Abstract
It is well known that the Hilbert matrix \(\mathcal {H}\) is bounded from the logarithmically weighted Bergman space \(A_{\log ^\alpha }^2\) into Bergman space \(A^2\) when \(\alpha >2\). In this paper, we calculate lower bound and upper bound for the norm of the Hilbert matrix operator \(\mathcal {H}\) from the logarithmically weighted Bergman space \(A_{\log ^\alpha }^2\) into Bergman space \(A^2\) when \(\alpha >2\). We also calculate lower bound and upper bound for the norm of the Hilbert matrix operator from \(A_{\log ^\alpha }^p\) into \(A^p\), for \(2<p<\infty \) and \(\alpha >1\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and Preliminaries
In recent years, the study for Hilbert matrix operator \(\mathcal {H}\)’s boundedness and norm on different analytic function spaces has been under active investigation (see [1,2,3,4,5,6,7,8]). Diamantopoulus and Siskakis [7] studied \(\mathcal {H}\) is bounded on Hardy space \(H^p(1<p<\infty )\) and also obtained an upper bound estimate for its norm. In [6], Diamantopoulus began consider the boundedness of \(\mathcal {H}\) on the Bergman spaces \(A^p(2<p<\infty )\), and obtained the upper bound estimate for the norm of \(\mathcal {H}\). Then Dostanić, Jevtić, Vukotić [8] established the precise value of the norm of \(\mathcal {H}\) in the Hardy space \(H^p(1<p<\infty )\) and also gave exact value of the norm of \(\mathcal {H}\) in the Bergman space \(A^p(4<p<\infty )\). In 2017, Boz̆in and Karapetrović [3] solved the question of exact value of the norm of \(\mathcal {H}\) in the Bergman space \(A^p\), for \(2<p<4\). The norm of Hilbert matrix operator \(\mathcal {H}\) has also been studied in other analytic function spaces like Korenblum spaces \(H^\infty _\alpha \) [5, 19].
The study of boundedness of \(\mathcal {H}\) on \(A^p_\alpha \) was initiated in [10] and some partial results were obtained. The boundedness of the Hilbert matrix on \(A^p_\alpha \) for \(1< 2 + \alpha <p\) was also studied by Jevtić M and Karapetrović B in [12]. In [13], Karapetrović obtained the exact norm of \(\mathcal {H}\) on \(A^p_\alpha \) when \(4 \le 2(2+\alpha ) \le p < \infty \). Additionally, he showed that the same lower bound holds for all \(p> 2+\alpha > 1\). He also conjectured that the upper bound for the norm of \(\mathcal {H}\) is the same as above also for the case \(1< 2+\alpha<p < 2(2+\alpha )\) in [13]. In [18] Lindström, Miihkinen and Wikman confirmed the conjecture in the positive for \(2 + \alpha + \sqrt{ \alpha ^2 + \frac{7}{2}\alpha + 3} \le p < 2(2 + \alpha )\). Recently Karapetrović generalized the work of [18] by showing that the conjecture holds for \(2 + \alpha + \sqrt{ (2 + \alpha )^2 - (\sqrt{2} - \frac{1}{2} )(2 + \alpha )} \le p < 2(2 + \alpha )\). In [2], Bralović and Karapetrović provide a new upper bound for the norm of the Hilbert matrix \(\mathcal {H}\) on the weighted Bergman spaces \(A^p_\alpha \) when \(-1<\alpha < 0\), which represents an improvement.
We also realized the Hilbert matrix operator is unbounded on \(A^2\) in [6]. And the situation is actually even worse: the series defining \(\mathcal {H}f (0)\) is divergent in [8]. Then in [16] Łanucha, Nowak and Pavlović considered \(\mathcal {H}\) acts as a bounded operator from \(A^2_{\log ^\alpha }\) to \(A^2\) for \(\alpha >3\) and this was improved in [11, Theorem 4.5], where it is proved that \(\mathcal {H}\) maps \(A^2_{\log ^\alpha }\) into \(A^2\) for \(\alpha >2\). The last result is also improved in [15, Theorem 3.2] by Karapetrović, where it is proved that \(\mathcal {H}\) maps \(A^2_{\log ^\alpha }\) into \(A^2_{\log ^{\alpha -2-\varepsilon }}\) for \(\alpha >2\) and \(0<\varepsilon \le \alpha -2\). In this paper, we obtained the upper bound for the norm from logarithmically weighted Bergman spaces \(A^2_{\log ^\alpha }\) into Bergman spaces \(A^2\) for \(\alpha >2\). We also find \(\mathcal {H}\) acts as a bounded operator from \(A^p_{\log ^\alpha }\) into \(A^p\) for \(2<p<\infty \) and \(\alpha >0\), and also calculate the upper bound for the norm of \(\mathcal {H}\).
Let \(\mathbb {D}\) denote the open unit disk of the complex plane \(\mathbb {C}\), and let \(H(\mathbb {D})\) denote the set of all analytic functions in \(\mathbb {D}\).
For \(0 < p \le \infty \), the Hardy space \(H^p\) is the space of all functions \(f \in H(\mathbb {D} )\) for which
where
For \(0< p < \infty \) the Bergman space \(A^p\) consists of those \(f \in H(\mathbb {D})\) such that
Here, dA stands for the area measure on \(\mathbb {D}\), normalized so that the total area of \(\mathbb {D}\) is 1. Thus \(dA(z) = \frac{1}{\pi }dxdy = \frac{1}{\pi }rdrd\theta \).
Then for \(0< p < \infty \) and \(\alpha >0\) the logarithmically weighted Bergman space \(A^p_{\log ^\alpha }\) consists of those \(f \in H(\mathbb {D})\) such that
The relation between these spaces we introduced above is well known that \(A^p_{\log ^\alpha } \subset A^p\).
The Hilbert matrix is an infinite matrix \(\mathcal {H}\) whose entries are \(a_{n,k} = \frac{1}{n+k+1} \), \(n,k \ge 0\). The Hilbert matrix \(\mathcal {H}\) can be also viewed as an operator on spaces of analytic functions by its action on their Taylor coefficients. Hence for those \(f \in H(\mathbb {D})\), \(f(z) = \sum _{k=0}^\infty a_kz^k\), then we define a transformation \(\mathcal {H}\) by
As usual, throughout this paper, C denotes a positive constant which depends only on the displayed parameters but not necessarily the same from one occurrence to the next.
2 Norm Estimates of the Hilbert Matrix \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^2 \rightarrow A^2}\)
In this section, we drive norm estimates for Hilbert matrix operator acting from \(A_{\log ^{\alpha }}^2\) into \(A^2\) for \(\alpha >2\).
According to [16, Lemma 4.2], we obtain that there exists a constant \(C > 0\) such that
for every \(f(z)= \sum _{k=0}^\infty a_k z^k\) that belongs to \(A_{\log ^\alpha }^2\), \(\alpha >2\). Then we obtain a well-defined analytic function \(\mathcal {H}f(z)\) on \(\mathbb {D}\). Hence, we have that
2.1 Upper Bound for the Norm \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^2 \rightarrow A^2}\)
We know that the Hilbert matrix operator \(\mathcal {H}\) has an integral representation in terms of weighted composition operators \(T_t\) (see [6]):
where
Theorem 2.1
Let \(\alpha >2\). Then the norm of the Hilbert matrix operator acting from \(A_{\log ^{\alpha }}^2\) into \(A^2\) satisfies the upper estimate
Proof
By Minkowski’s inequality, we have
Using linear fractional change of variable \(w = \phi _t(z), z \in \mathbb {D}\), we obtain that
Therefore
here \(D_t = \phi _t(\mathbb {D})\). It is easy to find that \(D_t = D\left( \frac{1}{2-t}, \frac{1-t}{2-t} \right) \), i.e. \(D_t\) is the Euclidean disc with center on \(\frac{1}{2-t}\) and of radius \(\frac{1-t}{2-t}\). It is easy to see that \(|w| \ge \frac{t}{ 2-t}\), for \(w \in D_t\), and \(D_t \subset E_t \), where \(E_t = \{w \in \mathbb {C}: \frac{t}{2-t}< |w| < 1\}\). Hence, we obtain
On the other hand, we also have
It is easy to find function \(r \rightarrow \frac{1}{r^2}\) is decreasing and function \(r \rightarrow rM^2_2(r,f )\) is increasing, by using Chebyshev’s inequality, we get
And by using (2.5), we have that
Since function \(w\rightarrow \left( \log \frac{2}{1-|w|^2}\right) ^{-\alpha }\) is decreasing, by simple calculation we found that,
Making the change of variable in (2.7), we obtain that
From (2.3) and (2.8), we have that
It means that
and the last integral converges for \(\alpha >2\).
This finishes the proof of the theorem. \(\square \)
This result improves Theorem 4.3 in [16], and we also give a new proof method of Theorem 4.5 in [11].
2.2 Lower Bound for the Norm \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^2 \rightarrow A^2}\)
Before we get lower bound for the norm \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^2 \rightarrow A^2}\), we need find a special function in \(A^2_{\log ^\alpha }\).
Lemma 2.1
Let \(\alpha >2\), \(b\ge 1\) and \(1<\gamma <2\). Then the function
belongs to \(A_{\log ^{\alpha }}^2\).
Proof
First we recall a well known result of Littlewood [17, pp.93–96]: Shows the function has an an integral mean with growth [9, p49]
Since
we have
Thus we can find the integral \(\int _0^1 M_2(r,f)^2\left( \log {\frac{2}{1-r^2}}\right) ^{\alpha } dr \) converges while \(\alpha >2\), \(b\ge 1\) and \(1<\gamma <2\), this shows that \(f (z) \in A_{\log ^{\alpha }}^2\). It is also easy to see that
\(\square \)
Corollary 2.1
Let \(\alpha >1\), \(b\ge 1\) and \(1<\gamma <2\). Then the function
belongs to \(A_{\log ^{\alpha }}^p(p>2)\).
Theorem 2.2
Let \(\alpha >2\). Then the norm of the Hilbert matrix operator acting from \(A_{\log ^{\alpha }}^2\) into \(A^2\) satisfies the lower estimate
where
Proof
Let \(\alpha >2\), we begin by selecting a family of test functions. Choose an arbitrary \(\gamma \) such that \(1<\gamma <2\). It is a standard exercise to check that the function
and Lemma 2.1 shows that \(f_\gamma (z) \in A_{\log ^{\alpha }}^2\). It is also easy to see that
And we let
belong to \(A^2\), and
we obtain a relationship between \(f_\gamma (z)\) and \(F_\gamma (z)\) by the proof of Lemma 2.1, that
Thus, we let \(C_{\alpha }=\limsup _{\gamma \rightarrow 2}\frac{\left\| (1-z)^{-\frac{\gamma }{2}}\right\| _{A^2}}{\left\| \left( \frac{1}{z}\log \frac{1}{1-z} \right) ^{-\frac{\alpha }{2}}(1 - z)^{-\frac{\gamma }{2}}\right\| _{A_{\log ^{\alpha }}^2}}\), and \(C_\alpha \) is a constant, depending only on \(\alpha \).
Using (2.1), we find that
Then making the change of variable \(w=(1-tz)/(1-t)\), we calculate that
we define
for every z in \(\mathbb {D}\), which shows \(\mathcal {H}f_\gamma (z)=F_\gamma (z)\phi _\gamma (z).\)
Knowing that in the definition (2.10) of the function \(\phi _\gamma \) is similar to the function \(\phi _\gamma \) defined in the [8, Theorm 4], we can let w to be a real number \(s \ge 1\). Thus we obtained that \(\phi _\gamma \) belongs to the disk algebra whenever \(\gamma \le 2\), (the case \(\gamma = 2\) will also be useful to us although \(f_2 \notin A_{\log ^{\alpha }}^2\)), we can view \(\phi _\gamma \) is an analytic function of z that
We will use the test function \(g_\gamma (z)=\frac{f_\gamma (z)}{\Vert F_\gamma \Vert _{A^2}}\) and \(G_\gamma (z)=\frac{F_\gamma (z)}{\Vert F_\gamma \Vert _{A^2}}\), we obtain that
Letting \(\gamma \rightarrow 2\), and by [8, Theorem 4] we get,
Then making the change of variable \(x=(s-1)/s\), we calculate that
Thus we obtained that,
where
and the last integral converges for \(\alpha >2\).
This concludes the proof. \(\square \)
Corollary 2.2
Let \(\alpha >2\). Then the norm of the Hilbert matrix operator acting from \(A_{\log ^{\alpha }}^2\) into \(A^2\) satisfies
where
3 Norm Estimates of the Hilbert Matrix \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^p \rightarrow A^p}\)
Then we consider the boundedness of Hilbert matrix from into \(A_{\log ^\alpha }^p\) into \(A^p\), for \(p>2\) and \(\alpha > 0\). That can easy obtain Lemma 3.1.
Lemma 3.1
If \(p>2\) and \(\alpha > 0\), then \(\mathcal {H}\) acts as a bounded operator from \(A_{\log ^\alpha }^p\) into \(A^p\).
Since \(A_{\log ^\alpha }^p \subset A^p\), that the lemma is obviously established.
Lemma 3.2
[3, 6] Let \(2<p<\infty \). Then the norm of the Hilbert matrix operator acting on \(A^p\) satisfies
Then we give the upper bound for the norm estimates of the Hilbert matrix \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^p \rightarrow A^p}\).
Theorem 3.1
Let \(2<p<\infty \) and \(\alpha >1\). Then the norm of the Hilbert matrix operator acting from \(A_{\log ^{\alpha }}^p\) into \(A^p\) satisfies
where
Proof
First, we establish the lower bound for the norm \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^p\rightarrow A^p}\). We also construct a family of test functions like Theorem 2.2. Choose an arbitrary \(\gamma \) such that \(1<\gamma <2\). It is a standard exercise to check that the function
After some elementary calculations, we can also establish that \(f^1_\gamma (z)\) belongs to \(A_{\log ^{\alpha }}^p\). It is also easy to observe that
And we let
belong to \(A^p\), have that
We also obtain a relationship between \(f^1_\gamma (z)\) and \(F^1_\gamma (z)\) by the Theorem 2.2 and Corollary 2.1, that
Thus, we let \(C_{\alpha ,p}=\limsup _{\gamma \rightarrow 2}\frac{\left\| (1-z)^{-\frac{\gamma }{p}}\right\| _{A^p}}{\left\| \left( \frac{1}{z}\log \frac{1}{1-z} \right) ^{-\frac{\alpha }{2}}(1 - z)^{-\frac{\gamma }{p}}\right\| _{A_{\log ^{\alpha }}^2}}\) and \(C_{\alpha ,p}\) is a constant, depending only on \(\alpha \) and p.
It can be seen from the proof of Theorem 2.2, we define
for every z in \(\mathbb {D}\), which shows \(\mathcal {H}f^1_\gamma (z)=F^1_\gamma (z)\phi ^1_\gamma (z).\)
According the proof of Theorem 2.2, we can let w to be a real number \(s \ge 1\). And we view \(\phi ^1_\gamma \) is an analytic function of z that
We will use the test function \(g^1_\gamma (z)=\frac{f^1_\gamma (z)}{\Vert F_\gamma \Vert _{A^p}}\) and \(G^1_\gamma (z)=\frac{F^1_\gamma (z)}{\Vert F^1_\gamma \Vert _{A^p}}\), we have that
Letting \(\gamma \rightarrow 2\), and by [8, Theorem 4] we get,
Then making the change of variable \(x=(s-1)/s\), we calculate that
Thus we obtained that,
where
On the other hand, we give the upper bound for the norm \(\Vert \mathcal {H}\Vert _{A_{\log ^{\alpha }}^p\rightarrow A^p}\). We using the method in the proof of Theorem 2.1 and Lemma 3.2, by simple calculation we found that
By Theorem 3.2 in [3] and Lemma 2 in [6] we can find when \(2<p<4\) and \(4\le p < \infty \), we have that
Hence, in this case, we conclude that
and this concludes the proof. \(\square \)
Data Availability
The authors declare that all data and material in this paper are available.
References
Aleman, A., Montes-Rodríguez, A., Sarafoleanu, A.: The eigenfunctions of the Hilbert matrix. Constr. Approx. 36(3), 353–374 (2012)
Bralović, D., Karapetrović, B.: New upper bound for the Hilbert matrix norm on negatively indexed weighted Bergman spaces. Bull. Malay. Math. Sci. Soc. 45(2), 1183–1193 (2022)
Boz̆in, V., Karapetrović, B.: Norm of the Hilbert matrix on Bergman spaces. J. Funct. Anal. 274(2), 525–543 (2018)
Brevig, O.F., Perfekt, K.M., Seip, K., Siskakis, A., Vukotić, D.: The multiplicative Hilbert matrix. Adv. Math. 302, 410–432 (2016)
Dai, J.: Norm of the Hilbert matrix operator on the Korenblum space. J. Math. Anal. Appl. 514(1), 126270 (2022)
Diamantopoulos, E.: Hilbert matrix on Bergman spaces. Ill. J. Math. 48(3), 1067–1078 (2004)
Diamantopoulos, E., Siskakis, A.G.: Composition operators and the Hilbert matrix. Stud. Math. 140(2), 191–198 (2000)
Dostanić, M., Jevtić, M., Vukotić, D.: Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type. J. Funct. Anal. 254(11), 2800–2815 (2008)
Duren, P.L., Schuster, A.: Bergman Spaces. American Mathematical Soc, Providence (2004)
Galanopoulos, P., Girela, D., Peláez, J.Á., Siskakis, A.G.: Generalized Hilbert operators. Ann. Acad. Sci. Fenn. Math. 39, 231–258 (2014)
Jevtić, M., Karapetrović, B.: Hilbert matrix operator on Besov spaces. Publ. Math. Debrecen. 90(3–4), 359–371 (2017)
Jevtić, M., Karapetrović, B.: Hilbert matrix on spaces of Bergman-type. J. Math. Anal. Appl. 453(1), 241–254 (2017)
Karapetrovtć, B.: Norm of the Hilbert matrix operator on the weighted Bergman spaces. Glasg. Math. J. 60(3), 513–525 (2018)
Karapetrović, B.: Hilbert matrix and its norm on weighted Bergman spaces. J. Geom. Anal. 31(6), 5909–5940 (2021)
Karapetrović, B.: Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces. Czech. Math. J. 68(2), 559–576 (2018)
Łanucha, B., Nowak, M., Pavlović, M.: Hilbert matrix operator on spaces of analytic functions. Ann. Acad. Sci. Fenn. Math. 37, 161–174 (2012)
Littlewood, J.E.: Lectures on the Theory of Functions, vol. 243. Oxford University Press, Oxford (1944)
Lindström, M., Miihkinen, S., Wikman, N.: On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces. Ann. Fenn. Math. 46(1), 201–224 (2021)
Lindström, M., Miihkinen, S., Wikman, N.: Norm estimates of weighted composition operators pertaining to the Hilbert matrix. Proc. Am. Math. Soc. 147(6), 2425–2435 (2019)
Acknowledgements
The authors thank the referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Additional information
Communicated by Harry Dym.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research was supported by National Natural Science Foundation of China (Grant No. 11671357) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LY23A010003).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ye, S., Feng, G. Norm of the Hilbert Matrix on Logarithmically Weighted Bergman Spaces. Complex Anal. Oper. Theory 17, 97 (2023). https://doi.org/10.1007/s11785-023-01403-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-023-01403-2