Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to F p , q , s Space on the Unit Ball

and Applied Analysis 3 It is interesting to characterize the boundedness and compactness of the product operator on all kinds of function spaces. Even on the disk of , some properties are not easily managed; see some recent papers in 18, 25–28 . Building on those foundations, the present paper continues this line of research and discusses the operator in high dimension. The remainder is assembled as follows: in Section 2, we state a couple of lemmas. In Section 3, we characterize the boundedness and compactness of the product ThCφ of extended Cesàro operator and composition operator from Lipschitz spaces to F p, q, s spaces on the unit ball of n . Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. The notation A B means that there is a positive constant C such that B/C ≤ A ≤ CB. 2. Some Lemmas To begin the discussion, let us state a couple of lemmas, which are used in the proofs of the main results. Lemma 2.1. Suppose that f, h ∈ H . Then, [ ThCφ ( f )] z f ( φ z ) h z . 2.1 Proof. The proof of this Lemma follows by standard arguments see, e.g., 9, 29, 30 . Lemma 2.2 see 2, 31 . If 0 < α < 1, then B1−α Lα ; furthermore, ∥f ∥∥ B1−α ∥f ∥∥ Lα 2.2 as f varies through Lα . The following criterion for compactness follows from standard arguments similar to the corresponding lemma in 6 . Hence, we omit the details. Lemma 2.3. Assume that h ∈ H and φ ∈ S . Suppose that X or Y is one of the following spaces Lα , F p, q, s . Then, ThCφ : X → Y is compact if and only if ThCφ : X → Y is bounded, and for any bounded sequence {fk}k∈N in X which converges to zero uniformly on compact subsets of as k → ∞, one has ‖ThCφfk‖Y → 0 as k → ∞. Lemma 2.4 see 4, 5 . If f ∈ Bα, then ∣f z ∣∣ ≤ C∥f∥ Ba , 0 < α < 1, 2.3 ∣f z ∣∣ ≤ C∥f∥ Ba ln e 1 − |z| , α 1, 2.3 ′ ∣f z ∣∣ ≤ C ∥f ∥∥ Ba ( 1 − |z| )α−1 , α > 1. 2.3 The next lemma was obtained in 32 . 4 Abstract and Applied Analysis Lemma 2.5. If a > 0, b > 0, then the elementary inequality holds a b p ≤ ⎧ ⎨ ⎩ a b, 0 < p < 1, 2p−1 a b , p ≥ 1. 2.4 It is obvious that Lemma 2.5 holds for the sum of finite number k, that is, a1 · · · ak p ≤ C ( a p 1 · · · a p k ) , 2.5 where a1, . . . , ak > 0 and C is a positive constant. Lemma 2.6 see 4, 5 . For 0 < p, s < ∞, −n − 1 < q < ∞, q s > −1, there exists C > 0 such that sup a∈ ∫ ( 1 − |w| )p |1 − 〈z,w〉| 1 q p ( 1 − |z| )q g z, a dν z ≤ C 2.6 for every ω ∈ . Lemma 2.7 see 4 . There is a constant C > 0 so that, for all t > −1 and z ∈ , one has ∫ ∣∣∣∣ln 1 1 − 〈z,w〉 ∣∣∣∣ 2 ( 1 − |w| )t |1 − 〈z,w〉| 1 t dν z ≤ C ( ln 1 1 − |z| )2 . 2.7 Lemma 2.8 see 4, 5 . Suppose that 0 < p, s < ∞, −n−1 < q < ∞, and q s > −1. If f ∈ F p, q, s , then f ∈ B n 1 q /p , and ‖f‖B n 1 q /p ≤ C‖f‖F p,q,s . Lemma 2.9. Let {fk}k∈N be a bounded sequence in F p, q, s which converges to zero uniformly on compact subsets of the unit ball , where n 1 q /p < 1. Then, limk→∞supz∈ |fk z | 0. Proof. It follows from Lemma 2.8 that F p, q, s ⊆ B n 1 q /p and ‖f‖B n 1 q /p ≤ C‖f‖F p,q,s for any f ∈ F p, q, s . So, when n 1 q /p < 1, the proof of this lemma is similar to that of Lemma 3.6 of 33 , hence the proof is omitted. 3. The Boundedness and Compactness of the Operator ThCφ : Lα → F p, q, s Theorem 3.1. Assume that α ∈ 0, 1 , 0 < p, s < ∞, −n − 1 < q < ∞, q s > −1, φ ∈ S , and h ∈ H . Then, ThCφ : Lα → F p, q, s is bounded if and only if h ∈ F p, q, s . Proof. Assume that h ∈ F p, q, s . Since 0 < 1 − α < 1, by Lemmas 2.2 and 2.4, for any f ∈ Lα, we have ∣f z ∣∣ ≤ C∥f∥B1−α ≤ C ∥f ∥∥ Lα . 3.1 Abstract and Applied Analysis 5 Since |ThCφf 0 | 0, by using Lemma 2.1 and relations 2.3 and 3.1 , we have ∥ThCφf ∥∥p F p,q,s sup a∈ ∫ ∣f ( φ z ) h z ∣∣p ( 1 − |z| )q g z, a dν zand Applied Analysis 5 Since |ThCφf 0 | 0, by using Lemma 2.1 and relations 2.3 and 3.1 , we have ∥ThCφf ∥∥p F p,q,s sup a∈ ∫ ∣f ( φ z ) h z ∣∣p ( 1 − |z| )q g z, a dν z ≤ C sup a∈ ∫ | h z | ( 1 − |z| )q g z, a dν z ∥f ∥∥p B1−α ≤ C‖h‖ F p,q,s ∥f ∥∥p Lα < ∞. 3.2 Thus ThCφ : Lα → F p, q, s is bounded. Conversely, suppose that ThCφ : Lα → F p, q, s is bounded. Taking the function f z 1 ∈ Lα, then ∥ThCφf ∥∥p F p,q,s ∣ThCφf 0 ∣∣p sup a∈ ∫ ∣∣ ( ThCφf ) z ∣∣p ( 1 − |z| )q g z, a dν z sup a∈ ∫ ∣f ( φ z ) h z ∣∣p ( 1 − |z| )q g z, a dν z sup a∈ ∫ | h z | ( 1 − |z| )q g z, a dν z ‖h‖pF p,q,s . 3.3 From which, the boundedness of ThCφ implies that h ∈ F p, q, s . This completes the proof of this theorem. Next, we characterize the compactness of ThCφ : Lα → F p, q, s . Theorem 3.2. Assume that α ∈ 0, 1 , 0 < p, s < ∞, −n − 1 < q < ∞, q s > −1, φ ∈ S , and h ∈ H . Then, ThCφ : Lα → F p, q, s is compact if and only if ThCφ : Lα → F p, q, s is bounded, and lim r→ 1 sup a∈ ∫ {|φ z |>r} | h z | ( 1 − |z| )q g z, a dν z 0. 3.4 Proof. Assume that ThCφ : Lα → F p, q, s is bounded and 3.4 holds. It follows from Theorem 3.1 that h ∈ F p, q, s . Now, let {fj}j∈N be a bounded sequence of functions inLα such that fj → 0 uniformly on the compact subsets of as j → ∞. Suppose that supj∈N‖fj‖Lα ≤ L. It follows from 3.4 that, for any ε > 0, there exists r0 ∈ 0, 1 such that, for every r0 < r < 1, sup a∈ ∫ {|φ z |>r} | h z | ( 1 − |z| )q g z, a dν z < ε. 3.5 6 Abstract and Applied Analysis Set r0 < r < 1, then ∥ThCφfj ∥∥p F p,q,s sup a∈ ∫ ∣fj ( φ z ∣p| h z | ( 1 − |z| )q g z, a dν z ≤ sup a∈ ∫ {|φ z |≤r} ∣fj ( φ z ∣p| h z | ( 1 − |z| )q g z, a dν z sup a∈ ∫ {|φ z |>r} ∣fj ( φ z ∣p| h z | ( 1 − |z| )q g z, a dν z