Abstract
Let σ be a Békollé weight function and ν be a weight function. In this paper, we characterize the boundedness and compactness of weighted composition operators acting from Bergman-type spaces \(A^{p}(\sigma)\) to Bloch-type spaces \(\mathcal{B}_{\nu}\) and \(\mathcal{B}_{\nu, 0}\), considerably extending some results in the literature.
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1 Introduction and preliminaries
Let \(\mathbb{D}\) denote the open unit disk in the complex plane \(\mathbb{C}\), \(H(\mathbb{D})\) the space of all holomorphic functions on \(\mathbb{D}\), and \(S({\mathbb{D}})\) the class of all holomorphic self-maps of \({\mathbb{D}}\). Let \(\psi\in H(\mathbb{D})\) and \(\varphi\in S(\mathbb{D})\), the weighted composition operator \(W_{\psi, \varphi}\) is a linear operator on \(H(\mathbb{D})\) defined by
for \(f \in H(\mathbb{D})\). It is of interest to provide function-theoretic characterizations when symbols φ and ψ induce a bounded or compact weighted composition operator between different function spaces. Recently, numerous authors have studied the boundedness and compactness of weighted composition operators on spaces of analytic functions on various domains (see, for example, [1–17] and the related references therein) as well as of some related operators involving composition ones on these spaces [18–26]. A joint feature for the majority of these papers is that they study the operators from or to Bloch-type or Bergman-type spaces. Paper [27] is one of the older papers that studied composition operators (case \(\psi(z)\equiv1\)) on Bloch-type spaces and served as a motivation to several authors.
A continuous function \(\nu:{\mathbb{D}}\to(0,\infty)\) is called weight. If \(\nu(z)=\nu(|z|)\), \(z\in{\mathbb{D}}\), the weight is called radial. If a weight ν is such that \(\lim_{|z|\to 1^{-}}\nu(z)=0\), we will call it a standard weight. Weight \(\nu(r)\), \(r=|z|\), is called normal if there exist positive numbers η and τ, \(0 < \eta< \tau\), and \(\delta\in[0 , 1)\) such that
The classical weights \(\nu_{\alpha}(r)=(1-r^{2})^{\alpha}\), \(\alpha>-1\), are obviously normal.
Let \(dA(z)=dx \, dy/\pi=r \, dr\, d\theta/\pi\) stand for the normalized area measure in \(\mathbb{D}\). As usual, a measurable function g on \({\mathbb{D}}\) is called Lebesgue integrable if
and it is written \(g\in L^{1}({\mathbb{D}})\). If \(g\in L^{1}({\mathbb {D}})\) is nonnegative, we will write \(g\in L^{1}_{+}({\mathbb{D}})\).
For \(0< p<\infty\) and σ a nonnegative Lebesgue integrable function on \({\mathbb{D}}\), we denote by \(A^{p}(\sigma)\) the Bergman-type space consisting of all functions \(f\in H({\mathbb{D}})\) such that
For \(\sigma(z)=(1-|z|^{2})^{\alpha}\), \(\alpha >-1\), the space becomes the (standard) weighted Bergman space \(A^{p}_{\alpha}({\mathbb{D}})=A^{p}_{\alpha}\).
The weights considered here are the so-called Békollé weights [28], which are Bergman spaces analogues of the Muckenhoupt classes used in harmonic analysis. For \(p_{0} > 1\) and \(\alpha> -1\), let \(dA_{\alpha}(z)=(\alpha+1)(1-|z|^{2})^{\alpha}\, dA(z)\), \(B_{p_{0}}(\alpha)\) be the class consisting of \(\sigma\in L^{1}_{+}({\mathbb{D}})\) with the property that there exists a constant \(C > 0\) such that
for any Carleson square
where \(1/p_{0}+1/p_{0}'= 1\). It is not difficult to see that all normal weights are the Békollé weights.
It is well known [29] that the following inclusions hold:
and
A function \(\sigma\in L^{1}_{+}({\mathbb{D}})\) belongs to class \(C_{p_{0}}\), \(p_{0}> 1\), if there is a constant \(C > 0\) such that
for every disk \(D_{\lambda}(r) = \{z \in\mathbb{D}: | z - \lambda| < r(1 - |\lambda|) \}\). Here \(r \in(0 , 1)\) is fixed, but the class \(C_{p_{0}}\) is actually independent of \(r \in(0 , 1)\). Moreover, \(B_{p_{0}}(\alpha) \subset C_{p_{0}}\) for every \(\alpha> -1\) and the inclusion is strict. For more about the classes \(B_{p_{0}}(\alpha)\) and \(C_{p_{0}}\) and the properties satisfied by the weights in these classes, we refer to [30] and [29] and the references therein.
Throughout this paper constants are denoted by C, they are positive and not necessarily the same at each occurrence. The notation \(A \lesssim B\) means that \(A\le CB\) for some \(C>0\) independent of the variables involved into these quantities. If \(A \lesssim B\) and \(B \lesssim A\), then we write \(A \asymp B\).
For functions in \(A^{p}(\sigma)\), when the weight σ is in \(C_{p_{0}}\), we have the following estimate, which easily follows by using a standard procedure, namely, from the Cauchy inequality applied to function \(f^{(k)}\), the subharmonicity of \(|f|^{p}\), \(p>0\), the integral Hölder inequality and the definition of class \(C_{p_{0}}\).
Lemma 1
Let \(r\in(0, 1)\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\cap C_{p_{0}}\), \(p_{0}>1\), \(p> 0\) and \(k\in{\mathbb{N}}_{0}\). Then there is a constant \(C>0\) independent of z such that
for every \(f \in A^{p}(\sigma)\).
The next lemma provides an asymptotic estimate for the norm of weighted Bergman kernel (see [30]).
Lemma 2
Let \(r \in(0, 1)\) be fixed, \(p > 0\), \(p_{0} > 1\) and \(\eta> -1\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is such that \({\sigma(z)}/{(1 - |z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\) and \(\gamma\geq(\eta+ 2)p_{0}/p -2\). Let
(the reproducing kernel of the weighted Bergman space \(A^{p}_{\gamma}\)). Then
Lemma 3
Let \(r \in(0, 1)\) be fixed, \(p > 0\), \(p_{0} > 1\) and \(\eta> -1\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is such that \({\sigma(z)}/{(1-|z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\). Then
is in \(A^{p}(\sigma)\). Moreover,
and \(f_{\lambda}\) converges to zero uniformly on compact subsets of \(\mathbb{D}\) as \(|\lambda |\to1^{-}\).
Proof
Let \(\gamma= (\eta+ 2)p_{0}/p -2\). Then by Lemma 2 we have
from which (2) follows. By Lemma 1 with \(k=0\) and Lemma 2, we have that
Thus
Using (3) in (1) it is easy to see that \(f_{\lambda}\) converges to zero uniformly on compact subsets of \(\mathbb{D}\), as \(|\lambda |\to1^{-}\). □
For a weight ν, the Bloch-type space \(\mathcal{B}_{\nu}\) on \(\mathbb{D}\) is the space of all holomorphic functions f on \(\mathbb{D}\) such that
The little Bloch-type space \(\mathcal{B}_{\nu, 0}\) consists of all \(f \in\mathcal{B}_{\nu}\) such that
Both spaces \(\mathcal{B}_{\nu}\) and \(\mathcal{B}_{\nu, 0}\) are Banach spaces with the norm
and \(\mathcal{B}_{\nu, 0}\) is a closed subspace of \(\mathcal{B}_{\nu}\).
Depending on the weight ν, various Bloch-type spaces are obtained. For \(\nu(z)=(1-|z|^{2})^{\alpha}\), \(\alpha >0\), the spaces are reduced to the so-called α-Bloch, that is, the little α-Bloch space, which is for \(\alpha =1\) reduced to the classical Bloch space. The reader could see that papers [1–6, 8, 10–12, 14, 15, 18–26] consider concrete operators on or to various Bloch-type spaces. Nowadays we know that if the image space is a Bloch-type, then usually it does not affect much on the boundedness and compactness, so we will here consider the spaces with as much as general weights.
The compactness of a closed subset \(L \subset\mathcal{B}_{\nu, 0}\) can be characterized as follows.
Lemma 4
Let ν be a standard weight. A closed set L in \(\mathcal{B}_{\nu,0}\) is compact if and only if it is bounded with respect to the norm \(\|\cdot\|_{\mathcal{B}_{\nu}}\) and satisfies
This result for the case \(\nu(z)=1-|z|^{2}\) was proved by Madigan and Matheson in [27]. By a slight modification of their proof in the case, Lemma 4 is proved.
Motivated by [3], as well as by [9], here we characterize the boundedness and compactness of the weighted composition operators acting from the Bergman-type spaces \(A^{p}(\sigma)\) to Bloch-type spaces \(\mathcal{B}_{\nu}\) and \(\mathcal{B}_{\nu, 0}\). Our results extend some in [3].
The following criterion for compactness follows by standard arguments which appeared for the first time in [31].
Lemma 5
Let \(\sigma\in L^{1}_{+}({\mathbb{D}})\) be such that \({\sigma(z)}/{(1-|z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\), ν be a standard weight and \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) be bounded. Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is compact if and only if for any bounded sequence \((f_{n})_{n \in\mathbb{N}}\) in \(A^{p}(\sigma)\) which converges to zero uniformly on compact subsets of \(\mathbb{D}\), we have
2 Boundedness and compactness of \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}(\mathcal{B}_{\nu, 0})\)
In this section we formulate and prove the main results in this paper.
Theorem 1
Let \(r \in(0, 1)\) be fixed, \(p > 0\), \(p_{0} > 1\), \(\alpha> -1\), ν be a weight, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Assume that \(p_{0} \geq p\) and \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is such that \({\sigma(z)}/{(1 - |z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\). Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to \mathcal{B}_{\nu}\) is bounded if and only if the following conditions are satisfied:
-
(i)
\(M_{1} = \sup_{z \in\mathbb{D}} {\nu (z)|\psi^{\prime}(z)|}{ (\int_{D_{\varphi(z)}(r)} \sigma \,dA )^{-1/p}} < \infty\);
-
(ii)
\(M_{2} = \sup_{z \in\mathbb{D}} \frac{\nu (z)| \psi(z) \varphi^{\prime}(z)|}{1 - |\varphi(z)|^{2}}{ (\int_{D_{\varphi(z)}(r)} \sigma \,dA )^{-1/p}} < \infty\).
Moreover, if \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded, then
Proof
First suppose that conditions (i) and (ii) hold. Then by Lemma 1 we have
Furthermore,
Using (i), (ii), (5) and (6), we see that
So, we have that \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded and
Conversely, suppose that \(W_{\psi, \varphi} : A^{p}(\sigma) \to \mathcal{B}_{\nu}\) is bounded. Then, by taking \(f(z)\equiv1\in A^{p}(\sigma)\), we have that
By taking \(f(z) = z\in A^{p}(\sigma)\), using the boundedness of φ and asymptotic estimate (8), we easily get
Let \(\lambda= \varphi(\zeta)\), \(\zeta\in{\mathbb{D}}\), and \(g_{\lambda}(z) = \tau_{\lambda}(z) f_{\lambda}(z)\), where \(f_{\lambda}\) is defined in (1) and \(\tau_{\lambda}\) is defined as
Then \(\tau_{\lambda}\in H^{\infty}\) as
Thus \(g_{\lambda}\in A^{p}(\sigma)\) and \(\sup_{z \in\mathbb{D}} \|g_{\lambda}\|_{A^{p}(\sigma)} \lesssim1\). Moreover,
We also have
and
Therefore \(g_{\lambda}(\lambda) = 0\) and, from (11) and (12), we have
Using this fact we obtain
Thus, for a fixed \(\delta\in(0 , 1)\), we obtain
By using (3) and (9), we have that
Hence from (16) and (17) we have
Let \(\lambda= \varphi(\zeta)\), \(\zeta\in{\mathbb{D}}\), and \(f_{\lambda}\) be defined in (1). Recall that \(\sup_{\lambda\in\mathbb{D}} \|f_{\lambda}\|_{A^{p}(\sigma)} \lesssim1\). Hence, by (12) and (13) we get
from which, along with the boundedness of φ, it follows that
Taking the supremum over \(\zeta\in\mathbb{D}\) in (19) and using (18), we get
Finally, from (7) and (21), (4) holds. □
Theorem 2
Let \(r\in(0, 1)\) be fixed, \(p > 0\), \(p_{0} > 1\), \(\alpha> -1\), ν be a standard weight, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is such that \({\sigma(z)}/{(1-|z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\) and \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded. Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is compact if and only if the following conditions are satisfied:
-
(i)
\(\lim_{|\varphi(z)| \to 1}{\nu(z)|\psi^{\prime}(z)|}{ (\int_{D_{\varphi(z)}(r)} \sigma \,dA )^{-1/p}} = 0\);
-
(ii)
\(\lim_{|\varphi(z)| \to1} \frac{\nu(z)| \psi(z) \varphi^{\prime}(z)|}{1 - | \varphi(z)|^{2}}{ (\int_{D_{\varphi(z)}(r)} \sigma \,dA )^{-1/p}} = 0\).
Proof
First suppose that conditions (i) and (ii) hold. Then by Lemma 5 it is sufficient to show that if \((f_{n})_{n \in\mathbb{N}}\) is a bounded sequence in \({A^{p}(\sigma)}\) that converges to zero uniformly on compact subsets of \(\mathbb{D}\), then \(\|W_{\psi, \varphi}f_{n}\|_{\mathcal{B}_{\nu}} \to0 \) as \(n \to\infty\). Let \((f_{n})_{n \in\mathbb{N}}\subset{A^{p}(\sigma)}\) be such a sequence. By conditions (i) and (ii), we have that for any \(\varepsilon> 0\), there is \(\delta\in(0,1)\) such that
and
whenever \(\delta< |\varphi(z)| < 1\).
Let \(K = \{z \in\mathbb{D} : |z| \leq\delta\}\). Clearly, K is a compact subset of \(\mathbb{D}\). We have
where we have used the fact that \(\psi\in\mathcal{B}_{\nu}\) and \(N = \sup_{\zeta\in\mathbb{D}}\nu(\zeta)| \psi(\zeta) \varphi^{\prime}(\zeta)| < \infty\).
Using (22) and (23) along with facts that
for some \(N_{0}\in\mathbb{N}\) and for all \(n \geq N_{0}\), in (24), we have \(\|W_{\psi, \varphi}f_{n}\|_{\mathcal{B}_{\nu}}< C\varepsilon\) for \(n \geq N_{0}\). Since \(\varepsilon> 0\) is arbitrary, we have that \(\|W_{\psi, \varphi}f_{n}\|_{\mathcal{B}_{\nu}} \to0 \) as \(n \to\infty\). Hence \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is compact.
Conversely, suppose that \(W_{\psi, \varphi} : A^{p}(\sigma) \to \mathcal{B}_{\nu}\) is compact. Let \((\zeta_{n})_{n\in{\mathbb{N}}}\) be a sequence in \(\mathbb{D}\) such that \(|\varphi(\zeta_{n})| \to1\) as \(n \to\infty\). If such a sequence does not exist, then (i) and (ii) are vacuously satisfied. Let \(g_{n}(z) = \tau_{\varphi(\zeta_{n})}(z) f_{\varphi(\zeta_{n})}(z)\), where \(f_{\lambda}\) is defined in (1) and \(\tau_{\lambda}\) is defined in (10). Then as in Theorem 1, \(\|\tau_{\varphi(\zeta_{n})}\|_{A^{p}(\sigma)} \lesssim1\) and by Lemma 3, \(\|f_{\varphi(\zeta_{n})}\|_{A^{p}(\sigma)} \lesssim 1\) and \((f_{\varphi(\zeta_{n})})_{n\in{\mathbb{N}}}\) converges to zero uniformly on compact subsets of \(\mathbb{D}\) as \(n \to\infty\). Thus \(\|g_{n}\|_{A^{p}(\sigma)} \lesssim1\) and \((g_{n})_{n \in\mathbb{N}}\) converges to zero uniformly on compact subsets of \(\mathbb{D}\) as \(n \to\infty\). Since \(W_{\psi, \varphi} : A^{p}(\sigma) \to \mathcal{B}_{\nu}\) is compact, we have that \(\|W_{\psi, \varphi}g_{n}\|_{\mathcal{B}_{\nu}} \to0 \) as \(n \to\infty\). On the other hand, from (15) we have
Using these two facts we have that
from which (i) follows.
Let \(f_{\lambda}\) be defined in (1). Then \(\sup_{n\in{\mathbb{N}}}\|f_{\varphi(\zeta_{n})}\|_{A^{p}(\sigma)} \lesssim1\) and \(f_{\varphi(\zeta_{n})}\) converges to zero uniformly on compact subsets of \(\mathbb{D}\) as \(n \to\infty\). Since \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is compact, we have that
From (19), we have
which along with (25) and (26) implies that
from which (ii) follows. □
Next we characterize the boundedness and compactness of \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\).
Lemma 6
Let \(r \in(0 , 1)\) be fixed, \(p > 0\), \(p_{0} > 1\), \(\alpha> -1\), ν be a standard weight, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is such that \({\sigma(z)}/{(1 - |z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\). Then
if and only if \(\psi\in\mathcal{B}_{\nu, 0}\) and
Proof
Suppose that (27) holds. Then, since \(\sigma\in L^{1}({\mathbb{D}})\), we have
as \(|z| \to1\). Hence \(\psi\in\mathcal{B}_{\nu, 0}\). On the other hand, since \(|\varphi(z)| \to1\) implies \(|z| \to1\), (28) automatically holds.
Conversely, suppose that \(\psi\in\mathcal{B}_{\nu, 0}\) and (28) hold. By (28), for every \(\varepsilon> 0\), there exists \(\delta\in(0 , 1)\) such that
when \(\delta< |\varphi(z)| < 1\).
On the other hand, since \(\psi\in\mathcal{B}_{\nu, 0}\), there exists \(\gamma\in(0, 1)\) such that
whenever \(\gamma< |z| < 1\).
Thus if \(\gamma< |z| < 1\) and \(\delta< |\varphi(z)| < 1\), we have that (29) holds, while if \(\gamma< |z| < 1\) and \(|\varphi(z)| \leq\delta\), then from (3) we have
Combining (29) and (30), we easily obtain that (27) holds. □
The following lemma is proved similarly. Hence we omit the proof.
Lemma 7
Let \(r \in(0 , 1)\) be fixed, \(p > 0\), \(p_{0} > 1\), \(\alpha> -1\), ν be a standard weight, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb {D}})\) is such that \({\sigma(z)}/{(1 - |z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\). Then
if and only if
and
Theorem 3
Let \(r \in(0 , 1)\) be fixed, \(p > 0\), \(p_{0} > 1\), \(\alpha> -1\), ν be a standard weight, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is radial and such that \({\sigma(z)}/{(1 - |z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\). Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is bounded if and only if \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded, \(\psi\in\mathcal{B}_{\nu, 0}\) and
Proof
First suppose that \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is bounded. Then it is obvious that \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is also bounded. By taking \(f(z)\equiv1\in A^{p}(\sigma)\), we have that \(\psi\in\mathcal{B}_{\nu, 0}\). By taking \(f(z) = z\in A^{p}(\sigma)\) and using the fact that \(\psi\in\mathcal{B}_{\nu, 0}\), we have that (31) holds.
Conversely, assume that \(W_{\psi, \varphi} : A^{p}(\sigma) \to \mathcal{B}_{\nu}\) is bounded, \(\psi\in\mathcal{B}_{\nu, 0}\) and (31) holds. Then, for each polynomial p, we have that
from which, along with \(\psi\in\mathcal{B}_{\nu, 0}\) and (31), it follows that \(W_{\psi, \varphi}p \in\mathcal{B}_{\nu, 0}\). Since the set of all polynomials is dense in \(A^{p}(\sigma)\), we have that for every \(f \in A^{p}(\sigma)\), there is a sequence of polynomials \((p_{n})_{n \in\mathbb{N}}\) such that \(\| f - p_{n}\|_{A^{p}(\sigma)} \to0\) as \(n \to\infty\). Hence, by the boundedness of \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\), we have
as \(n \to\infty\). Since \(\mathcal{B}_{\nu, 0}\) is a closed subspace of \(\mathcal{B}_{\nu}\), we have that \(W_{\psi, \varphi}(A^{p}(\sigma)) \subseteq\mathcal{B}_{\nu, 0}\) and so \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is bounded. □
Theorem 4
Let \(r \in(0 , 1)\) be fixed, \(p > 0\), \(p_{0} > 1\), \(\alpha> -1\), ν be a standard weight, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Assume that \(p_{0} \geq p\), \(\sigma\in L^{1}_{+}({\mathbb{D}})\) is radial and such that \({\sigma(z)}/{(1 - |z|^{2})^{\alpha}}\) belongs to \(B_{p_{0}}(\alpha)\) and \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is bounded. Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is compact if and only if the following conditions are satisfied:
-
(i)
\(\lim_{|z| \to1}{\nu(z)|\psi^{\prime}(z)|}{ (\int_{D_{\varphi(z)}(r)} \sigma \,dA )^{-1/p}} = 0\);
-
(ii)
\(\lim_{|z| \to1} \frac{\nu(z)| \psi (z) \varphi^{\prime}(z)|}{1 - |\varphi(z)|^{2}}{ (\int_{D_{\varphi(z)}(r)} \sigma \,dA )^{-1/p}} = 0\).
Proof
First suppose that \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is compact. By taking \(f(z)\equiv1\in A^{p}(\sigma)\), we have that \(\psi\in\mathcal{B}_{\nu, 0}\). By taking \(f(z) = z\in A^{p}(\sigma)\) and using the fact that \(\psi\in\mathcal{B}_{\nu, 0}\), we have that (31) holds. Thus if \(\|\varphi\|_{\infty} <1\), then from (3) and the fact that \(\psi\in\mathcal{B}_{\nu, 0}\) we have
Thus in this case conditions (i) and (ii) follow.
Now assume \(\|\varphi\|_{\infty} = 1\). Let \((z_{n})_{n \in\mathbb{N}}\subset{\mathbb{D}}\) be a sequence such that \(\lim_{n \to \infty}|\varphi(z_{n})| = 1\). By the proof of Theorem 2, we have
and
Using (32) and the fact that \(\psi\in\mathcal{B}_{\nu, 0}\), by Lemma 6, we get (i). Using (33) and the fact that \(\lim_{|z| \to1}{\nu(z)| \psi(z) \varphi^{\prime}(z)|} = 0\), by Lemma 7, we get (ii).
Conversely, by taking the supremum in (5) over all \(f \in A^{p}(\sigma)\) such that \(\|f\|_{A^{p}(\sigma)} \leq1\) and then letting \(|z| \to1\), we obtain that
Thus, by Lemma 4, we obtain that \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu , 0}\) is compact. □
Note that if σ is also a normal weight, then it is easy to see that for a fixed \(r \in(0 , 1)\), the following relationship holds:
Thus from (34) we have
Using (35) and Theorems 1-4, we obtain the following corollaries.
Corollary 1
Let \(p > 0\), ν and σ be normal weights, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded if and only if the following conditions are satisfied:
-
(i)
\(\widehat{M}_{1} = \sup_{z \in\mathbb{D}} \frac{\nu(z)|\psi^{\prime}(z)|}{\sigma(|\varphi(z)|)^{1/p} (1 - |\varphi(z)|^{2})^{2/p}} < \infty\);
-
(ii)
\(\widehat{M}_{2} = \sup_{z \in\mathbb{D}} \frac{\nu(z)|\psi(z)\varphi^{\prime}(z)|}{\sigma(|\varphi(z)|)^{1/p} (1 - |\varphi(z)|^{2})^{1+2/p}} < \infty\).
Moreover, if \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded, then
Corollary 2
Let \(p > 0\), ν and σ be normal weights, \(\psi\in H(\mathbb{D})\), \(\varphi\in S({\mathbb{D}})\) and \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) be bounded. Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is compact if and only if the following conditions are satisfied:
-
(i)
\(\lim_{|\varphi(z)| \to1} \frac{\nu (z)|\psi^{\prime}(z)|}{\sigma(|\varphi(z)|)^{1/p} (1 - |\varphi (z)|^{2})^{2/p}} = 0\);
-
(ii)
\(\lim_{|\varphi(z)| \to1} \frac{\nu (z)|\psi(z)\varphi^{\prime}(z)|}{\sigma(|\varphi(z)|)^{1/p} (1 - |\varphi(z)|^{2})^{1+2/p}} = 0\).
Corollary 3
Let \(p > 0\), ν and σ be normal weights, \(\psi\in H(\mathbb{D})\) and \(\varphi\in S({\mathbb{D}})\). Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is bounded if and only if \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu}\) is bounded, \(\psi\in\mathcal{B}_{\nu, 0}\) and
Corollary 4
Let \(p > 0\), ν and σ be normal weights, \(\psi\in H(\mathbb{D})\), \(\varphi\in S({\mathbb{D}})\) and \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) be bounded. Then \(W_{\psi, \varphi} : A^{p}(\sigma) \to\mathcal{B}_{\nu, 0}\) is compact if and only if the following conditions are satisfied:
-
(i)
\(\lim_{|z| \to1} \frac{\nu(z)|\psi ^{\prime}(z)|}{\sigma(|\varphi(z)|)^{1/p} (1 - |\varphi(z)|^{2})^{2/p}} = 0\);
-
(ii)
\(\lim_{|z| \to1} \frac{\nu(z)|\psi (z)\varphi^{\prime}(z)|}{\sigma(|\varphi(z)|)^{1/p} (1 - |\varphi (z)|^{2})^{1+2/p}} = 0\).
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Stević, S., Sharma, A.K. Weighted composition operators from weighted Bergman spaces with Békollé weights to Bloch-type spaces. J Inequal Appl 2015, 337 (2015). https://doi.org/10.1186/s13660-015-0858-2
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DOI: https://doi.org/10.1186/s13660-015-0858-2