We studied the exact norm of the differential Hilbert operator $ \mathcal{H}_{2} $ on several classical analytic function spaces, including the space of bounded analytic functions $ H^{\infty} $, the Korenblum spaces $ H^{\infty}_{\alpha} $, and the $ \alpha $-Bloch spaces $ \mathcal{B}^{\alpha} $. We established the operator norm of $ \mathcal{H}_{2}: H^{\infty}_{\alpha}\to H^{\infty}_{\beta} $ and explicitly determined its exact value when $ \beta = 1+\alpha $. The norms of $ \mathcal{H}_{2}:H^{\infty}_{\alpha} \to \mathcal{B}^{\alpha+2} $, of $ \mathcal{H}_{2}: H^{\infty}\to H^{\infty}_{1} $, and of $ \mathcal{H}_{2}: H^{\infty}\to\mathcal{B}^{2} $ were also obtained.
Citation: Xiangling Zhu, Songxiao Li. The differential Hilbert operator on some analytic function spaces[J]. AIMS Mathematics, 2026, 11(10): 34531-34545. doi: 10.3934/math.20261345
We studied the exact norm of the differential Hilbert operator $ \mathcal{H}_{2} $ on several classical analytic function spaces, including the space of bounded analytic functions $ H^{\infty} $, the Korenblum spaces $ H^{\infty}_{\alpha} $, and the $ \alpha $-Bloch spaces $ \mathcal{B}^{\alpha} $. We established the operator norm of $ \mathcal{H}_{2}: H^{\infty}_{\alpha}\to H^{\infty}_{\beta} $ and explicitly determined its exact value when $ \beta = 1+\alpha $. The norms of $ \mathcal{H}_{2}:H^{\infty}_{\alpha} \to \mathcal{B}^{\alpha+2} $, of $ \mathcal{H}_{2}: H^{\infty}\to H^{\infty}_{1} $, and of $ \mathcal{H}_{2}: H^{\infty}\to\mathcal{B}^{2} $ were also obtained.
| [1] |
V. Božin, B. Karapetrović, Norm of the Hilbert matrix on Bergman spaces, J. Funct. Anal., 274 (2018), 525–543. https://doi.org/10.1016/j.jfa.2017.08.005 doi: 10.1016/j.jfa.2017.08.005
|
| [2] |
G. L. Bao, L. Tian, H. Wulan, The norm of the Hilbert matrix operator on Bergman spaces, Canad. Math. Bull., 2026, 1–16. https://doi.org/10.4153/S0008439526101751 doi: 10.4153/S0008439526101751
|
| [3] |
G. L. Bao, H. Wulan, Hankel matrices acting on Dirichlet spaces, J. Math. Anal. Appl., 409 (2014), 228–235. https://doi.org/10.1016/j.jmaa.2013.07.006 doi: 10.1016/j.jmaa.2013.07.006
|
| [4] |
J. Dai, Norm of the Hilbert matrix operator on the Korenblum space, J. Math. Anal. Appl., 514 (2022), 126270. https://doi.org/10.1016/j.jmaa.2022.126270 doi: 10.1016/j.jmaa.2022.126270
|
| [5] |
J. Dai, On the norm of the Hilbert matrix operator on weighted Bergman spaces, J. Funct. Anal., 287 (2024), 110587. https://doi.org/10.1016/j.jfa.2024.110587 doi: 10.1016/j.jfa.2024.110587
|
| [6] |
E. Diamantopoulos, Hilbert matrix on Bergman spaces, Illinois J. Math., 48 (2004), 1067–1078. https://doi.org/10.1215/ijm/1258131071 doi: 10.1215/ijm/1258131071
|
| [7] |
M. Dostanić, M. Jevtić, D. Vukotić, Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type, J. Funct. Anal., 254 (2008), 2800–2815. https://doi.org/10.1016/j.jfa.2008.02.009 doi: 10.1016/j.jfa.2008.02.009
|
| [8] |
E. Diamantopoulos, A. G. Siskakis, Composition operators and the Hilbert matrix, Stud. Math., 140 (2000), 191–198. https://doi.org/10.4064/sm-140-2-191-198 doi: 10.4064/sm-140-2-191-198
|
| [9] |
P. Galanopoulos, D. Girela, J. Á. Peláez, A. G. Siskakis, Generalized Hilbert operators, Ann. Acad. Sci. Fenn. Math., 39 (2014), 231–258. https://doi.org/10.5186/aasfm.2014.3912 doi: 10.5186/aasfm.2014.3912
|
| [10] |
S. X. Li, Generalized Hilbert operator on the Dirichlet-type spaces, Appl. Math. Comput., 214 (2009), 304–309. https://doi.org/10.1016/j.amc.2009.03.073 doi: 10.1016/j.amc.2009.03.073
|
| [11] |
M. Lindström, S. Miihkinen, D. Norrby, Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces, Adv. Math., 408 (2022), 108598. https://doi.org/10.1016/j.aim.2022.108598 doi: 10.1016/j.aim.2022.108598
|
| [12] |
S. X. Li, S. Stević, Generalized Hilbert operator and Fejér-Riesz type inequalities on the polydisc, Acta Math. Sci., 29 (2009), 191–200. https://doi.org/10.1016/S0252-9602(09)60020-5 doi: 10.1016/S0252-9602(09)60020-5
|
| [13] |
M. Lindström, S. Miihkinen, N. Wikman, Norm estimates of weighted composition operators pertaining to the Hilbert matrix, Proc. Amer. Math. Soc., 147 (2019), 2425–2435. https://doi.org/10.1090/proc/14437 doi: 10.1090/proc/14437
|
| [14] |
J. Á. Peláez, J. Rättyä, Generalized Hilbert operators on weighted Bergman spaces, Adv. Math., 240 (2013), 227–267. https://doi.org/10.1016/j.aim.2013.03.006 doi: 10.1016/j.aim.2013.03.006
|
| [15] |
P. C. Tang, Operators of Hilbert type acting on some spaces of analytic functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 120 (2026), 110. https://doi.org/10.1007/s13398-026-01890-w doi: 10.1007/s13398-026-01890-w
|
| [16] | H. Wulan, M. M. Zhou, J. F. Zhu, Hilbert matrix norms on weighted Bergman spaces: even exponents and a counterexample to the beta formula, 2026, arXiv: 2607.23540. |
| [17] |
S. L. Ye, G. H. Feng, A derivative-Hilbert operator acting on Hardy spaces, Acta Math. Sci., 43 (2023), 2398–2412. https://doi.org/10.1007/s10473-023-0605-6 doi: 10.1007/s10473-023-0605-6
|
| [18] |
S. L. Ye, Z. H. Zhou, A derivative-Hilbert operator acting on the Bloch space, Complex Anal. Oper. Theory, 15 (2021), 88. https://doi.org/10.1007/s11785-021-01135-1 doi: 10.1007/s11785-021-01135-1
|
| [19] |
S. L. Ye, Z. H. Zhou, A derivative-Hilbert operator acting on Bergman spaces, J. Math. Anal. Appl., 506 (2022), 125553. https://doi.org/10.1016/j.jmaa.2021.125553 doi: 10.1016/j.jmaa.2021.125553
|
| [20] | K. H. Zhu, Operator theory in function spaces, 2 Eds., American Mathematical Society, 2007. https://doi.org/10.1090/surv/138 |
Xiangling Zhu, Songxiao Li. The differential Hilbert operator on some analytic function spaces[J]. AIMS Mathematics, 2026, 11(10): 34531-34545. doi: 10.3934/math.20261345
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