Czechoslovak Mathematical Journal, first online, pp. 1-18

Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

Boban Karapetrović

Received October 23, 2016.   First published March 6, 2018.

Abstract:  We show that if $\alpha>1$, then the logarithmically weighted Bergman space $A_{\log^{\alpha}}^2$ is mapped by the Libera operator $\mathcal{L}$ into the space $A_{\log^{\alpha-1}}^2$, while if $\alpha>2$ and $0<\varepsilon\leq\alpha-2$, then the Hilbert matrix operator $H$ maps $A_{\log^\alpha}^2$ into $A_{\log^{\alpha-2-\varepsilon}}^2$. We show that the Libera operator $\mathcal{L}$ maps the logarithmically weighted Bloch space $\mathcal{B}_{\log^{\alpha}}$, $\alpha\in\mathbb{R}$, into itself, while $H$ maps $\mathcal{B}_{\log^{\alpha}}$ into $\mathcal{B}_{\log^{\alpha+1}}$. In Pavlović's paper (2016) it is shown that $\mathcal{L}$ maps the logarithmically weighted Hardy-Bloch space $\mathcal{B}_{\log^{\alpha}}^1$, $\alpha>0$, into $\mathcal{B}_{\log^{\alpha-1}}^1$. We show that this result is sharp. We also show that $H$ maps $\mathcal{B}_{\log^{\alpha}}^1$, $\alpha\geq0$, into $\mathcal{B}_{\log^{\alpha-1}}^1$ and that this result is sharp also.
Keywords:  Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space
Classification MSC:  47B38, 47G10, 30H25
DOI:  10.21136/CMJ.2018.0555-16

PDF available at:  Institute of Mathematics CAS

[1] M. Jevtić, B. Karapetrović: Hilbert matrix operator on Besov spaces. Publ. Math. 90 (2017), 359-371. DOI 10.5486/PMD.2017.7518 | MR 3666637
[2] M. Jevtić, B. Karapetrović: Libera operator on mixed norm spaces $H_{\nu}^{p,q,\alpha}$ when $0<p<1$. Filomat 31 (2017), 4641-4650. MR 3730385
[3] M. Jevtić, D. Vukotić, M. Arsenović: Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces. RSME Springer Series 2, Springer, Cham (2016). DOI 10.1007/978-3-319-45644-7 | MR 3587910 | Zbl 1368.30001
[4] B. Łanucha, M. Nowak, M. Pavlović: Hilbert matrix operator on spaces of analytic functions. Ann. Acad. Sci. Fenn., Math. 37 (2012), 161-174. DOI 10.5186/aasfm.2012.3715 | MR 2920431 | Zbl 1258.47047
[5] M. Mateljević, M. Pavlović: $L^p$-behaviour of the integral means of analytic functions. Stud. Math. 77 (1984), 219-237. DOI 10.4064/sm-77-3-219-237 | MR 0745278 | Zbl 1188.30004
[6] M. Pavlović: Definition and properties of the libera operator on mixed norm spaces. The Scientific World Journal 2014 (2014), Article ID 590656, 15 pages. DOI 10.1155/2014/590656
[7] M. Pavlović: Function Classes on the Unit Disc. An introduction. De Gruyter Studies in Mathematics 52, De Gruyter, Berlin (2014). DOI 10.1515/9783110281903 | MR 3154590 | Zbl 1296.30002
[8] M. Pavlović: Logarithmic Bloch space and its predual. Publ. Inst. Math. (Beograd) (N.S.) 100(114) (2016), 1-16. DOI 10.2298/PIM1614001P | MR 3586678 | Zbl 06749634
[9] K. Zhu: Operator Theory in Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics 139, Marcel Dekker, New York (1990). MR 1074007 | Zbl 0706.47019
[10] A. Zygmund: Trigonometric Series. Vol. I, II. Cambridge University Press, Cambridge (1959). MR 1963498 | Zbl 1084.42003

Affiliations:   Boban Karapetrović, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia, e-mail:

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