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Bergman spaces with exponential type weights
Journal of Inequalities and Applications volume 2021, Article number: 193 (2021)
Abstract
For \(1\le p<\infty \), let \(A^{p}_{\omega }\) be the weighted Bergman space associated with an exponential type weight ω satisfying
where \(K_{z}\) is the reproducing kernel of \(A^{2}_{\omega }\). This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight \(\omega _{*}\). As an application, we prove the boundedness of the Bergman projection on \(L^{p}_{\omega }\), identify the dual space of \(A^{p}_{\omega }\), and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from \(A^{p}_{\omega }\) into \(A^{q}_{\omega }\), \(1\le p,q<\infty \), such as Toeplitz and (big) Hankel operators.
1 Introduction and results
Let \(H({\mathbb{D}})\) denote the space of all analytic functions on \({\mathbb{D}}\), where \({\mathbb{D}}\) is the open unit disk in the complex plane \(\mathbb{C}\). A weight is a positive function \(\omega \in L^{1}(\mathbb{D}, dA)\), where \(dA(z) = \frac{dx\,dy}{\pi }\) is the normalized area measure on \({\mathbb{D}}\). For \(0< p < \infty \), the weighted Bergman space \(A^{p}(\omega )\) consists of those functions \(f\in H({\mathbb{D}})\) for which
For \(p=\infty \), we introduce the growth space \(L^{\infty }(\omega ^{1/2})\) of all measurable functions f on \({\mathbb{D}}\) with
and we let
In this paper we study Bergman spaces with weights belonging to a certain class \(\mathcal{W}\), which we describe now. Decreasing weights ω satisfying conditions will be specified in what follows. The class \(\mathcal{W}\), considered previously in [24] and [12], consists of the radial decreasing weights of the form \(\omega (z)=e^{-2\varphi (z)}\), where \(\varphi \in C^{2}({\mathbb{D}})\) is a radial function such that \((\Delta \varphi (z) )^{-1/2}\asymp \tau (z)\) for some positive radial function \(\tau (z)\) that decreases to 0 as \(|z|\rightarrow 1^{-}\) and satisfies \(\lim_{r\to 1^{-}}\tau '(r)=0\). Here Δ denotes the standard Laplace operator. Furthermore, we assume that either there exists a constant \(C>0\) such that \(\tau (r)(1-r)^{-C}\) increases for r close to 1 or
The prototype is the exponential weight
For the weights ω in our class, the point evaluations \(L_{z}: f\longmapsto f(z)\) are bounded linear functionals on \(A^{2}(\omega )\) for each \(z\in {\mathbb{D}}\). In particular, the space \(A^{2}(\omega )\) is a reproducing kernel Hilbert space: for each \(z\in {\mathbb{D}}\), there are functions \(K_{z}\in A^{2}(\omega )\) with \(\|L_{z}\|=\|K_{z}\|_{A^{2}(\omega )}\) such that \(L_{z} f=f(z)=\langle f, K_{z} \rangle _{\omega }\), where
The function \(K_{z}\) is called the reproducing kernel for the Bergman space \(A^{2}(\omega )\) and has the property that \(K_{z}(\xi )=\overline{K_{\xi }(z)}\). The Bergman spaces with exponential type weights have attracted a lot of attention in recent years [9, 12, 16, 17, 24, 25] since new techniques, different from those used for standard Bergman spaces, are required. Consider the class \(\mathcal{E}\) that consists of the weights \(\omega \in \mathcal{W}\) satisfying
It has been proved in [7] that the exponential type weights \(\omega _{\sigma }\) given by (1.1) with \(\sigma =1\) satisfy the previous condition and, therefore, they are in the class \(\mathcal{E}\). Note that following the proof given in [7] with nontrivial modifications, we are able to show that every weight of the form (1.1), with \(0<\sigma <\infty \), is in the class \(\mathcal{E}\). Recently, Hu, Xiaofen, and Schuster proved that the prototype weights, considered in (1.1), satisfy (1.2) (see [14, Corollary 3.2]). This integral estimate allows to study other properties and operators, such as the Bergman projection which is given by
The boundedness of the Bergman projection \(P_{\omega }\) on \(L^{2}(\omega )\) is trivial from the general theory of Hilbert spaces. In contrast with the case of standard Bergman spaces (where the Bergman projection is bounded for \(1< p<\infty \)) in the case of exponential type weights, it turns out that the natural Bergman projection is not bounded on \(L^{p}(\omega )\) unless \(p=2\) (see [8] and [35]). At first glance, this may look surprising, but when one takes into account the similarities with Fock spaces, this seems to be more natural. It turns out that, similarly to the Fock space setting, when studying problems where reproducing kernels are involved, the most convenient setting is provided by the spaces \(A^{p}(\omega ^{p/2})\). As a consequence of condition (1.2), we get the right estimates for the norm of reproducing kernels in \(A^{p}(\omega ^{p/2})\) for \(1\le p\le \infty \). Also, we prove in Theorem 3.2 that, for weights in the class \(\mathcal{E}\), the Bergman projection \(P_{\omega }: L^{p}(\omega ^{p/2})\rightarrow A^{p}(\omega ^{p/2})\) is bounded for \(1\le p\le \infty \). A consequence of that result will be the identification of the dual space of \(A^{p}(\omega ^{p/2})\) with the space \(A^{p'}(\omega ^{{p'}/2})\) and \(A^{1}(\omega ^{1/2})\) with \(A^{\infty }(\omega ^{1/2})\) under the natural integral pairing \(\langle \cdot , \cdot \rangle _{\omega }\), where \(p'\) denotes the conjugate exponent of p. Afterwards, by using the duality and the estimates for the p-norms of reproducing kernels, we are going to obtain an atomic decomposition for Bergman spaces with exponential type weights: for weights \(\omega \in \mathcal{E}\), every function in the weighted Bergman space \(A^{p}(\omega ^{p/2})\), \(1\le p<\infty \), can be decomposed into a series of very nice functions (called atoms). These atoms are defined in terms of kernel functions and in some sense act as a basis for the space \(A^{p}(\omega ^{p/2})\). The atomic decomposition for Bergman space with standard weights was obtained by Coifman and Rochberg [6], and it has become a powerful tool in the study of weighted Bergman spaces. We refer to the books [26, 37, 38] for a modern proof of these results. The norm estimates for the reproducing kernels in \(A^{p}(\omega ^{p/2})\) for \(1\le p<\infty \) permit to extend the results on the boundedness and compactness of Toeplitz operators \(T_{\mu }\), to consider the action of \(T_{\mu }\) between different large weighted Bergman spaces, and to find a general description of when \(T_{\mu }: A^{p}(\omega ^{p/2})\longrightarrow A^{q}(\omega ^{q/2})\) is bounded or compact for all values of \(1\le p,q<\infty \). Furthermore, we also generalize the results obtained in [12] on the boundedness and compactness of big Hankel operators \(H_{\overline{g}}\) with conjugate analytic symbols to the non-Hilbert space setting, characterizing for all \(1< p,q<\infty \) the operators
that are bounded or compact. As mentioned earlier, one of the key tools consists in using the estimates for the ∂̅-equation obtained in Sect. 2.
In what follows we use the notation \(a \lesssim b\) to indicate that there is a constant \(C > 0\) with \(a \le C b\); and the notation \(a \asymp b\) means that \(a \lesssim b\) and \(b \lesssim a\). Also, respectively the expressions \(L^{p}_{\omega }\) and \(A^{p}_{\omega }\) mean \(L^{p}({\mathbb{D}},\omega ^{p/2}\,dA)\) and \(A^{p}(\omega ^{p/2})\) for \(1 le p<\infty \).
2 Preliminaries and basic properties
A positive function τ on \({\mathbb{D}}\) is said to be of class \(\mathcal{L}\) if it satisfies the following two properties:
-
(A)
There is a constant \(c_{1}\) such that \(\tau (z)\le c_{1} (1-|z|)\) for all \(z\in {\mathbb{D}}\);
-
(B)
There is a constant \(c_{2}\) such that \(|\tau (z)-\tau (\zeta )|\le c_{2} |z-\zeta |\) for all \(z,\zeta \in {\mathbb{D}}\).
We also use the notation
where \(c_{1}\) and \(c_{2}\) are the constants appearing in the previous definition. For \(a\in {\mathbb{D}}\) and \(\delta >0\), we use \(D(\delta \tau (a))\) to denote the Euclidean disc centered at a and having radius \(\delta \tau (a)\). It is easy to see from conditions (A) and (B) (see [24, Lemma 2.1]) that if \(\tau \in \mathcal{L}\) and \(z \in D(\delta \tau ((a)) \), then
for sufficiently small \(\delta > 0 \), that is, for \(\delta \in (0, m_{\tau })\). This fact will be used many times in this paper.
Definition 2.1
We say that a weight ω is of class \(\mathcal{L}^{*}\) if it is of the form \(\omega =e^{-2\varphi }\), where \(\varphi \in C^{2}({\mathbb{D}})\) with \(\Delta \varphi >0\) and \((\Delta \varphi (z) )^{-1/2}\asymp \tau (z)\) with \(\tau (z)\) is a function in the class \(\mathcal{L}\).
It is straightforward to see that \(\mathcal{W}\subset \mathcal{L}^{*}\). The following result is from [24, Lemma 2.2] and gives the boundedness of the point evaluation functional on \(A^{2}_{\omega }\).
Lemma A
Let \(\omega \in \mathcal{L}^{*}\), \(0< p<\infty \), and \(z\in {\mathbb{D}}\). If \(\beta \in \mathbb{R}\), there exists \(M \geq 1\) such that
for all \(f \in H(\mathbb{D})\) and all sufficiently small \(\delta > 0 \).
We also need a similar estimate for the gradient of \(|f| \omega ^{1/2}\).
Lemma 2.2
Let \(\omega \in \mathcal{L}^{*}\) and \(0< p < \infty \). For any \(\delta _{0} > 0\) sufficiently small, there exists a constant \(C(\delta _{0})> 0\) such that
for all \(f \in H(\mathbb{D})\).
Proof
We follow the method used in [22]. Without loss of generality we can assume \(z= 0\). Then, applying the Riesz decomposition (see for example [29]) of the subharmonic function φ in \(D(0,\frac{\delta _{0}}{2}\tau (0))\), we obtain
where \(r = \delta _{0} \tau (0)\), u is the least harmonic majorant of φ in \(D(0,\frac{r}{2})\), and G is the Green function defined for every \(\xi ,\eta \in D(0,r)\), \(\xi \neq \eta \), by
For \(\xi ,\eta \in D(0,\frac{r}{2})\), we have
Then
We pick a function \(h \in H(\mathbb{D})\) such that \(\operatorname{Re}(h) = u\). Also,
Therefore, since \(h'(0) = 2 \frac{\partial u}{\partial \xi }(0)\), we get
By (2.3) we have
This gives
It follows from Lemma A that
To deal with the other term appearing in (2.4), notice that if we use identity (2.2) with the function \(\phi (\xi ) = |\xi |^{2} -(r/2)^{2}\) (since \(\Delta \phi (\xi )= 4\) and its least harmonic majorant is \(u_{\phi } = 0\)), we obtain
Therefore, since \(\Delta \varphi (\eta )= \frac{1}{\tau (\eta )^{2}}\lesssim \frac{1}{\tau (0)^{2}}= \Delta \varphi (0) \) and the Green function \(G\leq 0\), we obtain, for every \(\xi \in D(0,\frac{r}{2})\),
This gives
Therefore
On the other hand, using Cauchy’s inequality, the fact that \(\varphi -u\le 0\), and Lemma A, we get
Finally, using \(\tau (\eta ) \asymp \tau (0)\), we obtain
Bearing in mind (2.6) this gives
Plugging this and (2.5) into (2.4), we get the desired result. □
The following lemma on coverings is due to Oleinik, see [21].
Lemma B
Let τ be a positive function on \({\mathbb{D}}\) of class \(\mathcal{L}\), and let \(\delta \in (0,m_{\tau })\). Then there exists a sequence of points \(\{z_{n}\}\subset {\mathbb{D}}\) such that the following conditions are satisfied:
-
(i)
\(z_{n} \notin D(\delta \tau (z_{k})) \), \(n \neq k \);
-
(ii)
\(\bigcup_{\substack{n}} D(\delta \tau (z_{n})) = \mathbb{D}\);
-
(iii)
\(\widetilde{D}(\delta \tau (z_{n}))\subset D(3\delta \tau (z_{n})) \), where \(\widetilde{D}(\delta \tau (z_{n})) = \bigcup_{ \substack{z\in D(\delta \tau (z_{n}))}} D(\delta \tau (z)) \), \(n = 1, 2 ,3 ,\ldots\) ;
-
(iv)
\(\lbrace D(3\delta \tau (z_{n}))\rbrace \) is a covering of \(\mathbb{D}\) of finite multiplicity N.
The multiplicity N in Lemma B is independent of δ, and it is easy to see that one can take, for example, \(N=256\). Any sequence satisfying the conditions in Lemma B will be called a \((\delta ,\tau )\)-lattice. Note that \(|z_{n}|\to 1^{-}\) as \(n\to \infty \). In what follows, the sequence \(\{z_{n}\}\) will always refer to the sequence chosen in Lemma B.
2.1 Integral estimates for reproducing kernels
We use the notation \(k_{z}\) for the normalized reproducing kernels in \(A^{p}_{\omega }\), that is,
The next result (see [4, 17, 24] for (a) when \(p=2\) and [18, Lemma 3.6] for part (b)) provides useful estimates involving reproducing kernels.
Theorem A
Let \(K_{z} \) be the reproducing kernel of \(A^{2}_{\omega }\). Then
-
(a)
For \(\omega \in \mathcal{W}\), one has
$$ \Vert K_{z} \Vert _{A^{2}_{\omega }} \asymp \omega (z)^{-1/2} \tau (z)^{-1}, \quad z\in {\mathbb{D}}. $$(2.7) -
(b)
For all sufficiently small \(\delta \in (0,m_{\tau })\) and \(\omega \in \mathcal{W}\), one has
$$ \bigl\vert K_{z}(\zeta ) \bigr\vert \asymp \Vert K_{z} \Vert _{A^{2}_{\omega }}\cdot \Vert K_{\zeta } \Vert _{A^{2}_{\omega }} ,\quad \zeta \in D_{\delta }(z). $$(2.8)
Lemma 2.3
Let \(\omega \in \mathcal{E}\). For each \(z\in {\mathbb{D}}\), we have
Proof
By Lemma A and condition (1.2), we have
which finishes the proof. □
Lemma 2.3 together with condition (1.2) allows us to obtain the following estimate for the norm of the reproducing kernel in \(A^{p}_{\omega }\).
Lemma 2.4
Let \(1\le p<\infty \), \(\omega \in \mathcal{E}\), and \(z\in {\mathbb{D}}\). Then
Proof
By (1.2) and Lemma 2.3, we have
On the other hand, by using statement (b) of Theorem A, we have
This completes the proof. □
2.2 Estimates for the ∂̅-equation
The following result, which provides more estimates on the solutions of the ∂̅-equation, will play a crucial role in describing the bounded Hankel operators acting from \(A^{p}_{\omega }\) to \(A^{q}_{\omega }\) when \(1\le p\le q <\infty \). Also, it can be of independent interest.
Theorem 2.5
Let \(\omega \in \mathcal{W}\), and consider the associated weight \(\omega _{*}(z) : = \omega (z) \tau (z)^{\alpha }\), \(z\in {\mathbb{D}}\), and \(\alpha \in \mathbb{R}\). Then there exists a solution u of the equation \(\overline{\partial }u= f\) such that
for all \(1\le p<\infty \), provided the right-hand side is finite. Moreover, one also has the \(L^{\infty }\)-estimate
Proof
We follow the method used in [5] where the case \(\alpha =0\) was proved. By Lemma 3.1 in [24], there are holomorphic functions \(F_{a}\) and some \(\delta _{0} \in (0, m_{\tau })\) such that
Let \(\delta _{1}< \delta _{0}\). Then there is a sequence \(\lbrace z_{n}\rbrace _{n\geq 1}\) such that \(\lbrace D(\delta _{1}\tau (z_{n}))\rbrace \) is a covering of \(\mathbb{D}\) of finite multiplicity N and satisfies the other statements of Lemma B. Let \(\chi _{n}\) be a partition of unity subordinate to the covering \(D(\delta _{1}\tau (z_{n}))\). Consider
Since \(F_{z_{n}}\) are holomorphic functions on \(\mathbb{D}\), by the Cauchy–Pompeiu formula, we have
Then
where
Since \(\chi _{n}\) is a partition of the unity, we have
on \({\mathbb{D}}\), so that Sf solves the equation \(\overline{\partial }S f(z) = f(z)\).
On the other hand, assume that
and
Then, by (2.13), it is straightforward that the \(L^{\infty }\)-estimate holds. Our next goal is to prove the inequality
Consider \(g(\xi ): = f(\xi ) \omega _{*}(\xi )^{1/2}\) and \(Tg(z): = \int _{\mathbb{D}}G(z,\xi ) g(\xi )\,dA(\xi )\). Then the last inequality takes the form
Therefore, using Hölder’s inequality and (2.13), we have
These estimates and Fubini’s theorem give
Now, using the expression of the kernel \(G(z,\xi )\) and the fact that \(\chi _{n}\) are supported in \(D(\delta _{1}\tau (z_{n}))\), we obtain
By (2.11) and using the fact that \(| F_{z_{n}}(\xi )| \asymp \omega (\xi )^{-1/2}\), \(\xi \in D( \delta _{1}\tau (z_{n})) \), it follows that
where the last inequality above follows from the fact that \(\{D(\delta _{\tau }(z_{n}))\}\) is a covering of \({\mathbb{D}}\) of finite multiplicity.
Now we are going to prove that
First we consider the covering of \(\lbrace \xi \in \mathbb{D} : |z-\xi | > \delta _{2} \tau (z)\rbrace \) given by
Let \(4 \delta _{1} <\delta _{2} < \frac{\delta _{0}}{5}\) and \(z\in \mathbb{D}\) be fixed. If \(\xi \in D(\delta _{2}\tau (z))\cap D(\delta _{1}\tau (z_{n}))\), using (2.1), we have
that implies \(z \in D(\delta _{0}\tau (z_{n}))\). Using (2.1) and property (i) of (2.9), it follows
Therefore, using again (2.1) and polar coordinates, we get
If \(\xi \in ({\mathbb{D}}\smallsetminus D(\delta _{2}\tau (z)) ) \cap D(\delta _{1}\tau (z_{n}))\), we show that \(z\notin D(\delta _{1}\tau (z_{n}))\). In fact, if not, \(z\in D(\delta _{1}\tau (z_{n}))\), then using (2.1) we have
this implies a contradiction with our assumption. Thus,
Also, using \(\tau (\xi )\asymp \tau (z_{n})\), we get
Then, again using \(\tau (\xi )\asymp \tau (z_{n})\) and property (ii) of (2.9) with
we have
Then
• If \(2+\alpha > 0\).
• If \(2+\alpha \leq 0\). Using condition \((B)\) in the definition of the class \(\mathcal{L}\), it follows that
So,
This together with (2.14) establishes (2.13).
Finally, it remains to prove inequality (2.11). We split this integral in two parts: one integrating over the disk \(D(\delta _{2}\tau (\xi ))\) and the other one over \(\mathbb{D}\smallsetminus D(\delta _{2}\tau (\xi ))\). We compute the first integral using (i) of (2.9), \(\tau (\xi )\asymp \tau (z_{n})\asymp \tau (z)\), \(z\in D(\delta _{2} \tau (\xi ))\), and by using polar coordinates, we obtain
Now we consider
For \(z\notin D(\delta _{2}\tau (\xi ))\),
Then, again by using \(\tau (\xi )\asymp \tau (z_{n})\), we obtain
This together with (2.9) taking \(M> \max (1,2+\alpha /2)\) gives
• Suppose first that \(\alpha \geq 0\). By \(\tau (z)\leq C 2^{k}\tau (\xi )\), for every \(z\in R_{k}(\xi )\), \(k=1,2,\dots \), we get
• If \(\alpha < 0\), then
This together with (2.16) establishes (2.11). The proof is complete. □
2.3 Carleson type measures
We are going to define (vanishing) q-Carleson measures for \(A^{p}_{\omega }\), \(0< p, q < \infty \), for weights ω in the class \(\mathcal{W}\) and give some essential theorems.
Definition 2.6
Given \(\omega \in \mathcal{W}\) and \(0< p,q\le \infty \), let μ be a positive measure on \({\mathbb{D}}\). We say that μ is a q-Carleson measure for \(A^{p}_{\omega }\) if there exists a positive constant C such that
for all \(f \in A^{p}_{\omega }\). Thus, by the definition, μ is q-Carleson for \(A^{p}_{\omega }\) when the inclusion \(I_{\mu }:A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\mu )\) is bounded.
Next, the following theorems were essentially proved in [24, Theorem 1]. They established necessary and sufficient conditions for \(I_{\mu }:A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\mu )\) to be bounded (compact) when \(0< p,q<\infty \).
Theorem B
Given \(\omega \in \mathcal{W}\) and \(0 < p \leq q < \infty \), let μ be a finite positive Borel measure on \(\mathbb{D}\). Then \(I_{\mu }:A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\mu )\) is bounded if and only if, for each sufficiently small \(\delta > 0\),
Moreover, in that case, \(K_{\mu ,\omega } \asymp \|I_{\mu }\|^{q}_{A^{p}_{\omega }\rightarrow L^{q}( \mathbb{D},d\mu )}\).
Theorem C
Given \(\omega \in \mathcal{W}\) and \(0 < p \leq q < \infty \), let μ be a finite positive Borel measure on \(\mathbb{D}\). Then \(I_{\mu }:A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\mu )\) is compact if and only if, for each sufficiently small \(\delta > 0\),
Theorem D
Given \(\omega \in \mathcal{W}\) and \(0 < q < p < \infty \), let μ be a finite positive Borel measure on \(\mathbb{D}\). The following conditions are equivalent:
-
(a)
\(I_{\mu }: A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\mu )\) is compact.
-
(b)
\(I_{\mu }:A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\mu )\) is bounded.
-
(c)
For each sufficiently small \(\delta > 0\), the function
$$ F_{\mu }(z)= \frac{1}{\tau (z)^{2}} \int _{D(\delta \tau (z))}{ \omega (\xi )^{-q/2}\,d\mu (\xi )} $$belongs to \(L^{\frac{p}{p-q}}(\mathbb{D},dA)\).
Moreover, one has
3 Bounded projections
The boundedness of Bergman projection is a fact of fundamental importance. In the case of the unit disc, the boundedness of Bergman projections is studied in [13, 38], and it immediately gives the duality between the Bergman spaces. The natural Bergman projection is not necessarily bounded on \(L^{p}_{\omega }\) unless \(p = 2\) (see [8] and [35] for more details). However, we are going to see next that \(P_{\omega }\) is bounded on \(L^{p}_{\omega }\) for weights ω in the class \(\mathcal{E}\). We first prove the boundedness of the sublinear operator \(P^{+}_{\omega }\) defined as
We mention here that, for the case of the exponential weight with \(\sigma =1\), the results of this section have been obtained recently in [7].
Theorem 3.1
Let \(1 \leq p < \infty \) and \(\omega \in \mathcal{E} \). The operator \(P^{+}_{\omega }\) is bounded on \(L^{p}_{\omega }\) and on \(L^{\infty }(\omega ^{1/2})\).
Proof
We first consider the easiest case \(p= 1\). By Fubini’s theorem and condition (1.2), we obtain
Next, we consider the case \(1 < p < \infty \). Let \(p'\) denote the conjugate exponent of p. By Hölder’s inequality and (1.2), we get
This together with Fubini’s theorem and another application of (1.2) gives
Finally, if \(f \in L^{\infty }(\omega ^{1/2})\), by condition (1.2) we get
This shows that \(P^{+}_{\omega }\) is bounded on \(L^{\infty }(\omega ^{1/2})\). The proof is complete. □
Theorem 3.2
Let \(1 \leq p < \infty \) and \(\omega \in \mathcal{E} \). The Bergman projection \(P_{\omega }: L^{p}_{\omega }\longrightarrow A^{p}_{\omega }\) is bounded. Moreover, \(P_{\omega }: L^{\infty }(\omega ^{1/2})\rightarrow A^{\infty }(\omega ^{1/2})\) is also bounded.
Proof
In view of Theorem 3.1, it remains to see that \(P_{\omega } f\) defines an analytic function on \({\mathbb{D}}\). This follows easily by the density of the polynomials, the boundedness of \(P^{+}_{\omega }\), and the completeness of \(A^{p}_{\omega }\). □
Corollary 3.3
Let \(\omega \in \mathcal{E}\). The reproducing formula \(f=P_{\omega } f\) holds for each \(f\in A^{1}_{\omega }\).
Proof
This is an immediate consequence of the boundedness of the Bergman projection and the density of the polynomials. □
4 Duality
As in the case of the standard Bergman spaces, one can use the result just proved on the boundedness of the Bergman projection \(P_{\omega }\) in \(L^{p}_{\omega }\) to identify the dual spaces of \(A^{p}_{\omega }\). As usual, if X is a Banach space, we denote its dual by \(X^{*}\). Next two results (Theorems 4.1 and 4.2) on the duality of Bergman spaces with exponential type weights appear also on [7].
Theorem 4.1
Let \(\omega \in \mathcal{E}\) and \(1 < p < \infty \). The dual space of \(A^{p}_{\omega }\) can be identified (with equivalent norms) with \(A^{p'}_{\omega }\) under the integral pairing
Here \(p'\) denotes the conjugate exponent of p, that is, \(p'= p/(p-1)\).
Proof
Let \(1 < p < \infty \) and let \(p' = p/(p-1)\) be its dual exponent. Given a function \(g\in A^{p'}_{\omega }\), Hölder’s inequality implies that the linear functional \(\psi _{g}: A^{p}_{\omega }\longrightarrow \mathbb{C}\) defined by
is bounded with \(\|\psi _{g}\|\leq \|g\|_{A^{p'}_{\omega }}\).
Conversely, let \(T \in (A^{p}_{\omega } )^{*}\). By the Hahn–Banach theorem, we can extend T to an element \(\widetilde{T}\in ( L^{p}_{\omega } )^{*}\) such that \(\|\widetilde{T}\|= \|T\|\). By the Riesz representation theorem, there exists \(H \in L^{p'}(\mathbb{D},\omega ^{p/2}\,dA)\) with \(\|H\|_{L^{p'}(\omega ^{p/2})}=\|\widetilde{T}\|= \|T\| \) such that
for every \(f\in A^{p}_{\omega }\). Consider the function \(h(\xi ) = H(\xi ) \omega (\xi )^{\frac{p}{2}-1}\). Then \(h\in L^{p'}_{\omega }\) with
and
Let \(g= P_{\omega }h\). By Theorem 3.2, the Bergman projection \(P_{\omega }: L^{p'}_{\omega }\longrightarrow A^{p'}_{\omega }\) is bounded. Thus \(g\in A^{p'}_{\omega }\) with
From Fubini’s theorem it is easy to see that \(P_{\omega }\) is self-adjoint. Indeed,
The interchange of the order of integration is well justified, because of the boundedness of the operator \(P^{+}_{\omega }\) (see Theorem 3.1) given by
Therefore, since \(f = P_{\omega }f\) for every \(f\in A^{p}_{\omega }\), according to Corollary 3.3, we get
Finally, the function g is unique. Indeed, if there is another function \(\widetilde{g}\in A^{p'}_{\omega }\) with \(T(f) = \psi _{g}(f) = \psi _{\widetilde{g}}(f)\) for every \(f\in A^{p}_{\omega }\), then by taking \(f= K_{a}\) for each \(a\in \mathbb{D}\) (that belongs to \(A^{p}_{\omega }\) due to Lemma 2.4) and using the reproducing formula, we obtain
Thus, any bounded linear functional T is of the form \(T = \psi _{g}\) for some unique \(g\in A^{p'}_{\omega }\) and, furthermore,
The proof is complete. □
Theorem 4.2
Let \(\omega \in \mathcal{E}\). The dual space of \(A^{1}_{\omega }\) can be identified (with equivalent norms) with \(A^{\infty }(\omega ^{1/2})\) under the integral pairing \(\langle f, g \rangle _{\omega }\).
Proof
Let \(g\in A^{\infty }_{\omega }\). The linear functional \(\psi _{g}: A^{1}_{\omega }\longrightarrow \mathbb{C}\) defined by \(\psi _{g}(f):= \langle f, g \rangle _{\omega }\) is bounded with \(\|\psi _{g}\| \leq \|g\|_{L^{\infty }(\omega ^{1/2})}\) since, for every \(f\in A^{1}_{\omega }\),
Conversely, let \(T\in (A^{1}_{\omega } )^{*}\). Consider the space X that consists of the functions of the form \(h = f \omega ^{1/2}\) with \(f\in A^{1}_{\omega }\). Clearly, X is a subspace of \(L^{1}(\mathbb{D}, dA)\) and \(F(h): = T(h\omega ^{-1/2}) = T(f)\) defines a bounded linear functional on X with \(\|F\|= \|T\|\). By the Hahn–Banach theorem, F has an extension \(\widetilde{F}\in (L^{1}(\mathbb{D}, dA) )^{*}\) with \(\|\widetilde{F}\|= \|F\|\). Hence, there is a function \(G\in L^{\infty }(\mathbb{D},dA)\) with \(\|G\|_{L^{\infty }(\mathbb{D},dA)}= \|F\|\) such that
or
Consider the function \(H(z) = \omega (z)^{-1/2} G(z)\). Then \(H \in L^{ \infty }(\omega ^{1/2})\) with
By Theorem 3.2, the function \(g = P_{\omega }H\) is in \(A^{\infty }_{\omega }\) with
Also, for \(f \in A^{1}_{\omega }\), by the reproducing formula, we have
Finally, as in the proof of Theorem 4.1, the function g is unique. □
Corollary 4.3
Let \(\omega \in \mathcal{E}\). The set E of finite linear combinations of reproducing kernels is dense in \(A^{p}_{\omega }\), \(1\leq p<\infty \).
Proof
Since E is a linear subspace of \(A^{p}_{\omega }\), by standard functional analysis and the duality results in Theorems 4.1 and 4.2, it is enough to prove that \(g\equiv 0\) if \(g\in A^{p'}_{\omega }\) satisfies \(\langle f,g \rangle _{\omega }=0\) for each f in E (with \(p'\) being the conjugate exponent of p, and \(g\in A^{\infty }(\omega ^{1/2})\) if \(p=1\)). But, taking \(f=K_{z}\) for each \(z\in {\mathbb{D}}\) and using the reproducing formula, we get \(g(z)=P_{\omega } g(z)=\langle g,K_{z} \rangle _{\omega }=0\) for each \(z\in {\mathbb{D}}\). This finishes the proof. □
Our next goal is to identify the predual of \(A^{1}_{\omega }\). For a given weight v, we introduce the space \(A_{0}(v)\) consisting of those functions \(f\in A^{\infty }(v)\) with \(\lim_{|z|\rightarrow 1^{-}} v(z) |f(z)|=0\). Clearly, \(A_{0}(v)\) is a closed subspace of \(A^{\infty }(v)\).
Theorem 4.4
Let \(\omega \in \mathcal{E}\). Under the integral pairing \(\langle f,g \rangle _{\omega }\), the dual space of \(A_{0}(\omega ^{1/2})\) can be identified (with equivalent norms) with \(A^{1}_{\omega }\).
Proof
If \(g\in A^{1}_{\omega }\), clearly \(\Lambda _{g}(f)= \langle f,g\rangle _{\omega }\) defines a bounded linear functional in \(A_{0}(\omega ^{1/2})\) with \(\|\Lambda _{g} \|\le \|g\|_{A^{1}_{\omega }}\). Conversely, assume that \(\Lambda \in (A_{0}(\omega ^{1/2}) )^{*}\). Consider the space X that consists of functions of the form \(h=f\omega ^{1/2}\) with \(f\in A_{0}(\omega ^{1/2})\). Clearly, X is a subspace of \(C_{0}({\mathbb{D}})\) (the space of all continuous functions vanishing at the boundary) and \(T(h)=\Lambda (\omega ^{-1/2} h)=\Lambda (f)\) defines a bounded linear functional on X with \(\|T\|=\|\Lambda \|\). By the Hahn–Banach theorem, T has an extension \(\widetilde{T}\in (C_{0}({\mathbb{D}}) )^{*}\) with \(\|\widetilde{T}\|=\|T\|\). Hence, by the Riesz representation theorem, there is a measure \(\mu \in \mathcal{M}({\mathbb{D}})\) (the Banach space of all complex Borel measures μ equipped with the variation norm \(\|\mu \|_{{\mathcal{M}}}\)) with \(\|\mu \|_{{\mathcal{M}}}=\|T\|\) such that
or
Consider the function g defined on the unit disk by
Clearly, g is analytic on \({\mathbb{D}}\) and, by Fubini’s theorem and condition (1.2), we have
proving that g belongs to \(A^{1}_{\omega }\). Now, since \(A_{0}(\omega ^{1/2})\subset A^{2}_{\omega }\), the reproducing formula \(f(\zeta )=\langle f, K_{\zeta } \rangle _{\omega }\) holds for all \(f\in A_{0}(\omega ^{1/2})\). This and Fubini’s theorem yield
By the reproducing formula, the function g is uniquely determined by the identity \(g(z)=\overline{\Lambda (K_{z})}\). This completes the proof. □
For the case of normal weights, the analogues of Theorems 4.2 and 4.4 were obtained by Shields and Williams in [32]. They also asked what happens with the exponential weights, problem that is solved in the present paper.
5 Atomic decomposition
For \(1\le p < \infty \), in this section we are going to obtain an atomic decomposition for the large weighted Bergman space \(A^{p}_{\omega }\), that is, we show that every function in the Bergman spaces \(A^{p}_{\omega }\) with ω in the class \(\mathcal{E}\) can be decomposed into a series of kernel functions. With the help of the duality results and the estimates for the p-norm of the reproducing kernels \(K_{z}\), we can reach our goal. Before stating the main theorem of this section, we need two auxiliary lemmas as follows.
Lemma 5.1
Let \(\omega \in \mathcal{W}\). There is a sequence \(\lbrace z_{n}\rbrace \subset {\mathbb{D}}\) such that
for all \(f \in A^{p}_{\omega }\) and \(1\le p <\infty \).
Proof
Let \(\{z_{k}\}\) be a \((\varepsilon , \tau )\)-lattice on \({\mathbb{D}}\) (that exists by Lemma B) with \(\varepsilon >0\) small enough to be specified later. Let \(f \in A^{p}_{\omega } \). We consider
We have
For \(z\in D(\varepsilon \tau (z_{k}))\), there exists \(\xi _{k,z}\in [z,z_{k}]\) such that
This together with Lemma 2.2, with \(\delta _{0}\in (0,m_{\tau })\) fixed, yields
Using that \(\tau (\xi _{k,z}) \asymp \tau (z_{k})\) and \(D(\delta _{0}\tau (\xi _{k,z}))\subset D(3\delta _{0} \tau (z_{k}))\) for \(z\in D(\varepsilon \tau (z_{k}))\), we obtain
Therefore,
By Lemma B, every point \(z\in \mathbb{D}\) belongs to at most \(C\varepsilon ^{-2}\) of the sets \(D(3\delta _{0}\tau (z_{k}))\), and therefore
Thus, taking \(\varepsilon >0\) so that \(C\varepsilon ^{p} < 1/2\), we get the desired result. □
We may rephrase Lemma 5.1 by saying that every \((\varepsilon ,\tau )\)-lattice, with \(\varepsilon >0\) small enough, is a sampling sequence for the Bergman space \(A^{p}_{\omega }\). Recall that \(\{z_{k}\}\subset {\mathbb{D}}\) is a sampling sequence for the Bergman space \(A^{p}_{\omega }\) if
for any \(f\in A^{p}_{\omega }\). Just note that Lemma 5.1 gives one inequality, and the other follows by standard methods using Lemma A and the lattice properties. Sampling sequences on the classical Bergman space were characterized by K. Seip [30] (see also the monographs [10] and [31]). For sampling sequences on large weighted Bergman spaces, we refer to [4].
Lemma 5.2
Let \(\omega \in \mathcal{E}\), \(1 \le p < \infty \), and \((z_{k})_{k\in \mathbb{N}}\subset {\mathbb{D}}\) be the sequence defined in Lemma B. The function given by
belongs to \(A^{p}_{\omega }\) for every sequence \(\lambda =\lbrace \lambda _{k}\rbrace \in \ell ^{p}\). Moreover,
Proof
By Hölder’s inequality, Lemma A, and Lemma 2.4, it is easy to see that the partial sums of the series in (5.1) converge uniformly on compact subsets of \({\mathbb{D}}\). Thus, F defines an analytic function on \({\mathbb{D}}\). Furthermore, for p = 1, using (1.2) we have
For the case \(p > 1\), consider
By Hölder’s inequality, we have
On the other hand, using Lemma A, Lemma B, and (1.2), we have
Therefore, applying again (1.2), we obtain
which completes the proof. □
Now we are ready to state our main result related to an atomic decomposition of large weighted Bergman spaces \(A^{p}_{\omega }\) for \(1\le p <\infty \). Recall that \(k_{p,z}\) is the normalized reproducing kernel in \(A^{p}_{\omega }\), that is,
Theorem 5.3
Let \(\omega \in \mathcal{E}\) and \(1 \le p < \infty \). There exists a τ-lattice \(\lbrace z_{n} \rbrace \subset \mathbb{D}\) such that:
-
(i)
For any \(\lambda =\{\lambda _{n}\}\in \ell ^{p}\), the function
$$ f(z)=\sum_{n} \lambda _{n} k_{p,z_{n}}(z) $$is in \(A^{p}_{\omega }\) with \(\|f\|_{A^{p}_{\omega }}\le C \|\lambda \|_{\ell ^{p}}\).
-
(ii)
For every \(f\in A^{p}_{\omega }\), there exists \(\lambda =\lbrace \lambda _{n} \rbrace \in \ell ^{p} \) such that
$$ f(z)=\sum_{n} \lambda _{n} k_{p,z_{n}}(z) $$and \(\|\lambda \|_{\ell ^{p}}\le C \|f\|_{A^{p}_{\omega }}\).
Proof
On the one hand, statement (i) is exactly Lemma 5.2. On the other hand, in order to prove (ii), we define a linear operator \(S: \ell ^{p} \longrightarrow A^{p}_{\omega }\) given by
By (i), the operator S is bounded. By the duality results obtained in the previous section, when \(1< p<\infty \), the adjoint operator \(S^{*}:A^{p'}_{\omega }\rightarrow \ell ^{p'}\), where \(p'\) is the conjugate exponent of p, is defined by
for every \(x\in \ell ^{p}\) and \(f \in A^{p'}_{\omega }\). To compute \(S^{*}\), let \(e_{n}\) denote the vector that equals 1 at the nth coordinate and equals 0 at the other coordinates. Then \(Se_{n} = k_{p,z_{n}} \), and using the reproducing formula, we get
Hence, \(S^{*}: A^{p'}_{\omega }\longrightarrow \ell ^{p'} \) is given by
We must prove that S is surjective in order to finish the proof of this case. By a classical result in functional analysis, it is enough to show that \(S^{*}\) is bounded below. By Lemma 5.1 and Lemma 2.4, we obtain
which shows that \(S^{*}\) is bounded below. Finally, once the surjectivity is proved, the estimate \(\|\lambda \|_{\ell ^{p}}\lesssim \|f\|^{p}_{A^{p}_{\omega }}\) is a standard application of the open mapping theorem. When \(p=1\), then \(S^{*}:A^{\infty }(\omega ^{1/2})\longrightarrow \ell ^{\infty }\) is given by
Hence we must show that
for \(f\in A^{\infty }(\omega ^{1/2})\). However, this can be proved with the same method as Lemma 5.1. Indeed, let \(z\in {\mathbb{D}}\). Then there is a point \(z_{n}\) with \(z\in D(\varepsilon \tau (z_{n}))\). By Lemma A, we have
As done in the proof of Lemma 5.1, we have
Thus, putting this in the previous estimate, we obtain
Finally, taking the supremum on z and \(\varepsilon >0\) small enough so that \(C_{4} \varepsilon \le 1/2\), we have
The proof is complete. □
6 Toeplitz operators
In this section we are going to extend the results given in [3, Theorem 1.1] to the non-Hilbert space setting, when the weight ω is in the class \(\mathcal{E}\). Concretely, we characterize the bounded and compact operators \(T_{\mu }\) acting from \(A^{p}_{\omega }\) to \(A^{q}_{\omega }\) when \(1\le p,q<\infty \). Recall that the Toeplitz operator \(T_{\mu }\) is defined by
Note that \(T_{\mu }\) is very loosely defined here, because it is not clear when the integrals above will converge, even if the measure μ is finite. We suppose that μ is a finite positive Borel measure that satisfies the condition
Then the Toeplitz operator \(T_{\mu }\) is well defined on a dense subset of \(A^{p}_{\omega }\), \(1\le p < \infty \). In fact, by Corollary 4.3 and Theorem 5.3, the set E of finite linear combinations of reproducing kernels is dense in \(A^{p}_{\omega }\). Therefore, it follows from condition (6.1) and the Cauchy–Schwarz inequality that \(T_{\mu }(f)\) is well defined for any \(f\in E\). Also, recall that, for \(\delta \in (0,m_{\tau })\), the averaging function of μ on \({\mathbb{D}}\) is given by
Theorem 6.1
Let \(\omega \in \mathcal{E}\), \(1\le p\leq q < \infty \), and μ be a finite positive Borel measure on \(\mathbb{D}\) satisfying (6.1). Then \(T_{\mu }: A^{p}_{\omega }\rightarrow A^{q}_{\omega }\) is bounded if and only if, for each \(\delta \in (0,m_{\tau })\) sufficiently small,
Moreover,
Proof
Since we have the estimate \(\|K_{z}\|_{A^{p}_{\omega }}\asymp \omega (z)^{-1/2} \tau (z)^{-2(p-1)/p}\), if we assume that the Toeplitz operator \(T_{\mu }: A^{p}_{\omega }\longrightarrow A^{q}_{\omega }\) is bounded, then we obtain (6.2) with the same argument as in the proof of Theorem 1.1 in [3].
Conversely, we suppose that (6.2) holds. We first prove that
Indeed, by Lemma A, we have
Then, by Fubini’s theorem, the fact that \(\tau (s)\asymp \tau (\xi )\) for \(s\in D(\delta \tau (\xi ))\), and condition (1.2), we get
This establishes (6.3). Now we proceed to prove that \(T_{\mu }\) is bounded. If \(q > 1\), by Hölder’s inequality, we obtain
Using (6.3), we have
If \(q=1\), this holds directly. By Fubini’s theorem and condition (1.2), we obtain
Consider the measure ν given by
Since (6.2) holds, by Theorem B, the identity \(I_{\nu } : A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D},d\nu )\) is bounded. Moreover, \(\|I_{\nu }\|\lesssim E(\mu )^{1/q}\). Therefore,
This finishes the proof. □
In order to describe the boundedness of \(T_{\mu }:A^{p}_{\omega }\rightarrow A^{q}_{\omega }\) when \(1\le q< p<\infty \), we need first an auxiliary result.
Proposition 6.2
Let \(\omega \in \mathcal{E}\) and \(1 < q < p < \infty \). If \(\widehat{\mu }_{\delta }\in L^{\frac{pq}{p-q}}(\mathbb{D},dA) \), then
for any \(f\in A^{p}_{\omega }\).
Proof
For \(z\in {\mathbb{D}}\) and \(\xi \in D(\delta \tau (z))\), by Lemma A, Lemma B, and (2.1), we obtain
This gives
Therefore,
Applying Hölder’s inequality, we get
On the other hand, by Fubini’s theorem and \(\tau (z)\asymp \tau (s)\), for \(s\in D(\delta \tau (z))\), we have
Combining this with (6.5), we get
The proof is complete. □
Theorem 6.3
Let \(\omega \in \mathcal{E}\), \(1\le q< p < \infty \), and μ be a finite positive Borel measure on \(\mathbb{D}\) satisfying (6.1). The following conditions are equivalent:
-
(i)
The Toeplitz operator \(T_{\mu }: A^{p}_{\omega }\rightarrow A^{q}_{\omega }\) is bounded.
-
(ii)
For each sufficiently small \(\delta > 0\), \(\widehat{\mu }_{\delta }\in L^{\frac{pq}{p-q}}(\mathbb{D},dA) \).
Moreover,
Proof
(i) ⟹ (ii) For an arbitrary sequence \(\lambda =\lbrace \lambda _{k}\rbrace \in \ell ^{p}\), we consider the function
where \(r_{k}(t)\) is a sequence of Rademacher functions (see [20] or Appendix A of [11] ) and \(\lbrace z_{k}\rbrace \) is the sequence given in Lemma B. Because of the norm estimate
given in Lemma 2.4, by part (i) of Theorem 5.3, we obtain
Thus, the boundedness of \(T_{\mu }: A^{p}_{\omega }\rightarrow A^{q}_{\omega }\) gives
In other words, we have
Integrating with respect to t from 0 to 1, applying Fubini’s theorem, and invoking Khinchine’s inequality (see [20]), we obtain
Let \(\chi _{k}\) denote the characteristic function of the set \(D(3\delta \tau (z_{k}))\). Since the covering \(\lbrace D(3\delta \tau (z_{k}))\rbrace \) of \({\mathbb{D}}\) has finite multiplicity N, we have
Now, using Lemma A yields
On the other hand, since for small \(\delta >0\) we have \(|K_{z_{k}}(z)|\asymp \|K_{z_{k}}\|_{A^{2}_{\omega }}\|K_{z}\|_{A^{2}_{\omega }}\) for every \(z \in D(\delta \tau (z_{k}))\), applying statement (a) of Theorem A and (2.1), we have
That is,
Therefore,
Then, using the duality between \(\ell ^{p/q}\) and \(\ell ^{\frac{p}{p-q}}\), we conclude that
that means
This is the discrete version of our condition. To obtain the continuous version, simply note that
Then
This finishes the proof of this implication.
(ii) ⟹ (i) First we begin with the easiest case \(q = 1 \). By Fubini’s theorem and condition (1.2), we have
Now, by using Theorem D with the measure given by
we obtain the desired result. Finally, we study the case \(1< q < \infty \). Let \(\lbrace z_{j}\rbrace \) be the sequence given in Lemma B. Applying Lemma A and Lemma B, we obtain
Applying Hölder’s inequality, we get
where
and
Furthermore, by Lemma B and condition (1.2), we have
Thus
This gives
where
which, by Fubini’s theorem and condition (1.2), gives
Combining this with using (2.1) and Proposition 6.2 shows that
This proves the desired result. □
Next we characterize compact Toeplitz operators on weighted Bergman spaces \(A^{p}_{\omega }\) for weights ω in the class \(\mathcal{E}\). We need first a lemma.
Lemma 6.4
Let \(1< p<\infty \), and let \(k_{p,z}\) be the normalized reproducing kernels in \(A^{p}_{\omega }\), with \(\omega \in \mathcal{E}\). Then \(k_{p,z} \rightarrow 0\) weakly in \(A^{p}_{\omega }\) as \(|z|\rightarrow 1^{-}\).
Proof
By duality and the reproducing kernel properties, we must show that \(|g(z)|/\|K_{z}\|_{A^{p}_{\omega }}\) goes to zero as \(|z|\rightarrow 1^{-}\) whenever g is in \(A^{p'}_{\omega }\), where \(p'\) denotes the conjugate exponent of p, but this follows easily by the density of the polynomials and Lemma A. □
Theorem 6.5
Let \(\omega \in \mathcal{E}\), \(1< p\leq q < \infty \), and μ be a finite positive Borel measure on \(\mathbb{D}\) satisfying (6.1). Then the Toeplitz operator \(T_{\mu }: A^{p}_{\omega }\rightarrow A^{q}_{\omega }\) is compact if and only if, for each \(\delta \in (0,m_{\tau })\) small enough, one has
Proof
First we assume that \(T_{\mu }\) is compact. Following the proof of the boundedness part and the fact that \(\|K_{a}\|^{-1}_{ A^{p}_{\omega }}\asymp \omega (a)^{1/2} \tau (a)^{ \frac{2(p-1)}{p}}\), we get the estimate
where \(k_{p,a}\) are the normalized reproducing kernels in \(A^{p}_{\omega }\). Since, by Lemma 6.4, \(k_{p,a}\) tends to zero weakly in \(A^{p}_{\omega }\) and \(T_{\mu }\) is compact, the result follows.
Conversely, we suppose that (6.8) holds. Let \(\lbrace f_{n}\rbrace \subset A^{p}_{\omega }\) be a bounded sequence converging to zero uniformly on compact subsets of \(\mathbb{D}\). By (6.4), we have
where \(I_{\nu }: A^{p}_{\omega }\longrightarrow L^{q}(\mathbb{D}, d\nu )\) with \(d\nu (\xi ) = \omega (\xi )^{q/2} \tau (\xi )^{2(q-1)(\frac{1}{p}- \frac{1}{q})}\,d\mu (\xi )\). By using \(\tau (a) \asymp \tau (\xi )\), for \(\xi \in D(\delta \tau (a))\), we have
By Theorem C, \(I_{\nu }\) is compact, and in view of (6.10), \(T_{\mu }\) is compact. □
Theorem 6.6
Let \(\omega \in \mathcal{E}\), \(1\le q< p < \infty \), and μ be a finite positive Borel measure on \(\mathbb{D}\) satisfying (6.1). The following conditions are equivalent:
-
(i)
The Toeplitz operator \(T_{\mu }: A^{p}_{\omega }\rightarrow A^{q}_{\omega }\) is compact.
-
(ii)
For each sufficiently small \(\delta > 0\),
$$ \widehat{\mu }_{\delta }\in L^{\frac{pq}{p-q}}( \mathbb{D},dA). $$(6.11)
Proof
If \(T_{\mu }\) is compact, then it is bounded, and by Theorem 6.3 we get the desired result. Conversely, if (6.11) holds, then by Theorem 6.3\(T_{\mu }\) is bounded. Since by Theorem 5.3 the spaces \(A^{p}_{\omega }\) and \(A^{q}_{\omega }\) are isomorphic to \(\ell ^{p}\), the result is a consequence of a general result of Banach space theory: it is known that, for \(1 \le q< p< \infty \), every bounded operator from \(\ell ^{p}\) to \(\ell ^{q}\) is compact (see [19, Theorem I.2.7, p. 31]). □
7 Hankel operators
One of the most important classes of operators acting on spaces of analytic functions is the Hankel operators. When acting on the classical Hardy spaces, their study presents [23, 27] a broad range of applications such as to control theory, approximation theory, prediction theory, perturbation theory, and interpolation problems. Furthermore, one can find an extensive literature on Hankel operators acting on other classical function spaces in one or several complex variables, such as Bergman spaces [1, 2, 15, 36, 37], Fock spaces [28], or Dirichlet spaces [33, 34]. In this section, we are going to study big Hankel operators acting on our large weighted Bergman spaces.
Definition 7.1
Let \(M_{g}\) denote the multiplication operator induced by a function g, and \(P_{\omega }\) be the Bergman projection, where ω is a weight in the class \(\mathcal{E}\). The Hankel operator \(H_{g}\) is given by
We assume that the function g satisfies
Under this assumption, the Hankel operator \(H_{g}^{\omega }\) is well defined on the set E of all finite linear combinations of reproducing kernels and, therefore, is densely defined in the weighted Bergman space \(A^{p}_{\omega }\), \(1\le p<\infty \). Also, for \(f\in E\), one has
We are going to study the boundedness and compactness when the symbol is conjugate analytic. In the Hilbert space case \(A^{2}_{\omega }\), and for weights in the class \(\mathcal{W}\), a characterization of the boundedness, compactness, and membership in Schatten classes of the Hankel operator \(H_{\bar{g}}:A^{2}_{\omega }\rightarrow L^{2}_{\omega }\) was obtained in [12]. In order to extend such results to the non-Hilbert space setting, we need estimates for the p-norm of the reproducing kernels, and it is here when condition (1.2) and the exponential type class \(\mathcal{E}\) enters in action. Before going to study the boundedness of the Hankel operator on \(A^{p}_{\omega }\) with conjugate analytic symbols, we need the following lemma.
Lemma C
Let \(1\leq p < \infty \), \(g\in H({\mathbb{D}})\), and \(a\in {\mathbb{D}}\). Then
Proof
See for example [12]. □
Now we are ready to characterize the boundedness of the Hankel operator with conjugate analytic symbols acting on large weighted Bergman spaces in terms of the growth of the maximum modulus of \(g'\). We begin with the case \(1\le p\le q<\infty \).
Theorem 7.2
Let \(\omega \in \mathcal{E}\), \(1 \leq p\le q< \infty \), and \(g \in H({\mathbb{D}})\) satisfying (7.1). The Hankel operator \(H_{\overline{g}}: A^{p}_{\omega }\longrightarrow L^{q}_{\omega }\) is bounded if and only if
Proof
Suppose first that \(H_{\overline{g}}: A^{p}_{\omega }\longrightarrow L^{q}_{\omega }\) is bounded. Thus
For each \(z\in {\mathbb{D}}\), consider the function \(g_{z} (\xi ): = (g(z) - g(\xi ) )K_{z}(\xi )\). Condition (7.1) ensures that \(g_{z}\in A^{1}_{\omega }\), and by the reproducing formula in Corollary 3.3, one has
Now, for δ small enough, we have \(|K_{z}(a)|\asymp \|K_{z}\|_{A^{2}_{\omega }}\|K_{a}\|_{A^{2}_{\omega }}\) for \(z\in D(\delta \tau (a))\). Hence, by the statement (a) of Theorem A and (2.1), we have
Because of the boundedness of the Hankel operator \(H_{\overline{g}}\), we have
Finally, by the estimates on the norm of \(K_{z}\) in Lemma 2.4 and Theorem A, we obtain
By Lemma C, this completes the proof of this implication.
Conversely, assume that (7.2) holds, and let \(1\le p \le q<\infty \). By Theorem 2.5, there exists a solution u of the equation \(\overline{\partial }u = f \overline{g'}\) in \(L^{q}(\omega ^{q/2})\) such that
Since any solution v of the ∂̅-equation has the form \(v = u- h\) with \(h\in H({\mathbb{D}})\), and because \(H_{\overline{g}}f\) is also a solution of ∂̅-equation, there is a function \(h\in H({\mathbb{D}})\) such that \(H_{\overline{g}}f = u - h \). As a result of \(P_{\omega } (H_{\overline{g}}f )=0 \), we have \(H_{\overline{g}}f= (I-P_{\omega } )u\), where I is the identity operator. Therefore, by the boundedness of \(P_{\omega }\) on \(L^{q}_{\omega }\) (see Theorem 3.2), we obtain
By our assumption (7.2), we have
On the other hand, by Lemma A,
Using the last pointwise estimate, we have
This completes the proof. □
Next we are going to characterize the boundedness of the Hankel operator with conjugate analytic symbols when \(1\le q < p < \infty \). Before that we prove the following lemma.
Lemma 7.3
Let \(\delta _{0} \in (0,m_{\tau })\) and \(0< r<\infty \). Then
for \(f\in H({\mathbb{D}})\).
Proof
By Cauchy’s integral formula and Lemma A, we get
An application of \(\tau (\eta ) \asymp \tau (z)\), for \(\eta \in D(\delta _{0}\tau (z)/2)\), gives
which proves the desired result. □
The following result gives the characterization of the boundedness of the Hankel operator going from \(A^{p}_{\omega }\) into \(L^{q}_{\omega }\) when \(1\le q < p < \infty \).
Theorem 7.4
Let \(\omega \in \mathcal{E}\), \(1 \le q< p< \infty \), and let \(g \in H({\mathbb{D}})\) satisfy (7.1). Then the following statements are equivalent:
-
(a)
The Hankel operator \(H_{\overline{g}}: A^{p}_{\omega }\longrightarrow L^{q}_{\omega }\) is bounded.
-
(b)
The function \(\tau g'\) belongs to \(L^{r}({\mathbb{D}}, dA)\), where \(\frac{1}{r}= \frac{1}{q}-\frac{1}{p}\).
Proof
Suppose that \(\tau g' \in L^{r}({\mathbb{D}}, dA)\). By (7.3), since \(p/q > 1\), a simple application of Hölder’s inequality, yields
This proves the boundedness of \(H_{\overline{g}}\).
Conversely, pick \(\varepsilon >0\) and let \(\lbrace z_{k}\rbrace \) be a \((\varepsilon ,\tau )\)- lattice on \({\mathbb{D}}\). For a sequence \(\lambda =\lbrace \lambda _{k}\rbrace \in \ell ^{p}\), we consider the function
where \(r_{k}(t)\) is a sequence of Rademacher functions. Because of the norm estimate for reproducing kernels given in Lemma 2.4, by part (i) of Theorem 5.3, we obtain
Thus, the boundedness of \(H_{\overline{g}}\) gives
Therefore,
Using the same method in (6.6), we obtain
where \(\chi _{k}\) is the characteristic function of the set \(D(3 \varepsilon \tau (z_{k}))\). Additionally, by applying both statements (a) and (b) of Theorem A and (2.1), we get
Putting this in (7.4) gives
Furthermore, by Lemma C, we obtain
Moreover, by the duality between \(\ell ^{p/q}\) and \(\ell ^{\frac{p}{p-q}}\), it follows
In order to finish the proof, we will justify that \(\|\tau g'\|^{r}_{L^{r}({\mathbb{D}},dA)} \lesssim I_{g'}\). For that, on the one hand, by Lemma 7.3 applied to \(g'\), we get
On the other hand, by Cauchy estimates, there exists \(\xi \in [z,z_{k}]\) such that
Using \(\tau (\xi )\asymp \tau (z)\asymp \tau (z_{k})\), for \(\xi ,z\in D(\varepsilon \tau (z_{k}))\), we have
By (7.5) and using again (2.1), we obtain
By Lemma B, every point \(z\in \mathbb{D}\) belongs to at most \(C\varepsilon ^{-2}\) of the sets \(D(3\delta _{0}\tau (z_{k}))\). Hence
Thus, taking ε so that \(C \varepsilon ^{r} < 1/2\), we get the desired result. □
Next we characterize the compactness of the Hankel operator with conjugate analytic symbol acting from \(A^{p}_{\omega }\) into \(L^{q}_{\omega }\), \(1\le p,q < \infty \). This characterization will be given in two theorems depending on the order of p and q. We begin with the case \(1\le p \le q < \infty \).
Theorem 7.5
Let \(\omega \in \mathcal{E}\), \(1 < p\le q< \infty \), and \(g \in H({\mathbb{D}})\) satisfying (7.1). Then, the Hankel operator \(H_{\overline{g}}: A^{p}_{\omega }\longrightarrow L^{q}_{\omega }\) is compact if and only if
Proof
Let \(a\in {\mathbb{D}}\) and \(1 < p\le q< \infty \). Recall that \(k_{a,p}\) is the normalized reproducing kernel in \(A^{p}_{\omega }\). By Lemma 6.4, \(k_{a,p}\rightarrow 0\) weakly. Thus, if \(H_{\overline{g}}\) is compact, then
Let δ be small enough such that \(|K_{a}(z)|\asymp \|K_{a}\|_{A^{2}_{\omega }} \|K_{z}\|_{A^{2}_{\omega }}\) for \(z\in D(\delta \tau (a))\). By Theorem A, using (2.1) and Lemma 2.4, we have
It follows from Lemma C that
This implies that
which completes the proof of this implication.
Conversely, let \(\lbrace f_{n} \rbrace \) be a bounded sequence in \(A^{p}_{\omega }\) such that \(f_{n} \rightarrow 0\) uniformly on compact subsets of \({\mathbb{D}}\). To show compactness, it is standard to see that it is enough to prove that \(\|H_{\bar{g}}f_{n}\|_{L^{q}_{\omega }}\rightarrow 0\). By the assumption, given any \(\varepsilon >0\), there is \(0 < r_{0} < 1\) such that
Since \(\lbrace f_{n}\rbrace \) converges to zero uniformly on compact subsets of \({\mathbb{D}}\), there exists an integer \(n_{0}\) such that
According to (7.3), we have
On the one hand, it is easy to see that
On the other hand, by Lemma A, we have the pointwise estimate
Applying this together with our assumption, we get
Combining this with (7.6) gives \(\lim_{\substack{n\rightarrow \infty }}\|H_{\overline{g}}f_{n} \|_{L^{q}_{\omega }} =0\). This shows that the Hankel operator \(H_{\overline{g}}:A^{p}_{\omega }\longrightarrow L^{q}_{\omega }\) is compact. □
The next theorem contains a compactness criterion for Hankel operators when \(1 \le q< p < \infty \).
Theorem 7.6
Let \(\omega \in \mathcal{E}\), \(1 \le q< p < \infty \), and \(g \in H({\mathbb{D}})\) satisfying (7.1). The following conditions are equivalent:
-
(a)
The Hankel operator \(H_{\overline{g}}: A^{p}_{\omega }\longrightarrow L^{q}_{\omega }\) is compact.
-
(b)
The function \(\tau g' \) belongs to \(L^{r}({\mathbb{D}}, dA)\), where \(\frac{1}{r}= \frac{1}{q}-\frac{1}{p}\).
Proof
\((a)\Longrightarrow (b)\) Assume that \(H_{\overline{g}}\) is compact. Then \(H_{\overline{g}}\) is bounded. Hence, by applying Theorem 7.4, we get the desired result.
\((b)\Longrightarrow (a)\) Suppose that \(\tau g' \) belongs to \(L^{r}({\mathbb{D}}, dA)\), where \(\frac{1}{r}= \frac{1}{q}-\frac{1}{p}\). By Theorem 7.4, the Hankel operator \(H_{\overline{g}} \) is bounded and, as a result of Theorem 5.3, the space \(A^{p}_{\omega }\) is isomorphic to \(\ell ^{p}\). In this case, \(H_{\overline{g}} \) is also compact, due to a general result of Banach space theory: For \(1 \le q< p < \infty \), every bounded operator from \(\ell ^{p}\) to \(\ell ^{q}\) is compact (see [19, Theorem I.2.7]). This finishes the proof. □
8 Concluding remarks
I believe that I have done a satisfactory work in order to get a better understanding of the function properties of large weighted Bergman spaces and the operators acting on them. I hope that this work is going to attract many other researchers to this area, and expect that the study of this function spaces is going to experience a period of intensive research in the next years. There is still plenty of work to do for a better understanding of the theory of large weighted Bergman spaces, and several natural problems are waiting for further study or a complete solution: atomic decomposition, coefficient multipliers, zero sets, etc. I hope that the methods developed here will be of some help in order to attach the previous mentioned problems.
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Arroussi, H. Bergman spaces with exponential type weights. J Inequal Appl 2021, 193 (2021). https://doi.org/10.1186/s13660-021-02726-4
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DOI: https://doi.org/10.1186/s13660-021-02726-4