CUBO, A Mathematical Journal Vol.22, N◦02, (215–231). August 2020 http://dx.doi.org/10.4067/S0719-06462020000200215 Received: 12 March, 2019 | Accepted: 14 July, 2020 D-metric Spaces and Composition Operators Between Hyperbolic Weighted Family of Function Spaces A. Kamal 1 and T.I.Yassen 2 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 mohamed1@yahoo.com, taha hmour@yahoo.com ABSTRACT The aim of this paper is to introduce new hyperbolic classes of functions, which will be called B∗α, log and F ∗ log(p, q, s) classes. Furthermore, we introduce D-metrics space in the hyperbolic type classes B∗α, log and F ∗ log(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 B∗α, log to F ∗ log(p, q, s) spaces. RESUMEN El objetivo de este art́ıculo es introducir nuevas clases hiperbólicas de funciones, que serán llamadas clases B∗α, log y F ∗ log(p, q, s). A continuación, introducimos D-espacios métricos en las clases de tipo hiperbólicas B∗α, log y F ∗ log(p, q, s). Mostramos que estas clases son espacios métricos completos con respecto a las métricas correspondientes. Más aún, damos condiciones necesarias y suficientes para que el operador composición Cφ sea acotado y compacto desde el espacio B ∗ α, log a F ∗ log(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. http://dx.doi.org/10.4067/S0719-06462020000200215 216 A. Kamal & T.I.Yassen CUBO 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 |ϕa(z)| , where ϕa(z) = a−z 1−āz is the Möbius 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 ||f‖Bα = sup z∈D (1 − |z|)α|f′(z)| < ∞. The little α-Bloch space Bα, 0 consisting of all f ∈ Bα so that lim |z|→1− (1 − |z|2)|f′(z)| = 0. Definition 1. [15] For an analytic function f on D and 0 < α < ∞, if ||f||Bα log = sup z∈D (1 − |z|2)α|f′(z)| ( log 2 1 − |z|2 ) < ∞, then, f belongs to the weighted α-Bloch spaces Bαlog. 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 a∈D ∫ D |f′(z)|p(1 − |z|2)qgs(z, a)dA(z) < ∞. Moreover, if lim |a|→1− ∫ D |f′(z)|p(1 − |z|2)qgs(z, a)dA(z) = 0, then f ∈ F0(p, q, s). El-Sayed and Bakhit [5] gave the following definition: CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 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 sup I⊂∂D ( log 2 |I| )p |I|s ∫ S(I) |f′(z)|p(1 − |z|2)q ( log 1 |z| )s dA(z) < ∞. Where |I| denotes the arc length of I ⊂ ∂D and S(I) is the Carleson box defined by (see [8, 6]) S(I) = {z ∈ D : 1 − |I| < |z| < 1, z |z| ∈ |I|}. The interest in the Flog(p, q, s)-spaces rises from the fact that they cover some well known function spaces. It is immediate that Flog(2, 0, 1) = BMOAlog and Flog(2, 0, p) = Q p log, where 0 < p < ∞. 2 Preliminaries Definition 4. [11] The hyperbolic Bloch space B∗α is defined as B∗α = {f : f ∈ B(D) and sup z∈D (1 − |z| 2 )αf∗(z) < ∞}. Denoting f∗(z) = |f′(z)| 1−|f(z)|2 , the hyperbolic derivative of f ∈ B(D). [7] The little hyperbolic Bloch space B∗α, 0 is a subspace of B ∗ α consisting of all f ∈ B ∗ α so that lim |z|→1− (1 − |z|2)αf∗(z) = 0. The space B∗α is Banach space with the norm defined as ||f||B∗ α = |f(0)| + sup z∈D (1 − |z|)α|f∗(z)|. Definition 5. For 0 < p, s < ∞, −2 < q < ∞, α = q+2 p and q + s > −1, a function f ∈ H(D) is said to belong to F ∗(p, q, s), if sup a∈D ∫ D (f∗(z))p(1 − |z|2)αp−2gs(z, a)dA(z) < ∞. Definition 6. For f ∈ B(D) and 0 < α < ∞, if ||f||B∗ α, log = sup z∈D (1 − |z|2)α(f∗(z)) ( log 2 1 − |z|2 ) < ∞, then f belongs to the B∗α, log. 218 A. Kamal & T.I.Yassen CUBO 22, 2 (2020) We must consider the following lemmas in our study: Lemma 2.1. [12] Let 0 < r ≤ t ≤ 1, then log 1 t ≤ 1 r (1 − t2) Lemma 2.2. [12] Let 0 ≤ k1 < ∞, 0 ≤ k2 < ∞, and k1 − k2 > −1, then C(k1, k2) = ∫ D ( log 1 |z| )k1 (1 − |z|2)−k2dA(z) < ∞. To study composition operators on B∗α, log and F ∗ log(p, q, s) spaces, we need to prove the fol- lowing result: Theorem 1. If 0 < p < ∞, 1 < s < ∞ and α = q+2 p with q + s > −1. Then the following are equivalent: (A) f ∈ B∗α, log. (B) f ∈ F ∗log(p, q, s). (C) sup a∈D ( log 2 1−|a|2 )p ∫ D (f∗(z))p(1 − |z|2)αp−2(1 − |ϕ(z)|2)sdA(z) < ∞, (D) sup a∈D ( log 2 1−|a|2 )p ∫ D (f∗(z))p(1 − |z|2|αp−2gs(z, a)dA(z) < ∞. Proof. Let 0 < p < ∞, −2 < q < ∞, 1 < s < ∞ and 0 < r < 1. By subharmonicity we have for an analytic function g ∈ D that |g(0)|p ≤ 1 πr2 ∫ D(0,r) |g(w)|pdA(w). For a ∈ D, the substitution z = ϕa(z) results in Jacobian change in measure given by dA(w) = |ϕ′a(z)| 2 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: ∫ D(0,r) f(ϕa(w))dA(w) = ∫ D(a,r) f(z)|ϕ′a(z)| 2dA(z). Let g = f ′◦ϕa 1−|f◦ϕa|2 then we have ( |f′(a)| 1 − |f(a)|2 )p = (f∗(a))p ≤ 1 πr2 ∫ D(0,r) ( |f′(ϕa(w))| 1 − |f(ϕa(w))|2 )p dA(w) = 1 πr2 ∫ D(a,r) (f∗(z))p|ϕ′a(z)| 2dA(z). CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 219 Since |ϕ′a(z)| = 1 − |ϕa(z)| 2 1 − |z|2 , and 1 − |ϕa(z)| 2 1 − |z|2 ≤ 4 1 − |a|2 a, z ∈ D. So we obtain that (f∗(a))p ≤ 16 πr2(1 − |a|2)2 ∫ D(a,r) (f∗(z))pdA(z). Again f ∈ B∗α, log, and (1 − |z| 2)2 ≈ (1 − |a|2)2 ≈ D(a, r), for z ∈ D(a, r). Thus, we have ( log 2 1−|a|2 )p (f∗(a))p(1 − |a|2)αp ≤ 16 πr2(1 − |a|2)2−αp × ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))pdA(z) ≤ 16 πr2 × ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))p(1 − |z|2)αp−2dA(z) ≤ 16 πr2 × ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))p(1 − |z|2)αp−2 × ( 1 − |ϕa(z)| 2 1 − |ϕa(z)|2 )s dA(z) ≤ 16 πr2(1 − r2)s × ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ M(r) × ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))p(1 − |z|2)αp−2(1 − |ϕ′a(z)| 2)sdA(z). 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 (1 − |ϕa(z)| 2) ≤ 2 log 1 |ϕa(z)| = 2g(z, a) for a, z ∈ D, we have ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ ( log 2 1 − |a|2 )p∫ D(a,r) (f∗(z))p(1 − |z|2)αp−2gs(z, a)dA(z). Hence, the quantity (C) is less than or equal to a constant times (D). By taking α = q+2 p , it follows f ∈ F ∗log(p, q, s). Thus, the quantity (C) is less than or equal to a constant times the quantity (B). 220 A. Kamal & T.I.Yassen CUBO 22, 2 (2020) Finally, from the following inequality, let z = ϕa(w) then w = ϕa(z). Hence, ( log 2 1 − |a|2 )p∫ D (f∗(ϕa(w))) p(1 − |ϕa(w)| 2)αp−2 ( log 1 |w| )s |ϕ′a(w)| 2 dA(w) = ( log 2 1 − |a|2 )p∫ D (f∗(ϕa(w))) p(1 − |ϕa(w)| 2)αp ( log 1 |w| )s |ϕ′a(w)| 2 (1 − |ϕa(w)|2)2 dA(w) = ( log 2 1 − |a|2 )p∫ D (f∗(ϕa(w))) p(1 − |ϕa(w)| 2)αp ( log 1 |w| )s 1 (1 − |w|2)2 dA(w) ≤ ||f|| p B∗ α, log ( log 2 1 − |a|2 )p∫ D ( log 1 |w| )s (1 − |w|2)−2dA(w) = C(s, 2)||f|| p B∗ α, log . By lemma 2.2, C(s, 2) = ∫ D ( log 1 |w| )s (1 − |w|2)−2dA(w) < ∞, for 1 < s < ∞. Thus, the quantity (D) is less than or equal to a constant times the quantity (A). Hence, it is proved. Let us we give the following equivalent definition for F ∗log(p, q, s). Definition 7. For 0 < p, s < ∞, −2 < q < ∞, α = q+2 p and q + s > −1, a function f ∈ H(D) is said to belong to F ∗log(p, q, s), if sup a∈D ( log 2 1 − |a|2 )p∫ D (f∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ∞. Definition 8. A composition operator Cφ : B ∗ α, log → F ∗ log(p, q, s) is said to be bounded if there is a positive constant C so that ||Cφf||F ∗ log (p, q, s) ≤ C||f||B∗ α, log for all f ∈ B∗p, α. Definition 9. A composition operator Cφ : B ∗ α, log → F ∗ log(p, q, s) is said to be compact if it maps any ball in B∗p, α onto a precompact set in F ∗(p, q, s). The following lemma follows by standard arguments similar to those outline in [13]. Hence, we omit the proof. Lemma 2.3. Assume φ is a holomorphic mapping from D into itself. Let 0 < p, s, α < ∞, −2 < q < ∞, then Cφ : B ∗ α, log → F ∗ log(p, q, s) is compact if and only if for any bounded sequence {fn}n∈N ∈ B ∗ α, log which converges to zero uniformly on compact subsets of D as n → ∞ we have lim n→∞ ||Cφfn||F ∗ log (p,q,s) = 0. 3 D-metric space Topological properties of generalized metric space called D- metric space was introduced in [1], see for example, ([2] and [3]). This structure of D-metric space is quite different from a 2-metric space and natural generalization of an ordinary metric space in some sense. CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 221 Definition 10. [4] Let X denote a nonempty set and R the set of real numbers. A function D : X×X×X → R is said to be a D-metric on X if it satisfies the following properties: (i) D(x, y, z) ≥ 0 for all x, y, z ∈ X and equality holds if and only if x = y = z (nonnegativ- ity), (ii) D(x, y, z) = D(x, z, y) = · · ·· (symmetry), (iii) D(x, y, z) ≤ D(x, y, a) + D(x, a, z) + D(a, y, z) for all x, y, z, a ∈ X (tetrahedral inequality). A nonempty set X together with a D-metric D is called a D-metric space and is represented by (X, D). The generalization of a D-metric space with D-metric as a function of n variables is provided in Dhage [2]. Example1.1: [4] Let (X, d) be an ordinary metric space and define a function D1 on X 3 by D1(x, y, z) = max{d(x, y), d(y, z), d(z, x)}, for all x, y, z ∈ X. Then, the function D1 is a D-metric on X and (X, D1) is a D-metric space. Example1.2: [4] Let (X, d) be an ordinary metric space and define a function D2 on X 3 by D2(x, y, z) = d(x, y) + d(y, z) + d(z, x) for x, y, z ∈ X. Then, D2 is a metric on X and (X, D2) is a D-metric space. Remark 1. Geometrically, the D-metric D1 represents the diameter of a set consisting of three points x, y and z in X and the D-metric D2(x, y, z) represents the perimeter of a triangle formed by three points x, y, z in X as its vertices. Definition 11. (Cauchy sequence , completeness)[10] For every m, n > N. A sequence (xn) in a metric space X = (X, d) is said to be-Cauchy if for every ε > 0 there is an N = N(ε) such that d(xm, xn) < ε. The space X is said to be complete if every Cauchy sequence in X converges (that is, has a limit which is an element of X ). The following theorem can be found in [4]: Theorem 2. [4] Let d be an ordinary metric on X and let D1 and D2 be corresponding associated D-metrics on X. Then, (X, D1) and (X, D2) are complete if and only if (X, d) is complete. 222 A. Kamal & T.I.Yassen CUBO 22, 2 (2020) 4 D-metrics in B∗α, log and F ∗ log(p, q, s) In this section, we introduce a D-metric on B∗α, log and F ∗ log(p, q, s). Let 0 < p, s < ∞, −2 < q < ∞, and 0 < α < 1. First, we can find a D-metric in B∗α, log, for f, g, h ∈ B∗α, log by defining D(f, g, h; B∗α, log) := DB∗α, log (f, g, h) + ||f − g||Bα, log + ||g − h||Bα, log + ||h − f||Bα, log +|f(0) − g(0)| + |g(0) − h(0)| + |h(0) − f(0)|, where DB∗ α, log (f, g, h) := dB∗ α, log (f, g) + dB∗ α, log (g, h) + dB∗ α, log (h, f) and DB∗ α, log (f, g, h) := ( sup z∈D |f∗(z) − g∗(z)| + sup z∈D |g∗(z) − h∗(z)| + sup z∈D |h∗(z) − f∗(z)| ) × ( (1 − |z|2)α ( log 2 1 − |z|2 )) . Also, for f, g, h ∈ F ∗log(p, q, s) we introduce a D-metric on F ∗ log(p, q, s) by defining D(f, g, h; F ∗log(p, q, s)) := DF ∗log(p,q,s)(f, g, h) + ||f − g||Flog(p,q,s) + ||g − h||Flog(p,q,s)+ ||h − f||Flog(p,q,s) + |f(0) − g(0)| + |g(0) − h(0)| + |h(0) − f(0)|, where DF ∗ log (p,q,s)(f, g, h) := dF ∗ log (p,q,s)(f, g) + dF ∗ log (p,q,s)(g, h) + dF ∗ log (p,q,s)(h, f) and dF ∗ log (p,q,s)(f, g) := ( sup z∈D ℓ p(a) ∫ D |f∗(z) − g∗(z)|p(1 − |z|2)q(1 − |ϕ(z)|2)sdA(z) ) 1 p . Proposition 1. The class B∗α, log equipped with the D-metric D(., .; B ∗ α, log) is a complete metric space. Moreover, B∗α, log, 0 is a closed (and therefore complete) subspace of B ∗ α, log. Proof. Let f, g, h, a ∈ B∗α, log. Then, clearly (i) D(f, g, h; B∗α, log) ≥ 0, for all f, g, h ∈ B ∗ α, log. CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 223 (ii)D(f, g, h; B∗α, log) = D(f, h, g; B ∗ α, log) = D(g, h, f; B ∗ α, log). (iii)D(f, g, h; B∗α, log) ≤ D(f, g, a; B ∗ α, log) + D(f, a, h; B ∗ α, log) + D(a, g, h; B ∗ α, log) for all f, g, h, a ∈ B∗α, log. (iv)D(f, g, h; B∗α, log) = 0 implies f = g = h. Hence, D is a D-metric on B∗α, log, and (B ∗ α, log, D) is D-metric space. To prove the completeness, we use Theorem 2, let (fn) ∞ n=1 be a Cauchy sequence in the metric space (B∗α, log, d), that is, for any ε > 0 there is an N = N(ε) ∈ N such that d(fn, fm; B ∗ α, log) < ε, for all n, m > N. Since (fn) ⊂ B(D), the family (fn) is uniformly bounded and hence normal in D. Therefore, there exists f ∈ B(D) and a subsequence (fnj ) ∞ j=1 such that fnj converges to f uniformly on compact subsets of D. It follows that fn also converges to f uniformly on compact subsets, and by the Cauchy formula, the same also holds for the derivatives. Now let m > N. Then, the uniform convergence yields ∣ ∣ ∣ ∣ f∗(z) − f∗m(z) ∣ ∣ ∣ ∣ (1 − |z|2)α ( log 2 1 − |z|2 ) = lim n→∞ ∣ ∣ ∣ ∣ f ∗ n(z) − f ∗ m(z) ∣ ∣ ∣ ∣ (1 − |z|2)α ( log 2 1 − |z|2 ) ≤ lim n→∞ d(fn, fm; B ∗ α, log) ≤ ε for all z ∈ D, and it follows that ||f||B∗ α, log ≤ ||fm||B∗ α, log +ε. Thus f ∈ B∗α, log as desired. Moreover, the above inequality and the compactness of the usual B∗α, log space imply that (fn) ∞ n=1 converges to f with respect to the metric d, and (B∗α, log, D) is complete D-metric space. Since lim n→∞ d(fn, fm; B ∗ α, log) ≤ ε, the second part of the assertion follows. Next we give characterization of the complete D-metric space D(., .; F ∗log(p, q, s)). Proposition 2. The class F ∗log(p, q, s) equipped with the D-metric D(., .; F ∗ log(p, q, s)) is a complete metric space. Moreover, F ∗log, 0(p, q, s) is a closed (and therefore complete) subspace of F ∗ log(p, q, s). Proof. Let f, g, h, a ∈ F ∗log(p, q, s). Then clearly (i) D(f, g, h; F ∗log(p, q, s)) ≥ 0, for all f, g, h ∈ F ∗ log(p, q, s). (ii)D(f, g, h; F ∗log(p, q, s)) = D(f, h, g; F ∗ log(p, q, s)) = D(g, h, f; F ∗ log(p, q, s)). 224 A. Kamal & T.I.Yassen CUBO 22, 2 (2020) (iii)D(f, g, h; F ∗log(p, q, s)) ≤ D(f, g, a; F ∗ log(p, q, s)) + D(f, a, h; F ∗ log(p, q, s)) +D(a, g, h; F ∗log(p, q, s)) for all f, g, h, a ∈ F ∗log(p, q, s). (iv)D(f, g, h; F ∗log(p, q, s)) = 0 implies f = g = h. Hence, D is a D-metric on F ∗log(p, q, s), and (F ∗ log(p, q, s), D) is D-metric space. For the complete proof, by using Theorem 2, let (fn) ∞ n=1 be a Cauchy sequence in the metric space (F ∗log(p, q, s), d), that is, for any ε > 0 there is an N = N(ε) ∈ N so that d(fn, fm; F ∗ log(p, q, s)) < ε, for all n, m > N. Since (fn) ⊂ B(D), such that fnj converges to f uniformly on compact subsets of D. It follows that fn also converges to f uniformly on compact subsets, now let m > N, and 0 < r < 1. Then, the Fatou’s yields ∫ D(0,r) ∣ ∣ ∣ ∣ f∗(z) − f∗m(z) ∣ ∣ ∣ ∣ p (1 − |z|2)q(1 − |ϕa(z)| 2)sdA(z) = ∫ D(0,r) lim n→∞ ∣ ∣ ∣ ∣ f ∗ n(z) − f ∗ m(z) ∣ ∣ ∣ ∣ p (1 − |z|2)q(1 − |ϕa(z)| 2)sdA(z) ≤ lim n→∞ ∫ D(0,r) ∣ ∣ ∣ ∣ f∗(z) − f∗m(z) ∣ ∣ ∣ ∣ p (1 − |z|2)q(1 − |ϕa(z)| 2)sdA(z) ≤ εp, and by taking r → 1 − , it follows that, ∫ D (f∗(z))p(1 − |z|2)q(1 − |ϕa(z)| 2)sdA(z) ≤ 2pεp + 2p ∫ D (f∗m(z)) p(1 − |z|2)q(1 − |ϕa(z)| 2)sdA(z). This yields ||f|| p F ∗ log (p,q,s) ≤ 2p||fm|| p F ∗ log (p,q,s) + 2pεp. And thus f ∈ F ∗log(p, q, s). We also find that fn → f with respect to the metric of (F ∗ log(p, q, s), D) and (F ∗log(p, q, s), D) is complete D-metric space. The second part of the assertion follows. 5 Composition operators of Cφ : B ∗ α, log → F ∗ log(p, q, s) In this section, we study boundedness and compactness of composition operators on B∗α, log and F ∗log(p, q, s) spaces. We need the following notation: Φφ(α, p, s; a) = ℓ p(a) ∫ D |φ′(z)|p (1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)|2)αp ( log 2 (1−|φ(z)|2) )p dA(z), CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 225 where ℓp(a) = ( log 2 1−|a|2 )p . For 0 < α < 1, we suppose there exist two functions f, g ∈ B∗α, log such that for some constant C, (|f∗(z)| + |g∗(z)|) ≥ C (1 − |z|2)α ( log 2 1−|a|2 )p > 0, for each z ∈ D. Now, we provide the following theorem: Theorem 3. Assume φ is a holomorphic mapping from D into itself and let 0 < p, 1 < s < ∞, 0 < α ≤ 1. Then the induced composition operator Cφ maps B ∗ α, log into F ∗ log(p, αp − 2, s) is bounded if and only if, sup z∈D Φφ(α, p, s; a) < ∞. (5.1) Proof. First assume that sup z∈D Φφ(α, p, s; a) < ∞ is held, and f ∈ B ∗ α, log with ||f||Bα, log ≤ 1, we can see that ||Cφf|| p F ∗ log (p,αp−2,s) = sup a∈D ℓp(a) ∫ D ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) = sup a∈D ℓp(a) ∫ D (f∗(φ(z)))p|φ′(z)|αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ ||f|| p B∗ α, log sup a∈D ℓp(a) ∫ D |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)2|)pα(log 2 1−|z|2 ) dA(z) = ||f|| p B∗ α, log Φφ(α, p, s; a) < ∞. For the other direction, we use the fact that for each function f ∈ B∗α, log, the analytic function 226 A. Kamal & T.I.Yassen CUBO 22, 2 (2020) Cφ(f) ∈ F ∗ log(p, αp − 2, s). Then, using the functions of lemma 1.2 2p { ||Cφf1|| p F ∗ log (p,αp−2,s) + ||Cφf2|| p F ∗ log (p,αp−2,s) } = 2p { sup a∈D ℓp(a) ∫ D [ ((f1 ◦ φ) ∗(z))p + ((f2 ◦ φ) ∗(z))p ] ×(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) } ≥ { sup a∈D ℓ p(a) ∫ D [ (f1 ◦ φ) ∗(z) + (f2 ◦ φ) ∗(z) ]p ×(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) } ≥ { sup a∈D ℓp(a) ∫ D [ (f∗1 (φ))(z) + (f ∗ 2 (φ))(z) ]p ×|φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) } ≥ C { sup a∈D ℓp(a) ∫ D |φ′(z)|p (1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)|2)αp ( log 2 (1−|φ(z)|2) )p dA(z) } ≥ C sup a∈D Φφ(α, p, s; a). Hence Cφ is bounded, the proof is completed. The composition operator Cφ : B ∗ α, log → F ∗ log(p, αp − 2, s) is compact if and only if for every sequence fn ∈ N ⊂ F ∗ log(p, αp − 2, s) is bounded in F ∗ log(p, αp − 2, s) norm andfn → 0, n → ∞, uniformly on compact subset of the unit disk (where N be the set of all natural numbers), hence, ||Cφ(fn)||F ∗ log (p,αp−2,s) → 0, n → ∞. Now, we describe compactness in the following result: Theorem 4. Let 0 < p, 1 < s < ∞, α < ∞. If φ is an analytic self-map of the unit disk, then the induced composition operator Cφ : B ∗ α, log → F ∗ log(p, αp − 2, s) is compact if and only if φ ∈ F ∗log(p, αp − 2, s), and lim r→1 sup a∈D Φφ(α, p, s; a) → 0. (5.2) Proof. Let Cφ : B ∗ α, log → F ∗ log(p, αp − 2, s) be compact. This means that φ ∈ F ∗log(p, αp − 2, s). Let U1r = {z : |φ(z)| > r, r ∈ (0, 1)}, CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 227 and U2r = {z : |φ(z)| ≤ r, r ∈ (0, 1)}. Let fn(z) = z n n if α ∈ [0, ∞) or fn(z) = z n n1−α if α ∈ (0, 1). Without loss of generality, we only consider α ∈ (0, 1). Since ||fn||B∗ α, log ≤ M and fn(z) → 0 as n → ∞, locally uniformly on the unit disk, then ||Cφ(fn)||F ∗ log (p,αp−2,s), n → ∞. This means that for each r ∈ (0, 1) and for all ε > 0, there exist N ∈ N so that if n ≥ N, then Nαp rp(1−N) sup a∈D ℓ p(a) ∫ U1 r |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ε. If we choose r so that N αp rp(1−N) = 1, then sup a∈D ℓp(a) ∫ U1 r |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ε. (5.3) Let now f be with ||f||B∗ α, log ≤ 1. We consider the functions ft(z) = f(tz), t ∈ (0, 1). ft → f uniformly on compact subset of the unit disk as t → 1 and the family (ft) is bounded on B ∗ α, log, thus ||(ft ◦ φ ) − (f ◦ φ )|| → 0. Due to compactness of Cφ, we get that for ε > 0 there is t ∈ (0, 1) so that sup a∈D ℓp(a) ∫ D |Ft(φ(z))| p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ε, where Ft(φ(z)) = [ (f ◦ φ )∗ − (ft ◦ φ ) ∗ ] . Thus, if we fix t, then sup a∈D ℓp(a) ∫ U1 r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ 2p sup a∈D ℓp(a) ∫ U1 r |Ft(φ(z))| p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) +2p sup a∈D ℓp(a) ∫ U1 r ((ft ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ 2pε + ||f∗t || p H∞ sup a∈D ℓ p(a) ∫ U1 r |φ′(z)|p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ 2pε + 2pε||f∗t || p H∞ . 228 A. Kamal & T.I.Yassen CUBO 22, 2 (2020) i.e, sup a∈D ℓp(a) ∫ U1r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ 2pε(1 + ||f∗t || p H∞ ), (5.4) where we have used (4). On the other hand, for each ||f||B∗ α, log ≤ 1 and ε > 0, there exists a δ depending on f and ε, so that for r ∈ [δ, 1), sup a∈D ℓp(a) ∫ U1 r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ε. (5.5) Since Cφ is compact, then it maps the unit ball of B ∗ α, log to a relatively compact subset of F ∗log(p, q, s). Thus, for each ε > 0, there exists a finite collection of functions f1, f2, ..., fn in the unit ball of B∗α, log so that for each ||f||B∗α, log, there is k ∈ {1, 2, 3, ..., n} so that sup a∈D ℓp(a) ∫ U1r |Fk(φ(z))| p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ε, where Fk(φ(z)) = [ (f ◦ φ )∗ − (fk ◦ φ ) ∗ ] . Also, using (5), we get for δ = max1≤k≤nδ(fk, ε) and r ∈ [δ, 1), that sup a∈D ℓp(a) ∫ U1r ((fk ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) < ε. Hence, for any f, ||f||B∗ α, log ≤ 1, combining the two relations as above, we get the following sup a∈D ℓp(a) ∫ U1r ((f ◦ φ)∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ 2pε. Therefore, we get that (2) holds. For the sufficiency, we use that φ ∈ F ∗log(p, αp − 2, s) and (2) holds. Let {fn}n∈N be a sequence of functions in the unit ball of B ∗ α, log so that fn → 0 as n → ∞, uniformly on the compact subsets of the unit disk. Let also r ∈ (0, 1). Then, ||fn ◦ φ|| p F ∗ log (p,αp−2,s) ≤ 2p|fn(φ(0))| +2p sup a∈D ℓp(a) ∫ U2r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) +2p sup a∈D ℓp(a) ∫ U1 r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) = 2p(I1 + I2 + I3). CUBO 22, 2 (2020) D-metric Spaces and Composition Operators 229 Since fn → 0 as n → ∞, locally uniformly on the unit disk, then I1 = |fn(φ(0))| goes to zero as n → ∞ and for each ε > 0, there is N ∈ N so that for each n > N, I2 = sup a∈D ℓp(a) ∫ U2r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ ε||φ|| p F ∗ log (p,αp−2,s) . We also observe that I3 = sup a∈D ℓp(a) ∫ U1r ((fn ◦ φ) ∗(z))p(1 − |z|2)αp−2(1 − |ϕa(z)| 2)sdA(z) ≤ ||f|| p B∗ α, log × sup a∈D ℓp(a) ∫ U1r |φ′(z)|p (1 − |z|2)αp−2(1 − |ϕa(z)| 2)s (1 − |φ(z)|2)αp ( log 2 (1−|φ(z)|2) )p dA(z). Under the assumption that (2) holds, then for every n > N and for every ε > 0, there exists r1 so that for every r > r1, I3 < ε. Thus, if φ(z) ∈ F ∗log(p, αp − 2, s), we get ||fn ◦ φ|| p F ∗ log (p,αp−2,s) ≤ 2p { 0 + ε||φ|| p F ∗ log (p,αp−2,s) + ε } ≤ Cε. Combining the above, we get ||Cφ(fn)|| p F ∗ log (p,αp−2,s) → 0 as n → ∞ which proves compactness. 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