Hilbert 矩阵算子在对数加权 Bergman 空间上的范数

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1405-1416.

Hilbert 矩阵算子在对数加权 Bergman 空间上的范数

胡浩(),叶善力^*()

浙江科技大学理学院杭州 310023

收稿日期:2024-12-07 修回日期:2025-03-28 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 叶善力,E-mail:slye@zust.edu.cn
作者简介:胡浩,E-mail:2716007899@qq.com
基金资助:
浙江省自然科学基金(LY23A010003)

Norm of the Hilbert Matrix on the Logarithmically Weighted Bergman Spaces

Hao Hu(),Shanli Ye^*()

School of Sciences, Zhejiang University of Science and Technology, Hangzhou

Received:2024-12-07 Revised:2025-03-28 Online:2025-10-26 Published:2025-10-14
Supported by:
Zhejiang Provincial Natural Science Foundation of China(LY23A010003)

摘要/Abstract

摘要：

设 $1 < p < \infty $, $\alpha>0$ 和 $\beta > -1$. 设函数 $f\in H(\mathbb{D})$, 若

$\|f\|_{A_{\beta,\log^\alpha}^p}\overset{\rm def}{=} \left(\int_\mathbb{D}|f(z)|^p(1-|z|^2)^{\beta} \left(\log\frac{2}{1-|z|^2} \right)^\alpha {\rm d}A(z) \right)^{1/p}< \infty,$

则称 $f$ 属于对数加权的 Bergman 空间 $A^p_{\beta,\log^\alpha}$. 该文计算了当 $\alpha>p$ 且 $1 < p < 2$ 时, Hilbert 矩阵算子 $\mathcal{H}$ 从对数加权 Bergman 空间 $A_{p-2,\log^{\alpha}}^p$ 到 Bergman 空间 $A^p$ 的范数的下界和上界, 同时作者也得到了 Hilbert 矩阵算子从对数加权 Bergman 空间 $A^p_{\beta,\log^\alpha}$ 到加权 Bergman 空间 $A^p_{\beta}$ ($\alpha>0$, $\beta\geq0$, 且 $2+\beta) 的范数估计.

关键词: 算子范数, Hilbert 矩阵算子, 加权 Bergman 空间

Abstract:

Let $1 < p < \infty $, $\alpha>0$ and $\beta > -1$. Let $A^p_{\beta,\log^\alpha}$ denote the logarithmic weighted Bergman space of those functions $f$ which are analytic in the unit disk D such that

$\|f\|_{A_{\beta,\log^\alpha}^p}\overset{\rm def}{=} \left(\int_\mathbb{D}|f(z)|^p(1-|z|^2)^{\beta} \left(\log\frac{2}{1-|z|^2} \right)^\alpha{\rm d}A(z) \right)^{1/p}< \infty.$

This paper computes the lower and upper bounds for the norm of the Hilbert matrix operator $\mathcal{H}$ acting from the logarithmically weighted Bergman space $A_{p-2,\log^{\alpha}}^p$ to the Bergman space $A^p$ when $\alpha > p$ and $1 < p < 2$. We also compute norm estimates for the Hilbert matrix operator acting from the logarithmically weighted Bergman space $A^p_{\beta,\log^\alpha}$ to the weighted Bergman space $A^p_{\beta}$.

Key words: operator norm, Hilbert matrix operator, logarithmically weighted Bergman space

中图分类号:

O174.5

胡浩, 叶善力. Hilbert 矩阵算子在对数加权 Bergman 空间上的范数[J]. 数学物理学报, 2025, 45(5): 1405-1416.

Hao Hu, Shanli Ye. Norm of the Hilbert Matrix on the Logarithmically Weighted Bergman Spaces[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1405-1416.

参考文献

[1]	Duren P L. Theory of $H^p$ Spaces. New York: Academic Press, 1970
[2]	Duren P L, Schuster A. Bergman Spaces. Providence, RI: American Mathematical Soc, 2004
[3]	Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman Spaces. Berlin: Springer, 2000
[4]	Zhu K. Operator Theory in Function Spaces. Providence, RI: American Mathematical Soc, 2007
[5]	Diamantopoulos E, Siskakis A G. Composition operators and the Hilbert matrix. Stud Math, 2000, 140(2): 191-198
[6]	Contreras M D, Hernandez-Diaz A G. Weighted composition operators in weighted Banach spaces of analytic functions. J Aust Math Soc, 2000, 69(1): 41-60
[7]	Diamantopoulos E. Hilbert matrix on Bergman spaces. Ill J Math, 2004, 48(3): 1067-1078
[8]	Aleman A, Montes-Rodríguez A, Sarafoleanu A. The eigenfunctions of the Hilbert matrix. Constr Approx, 2012, 36(3): 353-374
[9]	Bralovi? D, Karapetrovi? B. New upper bound for the Hilbert matrix norm on negatively indexed weighted Bergman spaces. Bull Malays Math Sci Soc, 2022, 45(2): 1183-1193
[10]	Bo?in V, Karapetrovi? B. Norm of the Hilbert matrix on Bergman spaces. J Funct Anal, 2018, 274(2): 525-543
[11]	Brevig O F, Perfekt K M, Seip K, et al. The multiplicative Hilbert matrix. Adv Math, 2016, 302: 410-432
[12]	Dostani? M, Jevti? M, Vukoti? D. Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type. J Funct Anal, 2008, 254(11): 2800-2815
[13]	Galanopoulos P, Girela D, Peláez, J á, Siskakis A G. Generalized Hilbert operators. Ann Acad Sci Fenn Math, 2014, 39: 231-258
[14]	Ye S, Feng G. A Derivative-Hilbert operator acting on Hardy spaces. Acta Math Sci, 2023, 43B(6): 2398-2412
[15]	Ye S, Feng G. Generalized Hilbert operators acting on weighted Bergman spaces and Dirichlet spaces. Banach J Math Anal, 2023, 17(3): Article 38
[16]	Ye S, Feng G. Norm of the Hilbert Matrix on logarithmically weighted Bergman spaces. Complex Anal Oper Theory, 2023, 17(6): Article 97
[17]	Ye S, Xu Y. A Derivative-Hilbert operator acting from logarithmic Bloch spaces to Bergman spaces. Acta Math Sci, 2024, 44B(5): 1916-1930
[18]	Ye S, Zhou Z. A Derivative-Hilbert operator acting on Bergman spaces, J Math Anal Appl, 2022, 506(1): Article 125553
[19]	Ye S, Zhou Z. A Derivative-Hilbert operator acting on the Bloch space. Complex Anal Oper Theory, 2021, 15(5): Article 88
[20]	叶善力, 周智慧. Bloch 型空间上的广义 Hilbert 算子. 数学学报, 2023, 66(3): 557-568
[20]	Ye S L, Zhou Z H. Generalized Hilbert operator acting on Blocch type spaces. Acta Math Sin, 2023, 66(3): 557-568
[21]	Xu Y, Ye S, Zhou Z. A derivative-Hilbert operator acting on Dirichlet spaces. Open Math, 2023, 21(1): Article 20220559
[22]	Xu Y, Ye S. A Derivative-Hilbert operator acting from Bergman spaces to Hardy spaces. AIMS Math, 2023, 8(4): 9290-9302
[23]	Hu H, Ye S. Norm estimates of the Hilbert matrix operator between some spaces of analytic functions. J Geom Anal, 2025, 35(6): Article 184
[24]	Jevti? M, Karapetrovi? B. Hilbert matrix on spaces of Bergman-type. J Math Anal Appl, 2017, 453(1): 241-254
[25]	Karapetrovt? B. Norm of the Hilbert matrix operator on the weighted Bergman spaces. Glasg Math J, 2018, 60(3): 513-525
[26]	Littlewood J E. Lectures on the Theory of Functions. Oxford: Oxford University Press, 1944
[27]	Lindstr?m M, Miihkinen S, Wikman N. On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces. Ann Fenn Math, 2021, 46: 201-224