Hindawi Journal of Mathematics Volume 2021, Article ID 4398397, 8 pages https://doi.org/10.1155/2021/4398397
Research Article Weighted Composition Operators from Derivative Hardy Spaces into n-th Weighted-Type Spaces
Nanhui Hu 1,2
1Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China 2Department of Mathematics, Jiaying University, Meizhou 514015, Guangdong, China
Correspondence should be addressed to Nanhui Hu; [email protected]
Received 15 April 2021; Accepted 3 June 2021; Published 17 June 2021
Academic Editor: Sei Ichiro Ueki
Copyright © 2021 Nanhui Hu. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. +e boundedness, compactness, and the essential norm of weighted composition operators from derivative Hardy spaces into n-th weighted-type spaces are investigated in this paper.
p p p ′ p 1. Introduction ‖f‖Sp � |f(0)| +‖f‖H . (4) Let H(D) denote the space of analytic functions on the open When p � 2, S2 is a Hilbert space. In [4], Roan started p unit disk D. Let S(D) denote the set of all analytic self-maps the study of composition operators on the space S . In [5], of D. Let φ ∈ S(D). +e composition operator C with the MacCluer investigated composition operators on the space φ p symbol φ is defined by S in terms of Carleson measure. +e boundedness and compactness of weighted composition operators on Sp were �Cφf �(z) � f(φ(z)), f ∈ H(D). (1) studied in [6]. See references [4–7] and references therein for more study of composition operators and weighted com- Let φ ∈ S(D) and ψ ∈ H(D). +e weighted composition p position operators on the space S . operator ψC is defined on H(D) by φ If μ is a radial, positive, and continuous function on D, �ψC f �(z) � ψ(z)f(φ(z)), f ∈ H(D). (2) then μ is called a radial weight. Let μ be a radial weight and φ n n ∈ N, the set of all positive integers. Let Wμ denote the n-th It is important to give function theoretic descriptions weighted space, which consists of all f ∈ H(D) such that when ψ and φ induce a bounded or compact weighted n− 1� � � � composition operator on various function spaces. See ref- � (k) � � (n) � ‖f‖Wn � ��f (0)� + sup μ(z)�f (z)� < ∞. (5) erences [1, 2] for more information of this research field. μ k�0 z∈D For 0 < p < ∞, the Hardy space, denoted by Hp, consists of all functions f ∈ H(D) such that (see [3]) It is a Banach space with the norm ‖·‖ n . When n � 1 Wμ 2π � � n � n p 1 � iθ �p and 2, Wμ becomes the Bloch-type space Bμ and the ‖f‖ p � sup � �f� re �� dθ < ∞. (3) H Zygmund-type space Zμ, respectively. Furthermore, when 0
j j jn− k+1 n! x1 1 x2 2 xn− k+1 Bn,k x1, x2, ... , xn− k+1� � � � � � � , ... , � � , (6) j1!j2!, ... , jn− k+1! 1! 2! (n − k + 1)!
where the sum taken over all sequences j1, j2, ... , jn− k+1 of 2. Boundedness nonnegative integers such that the following two conditions hold: In this section, the boundedness of weighted composition p n operators from S to Wμ is characterized. j1 + j2 + · · · + jn− k+1 � k (7) j + 2j + · · · +(n − k + 1)j � n. 1 2 n− k+1 Lemma 1. Suppose 1 < p < ∞ and k ∈ N. .en, there exists a See reference [14] for more information about Bell positive constant C such that polynomials. ‖f‖∞ ≤ C‖f‖Sp , In [15], Stevic´ studied the boundedness and compactness p n of the composition operator from Aα to W on the unit disk. � � C‖f‖ p (9) μ �f(k)(z)� S , In [12], Stevic´ studied the boundedness and compactness of � � ≤ k− 1+(1/p) − |z|2 weighted composition operators from H∞ and the Bloch �1 � space to Wn. See references [8–10] for more characteriza- μ for every f ∈ Sp. tions for weighted composition operators from H∞ and the n Bloch space to Wμ. In [16], Zhu and Du studied the boundedness, compactness, and essential norm of weighted Proof. +e first inequality follows from the fact that Sp are composition operators from weighted Bergman spaces with contained in the disk algebra for p > 1. In addition, it is well p n p doubling weight Aω to Wμ. Recall that the essential norm of known that for every f ∈ H , there exists a positive constant a bounded linear operator T: X ⟶ Y is its distance to the C such that set of compact operators K mapping X into Y, that is, � � � (k) � C‖f‖Hp �f (z)� ≤ , ‖T‖ � �‖T − K‖ : K a �. 2 k+(1/p) (10) e,X⟶Y inf X⟶Y is compact operator �1 − |z| � (8) which implies the second inequality. Here, X and Y are Banach spaces and ‖·‖X⟶Y denotes For any a ∈ D, 1 < p < ∞, and j ∈ {1, 2, ... , n + 1}, set the operator norm. In [17], Colonna and Tjani studied the j boundedness and compactness of weighted composition �1 − |a|2 � p f (z) � , z D. (11) operators from derivative Hardy spaces S to Bloch-type j,a j+(1/p)− 1 ∈ (1 − az) space Bμ. In this paper, we use Bell polynomials to study the After a calculation, for each j ∈ {1, 2, ... , n + 1}, boundedness and compactness of weighted composition p p f S operators from derivative Hardy spaces S to n-th weighted- j,a ∈ n � � (12) type spaces W . Moreover, we give an estimate for the � � μ supa D�fj,a� p < ∞. essential norm of weighted composition operators from Sp ∈ S n □ to Wμ. +roughout the paper, we denote by C a positive con- stant which may differ from one occurrence to the next. In Lemma 2. Let 1 < p < ∞ and 0 ≠ a ∈ D. For any i i addition, we say that A≲B if there exists a constant C such i ∈ {}1, ... , n , there exist constants c2, ... , cn+1 such that that A ≤ CB. +e symbol A ≈ B means that A≲B≲A.
n+1 i p vi,a � f1,a + � cjfj,a ∈ S , j�2
vi,a(a) � 0, (13) ⎧⎪ ai ⎪ , k � i, ⎨⎪ i+(1/p)− 1 (k) �1 − |a|2 � vi,a (a) �⎪ ⎪ ⎩⎪ 0, k ≠ i.
Moreover, vi,a converges to 0 uniformly in D. Journal of Mathematics 3
Proof. +e proof is similar to the proof of +eorem 1 in [16]. +is fact will be used in the proof of the following Hence, we omit the details of the proof. theorem. For the simplicity of this paper, we define Now, we are in a position to state and prove the first n result in this paper. □ n n (n− l) (l− i+1) Ii (z) � � � �ψ (z)Bl,i�φ′(z), φ″(z), ... , φ (z)�. l�i l Theorem 1. Let n ∈ N, 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), and (14) p n μ be a weight. .en, the operator ψCφ: S ⟶ Wμ is n From the definition, we see that, for example, bounded if and only if ψ ∈ Wμ and
B0,0 x1 � � 1, B1,0 x1, x2 � � 0, B , x � � x , 1 1 1 1 (15) B2,0 x1, x2, x3 � � 0, B2,1 x1, x2 � � x2, 2 B2,2 x1 � � x1.
� � � � � n � � n ⎛⎝ ⎞⎠ (n− l) ′ ″ (l− i+1) � μ(z)��l�i ψ (z)Bl,i�φ (z), φ (z), ... , φ (z)�� n � l � (16) � supz D < ∞. ∈ 2 i+(1/p)− 1 i�1 �1 − |φ(z)| �
Proof (sufficiency). Let f ∈ Sp. After a calculation, we have (see, e.g., [12])
n n n (n) (i) (n− l) (l− i+1) �ψCφf � (z) � � f (φ(z)) �� �ψ (z)Bl,i�φ′(z), ... , φ (z)�. (17) i�0 l�i l
(l+1) Since B0,0(φ′(z)) � 1 and Bl,0(φ′(z), ... , φ (z)) � From (18), for each j ∈ {0, 1, ... , n − 1}, n (n) 0(l ∈ N), we see that I0(z) � ψ (z). Hence, by (14) and Lemma 1, we get � � � (n) � μ(z)��ψCφf � (z)� ( ) n � �� � 18 � (i) �� n � ≤ μ(z) + μ(z) ��f (φ(z))��Ii (z)�, i�1 � � n � n � μ(z)�Ii (z)� ≲‖f‖Sp ‖ψ‖Wn +‖f‖Sp � supz D . μ ∈ 2 i+(1/p)− 1 i�1 �1 − |φ(z)| � (19)
� � j � (j) � � � � � � � � (j) � � (i) j � ��ψCφf � (0)� ≤ �f(φ(0))‖ ψ (0)� + ��f (φ(0))‖ Ii (0)� i�1 � � (20) � � j �Ij( )� � (j) � � i 0 � ≲‖f‖Sp �ψ (0)� +‖f‖Sp � . 2 i+(1/p)− 1 i�1 �1 − |φ(0)| � 4 Journal of Mathematics
n +erefore, from (19) and (20) and the fact that ψ ∈ Wμ, +erefore, by the triangle inequality and the bounded- p n we get that ψCφ: S ⟶ Wμ is bounded. ness of φ, we obtain � � C : Sp Wn � n � Necessity. Assume that ψ φ ⟶ μ is bounded. It is sup μ(z)�I1(z)� < ∞. (24) n z∈D clear that ψ ∈ Wμ, and for each j ∈ {}1, ... , n + 1 and a ∈ D, � � sup �ψC f � < ∞, Now, we assume that for 1 ≤ i ≤ j − 1(j ≤ n), a∈D� φ j,a�Wn (21) n μ supz∈Dμ(z)|Ii (z)| < ∞. To get the desired result, we only p n need to show that by the boundedness of ψCφ: S ⟶ Wμ. Next, we show that for i � 1, 2, ... , n, � � (z)�In(z)� . � n � sup μ � j � < ∞ (25) � � z D supz∈Dμ(z)�Ii (z)� < ∞. (22) ∈ j Applying the operator ψC for hj(z) � z , we obtain Applying the operator ψCφ for h1(z) � z, by (17) and φ (14), we obtain � � � � � n n � � (n) � sup μ(z)�I0(z)φ(z) + I1(z)� � sup μ(z)��ψCφh1 � (z)� z∈D z∈D � � � � ≤ �ψCφh1� n < ∞. Wμ (23)
� � � j � � � sup μ(z)�φj(z)In(z) + � j(j − 1), ... , (j − k + 1)(φ(z))j− k,In(z)� � 0 k � z∈D � k�1 � (26) � � � � ≤ �ψCφhj� n < ∞. Wμ
Hence, from the boundedness of φ and triangle in- For any i ∈ {}1, ... , n and φ(a) ≠ 0, by Lemma 2, equality again, we get the desired result.
� � i� n � � � μ(a)|φ(a)| �Ii (a)� � � ≤ sup �ψC v � i+(1/p)− 1 � φ i,φ(a)�Wn �1 − |φ(a)|2 � a∈D μ (27) � � n+1� � � � � � � i � � � ≤ sup �ψC f � + ��c � sup �ψC f � < ∞, � φ 1,a�Wn � j� � φ j,a�Wn a∈D μ j�2 a∈D μ
� � i n � n � where cj are independent of the choice of a. From the last μ(z)�Ii (z)� � sup < ∞. ( ) inequality and (22), for any i ∈ {}1, ... , n , we get 2 i+(1/p)− 1 29 i�1 z∈D �1 − |φ(z)| � � � � n � μ(a)�Ii (a)� sup i+(1/p)− 1 +e proof is complete. | (a)| ( ) 2 φ > 1/2 �1 − |φ(a)| � Let n � 1. We get the following result, which was first obtained in [17]. □ � � n+1� � � � ≲ sup �ψC f � + ��ci � sup �ψC f � < ∞, � φ 1,a�Wn � j� � φ j,a�Wn a∈D μ j�2 a∈D μ Corollary 1. Let 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), and μ be a � � � n � C : Sp B μ(a)�I (a)� � � weight. Assume that ψ φ ⟶ μ is bounded if and only i (a)�In(a)� . sup i+(1/p)− 1 ≲ sup μ i < ∞ if ψ ∈ Bμ and |φ(a)|≤(1/2) �1 − |φ(a)|2 � |φ(a)|≤(1/2) � � μ(z)�ψ(z)φ′(z)� (28) . sup (1/p) < ∞ (30) z D 2 +erefore, ∈ �1 − |φ(z)| � Journal of Mathematics 5
Let n � 2. We get the following result. Lemma 4 (see [17]). Let 1 < p < ∞. Every sequence in Sp bounded in norm has a subsequence which converges uni- Corollary 2. Let 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), and μ be a formly in D to a function in Sp. p weight. .en, the operator ψCφ: S ⟶ Zμ is bounded if The following result is a direct consequence of Lemma 3 and only if ψ ∈ Z , and Lemma 4. � μ � � � μ(z)�ψ(z)φ″(z) + 2ψ (z)φ′(z)� ′ n N p T sup (1/p) < ∞ Lemma 5. Let ∈ , 1 < < ∞, and μ be a weight. If is a z∈D �− | (z)|2 � p n 1 φ bounded linear operator from S into Wμ, then T is compact � � (31) if and only if ‖Tfn‖Wn ⟶ 0 as n ⟶ ∞ for any sequence � 2 � p μ μ(z)�ψ(z)φ′ (z)� �fn � in S bounded in norm which convergences to 0 uni- . sup 1+(1/p) < ∞ formly in D. z∈D �1 − |φ(z)|2 � Theorem 2. Let n ∈ N, 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), and C : Sp Wn 3. Essential Norm μ be a weight such that ψ φ ⟶ μ is bounded. .en, � � � � n μ(z)�In(z)� � � i In this section, we obtain some estimates for the essential �ψC � p n ≈ � limsup| (z)| . φ e,S ⟶ W φ ⟶1 2 i+(1/p)− 1 norm of the weighted composition operator μ i�1 �1 − |φ(z)| � ψC : Sp ⟶ Wn, and we need the following lemma. φ μ (32) Lemma 3 (see [17]). Let X be a Banach space that is con- tinuously contained in the disk algebra, and let Y be any Proof. First, we prove that Banach space of analytic functions on D. Suppose that � � n μ(z)�In(z)� � � Y i � � (1) .e point evaluation functionals on are continuous � limsup| (z)| ≲�ψC � p n. φ ⟶1 2 i+(1/p)− 1 φ e,S ⟶ W i�1 �− | (z)| � μ (2) For every sequence �fn � in the unit ball of X, there 1 φ f X �f � (33) exists ∈ and a subsequence nj such that
fn ⟶ f uniformly on D If supz∈D|φ(z)| < 1, there is a number δ ∈ (0, 1) such that j | (z)| (3) .e operator T: X ⟶ Y is continuous if X has the supz∈D φ < δ. In this case, (32) is vacuously satisfied and supremum norm and Y is given the topology of the desired result follows. Assume that sup |φ(z)| � 1. Let �z � be a sequence uniform convergence on compact sets z∈D j j∈N in D such that |φ(zj)| ⟶ 1 as j ⟶ ∞. Since .en, T is a compact operator if and only if, given a p n ψCφ: S ⟶ Wμ is bounded, for any compact operator �f � X f p n bounded sequence n in such that n ⟶ 0 uniformly on K: S ⟶ Wμ and i ∈ {}1, ... , n , by using Lemma 2 and D, then\ ‖Tfn‖Y ⟶ 0 as n ⟶ ∞. Lemma 5, we obtain
� � � � � � � � � � � � �ψCφ − K� p n > limsupj⟶∞�ψCφv � − limsupj⟶∞�Kv � S ⟶ W � i,φ zj �� n � i,φ zj �� n μ � Wμ Wμ � � � � � �i� n � (34) μ�zj ��φ�zj �� �Ii �zj �� > limsupj⟶∞ � � i+( p)− . � � �2 1/ 1 �1 − �φ�zj �� �
� � Hence, � � μ(z)�In(z)� � � � � �ψC � > limsup i , � �i� � � φ�e,Sp Wn φ(z)⟶1 i+(1/p)− 1 � � � � � n � ⟶ μ � 2 � � μ�zj ��φ�zj �� �Ii �zj �� �1 − |φ(z)| � �ψC � > limsup , � φ�e,Sp Wn j⟶∞ � � i+(1/p)− 1 ⟶ μ � � �2 (36) �1 − �φ�zj �� �
(35) as desired. which implies that Now, we show that 6 Journal of Mathematics
� � � � n μ(z)�In(z)� j ⟶ ∞. +en, for any positive integer j, the operator � C � � i p n �ψ φ� p n≲ limsup|φ(z)|⟶1 i+( p)− . ψC K : S ⟶ W is compact. By the definition of the e,S ⟶ Wμ 2 1/ 1 φ rj μ i�1 �1 − |φ(z)| � essential norm, we get ( ) � � � � 37 � C � � C − C K � . �ψ φ� p n ≤ limsupj⟶∞�ψ φ ψ φ rj� p n e,S ⟶Wμ S ⟶Wμ Let r ∈ [0, 1) and define Krf(z) � fr(z) � f(rz). +en, p p (38) Kr: S ⟶ S is compact and ‖Kr‖Sp ⟶Sp ≤ 1. It is clear that fr ⟶ f uniformly on compact subsets of D as r ⟶ 1. Hence, it is sufficient to show that Let �rj � ⊂ (0, 1) be a sequence such that rj ⟶ 1 as
� � � � n (z)�In(z)� � � μ i limsupj⟶∞�ψCφ − ψCφKr � p n≲ � limsup| (z)| . ( ) j S ⟶ W φ ⟶1 2 i+(1/p)− 1 39 μ i�1 �1 − |φ(z)| �
p For any f ∈ S such that ‖f‖Sp ≤ 1,
� � � � �� C − C K �f� � ψ φ ψ φ rj � n Wμ n− t � � n � � 1 � (i) �� � � (i) � ≤ � ���f − f � (φ(0))��It(0)� + sup μ(z) ���f − f � (φ(z))‖ In(z)� � rj � i z∈D � rj i � t�0 i�0 i�0 � � � � n− 1 t � (i) � � �� � � t � � �� n � ≤ � ���f − f � (φ(0))I (0)� + sup μ(z)� f − f �(z)��I (z)� � rj i � z∈D � rj � 0 �√√√√√√√√√√√√√��√√√√√√√√√√√√√�t�0 i�0 �√√√√√√√√√√√√√√��√√√√√√√√√√√√√√� Ω1 Ω0 (40) n � � � (i) � + sup μ(z) ���f − f � (φ(z))‖ In(z)� |φ(z)|≤rN � rj i � �√√√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√√√�i�1 Ω2 n � � � (i) � + sup μ(z) ���f − f � (φ(z))‖ In(z)�, |φ(z)|>rN � rj i � �√√√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√√√�i�1 Ω3
where N ∈ N such that rj ≥ (2/3) for all j ≥ N. Since for any From Lemma 4, nonnegative integer s, (f − f )(s) ⟶ 0 uniformly on � � rj � � D j lim Ω ≤ ‖ψ‖ n lim sup � f − f �(z)� � 0, compact subsets of as ⟶ ∞. It is clear that j⟶∞ 1 Wμ j⟶∞ z∈D� rj �
limsupj⟶∞Ω0 � 0 (42) (41) limsupj⟶∞Ω2 � 0. while
n � � n � � � (i) n � � i (i) n � Ω ≤ � sup| (z)| r μ(z)�f (φ(z))‖ Ii (z)� P + � sup| (z)| r μ(z)�rjf �rjφ(z) �‖ Ii (z)� Q . ( ) 3 φ > N i �√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√�φ > N i 43 i�1 �√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√�i�1
For any i ∈ {}1, ... , n , by Lemma 1 and Lemma 2, Journal of Mathematics 7
� � 2 i+(1/p)− 1� (i) � � � �1 − |φ(z)| � �f (φ(z))� |φ(z)|i�In(z)� P � sup μ(z) i i |φ(z)|>rN i i+(1/p)− 1 |φ(z)| �1 − |φ(z)|2 � � � � � ‖f‖ p � C v � ≲ S sup|φ(z)|>rN�ψ φ i,φ(z)� n Wμ (44) � � n+1 � � � � ≲ sup �ψC f � + � �ci �sup �ψC f � |a|>rN� φ 1,a�Wn � k� |a|>rN� φ k,a�Wn μ k�2 μ
n+1 � � ≲ � sup �ψC f � . |a|>rN� φ k,a�Wn k�1 μ � � � n � μ(z)�Ii (z)� N limsupj Qi≲limsup| (z)| . Taking the limit as ⟶ ∞, by Lemma 1, we get ⟶∞ φ ⟶1 2 i+(1/p)− 1 � � �1 − |φ(z)| � μ(z)�In(z)� P i . ( ) limsupj⟶∞ i≲limsup|φ(z)|⟶1 i+(1/p)− 1 46 �1 − |φ(z)|2 � Hence, by (40)–(43), (45), and (46), we obtain (45) By the same reason, we have
� � � C − C K � limsupj⟶∞�ψ φ ψ φ rj� p n S ⟶ Wμ � � ( ) � � n � n � 47 � � μ(z)�Ii (z)� � limsupj⟶∞sup‖f‖ p ≤ 1��ψCφ − ψCφKr �f� ≲ � limsup|φ(z)|⟶1 . S � j �Wn 2 i+(1/p)− 1 μ i�1 �1 − |φ(z)| �
Hence, from the last two inequalities, we obtain (39). +e Let n � 2. We get the following result. proof is complete. From +eorem 2 and the well-known result that Corollary 5. Let 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), and μ be a ‖T‖e,X⟶Y � 0 if and only if T: X ⟶ Y is compact, we p weight such that ψCφ: S ⟶ Zμ is bounded. .en, the obtain the following corollary. □ p operator ψCφ: S ⟶ Zμ is compact if and only if � � � � Corollary 3. Let n ∈ N, 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), μ(z)�ψ(z)φ″(z) + 2ψ (z)φ′(z)� p n ′ � and μ be a weight. Assume that ψCφ: S ⟶ Wμ is bounded, limsup|φ(z)|⟶1 (1/p) 0 p n �1 − |φ(z)|2 � then the operator ψCφ: S ⟶ Wμ is compact if and only if � � n μ(z)�In(z)� � � i � 2 � � limsup|φ(z)|⟶1 � 0. (48) μ(z)�ψ(z)φ′ (z)� 2 i+(1/p)− 1 � . i�1 �1 − |φ(z)| � limsup|φ(z)|⟶1 1+(1/p) 0 �1 − |φ(z)|2 � Let n � 1. We get the following result, which was first (50) obtained in [17]. Data Availability Corollary 4. Let 1 < p < ∞, φ ∈ S(D), ψ ∈ H(D), and μ be a p weight such that ψCφ: S ⟶ Bμ is bounded. .en, the No data were used to support this study. p operator ψCφ: S ⟶ Bμ is compact if and only if � � μ(z)�ψ(z)φ′(z)� � . limsup|φ(z)|⟶1 (1/p) 0 (49) Conflicts of Interest �1 − |φ(z)|2 � +e author declares that there are no conflicts of interest. 8 Journal of Mathematics
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