# Composition Operators from the Weighted Bergman Space to thenth Weighted Spaces on the Unit Disc

Authors: Stevo Stević
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

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


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## Body

## 1. Introduction

Let𝔻 be the open unit disc in the complex plane ℂ, dm(z) the Lebesgue area measure on 𝔻, dmα(z)=(1-|z|2)αdm(z), α>-1, and H(𝔻) the space of all analytic functions on the unit disc.The weighted Bergman spaceAαp(𝔻), where p>0 and α>-1, consists of all f∈H(𝔻) such that
(1.1)∥f∥Aαp(𝔻)p=(α+1)∫𝔻|f(z)|p(1-|z|2)αdm(z)<∞.
With this norm, Aαp(𝔻) is a Banach space when p≥1, while for p∈(0,1), it is a Fréchet space with the translation invariant metric
(1.2)d(f,g)=∥f-g∥Aαp(𝔻)p,f,g∈Aαp(𝔻).Letμ(z) be a positive continuous function on a set X⊂ℂ (weight) and n∈ℕ0 be fixed. The nth weighted-type space  on X, denoted by 𝒲μ(n)(X), consists of all f∈H(X) such that
(1.3)b𝒲μ(n)(X)(f):=supz∈Xμ(z)|f(n)(z)|<∞.Forn=0, the space becomes the weighted-type space Hμ∞(X), for n=1 the Bloch-type space ℬμ(X), and for n=2 the Zygmund-type space 𝒵μ(X).Forn∈ℕ, the quantity b𝒲μ(n)(X)(f) is a seminorm on the nth weighted-type space 𝒲μ(n)(X) and a norm on 𝒲μ(n)(X)/ℙn-1, where ℙn-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:
(1.4)∥f∥𝒲μ(n)(X)=∑j=0n-1|f(j)(a)|+b𝒲μ(n)(X)(f),
where a is an element in X. With this norm, the nth weighted-type space becomes a Banach space.ForX=𝔻 is obtained the space 𝒲μ(n)(𝔻), on which a norm is introduced as follows:
(1.5)∥f∥𝒲μ(n)(𝔻):=∑j=0n-1|f(j)(0)|+supz∈𝔻μ(z)|f(n)(z)|.
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 weighted-type space (see [8]).Assumeφ is a holomorphic self-map of 𝔻. The composition operator induced by φ is defined on H(𝔻) by
(1.6)(Cφf)(z)=f(φ(z)).A typical problem is to provide function theoretic characterizations whenφ 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 thenth 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 weighted-type 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].LetX and Y be topological vector spaces whose topologies are given by translation-invariant metrics dX and dY, 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
(1.7)dY(Tf,0)≤KdX(f,0)
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.IfY is a Banach space, then the quantity ∥Cφ∥Aαp(𝔻)→Y is defined as follows:
(1.8)∥Cφ∥Aαp(𝔻)→Y:=sup∥f∥Aαp(𝔻)≤1∥Cφf∥Y.
It is easy to see that this quantity is finite if and only if the operator Cφ:Aαp(𝔻)→Y is metrically bounded. For the case p≥1 this is the standard definition of the norm of the operator Cφ:Aαp(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 byC, 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.


## 2. Auxiliary Results

Here, we quote several auxiliary results. The first lemma is a direct consequence of a well-known estimate in [37, Proposition  1.4.10]. Hence, we omit its proof.Lemma 2.1.
Assumep>0, α>-1, n∈ℕ0, and w∈𝔻. Then the function
(2.1)gw,n(z)=(1-|w|2)n+(α+2)/p(1-w̅z)n+2((α+2)/p),
belongs to Aαp(𝔻). Moreover, supw∈𝔻∥gw,n∥Aαp<∞.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.
Assumep>0, α>-1, n∈ℕ0, and z∈𝔻. Then there is a positive constant C independent of f such that
(2.2)|f(n)(z)|≤C∥f∥Aαp(𝔻)(1-|z|)n+(α+2)/p.Proof.
By the subharmonicity of the function|f(n)(z)|p, p>0, applied on the disk:
(2.3)D(z,1-|z|2)={z∈ℂ∣|z-w|<1-|z|2},
and since
(2.4)1-|w|≍1-|z|,w∈D(z,1-|z|2),
we have that
(2.5)|f(n)(z)|p≤C(1-|z|)2+α+pn∫D(z,(1-|z|)/2)|f(n)(w)|pdmα+pn(w).
From (2.5) and in light of the following well-known asymptotic relation [38]:
(2.6)∫𝔻|f(z)|p(1-|z|2)αdm(z)≍∑j=0n-1|f(j)(0)|+∫𝔻|f(n)(z)|p(1-|z|2)α+npdm(z),
the lemma easily follows.Lemma 2.3.
Assumea>0 and
(2.7)Dn(a)=|11⋯1aa+1⋯a+n-1a(a+1)(a+1)(a+2)⋯(a+n-1)(a+n)…∏j=0n-2(a+j)∏j=0n-2(a+j+1)⋯∏j=0n-2(a+j+n-1)|.
Then Dn=∏j=1n-1j!.Proof.
By using elementary transformations, we have(2.8)Dn(a)=|10⋯0a1⋯1a(a+1)2(a+1)⋯2(a+n-1)…∏j=0n-2(a+j)(n-1)∏j=0n-3(a+j+1)⋯(n-1)∏j=0n-3(a+j+n-1)|,
from which it follows that
(2.9)Dn(a)=(n-1)!Dn-1(a+1),
which along with the fact D2(a+n-2)=1 implies the lemma.We will also need the classical Faà di Bruno's formula(2.10)(f∘φ)(n)(z)=∑n!k1!⋯kn!f(k)(φ(z))∏j=1n(φ(j)(z)j!)kj,
where k=k1+k2+⋯+kn and the sum is over all nonnegative integers k1,k2,…,kn satisfying k1+2k2+⋯+nkn=n. For a nice exposition related to this formula see, for example, [39].By using Bell polynomialsBn,k(x1,…,xn-k+1), (2.10) can be written in the following form:
(2.11)(f∘φ)(n)(z)=∑k=0nf(k)(φ(z))Bn,k(φ′(z),φ′′(z),…,φ(n-k+1)(z)).Remark 2.4.
SinceBn,0(x1,…,xn+1)=0 the summation in (2.11) is from 1 to k. Moreover, since Bn,1(x1,…,xn)=xn and Bn,n(x1)=x1n, (2.11) can be written in the following form:
(2.12)(f∘φ)(n)(z)=f′(φ(z))φ(n)(z)+∑k=2n-1f(k)(φ(z))Bn,k(φ′(z),…,φ(n-k+1)(z))+f(n)(φ(z))(φ′(z))n.


## 3. Main Result

Here, we formulate and prove our main result.Theorem 3.1.
Assumep>0, α>-1, n∈ℕ, μ is a weight on 𝔻 and φ is a holomorphic self-map of 𝔻. Then Cφ:Aαp(𝔻)→𝒲μ(n)(𝔻) is bounded if and only if
(3.1)Ik:=supz∈𝔻μ(z)|∑(n!/(k1!⋯kn!))∏j=1n(φ(j)(z)/j!)kj|(1-|φ(z)|2)k+(α+2)/p<∞,k=1,…,n,
where for each fixed k∈{1,…,n}, the sum is over all nonnegative integers k1,k2,…,kn such that k=k1+k2+⋯+kn and k1+2k2+⋯+nkn=n.
Moreover, if the operatorCφ:Aαp(𝔻)→𝒲μ(n)(𝔻)/ℙn-1 is bounded, then
(3.2)∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻)/ℙn-1≍∑k=1nIk.Remark 3.2.
Note that by (2.11) we see that the conditions in (3.1) can be written in the following form:
(3.3)Ik=supz∈𝔻μ(z)|Bn,k(φ′(z),φ′′(z),…,φ(n-k+1)(z))|(1-|φ(z)|2)k+(α+2)/p<∞,k=1,…,n.Proof.
First assume that conditions in (3.1) hold. By formula (2.10) and Lemma 2.2 we have
(3.4)∥Cφf∥𝒲μ(n)(𝔻)=∑j=0n-1|(f∘φ)(j)(0)|+supz∈𝔻μ(z)|(Cφf)(n)(z)|=∑j=0n-1|∑j!l1!⋯lj!f(l)(φ(0))∏s=1j(φ(s)(0)s!)ls|+supz∈𝔻μ(z)|∑n!k1!⋯kn!f(k)(φ(z))∏j=1n(φ(j)(z)j!)kj|≤∑j=0n-1∑l=0j|f(l)(φ(0))||∑j!l1!⋯lj!∏s=1j(φ(s)(0)s!)ls|+C∥f∥Aαp(𝔻)∑k=1nsupz∈𝔻μ(z)|∑(n!/(k1!⋯kn!))∏j=1n(φ(j)(z)/j!)kj|(1-|φ(z)|2)k+(α+2)/p.
From this, (2.2) with z=φ(0), and by conditions in (3.1), it follows that the operator Cφ:Aαp(𝔻)→𝒲μ(n)(𝔻) is bounded. Moreover, if we consider the space 𝒲μ(n)(𝔻)/ℙn-1, we have that
(3.5)∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻)/ℙn-1≤C∑k=1nsupz∈𝔻μ(z)|∑(n!/(k1!⋯kn!))∏j=1n(φ(j)(z)/j!)kj|(1-|φ(z)|2)k+(α+2)/p.
Now assume that the operatorCφ:Aαp(𝔻)→𝒲μ(n)(𝔻) is bounded. For a fixed w∈𝔻, and constants c1,…,cn, set
(3.6)gw(z)=∑j=1ncjn-2+j+2((α+2)/p)(1-|w|2)n-2+j+(α+2)/p(1-w̅z)n-2+j+2((α+2)/p).
Applying Lemma2.1 we see that gw∈Aαp(𝔻) for every w∈𝔻. Moreover, we have that
(3.7)supw∈𝔻∥gw∥Aαp(𝔻)≤C.
Now we show that for eachl∈{1,…,n}, there are constants c1,c2,…,cn, such that
(3.8)gw(l)(w)=w̅l(1-|w|2)l+(α+2)/p,gw(m)(w)=0,m∈{1,…,n}∖{l}.
Indeed, by differentiating function gw, for each l∈{1,…,n}, the system in (3.8) becomes
(3.9)c1+c2+⋯+cn=0,(n+2α+2p)c1+(n+1+2α+2p)c2+⋯+(2n-1+2α+2p)cn=0,⋮∏j=0l-2(n+j+2α+2p)c1+∏j=0l-2(n+1+j+2α+2p)c2+⋯+∏j=0l-2(2n-1+j+2α+2p)cn=1,⋮∏j=0n-2(n+j+2α+2p)c1+∏j=0n-2(n+1+j+2α+2p)c2+⋯+∏j=0n-2(2n-1+j+2α+2p)cn=0.
By using Lemma2.3 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 eachk∈{1,…,n}, we choose the corresponding family of functions which satisfy (3.8) and denote it by gw,k.
For eachk∈{1,…,n}, the boundedness of the operator Cφ:Aαp(𝔻)→𝒲μ(n)(𝔻) along with (2.10) and (3.7) implies that for each φ(w)≠0:
(3.10)μ(w)|φ(w)|k|∑(n!/(k1!⋯kn!))∏j=1n(φ(j)(w)/j!)kj|(1-|φ(w)|2)k+(α+2)/p≤supw∈𝔻∥Cφ(gφ(w),k)∥𝒲μ(n)(𝔻)≤C∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻),
where (for each fixed k∈{1,…,n}) the sum is over all nonnegative integers k1,k2,…,kn such that k=k1+k2+⋯+kn and k1+2k2+⋯+nkn=n.
From (3.10), it follows that for each k∈{1,…,n},
(3.11)sup|φ(z)|>1/2μ(z)|∑(n!/(k1!⋯kn!))∏j=1n(φ(j)(z)/j!)kj|(1-|φ(z)|2)k+(α+2)/p≤C∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻).
Now we use consecutively the test functions(3.12)hk(z)=zk∈Aαp(𝔻),k=1,…,n,
in order to deal with the case |φ(z)|≤1/2. Note that
(3.13)∥hk∥Aαp(𝔻)≤1,foreachk∈ℕ.
By applying (2.11) to the function f(z)=h1(z), we get
(3.14)(h1∘φ)(n)(z)=h1′(φ(z))Bn,1(φ′(z),…,φ(n)(z))=Bn,1(φ′(z),…,φ(n)(z)),
which along with the boundedness of the operator Cφ:Aαp(𝔻)→𝒲μ(n)(𝔻) and (3.13) implies that
(3.15)supz∈𝔹μ(z)|Bn,1(φ′(z),…,φ(n)(z))|≤∥Cφ(z)∥𝒲μ(n)(𝔻)≤∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻),
or equivalently φ∈𝒲μ(n)(𝔻) (see Remark 2.4).
Further, by applying formula (2.11) to the function f(z)=h2(z), we get
(3.16)(h2∘φ)(n)(z)=h2′(φ(z))Bn,1(φ′(z),…,φ(n)(z))+h2′′(φ(z))Bn,2(φ′(z),…,φ(n-1)(z)).
From the boundedness of Cφ:Aαp(𝔻)→𝒲μ(n)(𝔻) and (3.13), we get
(3.17)supz∈𝔹μ(z)|(h2∘φ)(n)(z)|≤∥Cφ(z2)∥𝒲μ(n)(𝔻)≤∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻).
From (3.16) and (3.17), and by using the triangle inequality it follows that
(3.18)2supz∈𝔹μ(z)|Bn,2(φ′(z),…,φ(n-1)(z))|≤∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻)+2supz∈𝔹μ(z)|φ(z)Bn,1(φ′(z),…,φ(n)(z))|.
Using the factsupz∈𝔻|φ(z)|≤1 and applying inequality (3.15) in (3.18) we get
(3.19)supz∈𝔹μ(z)|Bn,2(φ′(z),…,φ(n-1)(z))|≤32∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻).
Assume that we have proved the following inequalities:(3.20)supz∈𝔹μ(z)|Bn,j(φ′(z),…,φ(n-j+1)(z))|≤C∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻),
for j∈{1,…,k-1} and a k≤n.
Applying formula (2.11) to the function f(z)=hk(z),  k∈{1,…,n}, we have that
(3.21)(hk∘φ)(n)(z)=∑j=1khk(j)(φ(z))Bn,j(φ′(z),…,φ(n-j+1)(z))=∑j=1kk(k-1)⋯(k-j+1)(φ(z))k-jBn,j(φ′(z),…,φ(n-j+1)(z)).
From this, by using the boundedness of the operator Cφ:Aαp(𝔻)→𝒲μ(n)(𝔻), the boundedness of function φ, the triangle inequality, noticing that the coefficient at Bn,k(φ′(z),…,φ(n-k+1)(z)) is independent of z (it is equal k!), and finally using hypothesis (3.20), we easily obtain
(3.22)supz∈𝔹μ(z)|Bn,k(φ′(z),…,φ(n-k+1)(z))|≤C∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻).
Hence, by induction, we get that (3.22) holds for each k∈{1,…,n}.
From (3.22) and bearing in mind Remark 2.4, for each fixed k∈{1,…,n}, we have that
(3.23)sup|φ(z)|≤1/2μ(z)|∑(n!/(k1!⋯kn!))∏j=1n(φ(j)(z)/j!)kj|(1-|φ(z)|2)k+(α+2)/p≤supz∈𝔹μ(z)|Bn,k(φ′(z),…,φ(n-k+1)(z))|≤C∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻),
where as usual for a fixed k∈{1,…,n}, the sum is over all nonnegative integers k1,k2,…,kn such that k=k1+k2+⋯+kn and k1+2k2+⋯+nkn=n.
Hence from (3.11) and (3.23), we get
(3.24)∑k=1nIk≤C∥Cφ∥Aαp(𝔻)→𝒲μ(n)(𝔻).
From (3.5) and (3.24), we obtain asymptotic relation (3.2).



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