nacional Experimental de Guayana, Puerto Ordaz 8050, Estado Boĺıvar, Venezuela. 2 Departamento de Matemáticas, Universidad Nacional de Colombia, AP360354 Bogotá, Colombia. 3 Departamento de Matemáticas, Universidad de Oriente, Cumaná 6101, Estado Sucre, Venezuela. gabman@gmail.com, recastillo@unal.edu.co, jcramos@udo.edu.ve ABSTRACT Using certain maximal analytic functions, we obtain new characterizations of the con- tinuity and compactness of the weighted composition operators when acts between Korenblum spaces, α-Bloch spaces and when acts from certain weighted Banach spaces of analytic functions with a logarithmic weight into α-Bloch spaces. As consequence of our results, we obtain a new characterization of the continuity and compactness of composition operators acting between α-Zygmund spaces. RESUMEN Usando ciertas funciones anaĺıticas maximales, obtenemos nuevas caracterizaciones de la continuidad y compacidad de operadores de composición con pesos cuando actúan entre espacios de Korenblum, espacios α-Bloch y cuando actúan desde ciertos espa- cios de Banach de funciones anaĺıticas con un peso logaŕıtmico en espacios α-Bloch. Como consecuencia de nuestros resultados, obtenemos una nueva caracterización de la continuidad y la compacidad de operadores de composición actuando entre espacios α-Zygmund. Keywords and Phrases: Weighted Banach spaces of analytic functions, Bloch space, weighted composition operators. 2010 AMS Mathematics Subject Classification: 30D45, 47B33. 40 G. Antón Marval, R. Castillo, J. Ramos-Fernández CUBO 19, 1 (2017) 1 Introduction Over recent years, there has been a growing interest in the study of the properties of the weighted composition operators when acts between Banach spaces of analytic functions on D, the open unit disk of the complex plane C. For fixed holomorphic functions u : D → C and φ : D → D, we can define the linear operator Wu,φ : H(D) → H(D), where H(D) denotes the space of all holomorphic functions, by Wu,φ(f) := u · (f ◦ φ). Which is known as the weighted composition operator with symbols u and φ. Clearly, if u ≡ 1 we have W1,φ(f) = f ◦φ = Cφ(f), the composition operator Cφ, and if φ(z) = id(z) = z for all z ∈ D, we obtain Wu,id(f) = u · f = Mu(f), the multiplication operator Mu. Furthermore, we can see that Wu,φ is 1-1 on H(D) unless that u ≡ 0 or φ is a constant function. Montes-Rodŕıguez in [16] and Contreras and Hernández-Dı́az in [9] characterized the continu- ity, the compactness and calculated the essential norm of Wu,φ : H ∞ v → H ∞ w in terms of a quotient involving the associated weight of v, under the requirement that the weights (a weight function is a bounded, continuous and positive function defined on D) v and w are radial (v(|z|) = v(z) for all z ∈ D), non-increasing (i.e. v(r1) ≥ v(r2) for all 0 < r1 < r2 < 1) and typical (that is, lim|z|→1− v(z) = 0). Here, H ∞ v is the weighted Banach spaces of analytic functions or growth spaces which consist of functions f ∈ H(D) such that ∥f∥H∞ v = sup z∈D v(z)|f(z)| < ∞. (1.1) The associated weight of v, denoted by ṽ, is defined by ṽ(z) = ( sup ∥f∥H∞ v ≤1 {|f(z)|} ) −1 , z ∈ D. (1.2) The weighted Banach spaces of analytic functions or growth spaces H∞ v are natural general- izations of H∞, the space of all bounded analytic functions on the open unit disk D. It is known that H∞v is a Banach space with the norm defined in (1.1). Initially, interest in the weighted Banach spaces of analytic functions were oriented to studying the growth conditions of analytic functions and the duality of these spaces, being present in various areas such as complex analysis, Fourier analysis, spectral theory and partial differential equations. Some examples may be found in [3]. The associated weight ṽ of v was introduced by Anderson and Duncan in [1] and studied by Bierstedt, Bonet and Taskinen in [2]. The associated weight is a very important tool in the study of the weighted Banach spaces of analytic functions, it is known that the space H∞ !v is isometrically equal to H∞ v , that is ∥f∥ H∞ !v = ∥f∥ H∞ v for all f ∈ H∞ v and v is dominated by its associated weight, that is, ṽ ≥ v > 0 on D. We said that a weight v is essential if ṽ ∼ v, that is, if there is a constant Cv > 0 such that v(z) ≤ ṽ(z) ≤ Cvv(z) for all z ∈ D. We refer to the interested reader the works [2] and [4] for more properties of the associated weight and the space H∞v . CUBO 19, 1 (2017) Maximal functions and properties of the weighted composition . . . 41 The Korenblum’s spaces (also known as Bergman spaces) H∞ α are particular cases of the growth spaces, they are obtained when the weights are the functions vα(z) = ( 1 − |z|2 )α with z ∈ D, where the parameter α is positive and fixed. Clearly the weight vα is radial, typical, non-increasing and essential. We refer to [12] for more details about the Korenblum’s spaces. The Bloch-type space Bv are related with the growth spaces, consist of all analytic function f on D such that f′ ∈ H∞ v . Bv is a Banach space with the norm ∥f∥Bv = |f(0)| + ∥f ′∥H∞ v = |f(0)| + ∥f∥ !Bv, where, ∥f∥ !Bv = sup z∈D v(z)|f′(z)|. When v(z) = 1 − |z|2, the space Bv becomes the classical Bloch space and it is denoted by B, we refer to [23] for more details about Bloch’s space. When v(z) = ( 1 − |z|2 )α , where α > 0 is fixed, we get the α-Bloch spaces which is denoted by Bα (see [24]). The study of the properties of composition operators on Bloch-type spaces began with the celebrated work of Madigan and Matheson in [14], they characterized continuity and compactness for composition operators on the classical Bloch space B. Their results have been extended by a big numerous of authors (see [18] and the lot of references therein). However, the properties of the weighted composition operators acting on Bloch-type space is still in development and there is not much references about this subject, it is remarkable the work of Ohno, Stroethoff and Zhao [17], where they characterized the continuity and the compactness of the weighted composition operators acting between α-Bloch spaces in terms of the continuity and compactness of certain weighted composition operators acting between certain growth spaces. In [22], Tjani shows that the composition operator Cφ : B → B is compact if and only if ∥Cφ (αλ)∥B → 0 as |λ| → 1, where αλ(z) = (λ − z) / ( 1 − λz ) is the Mobius transformation of the unit disk. A similar result for weighted composition operators from Hardy spaces into logarithmic Bloch spaces was obtained by Colonna and Li in [8]. Also, Giménez, Malavé and Ramos-Fernández in [11] extended the Tjani’s result to composition operators Cφ : B → B µ , where the weight µ was taken to be a non-vanishing, complex valued holomorphic function satisfying a reasonable geometric condition on the Euclidean disk D(1, 1). More recently, by changing the functions αλ, used by Tjani, to certain maximal analytic functions σa with a ∈ D, Ramos-Fernández et. al. [15, 19, 6, 5] show that Tjani’s result can be extended to the case of composition operators acting on α-Bloch and weighted Bloch spaces. The main objective of this note is to show that the technical developed by Ramos-Fernández et. al. [15, 19, 6, 5], using certain maximal analytic functions, can be used to obtain new characteriza- tions of the continuity and compactness of the weighted composition operators acting on Korenblum and α-Bloch spaces. In fact, in Section 2, we use maximal functions in certain log-growth space and in Korenblum spaces to give a new characterization of the continuity and compactness of 42 G. Antón Marval, R. Castillo, J. Ramos-Fernández CUBO 19, 1 (2017) weighted composition operators on these spaces. As consequence of our results in Section 2, in Section 3, we give similar results for Wu,φ acting between α-Bloch spaces and in Section 4, we give new characterizations of the continuity and compactness of the composition operators acting between α-Zygmund spaces. 2 The case of weighted composition operators acting on cer- tain growth spaces We consider first the case of weighted composition operators Wu,φ from H ∞ vlog to H∞ β . Our goal is to find maximal functions ga ∈ H ∞ vlog with a ∈ D in the sense that |ga(a)| ≥ K sup ∥f∥ H∞ v log ≤1 {|f(a)|} (2.1) for some constant K > 0. Here vlog(z) = ( log ( e 1 − |z|2 )) −1 is a weight defined on D which is radial, typical and non-increasing. The weight vlog is related with the Bloch space B, in fact it is easy to see that B is continuously contained into H∞vlog . A calculation tell us that we have to consider the functions ga defined by ga(z) = 1 − |a|2 2 (1 − az) (1 − log (1 − az)) , (2.2) with z ∈ D, where log(w) = log |w| + iarg(w) denotes the logarithm whose imaginary part lies in the interval (−π,π], which is holomorphic in the open Euclidean disk with center at 1 and radius 1. Then, following the ideas in [15] and [5], we have obtained the following result: Theorem 2.1. Let β > 0 be fixed, φ : D → D and u : D → C holomorphic functions. Then (1) The weighted composition operator Wu,φ is continuous from H ∞ vlog to H∞ β if and only if sup a∈D ∥Wu,φ (ga)∥H∞ β < ∞. (2.3) (2) The operator Wu,φ : H ∞ vlog → H∞ β is compact if and only if lim |a|→1− ∥Wu,φ (ga)∥H∞ β = 0. (2.4) Proof. Suppose first that Wu,φ : H ∞ vlog → H∞ β is continuous. Observe that for each a ∈ D, we CUBO 19, 1 (2017) Maximal functions and properties of the weighted composition . . . 43 have ∥ga∥H∞ v log = sup z∈D 1 log ( e 1−|z|2 ) · 1 − |a|2 2 |1 − az| |1 − log (1 − az)| ≤ sup z∈D |1 − log |1 − az|| + |arg (1 − az)| log ( e 1−|z|2 ) ≤ sup z∈D 1 − log (1 − |z|) + π log ( e 1−|z|2 ) ≤ sup z∈D ⎛ ⎝ log ( e 1−|z|2 ) log ( e 1−|z|2 ) + log(1 + |z|) log ( e 1−|z|2 ) ⎞ ⎠ + π ≤ 1 + log(2) + π. Then, we can see that there exists a constant K > 0 such that ∥Wu,φ (ga)∥H∞ β ≤ ∥Wu,φ∥ ∥ga∥Hv log ≤ K ∥Wu,φ∥ and (2.3) follows. Conversely, if (2.3) is true, then for each s ∈ D we have (1 − |s|2)β vlog (φ(s)) |u(s)| = 2(1 − |s|2)β ∣ ∣gφ(s) (φ(s)) ∣ ∣ |u(s)| ≤ 2 sup a∈D ∥Wu,φ (ga)∥H∞ β and the item (1) follows from the continuity’s theorem of Montes-Rodŕıguez [16] (see also Contreras and Hernández-Dı́az [9]) since the weight vlog is essential. Now we are going to show the item (2). In virtue of Tjani’s lemma in [21] and since ga is a bounded sequence in H∞ vlog which converges to zero uniformly on compact subsets of D, we conclude that if the operator Wu,φ : H ∞ vlog → H∞ β is compact then the relation (2.4) holds. Conversely, if the relation (2.4) is true, then for every ε > 0 we can find r1 ∈ (1/2, 1) such that ∥Wu,φ (ga)∥H∞ β < ε whenever r1 < |a| < 1. Hence, if z ∈ D satisfies |φ(z)| > r1, then we can write (1 − |z|2)β vlog (φ(z)) |u(z)| = 2(1 − |z|2)β ∣ ∣gφ(z) (φ(z)) ∣ ∣ |u(z)| ≤ sup w∈D (1 − |w|2)β ∣ ∣gφ(z) (φ(w)) ∣ ∣ |u(w)| = ∥ ∥Wu,φ ( gφ(z) )∥ ∥ H∞ β < ε. This shows that lim |φ(z)|→1− (1 − |z|2)β vlog (φ(z)) |u(z)| = 0 and Wu,φ : H ∞ vlog → H∞ β is compact in virtue of a result due to Montes-Rodŕıguez [16] (see also Contreras and Hernández-Dı́az [9]) and the fact that vlog is an essential weight. ! Next, we are going to consider the case of the weighted composition operator Wu,φ acting from H∞ α to H∞ β , where α and β are positive parameters fixed. We need to find maximal functions σ (α) a ∈ H ∞ α in the sense of the relation (2.1), changing, of course, the space H ∞ vlog by H∞α . 44 G. Antón Marval, R. Castillo, J. Ramos-Fernández CUBO 19, 1 (2017) In this case, for a ∈ D, we consider functions σ (α) a defined by σ(α) a (z) = 1 − |a| (1 − az)α+1 , (2.5) where z ∈ D. Then, the argument in the proof of Theorem 2.1 allow us to show the following result: Theorem 2.2. Let α,β > 0 be fixed and u : D → C, φ : D → D holomorphic functions. (1) The weighted composition operators Wu,φ is bounded from H ∞ α to H ∞ β if and only if sup a∈D ∥ ∥ ∥Wu,φ ( σ(α)a )∥ ∥ ∥ H∞ β < ∞. (2.6) (2) The weighted composition operators Wu,φ : H ∞ α → H ∞ β is compact if and only if lim |a|→1− ∥ ∥ ∥Wu,φ ( σ(α) a )∥ ∥ ∥ H∞ β = 0. (2.7) Proof. The result follows arguing as in the proof of Theorem 2.1. Indeed, if (2.6) is true, then for any s ∈ D, we have (1 − |s|2)β ( 1 − |φ(s)| 2 )α |u(s)| = (1 − |s| 2)β (1 + |φ(s)|) ∣ ∣ ∣σ (α) φ(s) (φ(s)) ∣ ∣ ∣ |u(s)| ≤ 2 sup a∈D ∥ ∥ ∥Wu,φ ( σ(α) a )∥ ∥ ∥ H∞ β and the continuity of Wu,φ : H ∞ α → H∞ β follows from the continuity’s theorem of Montes-Rodŕıguez [16] (see also Contreras and Hernández-Dı́az [9]). Conversely, if Wu,φ : H ∞ α → H∞ β is continuous, then we have sup a∈D ∥ ∥ ∥Wu,φ ( σ(α) a )∥ ∥ ∥ H∞ β ≤ ∥Wu,φ∥ sup a∈D ∥ ∥ ∥σ(α)a ∥ ∥ ∥ H∞ α = ∥Wu,φ∥ sup a∈D sup z∈D ( 1 − |z|2 )α 1 − |a| |1 − az| α+1 ≤ 2α ∥Wu,φ∥ . This shows the equivalence in the item 1. Now, since σ (α) a converges to zero uniformly on compact subsets of D as |a| → 1 − and∥ ∥ ∥σ (α) a ∥ ∥ ∥ H∞ α ≤ 2α for all a ∈ D, Tjani’s lemma in [21] implies that if Wu,φ : H ∞ α → H∞ β is compact, then lim |a|→1− ∥ ∥ ∥Wu,φ ( σ(α) a )∥ ∥ ∥ H∞ β = 0. CUBO 19, 1 (2017) Maximal functions and properties of the weighted composition . . . 45 Conversely, if the relation (2.7) is true, then for every ε > 0 we can find r1 ∈ (1/2, 1) such that∥ ∥ ∥Wu,φ ( σ (α) a )∥ ∥ ∥ H∞ β < ε whenever r1 < |a| < 1. Hence, if z ∈ D satisfies |φ(z)| > r1, then we can write (1 − |z|2)β ( 1 − |φ(z)| 2 )α |u(z)| = 2(1 − |z| 2)β ∣ ∣ ∣σ (α) φ(z) (φ(z)) ∣ ∣ ∣ |u(z)| ≤ 2 sup w∈D (1 − |w|2)β ∣ ∣ ∣σ (α) φ(z) (φ(w)) ∣ ∣ ∣ |u(w)| = ∥ ∥ ∥Wu,φ ( σ (α) φ(z) )∥ ∥ ∥ H∞ β < ε. This shows that lim |φ(z)|→1− (1 − |z|2)β ( 1 − |φ(z)| 2 )α |u(z)| = 0 and Wu,φ : H ∞ α → H ∞ β is compact in virtue of a result due to Montes-Rodŕıguez [16] (see also Contreras and Hernández-Dı́az [9]). ! As a consequence of our results, we obtain a recent result about the continuity and compactness of composition operators acting between α-Bloch spaces due to Malavé and Ramos-Fernández [15]. For a ∈ D we set λ(α) a (z) = αa ∫ z 0 σ(α) a (s)ds = (1 − |a|) [ 1 (1 − az)α − 1 ] . (2.8) Then we have the following result: Corolary 1. Let α,β > 0 be fixed and φ : D → D an holomorphic function. Then (1) The operator Cφ from B α to Bβ is bounded if and only if sup a∈D ∥ ∥ ∥Cφ ( λ(α) a )∥ ∥ ∥ Bβ < ∞. (2) The operator Cφ : B α → Bβ is compact if and only if lim |a|→1− ∥ ∥ ∥Cφ ( λ(α) a )∥ ∥ ∥ Bβ = 0. Proof. This result follows from the fact that Cφ : B α → Bβ is continuous (resp. compact) if and only if Wφ′,φ : H ∞ α → H∞ β is continuous (resp. compact). ! 3 Application. A new criterion for the continuity and com- pactness of weighted composition operators acting be- tween α-Bloch spaces Now, we are going to use the functions σ (α) a , λ (α) a and ga defined in (2.5), (2.8) and (2.2) to charac- terize the continuity and compactness of the weighted composition operator Wu,φ acting between 46 G. Antón Marval, R. Castillo, J. Ramos-Fernández CUBO 19, 1 (2017) α-Bloch spaces. According to the continuity’s results in [17, Theorem 2.1] and [16, Theorem 2.1] (see also [9, Proposition 3.1]) we arrive to the following result: Theorem 3.1. Let α,β > 0 fixed, φ : D → D and u : D → C holomorphic functions. Then (1) If 0 < α < 1, then the operator Wu,φ from B α into Bβ is continuous if and only if u ∈ Bβ and sup a∈D ∥ ∥ ∥IuCφ ( λ(α) a )∥ ∥ ∥ Bβ < ∞. (2) If α = 1, the operator Wu,φ from B into B β is continuous if and only if sup a∈D ∥JuCφ (ga)∥Bβ < ∞ and sup a∈D ∥ ∥ ∥IuCφ ( λ(1) a )∥ ∥ ∥ Bβ < ∞. (3) If α > 1, then the operator Wu,φ from B α into Bβ is continuous if and only if sup a∈D ∥ ∥ ∥JuCφ ( σ(α−1) a )∥ ∥ ∥ Bβ < ∞ and sup a∈D ∥ ∥ ∥IuCφ ( λ(α) a )∥ ∥ ∥ Bβ < ∞. Where for u ∈ H(D) fixed and f ∈ H(D), we define the operators Iu(f) and Ju(f) by Iu(f)(z) = ∫ z 0 f′(w)u(w)dw and Ju(f)(z) = ∫ z 0 f(w)u′(w)dw, with z ∈ D. Example 1. If u ≡ 1, then Ju is the null operator and we obtain that Cφ : B α → Bβ is continuous if and only if sup a∈D ∥ ∥ ∥Cφ ( λ(α)a )∥ ∥ ∥ Bβ < ∞. Again, we have extended a result in [15]. The same argument in the paragraph before Theorem 3.1 allow us to show a result about the compactness of the weighted composition operators Wu,φ acting between α-Bloch spaces. In fact, by [17, Theorem 3.1], [16, Theorem 2.1] (or [9, Corollary 4.3]) and the item 2 in our Theorem 2.1 and Theorem 2.2, imply the following result: Theorem 3.2. Let α,β > 0 be fixed, φ : D → D and u : D → C holomorphic functions. Then (1) If 0 < α < 1, then Wu,φ : B α → Bβ is compact if and only if u ∈ Bβ and lim |a|→1− ∥ ∥ ∥IuCφ ( λ(α) a )∥ ∥ ∥ Bβ = 0. (2) If α = 1, then Wu,ϕ : B → B β is compact if and only if lim |a|→1− ∥JuCφ (ga)∥Bβ = 0 and lim |a|→1−i ∥ ∥ ∥IuCφ ( λ(1) a )∥ ∥ ∥ Bβ = 0. CUBO 19, 1 (2017) Maximal functions and properties of the weighted composition . . . 47 (3) If α > 1, then Wu,φ : B α → Bβ is compact if and only if lim |a|→1− ∥ ∥ ∥JuCφ ( σ(α−1) a )∥ ∥ ∥ Bβ = 0, and lim |a|→1− ∥ ∥ ∥IuCφ ( λ(α) a )∥ ∥ ∥ Bβ = 0. Example 2. If u ≡ 1, then we have that the composition operator Cφ : B α → Bβ is compact if and only if lim |a|→1− ∥ ∥ ∥Cφ ( λ(α)a )∥ ∥ ∥ Bβ = 0. This extend a result in [15]. 4 Application. Continuity and compactness of the compo- sition operators acting between α-Zygmund spaces For α > 0 fixed, the α-Zygmund space, denoted by Zα, consist of all holomorphic functions f on D such that ∥f∥ !Zα := sup z∈D ( 1 − |z|2 )α |f′′(z)| < ∞. Clearly, f ∈ Zα if and only if f ′ ∈ Bα. Also, it is easy to see that Zα is a Banach space with the norm ∥f∥Zα = |f(0)| + |f ′(0)| + ∥f∥ !Zα = |f(0)| + ∥f ′∥Bα . We will use our results in the above sections to characterize the continuity and the compactness of the composition operator Cφ acting between α-Zygmund spaces in terms of the composition of certain special functions in Zα. The key of our result repose in the following lemma: Lemma 4.1. Let α,β > 0 be fixed and φ : D → D an holomorphic function. Then (1) The operator Cφ from Zα to Zβ is bounded if and only if Wφ′,φ : B α → Bβ is bounded. (2) The operator Cφ : Zα → Zβ is compact if and only if Wφ′,φ : B α → Bβ is compact. Proof. Suppose first that the operator Cφ : Zα → Zβ is bounded. There exists a constant L > 0 such that ∥Cφ (g)∥Zβ ≤ L ∥g∥Zα for all g ∈ Zα. Thus, for any f ∈ B α , we have that g(z) = ∫z 0 f(s)ds, belongs to Zα and therefore, ∥Wφ′,φ (f)∥Bβ = ∥φ ′ · f ◦ φ∥Bβ = ∥φ ′ · g′ ◦ φ∥Bβ = ∥ ∥(g ◦ φ) ′ ∥ ∥ Bβ ≤ |(g ◦ φ) (0)| + ∥ ∥(g ◦ φ) ′ ∥ ∥ Bβ = ∥Cφ(g)∥Zβ ≤ L ∥g∥Zα = L ∥f∥Bα since g(0) = 0. This shows that Wφ′,φ : B α → Bβ is bounded. 48 G. Antón Marval, R. Castillo, J. Ramos-Fernández CUBO 19, 1 (2017) Conversely, if Wφ′,φ : B α → Bβ is bounded, then there exists a constant L > 0 such that ∥Wφ′,φ (g)∥Bβ ≤ L ∥g∥Bα for all g ∈ B α . Thus, for any f ∈ Zα, we have that g = f ′ ∈ Bα and hence ∥ ∥(f ◦ φ) ′ ∥ ∥ Bβ = ∥Wφ′,φ (g)∥Bβ ≤ L ∥f ′∥Bα . Furthermore, since f ∈ H(D), we can write |f (φ(0))| ≤ |f(0)| + |f′(0)| + ∫ φ(0) 0 ∫ s 0 |f′′(w)| |dw| |ds|. Therefore, multiplying and dividing by ( 1 − |w|2 )α inside the integral symbol, we can find a con- stant K > 0, which can depend on φ(0), such that |f (φ(0))| ≤ K ∥f∥Zα. We conclude then that ∥Cφ (f)∥Zβ = |f (φ(0))| + ∥ ∥(f ◦ φ) ′ ∥ ∥ Bβ ≤ (K + L) ∥f∥Zα and the operator Cφ : Zα → Zβ is bounded. This shows the item (1). Now we are going to show the item (2). Suppose first that Wφ′,φ : B α → Bβ is a compact operator and let {gn} be a bounded sequence in Zα, then the sequence {g ′ n} is a bounded sequence in Bα. Hence, by passing to a subsequence, we can suppose that {Wφ′,φ (g ′ n)} converges in B β to a function h ∈ Bβ. Also, {gn (φ(0))} is a bounded sequence in C and, by passing to a subsequence, we can suppose that {gn (φ(0))} goes to z0 ∈ C as n → ∞. Then, we can define the function g(z) = z0 + ∫z 0 h(s)ds, which belongs to Zβ and we have that ∥Cφ (gn) − g∥Zβ = |gn (φ(0)) − g(0)| + ∥Wφ ′,φ (g ′ n ) − h∥ Bβ → 0 as n → ∞. That is, Cφ : Zα → Zβ is a compact operator. Conversely, if Cφ : Zα → Zβ is a compact operator and {fn} is a bounded sequence in B α , then the sequence {gn} defined by gn(z) = ∫ z 0 fn(s)ds is bounded in Zα and hence, by passing to a subsequence, we can suppose that {Cφ (gn)} converges in Zβ to a function h ∈ Zβ. Therefore, we can write ∥Wφ′,φ (fn) − h ′∥Bβ = ∥Wφ′,φ (g ′ n) − h ′∥Bβ = ∥ ∥(Cφ (gn) − h) ′ ∥ ∥ Bβ ≤ ∥Cφ (gn) − h∥Zβ → 0 as n → ∞ and the result follows since h′ ∈ Bβ. ! As consequence of above lemma, [17, Theorem 2.1], [16, Theorem 2.1] (or [9, Proposition 3.1]) and our Theorem 2.1, Theorem 2.2 and Theorem 3.1, we have: CUBO 19, 1 (2017) Maximal functions and properties of the weighted composition . . . 49 Corolary 2. Let α,β > 0 be fixed and φ : D → D an holomorphic function. Then, for each case, the following propositions are equivalents: (1) Case 0 < α < 1. (a) The operator Cφ from Zα to Zβ is continuous, (b) φ ∈ Zβ and W(φ′)2,φ : H ∞ α → H ∞ β is continuous, (c) φ ∈ Zβ and sup w∈D (1 − |w|2)β (1 − |φ(w)| 2 )α |φ′(w)| 2 < ∞. (d) φ ∈ Zβ and sup a∈D ∥ ∥ ∥W(φ′)2,φ ( σ(α)a )∥ ∥ ∥ H∞ β < ∞. (e) φ ∈ Zβ sup a∈D ∥ ∥ ∥Iφ′Cφ ( λ(α) a )∥ ∥ ∥ Bβ < ∞. (2) Case α = 1. (a) The operator Cφ from Zα to Zβ is continuous, (b) Wφ′′,φ : H ∞ vlog → H∞ β and W(φ′)2,φ : H ∞ 1 → H∞ β are continuous, (c) sup w∈D (1 − |w|2)β vlog (φ(w)) |φ′′(w)| < ∞, and sup w∈D (1 − |w|2)β 1 − |φ(w)| 2 |φ′(w)| 2 < ∞. (d) sup a∈D ∥Wφ′′,φ (ga)∥H∞ β < ∞ and sup a∈D ∥ ∥ ∥W(φ′)2,φ ( σ(1) a )∥ ∥ ∥ H∞ β < ∞ (e) sup a∈D ∥Jφ′Cφ (ga)∥Bβ < ∞ and sup a∈D ∥ ∥ ∥Iφ′Cφ ( λ(1) a )∥ ∥ ∥ Bβ < ∞. (3) Case α > 1. (a) The operator Cφ from Zα to Zβ is continuous, (b) Wφ′′,φ : H ∞ α−1 → H∞ β and W(φ′)2,φ : H ∞ α → H∞ β are continuous, (c) sup w∈D (1 − |w|2)β (1 − |φ(w)| 2 )α−1 |φ′′(w)| < ∞, and sup w∈D (1 − |w|2)β (1 − |φ(w)| 2 )α |φ′(w)| 2 < ∞. (d) sup a∈D ∥ ∥ ∥Wφ′′,φ ( σ(α−1) a )∥ ∥ ∥ H∞ β < ∞ and sup a∈D ∥ ∥ ∥W(φ′)2,φ ( σ(α) a )∥ ∥ ∥ H∞ β < ∞ (e) sup a∈D ∥ ∥ ∥Jφ′Cφ ( σ(α−1) a )∥ ∥ ∥ Bβ < ∞ and sup a∈D ∥ ∥ ∥Iφ′Cφ ( λ(α) a )∥ ∥ ∥ Bβ < ∞. Similar results can be found for the compactness of the operator Cφ from Zα to Zβ. Also, it is possible to write a characterizations using the Zygmund’s norm and the fact that ∥f∥Zα = |f(0)| + ∥f′∥Bα. Here, we can use the operators Ĩuf(z) = ∫z 0 Iu(f(s))ds and J̃uf(z) = ∫z 0 Ju(f(s))ds. 50 G. Antón Marval, R. Castillo, J. Ramos-Fernández CUBO 19, 1 (2017) Acknowledgement. The authors would like to thank the referees for their valuable comments which helped to improve the manuscript. References [1] J. M. Anderson and J. Duncan, Duals of Banach spaces of entire functions. Glasgow Math. J. 32 (1990), no. 2, 215–220. [2] K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions. Studia Math. 127 (1998), 137–168. [3] K. D. Bierstedt, R. Meise and W. H. 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