Composition Operators from the Weighted Bergman Space to the n th Weighted Spaces on the Unit Disc

The boundedness of the composition operator from the weighted Bergman space to the recently introduced by the author, the 𝑛th weighted space on the unit disc, is characterized. Moreover, the norm of the operator in terms of the inducing function and weights is estimated.


Introduction
Let D be the open unit disc in the complex plane C, dm z the Lebesgue area measure on D, dm α z 1 − |z| 2 α dm z , α > −1, and H D the space of all analytic functions on the unit disc.
The weighted Bergman space A p α D , where p > 0 and α > −1, consists of all f ∈ H D such that With this norm, A p α D is a Banach space when p ≥ 1, while for p ∈ 0, 1 , it is a Fréchet space with the translation invariant metric Let μ z be a positive continuous function on a set X ⊂ C weight and n ∈ N 0 be fixed.The nth weighted-type space on X, denoted by W n μ X , consists of all f ∈ H X such that b W n μ X f : sup z∈X μ z f n z < ∞.

1.3
For n 0, the space becomes the weighted-type space H ∞ μ X , for n 1 the Bloch-type space B μ X , and for n 2 the Zygmund-type space Z μ X .
For n ∈ N, the quantity b W n μ X f is a seminorm on the nth weighted-type space W n μ X and a norm on W n μ X /P n−1 , where P n−1 is the set of all polynomials whose degrees are less than or equal to n − 1.A natural norm on the nth weighted-type space can be introduced as follows: where a is an element in X.With this norm, the nth weighted-type space becomes a Banach space.
For X D is obtained the space W n μ D , on which a norm is introduced as follows: Some information on Zygmund-type spaces on the unit disc and some operators on them can be found, for example, in 1-6 , for the case of the upper half-plane, see 7, 8 , while some information in the setting of the unit ball can be found, for example, in 9-13 .This considerable interest in Zygmund-type spaces motivated us to introduce the nth weightedtype space see 8 .Assume ϕ is a holomorphic self-map of D. The composition operator induced by ϕ is defined on H D by A typical problem is to provide function theoretic characterizations when ϕ induces bounded or compact composition operators between two given spaces of holomorphic functions.Some classical results on composition and weighted composition operators can be found, for example, in 14 , while some recent results can be found in 1, 5, 7, 15-34 see also related references therein .
Here we characterize the boundedness of the composition operator from the weighted Bergman space to the nth weighted space on the unit disc when n ∈ N. The case n 0 was previously treated in 16,22,24,31,35 .Hence we will not consider this case here.See also 36 for some good results on weighted composition operators between weightedtype spaces.The case n 1 was treated, for example, in 26, 32 .For some other results on weighted composition operators which map a space into a weighted or a Bloch-type space, see, for example, 15, 17-21, 23, 25, 33, 34 .Let X and Y be topological vector spaces whose topologies are given by translationinvariant metrics d X and d Y , respectively, and T : X → Y be a linear operator.It is said that T is metrically bounded if there exists a positive constant K such that for all f ∈ X.When X and Y are Banach spaces, the metrically boundedness coincides with the usual definition of bounded operators between Banach spaces.
If Y is a Banach space, then the quantity C ϕ A p α D → Y is defined as follows: It is easy to see that this quantity is finite if and only if the operator C ϕ : A p α D → Y is metrically bounded.For the case p ≥ 1 this is the standard definition of the norm of the operator C ϕ : A p α D → Y , between two Banach spaces.If we say that an operator is bounded, it means that it is metrically bounded.
Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other.The notation a b means that there is a positive constant C such that a ≤ Cb.Moreover, if both a b and b a hold, then one says that a b.

Auxiliary Results
Here, we quote several auxiliary results.The first lemma is a direct consequence of a wellknown estimate in 37, Proposition 1.4.10 .Hence, we omit its proof.Lemma 2.1.Assume p > 0, α > −1, n ∈ N 0 , and w ∈ D. Then the function The next lemma is folklore and was essentially proved in 38 .We will sketch a proof of it for the completeness and the benefit of the reader.Lemma 2.2.Assume p > 0, α > −1, n ∈ N 0 , and z ∈ D. Then there is a positive constant C independent of f such that Proof.By the subharmonicity of the function |f n z | p , p > 0, applied on the disk: we have that From 2.5 and in light of the following well-known asymptotic relation 38 : the lemma easily follows.

2.7
Then Proof.By using elementary transformations, we have from which it follows that which along with the fact D 2 a n − 2 1 implies the lemma.
We will also need the classical Faà di Bruno's formula where Remark 2.4.Since B n,0 x 1 , . . ., x n 1 0 the summation in 2.11 is from 1 to k.Moreover, since B n,1 x 1 , . . ., x n x n and B n,n x 1 x n 1 , 2.11 can be written in the following form: 2.12

Main Result
Here, we formulate and prove our main result.
where for each fixed k ∈ {1, . . ., n}, the sum is over all nonnegative integers k 1 , k 2 , . . ., k n such that Remark 3.2.Note that by 2.11 we see that the conditions in 3.1 can be written in the following form: Proof.First assume that conditions in 3.1 hold.By formula 2.10 and Lemma 2.2 we have

3.4
From this, 2.2 with z ϕ 0 , and by conditions in 3.1 , it follows that the operator

3.5
Now assume that the operator C ϕ : For a fixed w ∈ D, and constants c 1 , . . ., c n , set

3.6
Applying Lemma 2.1 we see that g w ∈ A Indeed, by differentiating function g w , for each l ∈ {1, . . ., n}, the system in 3.8 becomes 3.9 By using Lemma 2.3 with a n 2 2 α /p > 0, we obtain that the determinant of system 3.9 is different from zero from which the claim follows.Now for each k ∈ {1, . . ., n}, we choose the corresponding family of functions which satisfy 3.8 and denote it by g w,k .
For each k ∈ {1, . . ., n}, the boundedness of the operator C ϕ : A p α D → W n μ D along with 2.10 and 3.7 implies that for each ϕ w / 0:

3.11
Now we use consecutively the test functions k−j B n,j ϕ z , . . ., ϕ n−j 1 z .

3.21
From this, by using the boundedness of the operator C ϕ : A

Theorem 3 . 1 .
Assume p > 0, α > −1, n ∈ N, μ is a weight on D and ϕ is a holomorphic self-map of D. Then C ϕ : A p α D → W n μ D is bounded if and only if