CUBO, A Mathematical Journal ◦ Vol.22, N 02, (215–231). August 2020 Received: 12 March, 2019 | Accepted: 14 July, 2020 D-metric Spaces and Composition Operators Between Hyperbolic Weighted Family of Function Spaces A. Kamal1 and T.I.Yassen2 1 Port Said University, Faculty of Science, Department of Mathematics and Computer Science, Port Said, Egypt. 2 The Higher Engineering Institute in Al-Minya (EST-Minya) Minya , Egypt. alaa [email protected], taha [email protected] ABSTRACT The aim of this paper is to introduce new hyperbolic classes of functions, which will ∗ ∗ be called Bα, log and Flog(p,q,s) classes. Furthermore, we introduce D-metrics space ∗ ∗ in the hyperbolic type classes Bα, log and Flog(p,q,s). These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, necessary and sufficient conditions are given for the composition operator Cφ to be bounded and ∗ ∗ compact from Bα, log to Flog(p,q,s) spaces. RESUMEN El objetivo de este art´ıculo es introducir nuevas clases hiperb´olicas de funciones, que ∗ ∗ ser´an llamadas clases Bα, log y Flog(p,q,s). A continuaci´on, introducimos D-espacios ∗ ∗ m´etricos en las clases de tipo hiperb´olicas Bα, log y Flog(p,q,s). Mostramos que estas clases son espacios m´etricos completos con respecto a las m´etricas correspondientes. M´as a´un, damos condiciones necesarias y suficientes para que el operador composici´on ∗ ∗ Cφ sea acotado y compacto desde el espacio Bα, log a Flog(p,q,s). Keywords and Phrases: D-metric spaces, Logarithmic hyperbolic classes, Composition opera- tors. 2020 AMS Mathematics Subject Classification: 47B38, 46E15. c 2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. CUBO 216 A.Kamal&T.I.Yassen 22, 2 (2020) 1 Introduction Let φ be an analytic self-map of the open unit disk D = {z ∈ C : |z| < 1} in the complex plane C and let ∂D be its boundary. Let H(D) denote the space of all analytic functions in D and let B(D) be the subset of H(D) consisting of those f ∈ H(D) for which |f(z)| < 1 for all z ∈ D. Let the Green’s function of D be defined as g(z,a) = log 1 , where ϕ (z) = a−z is the |ϕa(z)| a 1−az¯ M¨obius transformation related to the point a ∈ D. A linear composition operator Cφ is defined by Cφ(f) = (f ◦ φ) for f in the set H(D) of analyticfunctions on D (see [9]). A function f ∈ B(D) belongs to α-Bloch space Bα, 0 <α< ∞, if α ′ ||fkBα = sup(1 − |z|) |f (z)| < ∞. z∈D The little α-Bloch space Bα, 0 consisting of all f ∈Bα so that lim (1 − |z|2)|f ′(z)| =0. |z|→1− Definition 1. [15] For an analytic function f on D and 0 <α< ∞, if 2 α ′ 2 ||f|| α = sup(1 − |z| ) |f (z)| log < ∞, Blog 2 z∈D 1 − |z| α then, f belongs to the weighted α-Bloch spaces Blog. If α = 1, the weighted Bloch space Blog is the set for all analytic functions f in D for which ||f||Blog < ∞. The expression ||f||Blog defines a seminorm while the norm is defined by ||f||Blog = |f(0)| + ||f||Blog . Definition 2. [14] For 0 < p,s < ∞, −2 <q< ∞ and q + s > −1, a function f ∈ H(D) is in F (p,q,s), if sup |f ′(z)|p(1 − |z|2)qgs(z,a)dA(z) < ∞. a∈D Z D Moreover, if lim |f ′(z)|p(1 − |z|2)qgs(z,a)dA(z)=0, |a|→1− Z D then f ∈ F0(p,q,s). El-Sayed and Bakhit [5] gave the following definition: CUBO 22, 2 (2020) D-metricSpacesandCompositionOperators 217 Definition 3. For 0 < p,s < ∞, −2 <q< ∞ and q + s > −1, a function f ∈ H(D) is said to belong to Flog(p,q,s), if p 2 log |I| s ′ p 2 q 1 sup s |f (z)| (1 − |z| ) log dA(z) < ∞. I⊂∂D |I| Z |z| S(I) Where |I| denotes the arc length of I ⊂ ∂D and S(I) is the Carleson box defined by (see [8, 6]) z S(I)= {z ∈ D :1 − |I| < |z| < 1, ∈ |I|}. |z| The interest in the Flog(p,q,s)-spaces rises from the fact that they cover some well known p function spaces. It is immediate that Flog(2, 0, 1) = BMOAlog and Flog(2, 0,p) = Qlog, where 0 <p< ∞. 2 Preliminaries ∗ Definition 4. [11] The hyperbolic Bloch space Bα is defined as ∗ D 2 α ∗ Bα = {f : f ∈ B( ) and sup(1 − |z| ) f (z) < ∞}. z∈D ′ ∗ |f (z)| D Denoting f (z)= 1−|f(z)|2 , the hyperbolic derivative of f ∈ B( ). [7] ∗ ∗ ∗ The little hyperbolic Bloch space Bα, 0 is a subspace of Bα consisting of all f ∈Bα so that lim (1 − |z|2)αf ∗(z)=0. |z|→1− ∗ The space Bα is Banach space with the norm defined as ∗ α ∗ ||f||Bα = |f(0)| + sup(1 − |z|) |f (z)|. z∈D q+2 D Definition 5. For 0 <p,s< ∞, −2 <q< ∞, α = p and q + s> −1, a function f ∈ H( ) is said to belong to F ∗(p,q,s), if sup (f ∗(z))p(1 − |z|2)αp−2gs(z,a)dA(z) < ∞. a∈D ZD Definition 6. For f ∈ B(D) and 0 <α< ∞, if α 2 ||f|| ∗ = sup(1 − |z|2) (f ∗(z)) log < ∞, Bα, log 2 z∈D 1 − |z| ∗ then f belongs to the Bα, log. CUBO 218 A.Kamal&T.I.Yassen 22, 2 (2020) We must consider the following lemmas in our study: Lemma 2.1. [12] Let 0 < r ≤ t ≤ 1, then 1 1 log ≤ (1 − t2) t r Lemma 2.2. [12] Let 0 ≤ k1 < ∞, 0 ≤ k2 < ∞, and k1 − k2 > −1, then k1 1 2 −k2 C(k1, k2)= log (1 − |z| ) dA(z) < ∞. ZD |z| ∗ ∗ To study composition operators on Bα, log and Flog(p,q,s) spaces, we need to prove the fol- lowing result: q+2 Theorem 1. If 0 <p< ∞, 1 <s< ∞ and α = p with q + s > −1. Then the following are equivalent: ∗ (A) f ∈Bα, log. ∗ (B) f ∈ Flog(p,q,s). p 2 ∗ p 2 αp−2 2 s (C) sup log 1−|a|2 D (f (z)) (1 − |z| ) (1 − |ϕ(z)| ) dA(z) < ∞, a∈D pR 2 ∗ p 2 αp−2 s (D) sup log a 2 D (f (z)) (1 − |z| | g (z,a)dA(z) < ∞. D 1−| | a∈ R Proof. Let 0 <p< ∞, −2 <q< ∞, 1 <s< ∞ and 0 <r< 1. By subharmonicity we have for an analytic function g ∈ D that 1 g p g w pdA w . | (0)| ≤ 2 | ( )| ( ) πr ZD(0,r) For a ∈ D, the substitution z = ϕa(z) results in Jacobian change in measure given by ′ 2 dA(w)= |ϕa(z)| dA(z). For a Lebesgue integrable or a non-negative Lebesgue measurable function f on D, we thus have the following change of variable formula: ′ 2 f(ϕa(w))dA(w)= f(z)|ϕa(z)| dA(z). ZD(0,r) ZD(a,r) ′ f ◦ϕa Let g = 2 then we have 1−|f◦ϕa| |f ′(a)| p 1 |f ′(ϕ (w))| p f ∗ a p a dA w 2 = ( ( )) ≤ 2 2 ( ) 1 − |f(a)| πr ZD(0,r)1 − |f(ϕa(w))| 1 f ∗ z p ϕ′ z 2dA z . = 2 ( ( )) | a( )| ( ) πr ZD(a,r) CUBO 22, 2 (2020) D-metricSpacesandCompositionOperators 219 Since 1 − |ϕ (z)|2 |ϕ′ (z)| = a , a 1 − |z|2 and 1 − |ϕ (z)|2 4 a ≤ a,z ∈ D. 1 − |z|2 1 − |a|2 So we obtain that 16 f ∗ a p f ∗ z pdA z . ( ( )) ≤ 2 2 2 ( ( )) ( ) πr (1 − |a| ) ZD(a,r) ∗ 2 2 2 2 D D Again f ∈Bα, log, and (1 − |z| ) ≈ (1 − |a| ) ≈ (a, r), for z ∈ (a, r). Thus, we have p log 2 (f ∗(a))p(1 − |a|2)αp 1−|a|2 16 2 p f ∗ z pdA z ≤ 2 2 2−αp × log 2 ( ( )) ( ) πr (1 − |a| ) 1 − |a| ZD(a,r) 16 2 p f ∗ z p z 2 αp−2dA z ≤ 2 × log 2 ( ( )) (1 − | | ) ( ) πr 1 − |a| ZD(a,r) 16 2 p 1 − |ϕ (z)|2 s f ∗ z p z 2 αp−2 a dA z ≤ 2 × log 2 ( ( )) (1 − | | ) × 2 ( ) πr 1 − |a| ZD(a,r) 1 − |ϕa(z)| 16 2 p f ∗ z p z 2 αp−2 ϕ z 2 sdA z ≤ 2 2 s × log 2 ( ( )) (1 − | | ) (1 − | a( )| ) ( ) πr (1 − r ) 1 − |a| ZD(a,r) 2 p M r f ∗ z p z 2 αp−2 ϕ′ z 2 sdA z . ≤ ( ) × log 2 ( ( )) (1 − | | ) (1 − | a( )| ) ( ) 1 − |a| ZD(a,r) Where M(r) is a constant depending on r. Thus, the quantity (A) is less than or equal to constant times the quantity (C). From the fact 2 1 (1 − |ϕa(z)| ) ≤ 2 log =2g(z,a) for a, z ∈ D, |ϕa(z)| we have 2 p f ∗ z p z 2 αp−2 ϕ z 2 sdA z log 2 ( ( )) (1 − | | ) (1 − | a( )| ) ( ) 1 − |a| ZD(a,r) p 2 ∗ p 2 αp−2 s ≤ log 2 (f (z)) (1 − |z| ) g (z,a)dA(z). 1 − |a| ZD(a,r) q+2 Hence, the quantity (C) is less than or equal to a constant times (D). By taking α = p , it follows ∗ f ∈ Flog(p,q,s).