A GENERALIZED HILBERT OPERATOR ACTING ON HARDY SPACES

Huiling CHEN; Shanli YE*

doi:10.1007/s10473-026-0110-9

Acta mathematica scientia, Series B >

2026 , Vol. 46 >Issue 1: 145 - 163

DOI: https://doi.org/10.1007/s10473-026-0110-9

A GENERALIZED HILBERT OPERATOR ACTING ON HARDY SPACES

Huiling CHEN ,
Shanli YE^*

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School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Huiling Chen, E-mail: HuillingChen@163.com

Received date: 2024-10-20

Revised date: 2025-02-08

Online published: 2026-05-22

Supported by

The first author was supported by the Zhejiang Province Natural Science Foundation of China (LY23A010003).

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Abstract

Let $\alpha>0$ and let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu,\alpha}=(\mu_{n,k,\alpha})_{n,k\ge0}$ with entries
$\mu_{n,k,\alpha}=\int_{[0,1)}^{}\frac{\Gamma(n+\alpha)}{\Gamma(n+1)\Gamma(\alpha)}t^{n+k}{\rm d}\mu(t)$
induces, formally, the generalized-Hilbert operator
$ \mathcal{H}_{\mu,\alpha}\left ( f \right ) \left ( z \right ) =\sum_{n=0}^{\infty} \left (\sum_{k=0}^{\infty} \mu_{n,k,\alpha}a_k \right )z^n,z\in\mathbb{D}, $
where $f(z)={\textstyle \sum_{k=0}^{\infty }} a_kz^k$ is an analytic function in $\mathbb{D}$. This article is devoted to study the measures $\mu$ for which $\mathcal{H}_{\mu,\alpha }$ is a bounded (resp., compact) operator from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$. We also study the analogous problem in the Hardy spaces $H^p(1\le p\le2)$. Finally, we obtain the essential norm of $\mathcal{H}_{\mu,\alpha}$ from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$.

Key words： Hilbert operator; Hardy space; Carleson measure; essential norm

Cite this article

Huiling CHEN , Shanli YE^* . A GENERALIZED HILBERT OPERATOR ACTING ON HARDY SPACES[J]. Acta mathematica scientia, Series B, 2026 , 46(1) : 145 -163 . DOI: 10.1007/s10473-026-0110-9

References

[1] Bao G, Wulan H. Carleson measure and the range of a Cesàro-like operator acting on $H^\infty$. Anal Math Phys, 2022, $\textbf{12}$: Article 142
[2] Chatzifountas C, Girela D, Peláez J Á. A generalized Hilbert matrix acting on Hardy spaces. J Math Anal Appl, 2014, $\textbf{413}$(1): 154-168
[3] Cowen C C, Maccluer B D.Composition Operators on Spaces of Analytic Functions. Boca Raton: CRC Press, 1995
[4] Cwikel M, Kalton N J. Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre. Proceedings of the Edinburgh Mathematical Society, 1995, $\textbf{38}$(2): 261-276
[5] Duren P L.Theory of $H^p$ Spaces. New York: Academic Press, 1970
[6] Galanopoulos P, Peláez J Á. A Hankel matrix acting on Hardy and Bergman spaces. Studia Math, 2010, $\textbf{200}$(3): 201-220
[7] Girela D.Analytic functions of bounded mean oscillation//Aulaskari R. Complex Function Spaces. Joensuu: Univ Joensuu, 2001: 61-170
[8] Girela D, Merchán N. A generalized Hilbert operator acting on conformally invariant spaces. Banach J Math Anal, 2018, $\textbf{12}$(2): 374-398
[9] Liu J, Lou Z, Xiong C. Essebtial norms of integral operators on spaces of analytic functions. Nonlinear Analysis, 2012, $\textbf{75}$(13): 5145-5156
[10] MacCluer B, Zhao R. Vanishing logarithmic Carleson measures. Illinois J Math, 2002, $\textbf{46}$(2): 507-518
[11] Merchán N. Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces. Rev Mat Comput, 2019, $\textbf{32}$: 799-822
[12] Pommerenke C, Clunie J, Anderson J. On Bloch functions and normal functions. J Reine Angew Math, 1974, $\textbf{270}$: 12-37
[13] Rudin W.Function Theory in the Unit Ball of $C^n$. New York: Springer, 1980
[14] Song X, Ji Z. Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space. AIMS Math, 2021, $\textbf{6}$(4): 3305-3318
[15] Tang P, Zhang X. Generalized integral type Hilbert operator acting on weighted Bloch space. Math Methods in the Appl Sci, 2023, $\textbf{46}$(17): 18458-18472
[16] Wang L, Ye S.Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces. arXiv: 2506.19338
[17] Xu Y, Ye S. A derivative-Hilbert operator acting from Bergman spaces to Hardy spaces. AIM Math, 2023, $\textbf{8}$: 9290-9302
[18] Xu Y, Ye S, Zhou Z. A derivative-Hilbert operator acting on Dirichlet spaces, Open Math.2023, $\textbf{21}$: Article 20220559
[19] Ye S, Feng G. A derivative-Hilbert operator acting on Hardy spaces. Acta Math Sci, 2023, $\textbf{43B}$: 2398-2412
[20] Ye S, Feng G. Generalized Hilbert operators acting on weighted Bergman spaces and Dirichlet spaces. Banach J Math Anal, 2023, $\textbf{17}$: Article 38
[21] Ye S, Xu Y.A derivative-Hilbert operator acting from logarithmic Bloch spaces to Bergman spaces. Acta Math Sci, 2024, $\textbf{44B}$: 1916-1930
[22] Ye S, Zhou Z. Generalized Hilbert operator acting on Bloch type spaces. Acta Math Sinica Ser A, 2023, $\textbf{66}$: 557-568
[23] Ye S, Zhou Z. A derivative-Hilbert operator acting on Bergman spaces. J Math Anal Appl, 2022, $\textbf{506}$(1): Article 125553
[24] Ye S, Zhou Z. A derivative-Hilbert operator acting on the Bloch space. Complex Anal Oper Theory, 2021, $\textbf{15}$: Article 88
[25] Zhao R. Essential norms of composition operators between Bloch type spaces. Proc Amer Math Soc, 2010, $\textbf{138}$: 2537-2546
[26] Zhao R. On logarithmic Carleson measures. Acta Sci Math, 2003, $\textbf{69B}$: 605-618
[27] Zhu K. Bloch Type spaces of analytic functions. Rocky Mountain J Math, 1993, $\textbf{23}$(3): 1143-1177
[28] Zhu K.Spaces of Holomorphic Functions in the Unit Ball. New York: Springer, 2005

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