Int. J. Anal. Appl. (2023), 21:81 Composition Operators on NK(p,q)-Type Spaces on the Unit Ball H. Gissyr, M. A. Bakhit∗ Department of Mathematics, Faculty of Science, Jazan university, Jazan 45142, Saudi Arabia ∗Corresponding author: mabakhit2020@hotmail.com Abstract. We describe the boundedness and compactness of the composition operators Cϕ acting in NK(p,q) on the open unit ball B. 1. Introduction For the unit ball B of Cn, HOl(B) denotes the class of all holomorphic functions on B while H∞ = H∞(B) denotes the class of all functions that are holomorphic u ∈HOl(B) equipped with the norm ‖u‖∞ = sup ζ∈B |u(ζ)|. For any d > 0, the weighted Banach space H∞d = H ∞ d (B) consists of all functions u ∈HOl(B) such that ‖u‖∞d := sup ζ∈B (1 −|ζ|)d|u(ζ)| < ∞. The space H∞d,0 = H ∞ d,0(B) indicate the closed subspace of H ∞ d such that lim |ζ|→1 |u(ζ)|(1 −|ζ|)d = 0. For further details about the properties of H∞d spaces see [10]). For ζ ∈B, we let dV be the Lebesgue measure on B with V (B) = ∫ B dV (ζ) = 1. Received: Jun. 27, 2023. 2020 Mathematics Subject Classification. 32A36, 46G20, 58B12, 47B33. Key words and phrases. Bergman spaces; NK(p,q)-type spaces; several complex variables; composition operators. https://doi.org/10.28924/2291-8639-21-2023-81 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-81 2 Int. J. Anal. Appl. (2023), 21:81 In addition, we let dω be the surface measure on S, normalized so that ω(S) ≡ 1. If u is a nonnegative Lebesgue measurable function on B, then the measures V and ω are related by∫ B u(ζ)dV (ζ) = 2n ∫ 1 0 t2n−1dt ∫ S u(tζ)dω(ζ). Moreover, the formulas for integration on S (see, [11]) as:∫ S udω = ∫ S dω(ζ) 1 2π ∫ 2π 0 u(eiθζ)dθ, for all 0 ≤ θ ≤ 2π. For any ψ ∈ Aut(B),u ∈ L1(B), the Möbius invariant on B (see e.g., [5]) such that∫ B u(ζ)dλ(ζ) = ∫ B u ◦ψ(ζ) dV (1 −|ζ|2)n+1 . The inner product of ζ = (ζ1, . . . ,ζn) and η = (η1, . . . ,ηn) in Cn, is given by 〈ζ,η〉 = n∑ i=1 ζiηi. For any ζ ∈B, we define the complex gradient and the radial derivative of the function u ∈HOl(B) respectively as follows: ∇u(ζ) = ( ∂u ∂ζ1 (ζ), · · · , ∂u ∂ζn (ζ) ) , Ru(ζ) = 〈∇u(ζ),ζ〉 = n∑ i=1 ζi ∂u ∂ζ1 (ζ). We know the Bloch space Bd = Bd(B) is the Banach space of functions u ∈ HOl(B) such that Ru ∈ H∞d which has the norm ‖u‖Bd := |f (0)| + ‖Ru‖ ∞ d . The involution automorphisms Ψb (the Möbius transformation of B) is define for ζ ∈ B and b ∈B−{0} as Ψb(ζ) = b− 〈ζ,b〉b|b|2 − √ 1 −|b|2 ( ζ − 〈ζ,b〉b|b|2 ) 1 −〈ζ,b〉 , where Ψ0(ζ) = −ζ, Ψb(0) = b, Ψb(b) = 0 and Ψb = Ψ−1b . It is well known that for any ζ ∈B 1 −|Ψb(ζ)|2 = (1 −|b|2)(1 −|ζ|2) |1 −〈b,ζ〉|2 . The Bergman metric and the Bergman metric ball on B, for ζ,η ∈B and M > 0 as follows: β(ζ,η) = 1 2 log 1 + |Ψζ(η)| 1 −|Ψζ(η)| , D(ζ,M) = {η ∈B : β(ζ,η) < M}. Int. J. Anal. Appl. (2023), 21:81 3 Let RC+ denote the set of all right-continuous nondecreasing functions K 6= 0 and K : [0,∞) → [0,∞). For K ∈ RC+ and p,q > 0, the weighted Banach type spaces NK(p,q) = NK(p,q)(B) consists of functions u ∈HOl(B) such that NK(p,q) := {u ∈ H(B) : ‖u‖ p NK(p,q) < ∞}, where ‖u‖pNK(p,q) = sup b∈B ∫ B |u(ζ)|p(1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ). This space was introduced first by Bakhit and Aljuaid in [1] who study several fundamental properties of NK(p,q)-type spaces and its closed subspaces NK,0(p,q), which are Banach spaces of functions that are analytic and their norms determined by a weighted function K ∈RC+, together with a Möbius transformation. Also in [1] the authors show that the norm of NK(p,q)-type space is equivalent to the norm ‖u‖pNK(p,q) = sup b∈B ∫ B |u(ζ)|p(1 −|ζ|2)qK ( G(b,ζ) ) dV (ζ) < ∞, where G(b,ζ) = log 1|Ψb(ζ)|. We set the integral JK,q(t) with q > n as: JK,q(t) = ∫ 1 0 t2n−1 (1 − t2)n+1−q K ( (1 − t2)n ) dt. (1.1) Throughout the paper, we suppose that JK,q(t) < ∞, then NK(p,q) contain all the polynomials, otherwise NK(p,q) consists only of zero functions. Let X and Y be two function spaces on B and consider ϕ be a holomorphic self-map of B. We define the composition operator Cϕ : X →Y by Cϕ(u)(ζ) = u ◦ϕ, ∀u ∈X . Recall that, for any two normed linear spaces X and Y , the linear operator T : X −→ Y is said to be bounded if there exists C > 0 such that ‖Tu‖Y ≤ C‖u‖X,∀u ∈ X. Furthermore, a linear operator T : X −→ Y is said to be compact if it maps every bounded set in X to a relatively compact set in Y (i.e., a set whose closure is compact) (see e.g., [12]). Studying the composition operators acting in different spaces is a quite classical topic since they arise in different problems; see the excellent monographs [2], [3] and [4]. Some of the earlier study on this topic is reflected in [9] descriptions of bounded and compact composition operators on F (p,q,s) spaces were provided [8]. This paper is organized as follows: in Section 2 we shortly give the preliminaries and background information. In Section 3 we establish proving our main results respectively. We use the notation a . b in what follows to mean that there is a constant C > 0 with a ≤ Cb. and the notation a � b means that a . b and b . a. 4 Int. J. Anal. Appl. (2023), 21:81 2. Preliminaries For 0 < t < ∞, we use the auxiliary function φK(t) = sup s∈(0,1] K(st) K(s) (see e.g., [6], [7]). The following constraints on φK(t) play a significant role in the study of any class of NK(p,q) spaces: JK(t) = ∫ 1 0 φK(t) dt t < ∞, (2.1) and ∫ ∞ 1 φK(t) dt t2 < ∞, (2.2) and more generally, ∫ ∞ 1 φK(t) dt t1+% < ∞, % > 0. (2.3) In the case that K satisfies condition (2.1), then K(2t) . K(t) ∀ 0 ≤ 2t ≤ 1. If we started with the property that K(t) = K(1) for t ≥ 1, then K(2t) ≈ K(t) for t > 0 (see, [6]). The following results will have an important role in the subsequent. The following lemma was proven in [1]. Lemma 2.1. Let K ∈RC+, p ≥ 1 and q > 0 then • NK(p,q) ⊆ H∞q/p(B). • NK(p,q) = H∞q/p(B) if IK(t) = ∫ 1 0 t2n−1 (1 − t2)n+1 K ( (1 − t2)n ) dt < ∞. (2.4) We can find the subsequent result in [11]. Lemma 2.2. Let δ ∈ (0, 1] then there is a sequence {ni}∈B such that • limi→∞ |ni| = 1. • B = ⋃∞ i=1 D(mi,δ). • Let N > 0 be an integer, then ζ ∈ ⋂N+1 k=1 D(mik, 4δ) and mik ∈ D(ζ, 4δ) for each ζ ∈ B, 1 ≤ k ≤ N + 1. Lemma 2.3. For any K ∈ RC+,δ > 0, let p,q > 0 and ζ,b ∈ B. Then there is a positive constant C, such that |u(ζ)|p ≤ (1 −|z|2)−q−n−1 K ( (1 −|Ψb(ζ)|2)n ) ∫ D(z,2δ) |u(w)|p(1 −|w|2)qK(1 −|Ψb(w)|2)dV (w), for all ζ ∈ D(z,δ) and u ∈HOl(B). Proof. By the result in Lemma 2.24 in [5], we obtain |u(ζ)|p ≤ 1 (1 −|ζ|2)n+1 ∫ D(ζ,δ) |u(w)|pdV (w), for all ζ ∈B and u ∈HOl(B). Int. J. Anal. Appl. (2023), 21:81 5 Now let ζ ∈ D(z,δ) and w ∈ D(ζ,δ), then obtain β(w,z) ≤ β(w,ζ) < 2δ. Thus, D(ζ,δ) ⊂ D(z, 2δ). From some results in [5], we obtain 1 −|ζ|2 � 1 −|z|2 � 1 −|w|2, |1 −〈b,w〉| � |1 −〈b,z〉|. Thus, |u(ζ)|p ≤ (1 −|z|2)−q−n−1 K ( (1 −|Ψb(ζ)|2)n ) ∫ D(z,2δ) |u(w)|p(1 −|w|2)qK ( (1 −|Ψb(w)|2)n ) dV (w). � Lemma 2.4. Let φ be a holomorphic self-map of B and b ∈ B. If u is a nonnegative Lebesgue measurable function on B, then∫ B u(ζ) dλK,q,ϕ(ζ) = ∫ B u(ϕ(ζ)) (1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ), where λK,q,ϕ = ∫ ϕ−1(E) (1 −|ζ|2)q K ( (1 −|Ψb(ζ)|2)n ) dV (ζ), for any Borel measurable set E ⊆B. Proof. Let u be a nonnegative simple Lebesgue measurable function. Assume that u(ζ) = n∑ i=1 bi κEi, where Ei is the measurable set on B. Then,∫ B u(ζ)dλK,q,ϕ(ζ) = n∑ i=1 biλK,q,ϕ(Ei ) = n∑ i=1 bi ∫ Ei dλK,q,ϕ(ζ) = n∑ i=1 bi ∫ φ−1(Ei ) (1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ) = ∫ B ( n∑ i=1 biκφ−1(Ei )∩B ) (1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ) = ∫ B u(ϕ(ζ))(1 −|ζ|2)q K ( (1 −|Ψb(ζ)|2)n ) dV (ζ). If u is a nonnegative Lebesgue measurable function, for ζ ∈ B then there is a monotone increasing simple measurable function sequence {uj} such that lim j→∞ uj(ζ) = u(ζ). Thus, lim j→∞ ∫ B uj(ζ) dλK,q,ϕ(ζ) = ∫ B u(ζ) dλK,q,ϕ(ζ). 6 Int. J. Anal. Appl. (2023), 21:81 Now let the function sequence {Uj(K,q,φ)} = {uj(ϕ(ζ))(1 − |ζ|2)q K ( (1 − |Ψb(ζ)|2)n ) }, then {Uj(K,q,φ)} is a monotone increasing measurable function. Moreover, lim j→∞ Uj(K,q,φ) = u(ϕ(ζ))(1 −|ζ|2)q K ( (1 −|Ψb(ζ)|2)n ) , which implies that ∫ B u(ζ) dλK,q,ϕ(ζ) = lim j→∞ ∫ B uj(ζ)dλK,q,ϕ(ζ) = lim j→∞ ∫ B uj(ϕ(ζ))(1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ) = lim j→∞ ∫ B u(ϕ(ζ))(1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ). This completes the proof. � Lemma 2.5. For K ∈RC+ and p > 0,q + n + 1 > 0. If (2.4) holds, then uw (ζ) ∈NK(p,q), where uw (ζ) = (1 −|w|2) (1 −〈ζ,w〉) q+n+1 p +1 . Proof. Firstly, suppose that (2.4) holds, to show that uw (ζ) ∈NK(p,q), it suffices to show that there is ε > 0, such that sup b∈B ∫ B (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p K ( (1 −|Ψb(ζ)|2)n ) dV (ζ) ≤ ε, ∀ z ∈B. Now we let 1√ 2 < |Ψb(ζ)| < 1, by the fact that (1 −|ζ|) ≤ |1 −〈ζ,b〉| and Theorem 1.4.10 in [5], therefore ∫ 1√ 2 <|Ψb(ζ)|<1 (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p K ( (1 −|Ψb(ζ)|2)n ) dV (ζ) ≤ ε ∫ B (1 −|ζ|2)−n−1K ( (1 −|Ψb(ζ)|2)n ) dV (ζ) ≤ ε ∫ 1 0 t2n−1 (1 − t2)n+1 K ( (1 − t2)n ) dt < ε. (2.5) At the same time, ∫ |Ψb(ζ)|≤ 1√2 (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p K ( (1 −|Ψb(ζ)|2)n ) dV (ζ) ≤ ∫ |w|≤1 2 (1 −|z|2)p(1 −|Ψb(w)|2)q(1 −|b|2)n+1 |1 −〈Ψb(w),z〉|n+1+q+p|1 −〈w,b〉|2n+2 K ( (1 −|w|2)n ) dV (w) Int. J. Anal. Appl. (2023), 21:81 7 ≤ ε ∫ |w|≤1 2 (1 −|b|2)n+1 (1 −|Ψb(w)|2)n+1|1 −〈w,b〉|2n+2 K ( (1 −|w|2)n ) dV (w) ≤ ε ∫ |w|≤1 2 K ( (1 −|w|2)n ) dV (w) |1 −〈w,b〉|n+1 ≤ ε ∫ |w|≤1 2 K ( (1 −|w|2)n ) dV (w) (1 −|w|2)n+1 ≤ ε ∫ B K ( (1 −|w|2)n ) dV (w) < ε. (2.6) Combining (2.5) and (2.6), it follows that sup b∈B ∫ B (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p K ( (1 −|Ψb(ζ)|2)n ) dV (ζ) ≤ ε, ∀ z ∈B. � 3. Main Results 3.1. Boundedness. Theorem 3.1. Let K ∈ RC+ and 0 < p,q < ∞. Then the operator Cϕ is bounded on NK(p,q) if and only if sup w,b∈B (1 −|w|2)p (∫ B (1 −|ζ|2)q |1 −〈ϕ(ζ),w〉|q+n+1 K(1 −|Ψb(ζ)|2)dV (ζ) ) < ∞. (3.1) Proof. Let Cϕ be the bounded operator in NK(p,q). Consider the function uw (ζ) = (1 −|w|2) (1 −〈ζ,w〉) q+n+1 p +1 . Then by Lemma 2.5, we obtain∫ B |uw (ζ)|p(1 −|ζ|2)qK(1 −|Ψb(ζ)|2) dV (ζ) ≤ ∫ B (1 −|w|2)p(1 −|ζ|2)q |1 −〈ζ,w〉|p+q+n+1 K(1 −|Ψb(ζ)|2)dV (ζ) ≤ ε, which exactly ‖Cϕ(uw )‖NK(p,q) ≤‖Cϕ‖‖uw‖NK(p,q) ≤ ε 1 p‖Cϕ‖. That is sup w,b∈B (1 −|w|2)p ∫ B (1 −|ζ|2)q |1 −〈ϕ(ζ),w〉|q+n+1 K(1 −|Ψb(ζ)|2)dV (ζ) ≤ ε‖Cϕ‖p. Conversely, suppose that (3.1) holds, then by Lemma (2.3), there exists a constant ε such that (1 −|w|2)p K(1 −|Ψb(w)|2) ∫ B dλK,q,ϕ(ζ) |1 −〈ζ,w〉|q+n+1 ≤ ε, ∀ w,b ∈B, 8 Int. J. Anal. Appl. (2023), 21:81 where λK,q,ϕ = ∫ ϕ−1(E) (1 −|ζ|2)q K ( (1 −|Ψb(ζ)|2)n ) dV (ζ), ∀ E ∈B. Fixed δ > 0, so that (1 −|w|2)p K(1 −|Ψb(w)|2) ∫ D(w,δ) dλK,q,ϕ(ζ) |1 −〈ζ,w〉|q+n+1 ≤ ε, ∀ w,b ∈B. Then, we have λK,q,ϕ(D(w,δ)) . (1 −|w|2)q+n+1K(1 −|Ψb(w)|2). If u ∈NK(p,q), then∫ B |u(ϕ(ζ))|p(1 −|ζ|2)qK(1 −|Ψb(ζ)|2)dV (ζ) = ∫ B |u(ζ)|pdλK,q,ϕ(ζ) ≤ ∞∑ j=1 ∫ D(wj,δ) |u(ζ)|pdλK,q,ϕ(ζ) ≤ ∞∑ j=1 sup ζ∈D(wj,δ) |u(ζ)|p ∫ D(wj,δ) dλK,q,ϕ(ζ) . ∞∑ j=1 sup ζ∈D(wj,δ) |u(ζ)|p{(1 −|wj|2)q+n+1K(1 −|Ψb(wj)|2)} . ∞∑ j=1 ∫ D(wj,4δ) |u(ζ)|p(1 −|ζ|2)qK(1 −|Ψb(ζ)|2)dV (ζ) . ‖u‖qNK(p,q). � 3.2. Compactness. Theorem 3.2. Let K ∈ RC+ and 0 < p,q < ∞. Then the operator Cϕ is compact on NK(p,q) if and only if lim |w|→1− sup b∈B (1 −|w|2)p (∫ B (1 −|ζ|2)q |1 −〈ϕ(ζ),w〉|q+n+1 K(1 −|Ψb(ζ)|2)dV (ζ) ) = 0. (3.2) Proof. Let Cϕ be compact on NK(p,q). Then, for any sequence {ξj}⊂B with limj→∞ |ξj| = 1. Take hj(ζ) = (1 −|ξj|) (1 −〈ζ,ξj〉) q+n+1 p . Since {hj} is bounded on NK(p,q) and converges uniformly to 0 on any compact subset of B. So, by the compactness of Cϕ, we obtain (1 −|w|2)p ∫ B (1 −|ζ|2)qK(1 −|Ψb(ζ)|2) dV (ζ) |1 −〈ϕ(ζ),w〉|q+p+1 = ‖Cϕ(hj)‖ p NK(p,q) → 0, as j →∞. Int. J. Anal. Appl. (2023), 21:81 9 Conversely, assume that (3.2) holds. Then, we can choose the sequence {wi} ∈ B from Lemma (2.2), such that sup b∈B (1 −|wi|2)p K(1 −|Ψb(wi )|2) ∫ B dλK,q,ϕ(ζ) |1 −〈ζ,wi〉|q+n+p+1 → 0, as i → 0. Thus, for any � > 0, there exists a positive integer N0 such that sup b∈B (1 −|wi|2)p K(1 −|Ψb(wi )|2) ∫ B dλK,q,ϕ(ζ) |1 −〈ζ,wi〉|q+n+p+1 < �, when i > N0. (3.3) In this case, by (3.3) for all a ∈B when j > N0, we have λK,q,ϕ(D(wi,δ) . � p(1 −|w|2)q+n+p+1K(1 −|Ψb(ζ)|n). (3.4) Now we let {uj} be any sequence that converges to 0 uniformly on any compact subset of B with ‖uj‖NK(p,q) ≤ C. Then, the sequence {uj} converges to 0 uniformly on M = ⋃N0 k=1 D(wk,δ). Thus, there exists a positive integer N0 such that sup ζ∈M |uj(ζ)| < � when j > N0. (3.5) Otherwise, λK,q,ϕ(B) ≤ ∫ B (1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ) ≤ C. (3.6) Therefore, when j > N0, by Lemma 2.2-2.4, (3.4)-(3.6), for all a ∈B we have∫ B |uj(ϕ(ζ))|p(1 −|w|2)qK(1 −|Ψb(ζ)|n)dV (ζ) = ∫ B|uj(ζ)|pdλK,q,ϕ ≤ ∞∑ k=1 ∫ D(wk,δ) |uj(ζ)|pdλK,q,ϕ ≤ N0∑ k=1 ∫ D(wk,δ) |uj(ζ)|pdλK,q,ϕ + ∞∑ k=N0+1 sup ζ∈D(wk,δ) |uj(ζ)|pλK,q,ϕ . N0 � p λK,q,ϕ(B) + �p ∞∑ k=N0+1 sup ζ∈D(wk,δ) |uj(ζ)|p(1 −|ζ|2)q+n+1K ( (1 −|Ψb(ζ)|2)n ) . N0 � p λK,q,ϕ(B) + �p ∫ D(wk,4δ) |uj|p(1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ) . N0 � p λK,q,ϕ(B) + �p ∫ B |uj|p(1 −|ζ|2)qK ( (1 −|Ψb(ζ)|2)n ) dV (ζ) . N0 � p λK,q,ϕ(B) + �p‖uj‖NK(p,q) . � p, which exactly lim k→∞ ‖Cϕ(uj)‖NK(p,q) = 0. In this case, the operator Cϕ is compact on NK(p,q), which completed the proof. � 10 Int. J. Anal. Appl. (2023), 21:81 Data Availability: The research conducted in this paper does not make use of separate data. 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Zhu, Operator Theory in Function Spaces, Second Edition, Mathematical Surveys and Monographs, Vol. 138, American Mathematical Society, Providence, Rhode Island, 2007. https://doi.org/10.1080/27684830.2022.2091724 https://doi.org/10.1016/s0252-9602(13)60127-7 https://doi.org/10.1007/s10114-021-0480-9 https://doi.org/10.1007/978-1-4612-0887-7 https://doi.org/10.1007/978-1-4612-0887-7 https://doi.org/10.1007/978-3-540-68276-9 https://doi.org/10.1007/978-3-540-68276-9 https://doi.org/10.1155/2006/910813 https://doi.org/10.1155/2006/910813 https://doi.org/10.1016/j.jmaa.2006.10.082 https://doi.org/10.1080/17476933.2022.2142783 https://doi.org/10.1007/s11785-016-0610-z https://doi.org/10.1080/10652460701210250 https://doi.org/10.1007/0-387-27539-8 https://doi.org/10.1007/0-387-27539-8 1. Introduction 2. Preliminaries 3. Main Results 3.1. Boundedness 3.2. Compactness References